
Class 1 d | S \ 
Book ,K f- 

££ffi¥RIGHT DEPOSm 



THE I 

MECHANICAL ENGINEER'S 
POCKET-BOOK. 



A REFERENCE-BOOK OF RULES, TABLES, DATA, 

AND FORMULAE, FOR THE USE OF 

ENGINEERS, MECHANICS, 

AND STUDENTS. 

8 1189J 



WILLIAM KENT, A.M., M.E., 

Consulting Engineer, 
Member Amer. Soc'y Mechl. Engrs. and Amer Inst. Mining Engrs. 



FIRST THOUSAND. 

FIRST EDITION. 



NEW YORK : 

JOHN WILEY & SONS, 

53 East Tenth Street. 

1895. 



*> 



<^w 



*\%K 



Copyright, 1895, 

BY 

WILLIAM KENT. 



#' 



ROBERT DRUMMOND, ELECTROTYPER ANP PRINTER, NEW YORK. 



PBEFACE. 



More than twenty years ago the author began to follow 
the advice given by Nystrom : " Every engineeer should 
make his own pocket-book, as he proceeds in study and 
practice, to suit his particular business." The manuscript 
pocket-book thus begun, however, soon gave place to more 
modern means for disposing of the accumulation of engi- 
neering facts and figures, viz., the index rerum, the scrap- 
book, the collection of indexed envelopes, portfolios and 
boxes, the card catalogue, etc. Four years ago, at the re- 
quest of the publishers, the labor was begun of selecting 
from this accumulated mass such matter as pertained to 
mechanical engineering, and of condensing, digesting, and 
arranging it in form for publication. In addition to this, a 
careful examination was made of the transactions of engi- 
neering societies, and of the most important recent works 
on mechanical engineering, in order to fill gaps that might 
be left in the original collection, and insure that no impor- 
tant facts had been overlooked. 

Some ideas have been kept in mind during the prepara- 
tion of the Pocket-book that will, it is believed, cause it to 
differ from other works of its class. In the first place it 
was considered that the field of mechanical engineering was 
so great, and the literature of the subject so vast, that as 
little space as possible should be given to subjects which 
especially belong to civil engineering. While the mechan- 
ical engineer must continually deal with problems which 
belong properly to civil engineering, this latter branch is 
so well covered by Trautwine's " Civil Engineer's Pocket- 
book " that any attempt to treat it exhaustively would not 
only fill no " long-felt want," but would occupy space 
which should be given to mechanical engineering. 



IV PREFACE. 

Another idea prominently kept in view by the author has 
been that he would not assume the position of an " au- 
thority " in giving rules and formulae for designing, but 
only that of compiler, giving not only the name of the 
originator of the rule, where it was known, but also the 
volume and page from which it was taken, so that its 
derivation may be traced when desired. When different 
formulas for the same problem have been found they have 
been given in contrast, and in many cases examples have 
been calculated by each to show the difference between 
them. In some cases these differences are quite remark- 
able, as will be seen under Safety-valves and Crank-pins. 
Occasionally the study of these differences has led to the 
author's devising a new formula, in which case the deriva- 
tion of the formula is given. 

Much attention has been paid to the abstracting of data 
of experiments from recent periodical literature, and numer- 
ous references to other data are given. In this respect 
the present work will be found to differ from other Pocket- 
books. 

The author desires to express his obligation to the many 
persons who have assisted him in the preparation of the 
work, to manufacturers who have furnished their cata- 
logues and given permission for the use of their tables, 
and to many engineers who have contributed original data 
and tables. The names of these persons are mentioned in 
their proper places in the text, and in all cases it has been 
endeavored to give credit to whom credit is due. The 
thanks of the author are also due to the following gentle- 
men who have given assistance in revising manuscript or 
proofs of the sections named : Prof. De Volson Wood, 
mechanics and turbines ; Mr. Frank Richards, compressed 
air ; Mr. Alfred R. Wolff, windmills ; Mr. Alex. C. 
Humphreys, illuminating gas ; Mr. Albert E. Mitchell, 
locomotives ; Prof. James E. Denton, refrigerating-ma- 
chinery ; Messrs. Joseph Wetzler and Thomas W. Varley, 
electrical engineering ; and Mr. Walter S. Dix, for valu- 
able contributions on several subjects, and suggestions as 
to their treatment. Wm. Kent. 

Passaic, N. J., April, 1895. 






CONTENTS. 



(For Alphabetical Index see page 1075.) 



MATHEMATICS. 

Arithmetic. 

PAGE 

Arithmetical and Algebraical Signs 1 

Greatest Common Divisor 2 

Least Common Multiple 2 

Fractions 2 

Decimals 3 

Table. Decimal Equivalents of Fractions of One Inch 3 

Table. Products of Fractions expressed in Decimals 4 

Compound or Denominate Numbers 5 

Reduction Descending and Ascending 5 

Ratio and Proportion 5 

Involution, or Powers of Numbers 6 

Table. First Nine Powers of the First Nine Numbers 7 

Table. First Forty Powers of 2 7 

Evolution. Square Root 7 

Cube Root _ 8 

Alligation 10 

Permutation 10 

Combination 10 

Arithmetical Progression 11 

Geometrical Progression 11 

Interest 13 

Discount 13 

Compound Interest 14 

Compound Interest Table, 3, 4, 5, and 6 per cent 14 

Equation of Payments 14 

Partial Payments . 15 

Annuities 16 

Tables of Amount, Present Values, etc., of Annuities 16 

"Weights and Measures. 

Long Measure 

Old Land Measure 

Nautical Measure , 

Square Measure , 

Solid or Cubic Measure 

Liquid Measure 

The Miners' Inch 

Apothecaries' Fluid Measure 

Dry Measure 

Shipping Measure 

Avoirdupois Weight 

Troy Weight 

Apothecaries' Weight , 

To Weigh Correctly on an Incorrect Balance , 

Circular Measure au 

Measure of time 20 



PAGE 

Board and Timber Measure 20 

Table. Contents in Feet of Joists, Scantlings, and Timber 20 

French or Metric Measures 21 

British and French Equi valents 21 

Metric Conversion Tables . . 23 

Compound Units. 

of Pressure and Weight 27 

of Water, Weight, and Bulk 27 

of Work, Power, and Duty 27 

of Velocity 27 

of Pressure per unit area 27 

Wire and Sheet Metal Gauges 28 

Twist-drill and Steel-wire Gauges. 28 

Music-wire Gauge 29 

Circular- mil Wire Gauge 30 

New U. S. Standard Wire and Sheet Gauge, 1893 30 

Algebra. 

Addition. Multiplication, etc 33 

Powers of Num bers 33 

Parentheses, Division 34 

Simple Equations and Problems 34 

Equations containing two or more Unknown Quantities 3D 

Elimination 35 

Quadratic Equations 35 

Theory of Exponents 36 

Binomial Theorem 36 

Geometrical Problems of Construction 37 

of Straight Lines 37 

ofAngles 38 

of Circles 39 

of Triangles 41 

of Squares and Polygons 42 

of the Ellipse 45 

of the Parabola 48 

of the Hyperbola 49 

of the Cycloid 49 

of the Tractrix or Schiele Anti-friction Curve 50 

of the Spiral 50 

of the Catenary -. , - 51 

of the Involute 52 

Geometrical Propositions 53 

Mensuration, Plane Surfaces. 

Quadrilateral, Parallelogram, etc 54 

Trapezium and Trapezoid 54 

Triangles .... 54 

Polygons. Table of Polygons 55 

Irregular Figures 55 

Properties of the Circle 57 

Values of n and its Multiples, etc 57 

Relations of arc, chord, etc 58 

Relations of circle to inscribed square, etc 58 

Sectors and Segments 59 

Circular Ring 59 

The Ellipse 59 

The Helix 60 

The Spiral 60 

Mensuration, Solid Bodies. 

Prism 60 

Pyramid 60 

Wedge 61 

The Prismoidal Formula 62 

Rectangular Prismoid 61 

Cylinder 61 

Cone CI 



CONTENTS. \11 

PAGE 

Sphere 61 

Spherical Triangle 61 

Spherical Polygon 61 

Spherical Zone 62 

Spherical Segment 62 

Spheroid or Ellipsoid , 63 

Polyedron 62 

Cylindrical Ring 62 

Solids of Revolution 62 

Spindles 63 

Frustrum of a Spheroid 63 

Parabolic Conoid 64 

Volume of a Cask 64 

Irregular Solids : 64 

Plane Trigonometry. 

Solution of Plane Triangles 65 

Sine, Tangent, Secant, etc 65 

Signs of the Trigonometric Functions 66 

Trigonometrical Formulae 66 

Solution of Plane Right-angled Triangles 68 

Solution of Oblique-angled Triangles 68 

Analytical Geometry. 

Ordinates and Abscissas 69 

Equations of a Straight Line, Intersections, etc 69 

Equations of the Circle , 70 

Equations of the Ellipse , 70 

Equations of the Parabola 70 

Equations of the Hyperbola 70 

Logarithmic Curves 71 

Differential Calculus. 

Definitions 72 

Differentials of Algebraic Functions 72 

Formulae for Differentiating 73 

Partial Differentials 73 

Integrals , 73 

Formulae for Integration 74 

Integration between Limits 74 

Quadrature of a Plane Surface. 74 

Quadrature of Surfaces of Revolution 75 

Cubature of Volumes of Revolution 75 

Second, Third, etc., Differentials 75 

Maclaurin's and Taylor's Theorems „ 76 

Maxima and Minima 76 

Differential of an Exponential Function 77 

Logarithms 77 

Differential Forms which have Known Integrals 78 

Exponential Functions 78 

Circular Functions 78 

The Cycloid 79 

Integral Calculus , 79 

Mathematical Tables. 

Reciprocals of Numbers 1 to 2000 80 

Squares, Cubes, Square Roots, and Cube Roots from 0.1 to 1600 86 

Squares and Cubes of Decimals 101 

Fifth Roots and Fifth Powers 102 

Circumferences and Areas of Circles, Diameters 1 to 1000 103 

Circumferences and Areas of Circles, Advancing by Eighths from B V to 

100 108 

Decimals of a Foot Equivalent to Inches and Fractions of an Inch 112 

Circumferences of Circles in Feet and Inches, from 1 inch to 32 feet 11 

inches in diameter 113 

Lengths of Circular Arcs, Degrees Given 114 

Lengths of Circular Arcs, Height of Arc Given 115 

Areas of the Segments of a Circle , 116 



CONTENTS. 



PARE 



Spheres 118 

Contents of Pipes and Cylinders, Cubic Feet and Gallons 120 

Cylindrical Vessels, Tanks, Cisterns, etc 121 

Gallons in a Number of Cubic Feet 122 

Cubic Feet in a Number of Gallons 122 

Square Feet in Plates 3 to 32 feet long and 1 inch wide 123 

Capacities of Rectangular Tanks in Gallons 125 

Number of Barrels in Cylindrical Cisterns and Tanks. 126 

Logarithms . 127 

Table of Logarithms 129 

Hyperbolic Logarithms 156 

Natural Trigonometrical Functions 159 

Logarithmic Trigonometrical Functions 162 

MATERIALS. 

Chemical Elements 163 

Specific Gravity and Weight of Materials 163 

Metals, Properties of 164 

The Hydrometer 165 

Aluminum 166 

Antimony 166 

Bismuth 166 

Cadmium 167 

Copper 167 

Gold 167 

Iridium 167 

Iron , 167 

Lead 167 

Magnesium 168 

Manganese 168 

Mercury 168 

Nickel 168 

Platinum 168 

Silver 168 

Tin 168 

Zinc 168 

Miscellaneous Materials. 

Order- of Malleability, etc., of Metals 169 

Formulas and Table for Calculating' Weight of Rods, Plates, etc 169 

Measures and Weights of Various Materials 169 

Commercial Sizes of Iron Bars 170 

Weights of Iron Bars 171 

of Flat Rolled Iron .* 172 

of Iron and Steel Sheets 174 

of Plate Iron 175 

of Steel Blooms 176 

of Structural Shapes 177 

Sizes and Weights of Carnegie Deck Beams 177 

- " •' " Steel Channels 178 

" ZBars 178 

" " Pencoyd Steel Angles 179 

Tees ." 180 

" " " Channels 180 

" " Roofing Materials 181 

" " Terra-cotta 181 

" " Tiles: 181 

" " Tin Plates ...181 

" " Slates 183 

" " Pine Shingles 183 

" " Sky-light Glass ...184 

Weights of Various Roof-coverings 184 

Cast-iron Pipes or Columns.. , 185 

" " " '' 12-ft. lengths 186 

'• " Pipe fittings 187 

" " " Water and Gas-pipe 188 

" and thickness of Cast-iron Pipes 189 

Safe Pressures on Cast Iron Pipe 189 



CONTENTS. 



Sheet-iron Hydraulic Pipe 191 

Standard Pipe Flanges 192 

Pipe Flanges and Cast-iron Pipe 193 

Standard Sizes of Wrought-iron Pipe 194 

Wrought-iron Welded Tubes 196 

Riveted Iron Pipes 197 

Weight of Iron for Riveted Pipe 197 

Spiral Riveted Pipe 198 

Seamless Brass Tubing 198, 199 

Coiled Pipes 199 

Brass, Copper, and Zinc Tubing 200 

Lead and Tin-lined Lead Pipe 201 

Weight of Copper and Brass Wire and Plates 202 

" Round Bolt Copper 203 

" Sheet and Bar Brass 203 

Composition of Rolled Brass 203 

Sizes of Shot 204 

Screw-thread, U. S. Standard 204 

Limit-gauges for Screw-threads 205 

Size of Iron for Standard Bolts 20G 

Sizes of Screw-threads for Bolts and Taps 207 

Set Screws and Tap Screws 208 

Standard Machine Screws 209 

Sizes and Weights of Nuts .- 209 

Weight of Bolts with Heads 210 

Track Bolts 210 

Weights of Nuts and Bolt-heads 211 

Rivets 211 

Sizes of Turnbuckles 211 

Washers 212 

Track Spikes 212 

Railway Spikes 212 

Boat Spikes 212 

Wrought Spikes 213 

Wire Spikes. 213 

CutNails 213 

Wire Nails 214, 215 

Iron Wire, Size, Strength, etc 216 

Galvanized lion Telegraph Wire 217 

Tests of Telegraph Wire 217 

Copper Wire Table, B. W. Gauge 218 

" " " Edison or Circular Mil Gauge 219 

" B.&S.Gauge 220 

Insulated Wire 221 

Copper Telegraph Wire 221 

Electric Cables 221, 222 

Galvanized Steel-wire Strand 223 

Steel-wire Cables for Vessels 223 

Specifications for Galvanized Iron Wire 224 

Strength of Piano Wire , 224 

Plough-steel Wire 224 

Wires of different metals , 225 

Specifications for Copper Wire 225 

Cable-traction Ropes 226 

Wire Ropes 226, 227 

Plough-steel Ropes 227, 228 

Galvanized Iron Wire Rope 228 

Steel Hawsers 223, 229 

Flat Wire Ropes , 2-,'9 

Galvanized Steel Cables 230 

Strength of Chains and Ropes 230 

Notes on use of Wire Rope 231 

Locked Wire Rope 231 

Crane Chains 232 

Weights of Logs, Lumber, etc 232 

Sizes of Fire Brick 233 

Fire Clay, Aualysis , 234 

Magnesia Bricks 235 

Asbestos.... 235 



X CONTENTS. 

Strength of Materials. 

PAGE 

Stress and Strain 236 

Elastic Limit 236 

Yield Point 237 

Modulus of Elasticity 237 

Resilience 238 

Elastic Limit and Ultimate Stress 238 

Repeated Stresses 238 

Repeated Sh< >eks 240 

Stresses due to Sudden Shocks 241 

Increasing Tensile Strength of Bars by Twisting 241 

Tensile Strength 242 

Measurement of Elongation 243 

Shapes of Test Specimens 243 

Compressive Strength 244 

Columns, Pillars, or Struts 246 

Hodgkinson's Formula 246 

Gordon's Formula 247 

Mom ent of Inertia 247 

Radius of Gyration 247 

Elements of Usual Sections 248 

Solid Cast-iron Columns 250 

Hollow Cast-iron Columns 250 

Wrought-iron Columns 251 

Safe load of Cast-iron Columns 253 

Eccentric loading of Columns 255 

Built Columns 256 

Phcenix Columns 257 

Working Formulas for Struts 259 

Merriman's Formula for Columns 260 

Working Strains in Bridge Members 262 

Working Stresses for Steel 263 

Resistance of Hollow Cylinders to Collapse 264 

Collapsing Pressure of Tubes or Flues. 265 

Formula for Corrugated Furnaces 266 

Transverse Strength 266 

Formulas for Flexure of Beams 267 

Safe Loads on Steel Beams 269 

Elastic Resilience 270 

Beams of Uniform Strength 271 

Properties of Rolled Structural Shapes 272 

Spacing of I Beams 273 

Properties of Steel I Beams 274 

" " " Channels , 275 

" ZBars 276 

Iron Beams and Channels 277 

Trenton Angle Bars 279 

Tee Bars 280 

Size of Beams for Floors 280 

Flooring Material 281 

Tie Rods for Brick Arches 281 

Torsional Strength 281 

Elastic Resistance to Torsion 282 

Combined Stresses 282 

Stress due to Temperature 283 

Strength of Flat Plates 283 

Strength of Unstayed Flat Surfaces 284 

Unbraced Heads of Boilers 285 

Thickness of Flat Cast-iron Plates 286 

Strength of Stayed Surfaces 286 

Spherical Shells and Domed Heads 286 

Stresses in Steel Plating under Water Pressure 287 

Thick Hollow Cylinders under Tension — 287 

Thin Cylinders under Tension 289 

Hollow Copper Balls 289 

Holding Power of Nails, Spikes, Bolts, and Screws 289 

Cut versus Wire Nails 290 

Strength of Wrought-iron Bolts 292 



CONTENTS. XI 

PAGE 

Initial Strain on Bolts 292 

Stand Pipes and their Design 292 

Riveted Steel Water-pipes 295 

Mannesmann Tubes 296 

Kirkaldy's Tests of Materials 296 

Cast Iron 296 

Iron Castings 297 

Iron Bars, Forgings, etc 297 

Steel Rails and Tires 298 

Steel Axles, Shafts, Spring Steel 299 

Riveted Joints 299 

Welds 300 

Copper, Brass, Bronze, etc 300 

Wire, Wire-rope 301 

Ropes, Hemp, and Cotton 301 

Belting, Canvas . 302 

Stones, Brick, Cement 302 

Tensile Strength of Wire « 303 

Watertown Testing-machine Tests 303 

Riveted Joints 303 

W rough t-iron Bars, Compression Tests 304 

Steel Eye-bars 304 

Wrought-iron Columns 305 

Cold Drawn Steel 305 

American Woods 306 

Shearing Strength of Iron and Steel , 306 

Holding Power of Boiler-tubes 307 

Chains, Weight, Proof Test, etc 307 

Wrought-iron Chain Cables 308 

Strength of Glass 308 

Copper at High Temperatures 309 

Strength of Timber 309 

Expansion of Timber 311 

Shearing Strength of Woods 312 

Strength of Brick, Stone, etc , 312 

" Flagging 313 

" " Lime and Cement Mortar 313 

Moduli of Elasticity of Various Materials 314 

Factors of Safety 314 

Properties of Cork 316 

Vulcanized India-rubber 316 

Xylolith or Woodstone 316 

Aluminum, Properties and Uses 317 

Alloys. 

Alloys of Copper and Tin, Bronze 319 

Copper and Zinc, Brass 321 

Variation in Strength of Bronze 321 

Copper-tin-zinc Alloys 322 

Liquation or Separation of Metals 323 

Alloys used in Brass Foundries 325 

Copper-nickel Alloys 326 

Copper-zinc-iron Alloys 326 

Tobin Bronze 327 

Phosphor Bronze 327 

Aluminum Bronze 328 

Aluminum Brass 329 

Caution as to Strength of Alloys 329 

Aluminum hardened 330 

Alloys of Aluminum, Silicon, and Iron 330 

Tungsten-aluminum Alloys 331 

Aluminum-tin Alloys 331 

Manganese Alloys t 331 

Manganese Bronze 331 

German Silver ? 332 

Alloys of Bismuth 332 

Fusible Alloys 333 

Bearing Metal Alloys 333 



CONTENTS. 



PAGE 

Alloys containing Antimony 336 

White-metal Alloys 336 

Type-metal 336 

Babbitt metals 336 

Solders 338 

Ropes and Chains. 

Strength of Hemp, Iron, and Steel Ropes 33^ 

Flat Ropes , 339 

Working Load of Ropes and Chains 339 

Strength of Ropes and Chain Cables 340 

Rope for Hoisting or Transmission 340 

Cordage, Technical terms of 341 

Splicing of Ropes 341 

Coal Hoisting 343 

Manila Cordage, Weight, etc 344 

Knots, how to make 344 

Splicing Wire Ropes 346 

Springs. 

Laminated Steel Springs 347 

Helical Steel Springs 347 

Carrying Capacity of Springs 349 

Elliptical Springs 352 

Phosphor-bronze Springs 352 

Springs to Resist Torsional Force 352 

Helical Springs for Cars, etc 353 

Riveted Joints. 

Fairbairn's Experiments , 354 

Loss of Strength by Punching 354 

Strength of Perforated Plates 354 

Hand vs. Hydraulic Riveting 355 

Formulae for Pitch of Rivets 357 

Proportions of Joints 358 

Efficiencies of Joints 359 

Diameter of Rivets 360 

Strength of Riveted Joints 361 

. Riveting Pressures 362 

Shearing Resistance of Rivet Iron 363 

Iron and Steel. 

Classification of Iron and Steel 364 

Grading of Pis: Iron 365 

Influence of Silicon Sulphur, Phos. and Mn on Cast Iron 365 

Tests of Cast Iron 369 

Chemistry of Foundry Iron 370 

Analyses of Castings 373 

Strength of Cast Iron 374 

Specifications for Cast Iron 374 

Mixture of Cast Iron with Steel 375 

Bessemerized Cast Iron 375 

Bad Cast Iron 375 

Malleable Cast Iron 375 

Wrought Iron 377 

Chemistry <»f Wrought Iron 377 

Influence of Rolling on Wrought Iron 377 

Specifications for Wrought Iron 378 

Stay-holt Iron 379 

Formula} for Unit Strains in Structures 379 

Permissible Stresses in Structures 381 

Proportioning Materials in Memphis Bridge 382 

Tenacity of Iron at High Temperatures 382 

Effect of Cold on Strength of Iron 383 

Expansion of Iron by Heat 385 

Durability of Cast Iron 385 

Corrosion of Iron and Steel 386 

Manganese Plating of Iron 387 



CONTENTS. 



Non-oxidizing Process of Annealing 387 

Painting Wood and Iron Structures. . 388 

Qualities of Paints 389 

Steel. 

Relation between Chem. and Phys. Properties 389 

Variation in Strength 391 

Open-hearth 392 

Bessemer 392 

Hardening Soft Steel 393 

Effect of Cold Rolling 393 

Comparison of Full-sized and Small Pieces 393 

Treatment of Structural Steel 394 

Influence of Annealing upon Magnetic Capacity 396 

Specifications for Steel 397 

Boiler, Ship and Tank Plates 399 

Steel for Springs, Axles, etc 400 

May Carbon be Burned out of Steel? . . 402 

Recalescence of Steel 402 

Effect of Nicking a Bar 402 

Electric Conductivity 403 

Specific Gravity -. 403 

Occasional Failures 403 

Segregation in Ingots 404 

Earliest Uses for Structures 405 

Steel Castings 405 

Manganese Steel 407 

Nickel Steel 407 

Aluminum Steel , 409 

Chrome Steel 409 

Tungsten Steel , 409 

Compressed Steel 410 

Crucible Steel 410 

Effect of Heat on Grain „ 412 

" " Hammei-ing, etc 412 

Heating and Forging 412 

Tempering Steel . 413 

MECHANICS. 

Force, Unit of Force 415 

Inertia 415 

Newton's Laws of Motion 415 

Resolution of Forces 415 

Parallelogram of Forces 416 

Moment of a Force 416 

Statical Moment, Stability 417 

Stability of a Dam 417 

Parallel Forces ., 417 

Couples 418 

Equilibrium of Forces 418 

Centre of Gravity 418 

Moment of Inertia 419 

Centre of Gyration •. 420 

Radi us of Gyration .... 420 

Centre of Oscillation 421 

Centre of Percussion 422 

The Pendulum 422 

Conical Pendulum 423 

Centrifugal Force 423 

Acceleration , 423 

Falling Bodies.. 424 

Value of g 424 

Angular Velocity . . 425 

Height due to Velocity 425 

Parallelogram of Velocities 426 

Mass 427 

Force of Acceleration . . 427 

Motion on Inclined Planes 428 

Momentum . ..... 428 



XIV CONTENTS. 

PAGE 

Vis Viva 428 

Work, Foot-pound 428 

Power, Horse-power 429 

Energy 429 

Work of Acceleration 430 

Force of a Blow , 430 

Impact of Bodies 431 

Energy of Recoil of Guns 431 

Conservation of Energy 432 

Perpetual Motion 432 

Efficiency of a Machine 432 

Amiual-power, Man-power 433 

Work of a Horse 434 

Man-wheel 434 

Horse-gin 434 

Resistance of Vehicles 435 

Elements of Machines. 

The Lever 435 

The Bent Lever 436 

The Moving Strut 436 

The Toggle-joint 436 

The Inclined Plane 437 

The Wedge 437 

The Screw 437 

The Cam 438 

The Pulley 438 

Differential Pullev 439 

Differential Windlass 439 

Differential Screw 439 

Wheel and Axle ... 439 

Toothed-wheel Gearing 439 

Endless Screw 440 

Stresses in Framed Structures. 

Cranes and Derricks 440 

Shear Poles and Guys 442 

King Pest Truss or Bridge 442 

Queen Post Truss 442 

Burr Truss 443 

Pratt or Whipple Truss 443 

Howe Truss 445 

Warren Girder 445 

Roof Truss 446 

HEAT. 

Thermometers and Pyrometers 448 

Centigrade and Fahrenheit degrees compared 449 

Copper-ball Pyrometer 451 

Thermo-electric Pyrometer 451 

Temperatures in Furnaces 451 

Wiborgh Air Pyrometer 453 

Seegers Fire-clay Pyrometer 453 

Mesur6 and Nouel's Pyrometer 453 

Uehling and Steinhart's Pyrometer 453 

Air-thermometer 454 

High Temperatures judged by Color 454 

Boiling-points of Substances 455 

Melting-points 455 

Unit of Heat 455 

Mechanical Equivalent of Heat 456 

Heat of Combustion 456 

Specific Heat 457 

Latent Heat of Fusion 459, 461 

Expansion by Heat, 460 

Absolute Temperature 461 

A bsolute Zero 461 



CONTENTS. XV 

PAGE 

Latent Heat 461 

Latent Heat of Evaporation 462 

Total Heat of Evaporation 462 

Evaporation and Drying 462 

Evaporation from Reservoirs 463 

Evaporation by the Multiple System 463 

Resistance to Boiling 463 

Manufacture of Salt 464 

Solubility of Salt and Sulphate of Lime 464 

Salt Contents of Brines 464 

Concentration of Sugar Solutions 465 

Evaporating by Exhaust Steam 465 

Drying in Vacuum 466 

Radiation of Heat 467 

Conduction and Convection of Heat 468 

Steam-pipe Coverings 470 

Rate of External Conduction 471 

Transmission through Plates 473 

" in Condenser Tubes 473 

" " Cast-iron Plates 474 

" from Air or Gases to Water . . 474 

" from Steam or Hot Water to Air 475 

" through Walls of Buildings 478 

Thermodynamics 478 

PHYSICAL PROPERTIES OF GASES. 

Expansion of Gases 479 

Boyle and Marriotte's Law 479 

Law of Charles, Avogadro's Law 479 

Saturation Point of Vapors 480 

Law of Gaseous Pressure 480 

Flow of Gases 480 

Absorption by Liquids 480 



AIR. 

Properties of Air 481 

Air-manometer 481 

Pressure at Different Altitudes 481 

Bai-ometric Pressures 482 

Levelling by the Barometer and by Boiling Water 482 

To find Difference in Altitude 483 

Moisture in Atmosphere 483 

Weight of Air and Mixtures of Air and Vapor 484 

Specific Heat of Air 484 

Flow of Air. 

Flow of Air through Orifices 484 

Flow of Air in Pipes 485 

Effect of Bends in Pipe 488 

Flow of Compressed Air 488 

Tables of Flow of Air 489 

Anemometer Measurements 491 

Equalization of Pipes 492 

Loss of Pressure in Pipes -. 493 

Wind. 

Force of the Wind ; 493 

Wind Pressure in Storms 495 

Windmills 495 

Capacity of Windmills 497 

Economy of Windmills 498 

Electric Power from Windmills 499 

Compressed Air. 

Heatiner of Air by Compression 499 

Loss of Energy in Compressed Air 499 

Volumes and Pressures 500 



XVI CONTENTS. 

PAGE 

Loss due to Excess of Pressure 501 

Horse-power Required for Compression 5(1 

Table for Adiabatic Compression 5G2 

Mean Effective Pressures 502 

Mean and Terminal Pressures . . 503 

Air-compressors . . 503 

Practical Results 505 

Efficiency of Compressed-air Engines 506 

Requirements of Rock-drills 506 

Popp Compressed-air System 507 

Small Compressed-air Motors 507 

Efficiency of Air-heating Stoves 507 

Efficiency of Compressed-air Transmission 50* 

Shops Operated by Compressed Air 509 

Pneumatic Postal Transmission 509 

Mekarski Compressed-air Tramways 509 

Compressed Air Working Pumps in Mines 511 

Fans and Blowers. 

Centrifugal Fans 511 

Rest Proportions of Fans 512 

Pressure due to Velocity 513 

Experiments with Blowers 514 

Quantity of Air Delivered 514 

Efficiency of Fans and Positive Blowers 516 

Capacity of Fans and Blowers 517 

Table of Centrifugal Fans 518 

Engines, Fans, and Steam-coils for the Blower System of Heating 519 

Sturtevant Steel Pressure-blower 519 

Diameter of Blast-pipes 519 

Centrifugal Ventilators for Mines 521 

Experiments on Mine Ventilators 522 

DiskFans 524 

Air Removed by Exhaust Wheel. 525 

Efficiency of Disk Fans 525 

Positive Rotary Blowers 526 

Blowing Engines . . . . . . 526 

Steam-jet Blowers 527 

Steam-jet for Ventilation 527 

HEATING AND VENTILATION. 

Ventilation 528 

Quantity of Air Discharged through a Ventilating Duct 530 

Artificial Cooling of Air 531 

Mine-ventilation 531 

Friction of Air in Underground Passages 531 

Equivalent Orifices 533 

Relative Efficiency of Fans and Heated Chimneys 533 

Heating and Ventilating of Large Buildings 534 

Rules for Computing Radiating Surfaces 536 

Overhead Steam-pipes 537 

Indirect Heating-surface 537 

Boiler Heating-surface Required 538 

Proportion of Grate-surface to Radiator-surface 538 

st cam consumption in Car-heating 538 

Diameters of Steam Supply Mains 539 

Registers and Cold-air Ducts 539 

Physical Properties of Steam and Condensed Water 540 

Size of Steam-pipes for Heating 540 

Heating a Greenhouse by Steam 541 

Heating a Greenhouse by Hot Water 543 

Hot-water Heating 542 

Law of Velocity of Flow 542 

Proportions of Radiating Surfaces to Cubic Capacities 543 

Diameter of Main and Branch Pipes 543 

Rules for Hot- water Heating 544 

Arrangements of Mains 544 



CONTENTS. XV11 

PAGE 

Blower System of Heating and Ventilating. . 545 

Experiments with Radiators . . . 545 

Heating a Building to 70° F... 545 

Heating bj r Electricity 546 

WATER. 

Expansion of Water — 547 

Weight of Water at different temperatures ■ 547 

Pressure of Water due to its Weight 549 

Head Corresponding to Pressures 549 

Buoyancy 550 

Boiling-point 550 

Freezing-point 550 

Sea-water 549,550 

Ice and Snow 550 

Specific Heat of Water 550 

Compressibility of Water 551 

Impurities of Water 551 

Causes of Incrustation 551. 

Means for Preventing Incrustation 552 

Analyses of Boiler-scale 552 

Hardness of Water 553 

Purifying Feed-water 554 

Softening Hard Water 555 

Hydraulics. Flow of Water. 

Fomulae for Discharge through Orifices 555 

Flow of Water from Orifices 555 

Flow in Open and Closed Channels 557 

General Formulae for Flow 557 

Table Fall of_Feet per mile, etc 558 

Values of S/r for Circular Pipes 559 

Kutter's Formula . . 559 

Molesworth's Formula 562 

Bazin's Formula 563 

D'Arcy's Formula 563 

Older Formulae 564 

Velocity of Water in Open Channels 564 

Mean, Surface and Bottom Velocities 564 

Safe Bottom and Mean Velocities 565 

Resistance of Soil to Erosion 565 

Abrading and Transporting Power of Water 565 

Grade of Sewers 566 

Relations of Diameter of Pipe to Quantity discharged 566 

Flow of Water in a 20-inch Pipe 566 

Velocities in Smooth Cast-iron Water-pipes 567 

Table of Flow of Water in Circular Pipes 568-573 

Loss of Head 573 

Frici ional Heads at given rates of discharge 577 

Effect of Bend and Curves 578 

Hydraulic Grade-line 578 

Flow of Water in House-service Pipes 578 

Air-bound Pipes 579 

VerticalJets , 579 

Water Delivered through Meters 579 

Fire Streams 579 

Friction Losses in Hose 580 

Head and Pressure Losses by Friction . . 580 

Loss of Pressure in smooth 214-inch Hose . , 580 

Rated capacity of Steam Fire-engines 580 

Pressures required to throw water through Nozzles 581 

The Siphon 581 

Measurement of Flowing Water 582 

Piezometer 582 

Pitot Tube Gauge . 583 

The Venturi Meter 583 

Measurement of Discharge by means of Nozzles 584 



-Will CONTENTS. 

PAGE 

Flow through Rectangular Orifices 584 

Measurement of an Open Stream 584 

Miners' 1 Inch Measurements 585 

Flow of Water over Weirs 586 

Francis's Formula for Weirs 586 

Weir Table 587 

Baziu*s Experiments 587 

Water-power. 

Power of a Fall of Water 588 

Horse-power of a Running Stream 589 

Current Motors 589 

Horse-power of Water Flowing in a Tube 589 

Maximum Efficiency of a Long Conduit 589 

Mill-power 589 

Value of Water-power 590 

The Power of Ocean Waves 599 

Utilization of Tidal Power 600 

Turbine Wheels. 

Proportions of Turbines 591 

Tests of Turbines 596 

Dimensions of Turbines 597 

The Pelton Water-wheel 597 

Pumps. 

Theoretical capacity of a pump 601 

Depth of Suction 602 

Amount oi W r ater raised by a Single-acting Lift-pump 602 

Proportioning the Steam cylinder of a Direct-acting Pump 602 

Speed of Water through Pipes and Pump -passages 602 

Sizes of Direct-acting Pumps 603 

The Deane Pump 603 

Efficiency of Small Pumps 603 

The Wor thington Duplex Pump 604 

Speed of Piston 605 

Speed of Water through Valves 605 

Boiler feed Pumps 605 

Pump Valves 606 

Centrifugal Pumps 606 

Lawrence Centrifugal Pumps 607 

Efficiency of Centrifugal and Reciprocating Pumps 608 

Vanes of Centrifugal Pumps 609 

The Centrifugal Pump used as a Suction Dredge 609 

Duty Trials of Pumping Engines 609 

Leakage Tests of Pumps 611 

Vacuum Pumps 612 

The Pulsometer 612 

The Jet Pump 614 

The Injector 614 

Air-lift Pump 6l4 

The Hydraulic Ram 614 

Quantity of Water Delivered by the Hydraulic Ram 615 

Hydraulic Pressure Transmission. 

Energy of Water under Pressure 616 

Efficiency of Apparatus 616 

Hydraulic Presses 617 

Hydraulic Power in London 617 

Hydraulic Riveting Machines . 618 

Hydraulic Forging 618 

The Aiken Intensifier 019 

Hydraulic Engine 619 

FUEL. 

Theory of Combustion 620 

Total Heat of Combustion 621 



CONTENTS. XIX 

PAGE 

Analyses of Gases of Combustion , — 622 

Temperature of the Fire 622 

Classification of Solid Fuel 623 

Classification of Coals 624 

Analyses of Coals „ , 624 

Western Lignites 631 

Analyses of Foreign Coals 631 

Nixon's Navigation Coal 632 

Sampling Coal for Analyses 632 

Relative Value of Fine Sizes 632 

Pressed Fuel 632 

Relative Value of Steam Coals 633 

Approximate Heating Value of Coals 634 

Kind of Furnace Adapted for Different Coals 635 

Downward-draught Furnaces 635 

Calorimetric Tests of American Coals 636 

Evaporative Power of Bituminous Coals 636 

Weathering of Coal 637 

Coke 637 

Experiments in Coking 637 

Coal Washing 63S 

Recovery of By-products in Coke manufacture 638 

Making Hard Coke 638 

Generation of Steam from the Waste Heat and Gases from Coke-ovens. 638 

Products of the Distillation of Coal 639 

Wood as Fuel 639 

Heating Value of Wood , 639 

Composition of Wood 640 

Charcoal , 640 

Yield of Charcoal from a Cord of Wood 641 

Consumption of Charcoal in Blast Furnaces 641 

Absorption of Water and of Gases by Charcoal 641 

Composition of Charcoals 642 

Miscellaneous Solid Fuels 642 

Dust-fuel— Dust Explosions i 642 

Peat or Turf 643 

Sawdust as Fuel 643 

Horse-manure as Fuel 643 

Wet Tan-bark as Fuel 643 

Straw as Fuel 643 

Bagasse as Fuel in Sugar Manufacture , 643 

Petroleum. 

Products of Distillation 645 

Lima Petroleum 645 

Value of Petroleum as Fuel 645 

Oil vs. Coal as Fuel 646 

Fuel Gas. 

Carbon Gas 646 

Anthracite Gas , 647 

Bituminous Gas , 647 

Water Gas 648 

Prod ucer-gas from One Ton of Coal 649 

Natural Gas in Ohio and Indiana 649 

Combustion of Producer-gas 650 

Use of Steam in Producers 650 

Gas Fuel for Small Furnaces 651 

Illuminating Gas. 

Coal-gas 651 

Water-gas 652 

Analyses of Water-gas and Coal gas 653 

Calorific Equivalents of Constituents 654 

Efficiency of a Water-gas Plant 654 

Space Required for a Water-gas Plant 650 

Fuel-value of Illuminating-gas 656 



XX CONTENTS. 

PAGE 

Flow of Gas in Pipes 657 

Service for Lamps 658 

STEAM. 

Temperature and Pressure 659 

Total Heat 659 

Latent Heat of Steam 659 

Latent Heat of Volume 660 

Specific Heat of Saturated Steam 660 

Density and Volume 660 

Superheated Steam „ 661 

Regnault's Experiments 661 

Table of the Properties of Steam 662 

Flow of Steam. 

Napier's Approximate Rule 669 

Flow of Steam in Pipes 669 

Loss of Pressure Due to Radiation 671 

Resistance to Flow by Bends 672 

Sizes of Steam-pipes for Stationary Engines 673 

Sizes of Steam-pipes for Marine Engines 674 

Steani Pipes. 

Bursting-tests of Copper Steam-pipes 674 

Thickness of Copper Steam-pipes.. 675 

Reinforcing Steam-pipes 675 

Wire-wound Steam-pipes 675 

Riveted Steel Steam pipes 675 

Valves in Steam-pipes 675 

Flanges for Steam-pipe 676 

The Steam Loop 676 

Loss from an Uncovered Steam-pipe 676 

THE STEAM BOILER. 

The Horse- power of a Steam -boiler 677 

Measures for Comparing the Duty of Boilers 678 

Steam-boiler Proportions 678 

Heating-surface 678 

Horse-power, Builders' Rating 679 

Grate-surface 680 

Areas of Flues 680 

Air-passages Through Grate-bars * 681 

Performance of Boilers 681 

Conditions which Secure Economy 682 

Efficiency of a Boiler 683 

Tests of Steam-boilers 685 

Boilers at the Centennial Exhibition 685 

Tests of Tubulous Boilers 686 

High Rates of Evaporation — 687 

Economy Effected by Heating the Air 687 

Results of Tests with Different Coals 688 

Maximum Boiler Efficiency with Cumberland Coal 689 

Boilers Using Waste Gases 689 

Boilers for Blast Furnaces 689 

Rules for Conducting Boiler Tests 690 

Table of Factors of Evaporation 695 

Strength of Steam-boilers. 

Rules for Construction 700 

Shell-plate Formulae 701 

Rules for Flat Plates 701 

Furnace Formulae 702 

Material for Stays 703 

Loads allowed on Stays 703 

Girders 703 

Rules for Construction of Boilers in Merchant Vessels in U. S 705 



CONTENTS. XXI 

PAGE 

U. S. Rule for Allowable Pressures 706 

Safe-working Pressures 707 

Rules Governing Inspection of Boilers in Philadelphia 708 

Flues and Tubes for Steam Boilers 709 

Flat-stayed Surfaces 709 

Diameter of Stay-bolts.. 710 

Strength of Stays ." 710 

Stay-bolts in Curved Surfaces 710 

Boiler Attachments, Furnaces, etc. 

Fusible Plugs 710 

Steam Domes 711 

Height of Furnace 711 

Mechanical Stokers 711 

The Hawley Down-draught Furnace 712 

Under-feed Stokers 712 

Smoke Prevention , < 712 

Gas-fired Steam-boilers 714 

Forced Combustion 714 

Fuel Economizers 715 

Incrustation and Scale 716 

Boiler-scale Compounds 717 

Removal of Hard Scale 718 

Corrosion in Marine Boilers 719 

Use of Zinc 720 

Effect of Deposit on Flues 720 

Dangerous Boilers 720 

Safety Valves. 

Rules for Area of Safety-valves , 721 

Spring-loaded Safety-valves 724 

The Injectoi*. 

Equation of the Injector 725 

Performance of Injectors 726 

Boiler-feeding Pumps 726 

Feed -water Heaters. 

Strains Caused by Cold Feed-water „ 727 

Steam Separators. 

Efficiency of Steam Separators 728 

Determination of Moisture in Steam. 

Coil Calorimeter 729 

Throttling Calorimeters. 729 

Separating Calorimeters 730 

Identification of Dry Steam 730 

Usual Amount of Moisture in Steam 731 

Chimneys. 

Chimney Draught Theory 731 

Force or Intensity of Draught 732 

Rate of Combustion Due to Height of Chimney 733 

High Chimneys not Necessary : 734 

Heights of Chimneys Required for Different Fuels 734 

Table of Size of Chimneys 734 

Protection of Chimney from Lightning 736 

Some Tall Brick Chimneys 737 

Stability of Chimneys 738 

Weak Chimneys 739 

Steel Chimneys ... 740 

Sheet-iron Chimneys 741 

THE STEAM ENGINE. 

Expansion of Steam 742 

Mean and Terminal Absolute Pressures 743 



XXll CONTENTS. 

PAGE 

Calculation of Mean Effective Pressure 744 

Work of Steam in a Single Cylinder. . . 746 

Measures for Comparing the Duty of Engines 748 

Efficiency, Thermal Units per Minute 749 

Real Ratio of Expansion 750 

Effect of Compression 751 

clearance in Low and High Speed Engines 751 

Cylinder- condensation 752 

Water-consumption of Automatic Cut-off Engines 753 

Experiments on Cylinder-condensation 753 

Indicator Diagrams 754 

Indicated Horse-power 755 

Rules for Estimating Horse-power 755 

Horse-power Constant "756 

Errors of Indicators 756 

Table of Engine Constants 756 

To Draw Clearance on Indicator-diagram 759 

To Draw Hyperbolic Curve on Indicator-diagram 759 

Theoretical Water Consumption . . 700 

Leakage of Steam 761 

Compound Engines. 

Advantages of Compounding 762 

Woolf and Receiver Types of Engines 762 

Combined Diagrams 764 

Proportions of Cylinders inCompound Engines 7(55 

Receiver Space 766 

Formula for Calculating Work of Steam 767 

Calculation of Diameters of Cylinders 768 

Triple-expansion Engines 769 

Proportions of Cylinders 769 

Annular Ring Method 769 

Rule for Proportioning Cylinders . 771 

Types of Three-stage Expansion Engines 771 

Sequence of Cranks 772 

Velocity of Steam Through Passages 772 

Quadruple Expansion Engines 772 

Diameters of Cylinders of Marine Engines 773 

Progress in Steam-engines 773 

A Double-tandem Triple-expansion Engine 773 

Principal Engines, World's Columbian Exhibition, 1893 774 

Steam Engine Economy. 

Economic Performance of Steam Engines — 775 

Feed-water Consumption of Different Types 775 

Sizes and Calculated Performances of Vertical High-speed Engines 777 

Most Economical Point of Cut-off 777 

Type of Engine Used when Exhaust-steam is used for Heating 780 

Comparison of Compound and Single cylinder Engines 780 

Two-cylinder and Three-cylinder Engines 781 

Effect of Water in Steam on Efficiency . . 781 

Relative Commercial Economy of Compound and Triple-expansion 

Engines 781 

Triple-expansion Pumping-engines 782 

Test of a Triple-expansion Engine with and without Jackets 783 

Relative Economy of Engines under Variable Loads 783 

Efficiency of Non-condensing Compound Engines 784 

Economy of Engines under Varying Loads 784 

Steam Consumption of Various Sizes 785 

Steam Consumption in Small Engines 786 

Steam Consumption at Various Speeds 786 

Limitation of Engine Speed 787 

Influence of the Steam Jacket 7S7 

Counterbalancing Engines 788 

Preventing Vibrations of Engines 789 

Foundations Embedded in Air 789 

Cost of Coal for Steam-power 789 



COKTEKTS. 



Storing Steam Heat. ............. 789 

Cost of Steam-power 790 

Rotary Steam-engines. 

Steam Turbines 791 

The Tower Spherical Engine 792 

Dimensions of Parts of Engines. 

Cylinder , 792 

Clearance of Piston 792 

Thickness of Cylinder 792 

Cylinder Heads 794 

Cylinder-head Bolts 795 

The Piston 795 

Piston Packing-rings 796 

Fit of Piston-rod .. : 790 

Diameter of Piston-rods 797 

Piston-rod Guides 798 

The Connecting-rod 799 

Connecting-rod Ends 800 

Tapered Connecting-rods » 801 

The Crank-pin 801 

Crosshead-pin or Wrist-pin 804 

The Crank-arm 805 

The Shaft, Twisting Resistance ... . 806 

Resistance to Bending 808 

Equivalent Twisting Moment 808 

Fly-wheel Shafts 809 

Length of Shaft-bearings 810 

Crank-shafts with Centre-crank and Double-crank Arms 813 

Crank-shaft with two Cranks Coupled at 90°.... 814 

Valve-stem or Valve-rod 815 

Size of Slot-link 815 

The Eccentric 816 

The Eccentric-rod 816 

Reversing-gear ., 816 

Engine-frames or Bed-plates , 817 

Fly-wheels. 

Weight of Fly-wheels 817 

Centrifugal Force in Fly-wheels 820 

Arms of Fly-wheels and Pulleys 820 

Diameters for Various Speeds 821 

Strains in the Rims 822 

Thickness of Rims S23 

A Wooden Rim Fly-wheel 824 

Wire-wound Fly-wheels 824 

The Slide-valve. 

Definitions, Lap, Lead, etc 824 

Sweet's Valve-diagram ,. 826 

The Zeuner Valve-diagram 827 

Port Opening 828 

Lead 829 

Inside Lead 829 

Ratio of Lap and of Port-opening to Valve-travel 829 

Crank Angles for Connecting-rods of Different Lengths 830 

Relative Motions of Crosshead and Crank 831 

Periods of Admission or Cut-off for Various Laps and Travels 831 

Diagram for Port-opening, Cut-off, and Lap 832 • 

Piston-valves 834 

Setting the Valves of an Engine 834 

To put an Engine on its Centre 834 

Link-motion 834 

Governors. 

Pendulum or Fly-ball Governors 836 

To Change the Speed of an Engine 837 



XXIV CONTENTS. 

PAGE 

Fly-wheel or Shaft-governors 83S 

Calculation of Springs for Shaft-governors 838 

Condensers, Air-pumps, Circulating-pumps, etc. 

The Jet Condenser 839 

Ejector Condensers 840 

The Surface Condenser . 8-10 

Condenser Tubes 840 

Tube-plates 841 

Spacing of Tubes 841 

Quantity of Cooling Water 841 

Air-pump 841 

Area through Valve-seats 842 

Circulating-pump 843 

Feed-pumps for Marine-engines 843 

An Evaporative Surface Condenser 844 

Continuous Use of Condensing Water 844 

Increase of Power by Condensers 846 

Evaporators and Distillers 847 

GAS, PETROLEUM, AND HOT-AIR ENGINES. 

Gas-engines 847 

Efficiency of the Gas-engine 848 

Tests of the Simplex Gas Engine 848 

A 320-H.P. Gas-engine , 848 

Test of an Otto Gas-engine 849 

Temperatures and Pressures Developed 849 

Test of the Clerk Gas-engine 849 

Combustion of the Gas in the Otto Engine 849 

Use of Carburetted Air in Gas-engines 849 

The Otto Gasoline-engine 850 

The Priest-man Petroleum-engine 850 

Test of a 5-H.P. Priestman Petroleum-engine 850 

Naptha-engines 851 

Hot-air or Caloric-engines : 851 

Test of a Hot-air Engine 851 

LOCOMOTIVES. 

Efficiency of Locomotives and Resistance of Trains 851 

Inertia and Resistance at Increasing Speeds 853 

Efficiency of the Mechanism of a Locomotive 854 

Size of Locomotive Cylinders 854 

Size of Locomotive Boilers 855 

Qualities Essential for a Free-steaming Locomotive 855 

Wootten's Locomotive 855 

Grate-surface, Smoke-stacks, and Exhaust-nozzles for Locomotives 855 

Exhaust Nozzles 856 

Fire-brick Arches 856 

Size, Weight, Tractive Power, etc 856 

Leading American Types 858 

Steam Distribution for High Speed 858 

Speed of Railway Train's 859 

Dimensions of Some American Locomotives 859-862 

Indicated Water Consumption 862 

Locomotive Testing Apparatus 863 

Waste of Fuel in Locomotives 863 

Advantages of Compounding 863 

Counterbalancing Locomotives 864 

Maximum Safe Load on Steel Rails . . 865 

.Narrow-guage Railways 865 

Petroleum-burning Locomotives 865 

Fireless Locomotives 866 

SHAFTING. 

Diameters Resist Torsional Strain 867 

Deflection of Shafting 868 

Horse-power Transmitted by Shafting 869 

Table fur Laying Out Shafting 871 



CONTENTS. XXV 

PULLEYS. 

PAGE 

Proportions of Pulleys 873 

Convexity of Pulleys 874 

Cone or Step Pulleys 874 

BELTING. 

Theory of Belts and Bands 876 

Centrifugal Tension 876 

Belting Practice, Formulae for Belting 877 

Horse-power of a Belt one inch wide 878 

A. F. Nagle's Formula 878 

Width of Belt for Given Horse-power , . . . , . . . 879 

Taylor's Rules for Belting. . . 880 

Notes on Belting .■ 882 

Lacing of Belts 883 

Setting a Belt on Quarter-twist . . . . . 883 

To Find the Length of Belt 884 

To Find the Angle of the Arc of Contact 884 

To Find the Length of Belt when Closely Rolled 884 

To Find the Approximate Weight of Belts 884 

Relations of the Size and Speeds of Driving and Driven Pulleys 884 

Evils of Tight Belts 885 

Sag of Belts 885 

Arrangements of Belts and Pulleys 885 

Care of Belts , 886 

Strength of Belting 886 

Adhesion, Independent of Diameter. 886 

Endless Belts 886 

Belt Data 886 

Belt Dressing 887 

Cement for Cloth or Leather : 887 

Rubber Belting 887 

GEARING. 

Pitch, Pitch-circle, etc 887 

Diametral and Circular Pitch 888 

Chordal Pitch 889 

Diameter of Pitch-line of Wheels from 10 to 100 Teeth 889 

Proportions of Teeth 889 

Proportion of Gear-wheels 891 

Width of Teeth 891 

Rules for Calculating the Speed of Gears and Pulleys 891 

Milling Cutters for Interchangeable Gears 892 

Forms of the Teeth. 

The Cycloidal Tooth 892 

The Involute Tooth 894 

Approximation by Circular Arcs .896 

Stepped Gears 897 

Twisted Teeth 897 

Spiral Gears 897 

Worm Gearing 897 

Teeth of Bevel-wheels ... 898 

Annular and Differential Gearing 898 

Efficiency of Gearing 899 

Strength of Gear Teeth. 

Various Formulae for Strength 900 

Comparison ot Formulae — , 903 

Maximum Speed of Gearing „„. , 905 

A Heavy Machine-cut Spur-gear 905 

Frictional Gearing 905 

Frictional Grooved Gearing 906 

HOISTING. 

Weight and Strength of Cordage 906 

Working Strength of Blocks 900 



CONTENTS. 



Efficiency of Chain-blocks 907 

Proportions of Hooks 907 

Power of Hoisting Engines 908 

Effect of Slack Pope on Strain in Hoisting 908 

Limit of Depth for Hoisting 908 

Large Hoisting Records 908 

Pneumatic Hoisting 909 

Counterbalancing of Winding-engines 909 

Belt Conveyors 911 

Bands for Carrying Grain 911 

Cranes. 

Classification of Cranes 911 

Position of the Inclined Brace in a Jib Crane '. 912 

A Large Travelling-crane 912 

A 150-ton Pillar Crane 912 

Compressed-air Travelling Cranes 912 

Wire-rope Haulage. 

Self-acting Inclined Plane 913 

Simple Engine Plane 913 

Tail-rope System 913 

Endless Rope System 914 

Wire-rope Tramways 914 

Suspension Cableways and Cable Hoists 915 

Stress in Hoisting-ropes on Inclined Planes 915 

Tension Required to Prevent Wire Slipping on Drums 916 

Taper Ropes of Uniform Tensile Strength 916 

Effect of Various Sized Drums on the Life of Wire Ropes 917 

WIRE-ROPE TRANSMISSION. 

The Driving Wheels 918 

Horse-power of Wire-rope Transmission 919 

Durability of Wire Ropes 919 

Inclined Transmissions 919 

The Wire-rope Catenary 919 

Diameter and Weight of Pulleys for Wire-rope 921 

Table of Transmission of Power by Wire Ropes 921 

Long-distance Transmissions 921 

ROPE DRIVING. 

Formulae for Rope Driving 922 

Horse-power of Transmission at Various Speeds 924 

Sag of the Rope Between Pulleys 925 

Tension on the Slack Part of the Rope 925 

Miscellaneous Notes on Rope-driving 926 

FRICTION AND LUBRICATION. 

Coefficient of Friction 928 

Rolling Friction 988 

Friction of Solids 928 

Friction of Rest 928 

Laws of Unlubricated Friction 928 

Friction of Sliding Steel Tires 928 

Coefficient of Polling Friction 929 

Laws of Fluid Friction 929 

Angles of Repose 929 

Friction of Motion 929 

Coefficient of Friction of Journal 930 

Experiments on Friction of a Journal 931 

( ioefficients of Friction of Journal with Oil Bath 932 

( ^efficients of Friction of Motion and of Rest 932 

Value of Anti-friction Metals 932 

Cast-iron for Bearings 938 

Friction of Metal Under Steam-pressure 933 

Moi in's Laws of Friction , 933 



CONTENTS. XXVll 

PAGE 

Laws of Friction of well-lubricated Journals 934 

Allowable Pressures on Bearing-surface 935 

Oil-pressure in a Bearing 937 

Friction of Car-journal Brasses 937 

Experiments on Overheating of Bearings 938 

Moment of Friction and Work of Friction 938 

Pivot Bearings 939 

The Schiele Curve 939 

Friction of a Flat Pivot-bearing 939 

Mercury-bath Pivot 940 

Ball Bearings 940 

Friction Rollers 940 

Bearings for Very High Rotative Speed 941 

Friction of Steam-engines.. 941 

Distribution of the Friction of Engines 941 

Lubrication. 

Durability of Lubricants 942 

Qualifications of Lubricants 943 

Amount of Oil to run an Engine :•,, 943 

Examination of Oils 944 

Penna. R. R. Specifications 944 

Solid Lubricants 945 

Graphite, Soapstone, Metaline 945 

THE FOUNDRY. 

Cupola Practice 946 

Charging a Cupola 948 / 

Charges in Stove Foundries 949 / 

Results of Increased Driving 949 

Pressure Blowers 950 j 

Loss of Iron in Melting 950 

Use of Softeners 950 

Shrinkage of Castings 951 

Weight of Castings from Weight of Pattern 952 

Moulding Sand 952 

Foundry Ladles 952 \ 

THE MACHINE SHOP. 

Speed of Cutting Tools 953 

Table of Cutting Speeds 954 

Speed of Turret Lathes 954 

Forms of Cutting Tools 955 

Rule for Gearing Lathes , 955 

Change-gears for Lathes 956 

Metric Screw-threads. . 956 

Setting the Taper in a Lathe 956 

Speed of Drilling Holes 956 

Speed of Twist-drills 957 

Milling Cutters 957 

Speed of Cutters . .. 95S 

Results with Milling-machines 959 

Milling with or Against Feed l 900 

Milling-machine vs. Planer 960 

Power Required for Machine Tools 960 

Heavy Work on a Planer 960 

Horse-power to run Lathes 961 

Power used by Machine Tools 963 

Power Required to Drive Machinery 964 

Power used in Machine-shops 965 

Abrasive Processes. 

The Cold Saw 966 

Reese's Fusing-disk 966 

Cutting Stone with Wire. 966 

The Sand-blast 966 

Emery-wheels 967-969 

Grindstones 968-970 



XXVI 11 CONTENTS. 

Various Tools and Processes. 

PAGE 

Taps for Machine-screws 970 

Tap Drills 971 

Taper Bolts, Pins, Reamers, etc 972 

Punches, Dies, Presses 972 

Clearance Between Punch and Die 972 

Size of Blanks for Drawing-press 973 

Pressure of Drop-press 973 

Flow of Metals 973 

Forcing and Shrinking Fits 973 

Efficiency of Screws 974 

Powell's Screw-thread 975 

Proportioning Parts of Machine 975 

Keys for Gearing, etc 975 

Holding-power of Set-screws 977 

Holding-power of Keys.. 978 

DYNAMOMETERS. 

Traction Dynamometers 978 

The Prony Brake. 978 

The Alden Dynamometer 979 

Capacity of Friction-brakes 980 

Transmission Dynamometers 980 

ICE MAKING OR REFRIGERATING MACHINES. 

Operations of a Refrigerator-machine 981 

Pressures, etc., of Available Liquids 982 

Ice-melting Effect 983 

Ether-machines 983 

Air-machines. 983 

Ammonia Compression-machines 983 

Ammonia Absorption-machines 984 

Sulphur-dioxide Machines. 985 

Performance of Ammonia Compression-machines 986 

Economy of Ammonia Compression-machines 987 

Machines Using Vapor of Water 988 

Efficiency of a Refrigerating machine 988 

Test Trials of Ref rigerating-machines 990 

Temperature Range. , 991 

Metering the Ammonia 992 

Properties of Sulphur Dioxide and Ammonia Gas 992 

Properties of Brine used to absorb Refrigerating Effect 994 

Chloride-of-calciurn Solution 994 

Actual Performances of Refrigerating Machines. 

Performance of a 75-ton Refrigerating-machine. 994, 998 

Cylinder-heating 997 

Tests of Ammonia Absorption-machine 997 

Ammonia Compression-machine, Results of Tests 999 

Means for Applying the Cold 999 

Artificial Ice-manufacture. 

Test of the New York Hygeia Ice-making Plant 1000 

MARINE ENGINEERING. 

Rules for Measuring Dimensions and Obtaining Tonnage of Vessels 1001 

The Displacement of a Vessel 1001 

Coefficient of Fineness 1002 

Coefficient of Water-lines . . 1002 

Resistance of Ships 1002 

Coefficient of Performance of Vessels 1003 

Defects of the Common Formula for Resistance. . . 1003 

Rankine's Formula 1003 

Dr. Kirk's Method 1004 

To find the I. H. P. from the Wetted Surface 1005 

E. R. Mumford's Method 100G 

Relative Horse-power required for different Speeds of Vessels 1000 



CONTENTS. XXIX 

PAGE 

Resistance per Horse-power for different Speeds 100(5 

Results of Trials of Steam-vessels of Various Sizes 1007 

Speed on Canals, 1008 

Results of Progressive Speed-trials in Typical Vessels 1008 

Estimated Displacement, Horse-power, etc., of Steam-vessels of Various 

Sizes 1009 

The Screw-propeller. 

Size of Screw 1010 

Propeller Coefficients 1011 

Efficiency of the Propeller 1012 

Pitch-ratio and Slip for Screws of Standard Form 1012 

Results of Recent Researches 1013 

The Paddle-wheel. 

Paddle-wheel with Radial Floats , 1013 

Feathering Paddle-wheels 1013 

Efficiency of Paddle-wheels, 1014 

Jet-propulsion. 

Reaction of a Jet 1015 

Recent Practice in Marine Engines. 

Forced Draught 1015 

Boilers.. 1015 

Piston-valves 1016 

Steam-pipes 1016 

Auxiliary Supply of Fresh-water Evaporators 1016 

Weir's Feed-water Heater 1016 

Passenger Steamers fitted with Twin-screws 1017 

Comparative Results of Working of Marine-engine, 1872, 1881, and 1891.. 1017 

Weight of Three-stage Expansion-engines 1017 

Particulars of Three-stage Expansion-engines 1018 

CONSTRUCTION OF BUILDINGS 

Walls of Warehouses, Stores, Factories, and Stables 1019 

Strength of Floors, Roofs, and Supports 1019 

Columns and Posts 1019,1022 

Fireproof Buildings , 1020 

Iron and Steel Columns , 1020 

Lintels. Bearings, and Supports 1020 

Strains on Girders and Rivets 1020 

Maximum Load on Floors 1021 

Strength of Floors 1021 

Safe Distributed Loads on Southern-pine Beams 1023 

ELECTRICAL ENGINEERING. 

Standards of Measurement. 

C. G. S. System of Physical Measurement 1024 

Practical Units used in Electrical Calculations 1024 

Relations of Various Units 1025 

Equivalent Electrical and Mechanical Units 1026 

Analogies between Flow of Water and Electricity 1027 

Analogy between the Ampere and Miner's Inch 1027 

Electrical Resistance. 

Laws of Electrical Resistance 1028 

Equivalent Conductors . 1028 

Electrical Conductivity of Different Metals and Alloys 1028 

Relative Conductivity of Different Metals 1029 

Conductors and Insulators 1029 

Resistance Varies with Temperature 1029 

Annealing 1029 

Standard of Resistance of Copper Wire 1030 

Electric Currents. 

Ohm's Law 1030 

Divided Circuits . . . io31 



CONTENTS. 



Conductors in Series 10; 

Internal Resistance 10: 

Joint Resistance of Two Branches 10: 

KirchhofTs Laws ....... 10: 

Power of the Circuit 10: 

Heat Generated by a Current 10: 

Heating of Conductors 10.' 

Heating of Wires of Cables 10' 

Copper-wire Table 1034, 10, 

Heating of Coils 10; 

Fusion of Wires 10J 

Electric Transmission. 

Section of Wire required for a Given Current 10, c 

Constant Pressure 101: 

Three-wire Feeder 10.- 

Short-circuiting 10> 

Economy of Electric Transmission 10; 

Table of Electrical Horse-powers 10-) 

Wiring Formulae for Incandescent Lighting 104 

Wire Table for 100 and 500 Volt Circuits 104 

Cost of Copper for Long-distance Transmission 104 

Graphical Method of Calculating Leads '. 104 

Weight of Copper for Long-distance Transmission 104 

Efficiency of Long-distance Transmission 104 

Efficiency of a Combined Engine and Dynamo 104 

Electrical Efficiency of a Generator and Motor 104 

Efficiency of an Electrical Pumping Plant 104 

Electric Railways. 

Test of a Street Railway Plant 105 

Proportioning Boiler, Engine, and Generator for Power Stations 105 

Electric Lighting. 

Quantity of Energy Required to Produce Light 105 

Life of Incandescent Lamps 105 

Life and Efficienc}- Tests of Lamps - 105 

Street Lighting 105.' 

Lighting-power of Arc-lamps 105: 

Candle-power of the Arc light 105'. 

Electric Welding 105: 

Electric Heaters 105< 

Electric Accumulators or Storage-batteries. 

Use of Storage-batteries in Power and Light Stations. 105( 

Working Current of a Storage-cell — 105G 

Electro-chemical Equivalents 1051 

Electrolysis 1051 

Electro-magnets. 

Units of Electro-magnetic Measurement 105? 

Lines of Loops of Force 10.".'. 

Strength of an Electro-magnet 105J1 

Force in the Gap between Two Poles of a Magnet 106C 

The Magnetic Circuit 1060 

Determining the Polarity of Electro-magnets 1000 

Dynamo-Electric Machines. 

Kinds of Dynamo-electric Machines as regards Manner of Winding.. . . 1061 

( !urrent ( ie'nerated by a Dynamo-electric Machine 1061 

Torque of an Armature 1062 

Electro-motive Force of the Armature Circuit 100:2 

Si rength of the Magnetic Field 1063 

Application to Designing of Dynamos 1064 

Permeability '066 

Permissible Amperage for Magnets with Cotton-covered Wire 10(16, HK>K 

Formula- <>f Efficiency of Dynamos 1060 

The Electric Motor 10™ 



NAMES AND ABBREVIATIONS OF PERIODICALS 
AND TEXT-BOOKS FREQUENTLY REFERRED TO 
IN THIS WORK. 



Am. Mach. American Machinist. 

Bull. I. & S. A. Bulletin of the American Iron and Steel Association 
(Philadelphia). 

Burr's Elasticity and Resistance of Materials. 

Clark, R, T. D. D. K. Clark's Rules, Tables, and Data for Mechanical En- 
gineers. 

Clark, S. E. D. K. Clark's Treatise on the Steam-engine. 

Engg. Engineering (London). 

Eng. News. Engineering News. 

Engr. The Engineer (London). 

Fairbairn's Useful Information for Engineers. 

Flynn's Irrigation Canals and Flow of Water. 

Jour. A. C. I. W. Journal of American Charcoal Iron Workers' Association. 

Jour. F. I. Journal of the Franklin Institute. 

Kapp's Electric Transmission of Energy. 

Merriman's Strength of Materials. 

Lanza's Applied Mechanics. 

Proc. Inst. C. E. Proceedings Institution of Civil Engineers (London). 

Proc. Inst. M. E. Proceedings Institution of Mechanical Engineers (Lon- 
don). 

Peabody's Thermodynamics, 

Proceedings Engineers' Club of Philadelphia. 

Rankine, S. E. Rankine'sThe Steam Engine and other Prime Movers. 

Rankine's Machinery and Millwork. 

Rankine, R. T. D. Rankine's Rules, Tables, and Data. 

Reports of U. S. Test Board. 

Reports of U. S. Testing Machine at Watertown, Massachusetts. 

Rontgen's Thermodynamics. 

Seaton's Manual of Marine Engineering. 

Hamilton Smith, Jr.'s Hydraulics. 

The Stevens Indicator. 

Thompson's Dynamo-electric Machinery. 

Thurston's Manual of the Steam Engine. 

Thurston's Materials of Engineering. 

Trans. A. I. E. E. Transactions American Institute of Electrical Engineers. 

Trans. A. I. M. E. Transactions American Institute of Mining Engineers. 

Trans. A. S. C E. Transactions American Society of Civil Engineers. 

Trans. A. S. M. E. Transactions American Soc'ty of Mechanical Engineers . 

Trautwine's Civil Engineer's Pocket Book. 

The Locomotive (Hartford, Connecticut). 

Unwin's Elements of Machine Design. 

Weisbach's Mechanics of Engineering. 

Wood's Resistance of Materials, 

Wood's Thermodynamics. 

xxxi 



MATHEMATICS. 



Arithmetical and Algebraical Signs and Abbreviations. 

Z. angle. 

L right angle. 

_L perpendicular to. 

sin., sine. 

cos., cosine. 

tang., or tan., tangent. 

sec, secant. 

versin., versed sine. 

cot., cotangent. 

cosec, cosecant. 

covers., co- versed sine. 

In Algebra, the first letters of the 
^alphabet, a, b, c, d, etc., are gener- 
ally used to denote known quantities, 
and the last letters, iv, x, y, z, etc., 
unknown quantities. 

Abbreviations and Symbols com- 
monly used. 
d, differential (in calculus). 
/, integral (in calculus). 

/ , integral between limits a and 6. 

A, delta, difference. 

2. sigma, sign of summation. 

>7r, pi, ratio of circumference of circle 

to diameter = 3. 14159. 
g, acceleration due to gravity = 32.16 

ft. per sec. 

Abbreviations frequently used in 

this Book. 
L., 1., length in feet and inches. 
B., b., breadth in feet and inches. 
D., d., depth or diameter. 
H., h., height, feet and inches. 
T., t., thickness or temperature. 
V., v., velocity. 
F., force, or factor of safety, 
f., coefficient of friction. 
E., coefficient of elasticity. 
R., r., radius. 
W., w., weight. 
P., p., pressure or load. 
H.P., horse-power. 
I.H.P., indicated horse-power. 
B.H.P., brake horse-power. 
h. p., high pressure, 
i. p., intermediate pressure. 
1. p., low pressure. 
A.W. G., American Wire Gauge 

(Brown & Sharpe). 
B.W.G., Birmingham Wire Gauge. 
r. p. m., or revs, per miu., revolutions 
per minute. 



~r 


plus (addition). 


+ positive. 


— 


minus (subtraction). 


— 


negative. 


± 


plus or minus. 


T 


minus or plus. 


= 


equals. 


X 


multiplied by. 


al 


or a.b = a x 6. 


-5- 


divided by. 


/ 


divided by. 


b 


or a-b = a/b = a -s- b. 




2 2 


.2 


= — : .002= — -. 




10' 1000 


V 


square root. 


V 


cube root. 


V 


4th root. 




is to, :: so is, : to (proportion). 


2 


: 4 :: 3 : 6, as 2 is to 4 so is 3 to 6. 




ratio; divided by. 


2 


: 4, ratio of 2 to 4 = 2/4. 




therefore. 


> 


greater than. 


< 


less than. 


D 


square . 





round. 


° 


degrees, arc or thermometer. 


' 


minutes or feet. 


" 


seconds or inches. 


/// 


"' accents to distinguish letters, as 




a', a", a'". 


a l 


« 2 - <*3. «?,' a c- rea( i a SUD 1> « sub b, 




etc. ° 


C ) [ ] \ vincula, denoting 




that the numbers enclosed are 




to be taken together ; as, 




(a + b)c = 4 + 3 x 5 = 35. 


a 2 


a 3 , a squared, a cubed. 


u n 


a raised to the nth power. 


al 


= v<**i a * = v~" 3 - 



a-i = -,a- 2 = ~ 
a a 2 

10 9 = 10 to the 9th power = 1,000,000,- 

000. 
sin. a = the sine of a. 
sin.— 1 a = the arc whose sine is a. 
1 

sin. a- 1 = — 

sin. a. 
log. = logarithm. 

log. or hyp. log. = hyperbolic loga- 
rithm. 



MATHEMATICS. 



ARITHMETIC. 

The user of this book is supposed to have had a training in arithmetic as 
well as in elementary algebra. Only those rules are given here which are 
apt to be easily forgotten. 

GREATEST COMMON MEASURE, OR GREATEST 
COMMON DIVISOR OF TWO NUMBERS. 

Rule.- Divide the greater number by the less ; then divide the divisor 
by the remainder, and so on, dividing always the last divisor by the last 
remainder, until there is no remainder, and the last divisor is the greatest 
common measure required. 

LEAST COMMON MUL.TIFUE OF TWO OR MORE 
NUMBERS. 

Rule.— Divide the given numbers by any number that will divide the 
greatest number of them without a remainder, and set the quotients with 
the undivided numbers in a line beneath. 

Divide the second line as before, and so on, until there are no two numbers 
that can be divided ; then the continued product of the divisors and last 
quotients will give the multiple required. 

FRACTIONS. 

To reduce a common fraction to its lowest terms.— Divide 
both terms by their greatest common divisor: j| = f 
To change an improper fraction to a mixed number.— 

Divide the numerator by the denominator; the quotient is the whole number, 
and the remainder placed over the denominator is the fraction: - 3 5 9 = 9|. 

To change a mixed number to an improper fraction.— 
Multiply the whole number by the denominator of the fraction; to the prod- 
uct add the numerator; place the sum over the denominator: ]| = % 5 '. 

To express a whole number in the form of a fraction 
with a given denominator.— Multiply the whole number by the 
given denominator, and place the product over that denominator: 13 = 5 3 9 . 

To reduce a compound to a simple fraction, also to 
multiply fractions.— Multiply the numerators together for a new 
numerator and the denominators together for a new denominator: 

2 c 4 8 . 2 4 8- 
§ of- = -, also § Xg = - g . 

To reduce a complex to a simple fraction,— The numerator 
and denominator must each first be given the form of a simple fraction; 
then multiply the numerator of the upper fraction by the denominator of 
the lower for the new numerator, and the denominator of the upper by the 
numerator of the lower for the new denominator: 

1 = I = 1 - 1 
H | 12" 2' 

To divide fractions.— Reduce both to the form of simple fractions, 
invert the divisor, and proceed as in multiplication: 

3 ' 3 3 ' 3 3 4 12* 

Cancellation of fractions.— In compound or multiplied fractions, 
divide any numerator and any denominator by any number which will 
divide them both without remainder, striking out the numbers thus divided 
and setting down the quotients in their stead. 

To reduce fractions to a common denominator.— Reduce 
each traction to the form of a simple ft action; then multiply each numera- 



DECIMALS. 6 

tor by all the denominators except its own for the new numerators, and all 
the denominators together for the common denominator: 

1 1 3 _ 21 14 '18 

2 1 3' 7~ 42' 42' 42' 

To add fractions.— Reduce them to a common denominator, then 
add the numerators and place their sum over the common denominator: 



3 



- 21 + 14 + 18 _ 53 _ 

42 ~~ 42 " 



To subtract fractions. — Reduce them to a common denominator, 
subtract the numerators and place the difference over the common denomi- 
nator: 

1 _ 3 _ 7-6 _ J_ 

2 7 ~ 14 14" 



DECIMALS. 

To add decimals.— Set down the figures so that the decimal points 
are one above the other, then proceed as in simple addition: 18.75 -f- .012 = 
18.762. 

To subtract decimals.— Set down the figures so that the decimal 
points are one above the other, then proceed as in simple subtraction: 18.75 
- .012 = 18.738. 

To multiply decimals.— Multiply as in multiplication of whole 
numbers, then point off as many decimal places as there are in multiplier 
and multiplicand taken together: 1.5 X .02 = .030 = .03. 

To divide decimals.— Divide as in whole numbers, and point off in 
the quotient as many decimal places as those in the dividend exceed those 
in the divisor. Ciphers must be added to the dividend to make its decimal 
places at least equal those in the divisor, and as many more as it is desired 
to have in the quotient: 1.5 -=- .25 = 6. 0.1 -=- 0.3 = 0.10000 -*- 0.3 = 0.3333 + 

Decimal Equivalents of Fractions of One Incb. 



1-64 
1-32 
3-64 
1-16 



3-32 
7-64 

1-8 

9-64 
5-32 
11-64 
3-16 

13-64 
7-32 

15-64 
1-4 



015625 


17-64 


03125 


9-32 


046875 


19-64 


0625 


£-16 


078125 


21-64 


09375 


11-32 


109375 


23-64 


.125 


3-8 


.140625 


2.i-64 


.15625 


13-32 


.171875 


27-64 


.1875 


7-16 


.203125 


29-64 


.21875 


15-32 


.234375 


31-64 


.25 


1-2 



.265625 


33-64 


.28125 


17-32 


.296875 


35-64 


.3125 


9-16 


.328125 


37-64 


.34375 


19-32 


.359375 


39-64 


.375 


5-8 


.390625 


41-64 


.40625 


21-32 


.421875 


43-64 


.4375 


11-16 


.453125 


45-64 


.46875 


23-32 


.484375 


47-64 


.50 


3-4 



.515625 


49-64 


.53125 


25-32 


.546875 


51-64 


.5625 


13-16 


.578125 


53-64 


.59375 


27-32 


.609375 


55-64 


625 


7-8 


.640625 


57-64 


.65625 


29-32 


.671875 


59-64 


.6875 


15-16 


.703125 


61-64 


.71875 


31-32 


.734375 


63-64 


.75 


1 



.796875 
.8125 

.828125 
.84375 
.859375 

.875 

.890625 
.90625 
.921875 
.9375 

.953125 
.96875 
.984375 



To convert a common fraction into a decimal.— Divide the 

numerator by the denominator, adding to the numerator as many ciphers 
prefixed by a decimal point as are n^c^ssary to give the number of decimal 
places desired in the result: % = 1.00U0h-3 = 0.3333 -f. 

To convert a decimal into a common fraction.— Set down 
the decimal as a numerator, and place as the denominator 1 with as many 
ciphers annexed as there are decimal places in the numerator; erase the 



ARITHMETIC. 



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COMPOUND NUMBERS. 

decimal poiut in the numerator, and reduce the fraction thus formed to its 
lowest terms: 

M 25 1 ooco 3333 1 
■ 25 = 100 = 4 ; - 3333 = 1000() = 3' nearly - 

To reduce a recurring decimal to a common fraction.— 

Subtract the decimal figures that do not recur from the whole decimal in- 
cluding one set of recurring figures; set down the remainder as the numer- 
ator of the fraction, and as many nines as there are recurring figures, fol- 
lowed by as many ciphers as there are non-recurring figures, in the denom- 
inator. Thus: 

.79054054, the recurring figures being 054. 
Subtract 79 

78975 . . . t ... . . ,117 

n = (reduced to its lowest terms) — . 

COMPOUND OR DENOMINATE NUMBERS. 

Reduction descending - .— To reduce a compound number to a lower 
denomination. Multiply the number by as many units of the lower denomi- 
nation as makes one of the higher. 

3 yards to inches: 3 X 36 = 108 inches. 

.04 square feet to square inches: .04 X 144 = 5.76 sq. in. 

If the given number is in more than one denomination proceed in steps 
from the highest denomination to the next lower, and so on to the lowest, 
adding in the units of each denomination as the oper .tion proceeds. 

3 yds. 1 ft. 7 in. to inches: 3x3 = 9, + 1 = 10, 10 X 12 = 120, -f 7 = 127 in. 

Reduction ascending.— To express a number of a lower denomi- 
nation in terms of a higher, divide the number by the numb r of units of 
the lower denomination contained in one of the next higher; the quotient is 
in the higher denomination, and the remainder, if any, in the lower. 

127 inches to higher denomination. 

127-5-12 = 10 feet + 7 inches; 10 feet -=-3 = 3 yards + 1 foot. 

Ans. 3 yds. 1 ft. 7 in. 

To express the result in decimals of the higher denomination, divide the 
given number by the number of units of the given denomination contained 
in one of the required denomination, carrying the result to as many places 
of decimals as may be desired. 

127 inches to yards: 127 -s- 36 = 3J| = 3.5277 + yards. 
RATIO AND PROPORTION. 

Ratio is the relation of one number to another, as obtained by dividing 
one by the other. 

Ratio of 2 to 4, or 2 : 4 = 2/4 = 1/2. 
Ratio of 4 to 2, or 4 : 2 = 2. 

Proportion is the equality of two ratios. Ratio of 2 to 4 equals ratio 
of 3 to 6, 2/4 = 3/6: expressed thus, 2 : 4 : : 3 : 6; read, 2 is to 4 as 3 is to 6. 

The first and fourth terms are called the extremes or outer terms, the 
second and third the means or inner terms. 

The product of the means equals the product of the extremes: 

2 : 4 : : 3 : 6; 2 X 6 = 12; 3 X 4 = 12. 

Hence, given the first three terms to find the fourth, multiply the second 
and third terms together and divide by the first. 

4X3 
2 : 4 : : 3 : what number ? Ans. — - — = 6. 



ARITHMETIC. 



Algebraic expression of proportion.— a : b : : c : d; r = -,,ad 

. be, be , ad ad 

= be; from which a = — : d= — ; b = — ; c = -=—. 
<i a c b 

Mean proportional between two given numbers, 1st and 2d, is such 
a number that the ratio which the first bears to it equals the ratio which it- 
bears to the second. Thus, 2 : 4 : : 4 : 8; 4 is a mean proportional between 
2 and 8. To find the mean proportional between two numbers, extract the 
square root of their product. 

Mean proportional of 2 and 8 = V-> x 8 = 4. 

Single Rule of Three ; or, finding the fourth term of a proportion 
when three terms are given.— Rule, as above, when the terms are stated in 
their proper order, multiply the second by the third and divide by the first. 
The difficulty is to state the terms in their proper order. The term which is 
of the same kind as the required or fourth term is made the third : the first 
and second must be like each other in kind and denomination. To deter-, 
mine which is to be made second and which first requires a little reasoning. 
If an inspection of the problem shows that the answer should be greater 
than the third term, then the greater of the other two given terms should 
be made the second term — otherwise the first. Thus, 3 men remove 54 cubic 
feet of rock in a day; how many men will remove in the same time 10 cubic 
yards ? The answer is to be men— make men third term; the answer is to 
be more than three men, therefore make the greater quantity, 10 cubic 
yards, the second term ; but as it is not the same denomination as the other 
term it must be reduced, = 270 cubic feet. The proportion is then stated: 



54 : 270 : : 3 : x (the required number) ; 



'A 



The problem is more complicated if we increase the number of given 
terms. Thus, in the above question, substitute for the words " in the same 
time " the words " iu 3 days." First solve it as above, as if the work were 
to be done in the same time; then make another proportion, stating it thus: 
If 15 men do it in the same time, it will take fewer men to do it in 3 days; 
make 1 day the 2d term and 3 days the first term. 3 : 1 : : 15 men : 5 men. 

Compound Proportion, or Double Rule of Three.— By this 
rule are solved questions like the one just given, in which two or more stat- 
ings are required by the single rule of three. In it as in the single rule, 
there is one third term, which is of the same kind and denomination as the 
fourth or required term, but there may be two or more first and second 
terms. Set down the third term, take each pair of terms of the same kind 
separately, and arrange them as first and second by the same reasoning as 
is adopted in the single rule of three, making the greater of the pair the 
second if this pair considered alone should require the answer to be 
greater. 

Set down all the first terms one under the other, and likewise all the 
second terms. Multiply all the first terms together and all the second terms 
together. Multiply the product of all the second terms by the third term, and 
divide this product by the product of all the first terms. Example: If 3 men 
remove 4 cubic yards in one day, working 12 hours a day, how many men 
working 10 hours a day will remove 20 cubic yards in 3 days ? 

Yards 4 : 201 

Davs 3 : 1 : : 3 men. 

Hours 10 : 12 | 

Products 120 : 240 : : 3 : 6 men. Ans. 

To abbreviate by cancellation, any one of the first terms may cancel 
either the third or any of the second terms; thus. 3 in first cancels 3 in third, 
making it 1, 10 cancels into 20 making the latter 2, which into 4 makes it 2, 
which into 12 makes it 0, and the figures remaining are only 1 : 6 : : 1 : 6. 

INVOLUTION, OR POWERS OF NUMBERS. 

Involution is the continued multiplication of a number by itself a 
given number of times. The number is called the root, or first power, and 
the products are called powers. The second power is called the square and 



POWERS OF LUMBERS. 



the third power the cube. The operation may be indicated without being 
performed by writing a small figure called the index or exponent to the 
right of and a little above the root; thus, 3 3 = cube of 3, = 27. 

To multiply two or more powers of the same number, add their exponents; 
thus, 22 X 2 3 ' = 2 5 , or 4 X 8 = 32 = 2 5 . 

To divide two powers of the same number, subtract their exponents; thus, 



-22 = 2! =2; 22-4-2" = i 



~ 2* ' 



- -. The exponent may thus be nega- 
tive. 2 3 ■+- 2 3 = 2° = 1, whence the zero power of any number = 1. The 
first power of a number is the number itself. The exponent may be frac- 
tional, as 2*, 2s, which means that the root is to be raised to a power whose 
exponent is the numerator of the fraction, and the root whose sign is the 
denominator is to be extracted (see Evolution). The exponent may be a 
decimal, as 2 ' 5 , 2 1-5 ; read, two to the five-tenths power, two to the one and 
five-tenths power. These powers are solved by means of Logarithms (which 
see,). 

First Nine Powers of the First Nine Numbers. 



1st 


2d 


3d 


4th 


5th 


6th 


7th 


8th 


9th 


Pow'r 


Pow'r 


Power. 


Power. 


Power. 


Power. 


Power. 


Power. 


Power. 


1 


1 


1 


1 


1 


1 


1 


1 


1 


2 


4 


8 


16 


32 


64 


128 


256 


512 


3 


9 


27 


81 


243 


729 


2187 


6561 


19683 


4 


16 


64 


256 


1024 


4096 


16384 


65536 


262144 


5 


25 


125 


625 


3125 


15625 


78125 


390625 


1953125 


6 


36 


216 


1296 


7776 


46656 


279936 


1679616 


10077696 


7 


49 


343 


2401 


16807 


117649 


823543 


5764801 


40353607 


8 


64 


512 


4096 


32768 


262144 


2097152 


16777216 


134217728 


9 


81 


729 


6551 


59049 


531441 


4782969 


43046721 


387420489 



The First Forty Powers of 2. 



© 


1 


3 


© 


6 

3 












J 33 




> 


Ph 


> 


P- 


> 


Ph 


1 


9 


512 


18 


262144 


27 


2 


10 


1024 


19 


524288 


28 


4 


11 


2048 


20 


1048576 


29 


8 


12 


4096 


21 


2097152 


30 


16 


13 


8192 


22 


4194304 


31 


32 


14 


16384 


23 


8388608 


32 


64 


15 


32768 


24 


16777216 


33 


128 


IS 


65536 


25 


33554432 


34 


256 


17 


131072 


26 


67108864 


35 



©* 
3 


© 

is 




o 


£* 


Ph 


134217728 


36 


268435456 


37 


536870912 


38 


1073741824 


39 


2147483048 


40 


4294967296 




8589934592 




17179869184 




34350738368 





68719476736 
137438953472 



549755813888 
1099511627776 



EVOLUTION. 

Evolution is the finding of the root (or extracting the root) of any 
number the power of which is given. 

The sign \/ indicates that the square root is to be extracted: \' V V^ the 
cube root, 4th root, nth root. 

A. fractional exponent with 1 for the numerator of the fraction is also 
used to indicate that the operation of extracting the root is to be performed ; 
thus, 2*, 2^= V2,.V2. 

When the power of a number is indicated, the involution not bein^ per- 
formed, the extraction of any root of that power may also be indicated by 



8 ARITHMETIC. 

dividing the index of the power by the index of the root, indicating the 
division by a fraction. Thus, extract the square root of the 6th power of 2: 

4/06 = 2* = 2 ? = 2 U = 8. 

The Gth power of 2, as in the table above, is G4 ; |/64 = 8. 

Difficult problems in evolution are performed by logarithms, but the 
square root and the cube root may be extracted directly according to the 
rules given below. The 4th root is the square root of the square root. The 
Gth root is the cube root of the square root, or the square root of the cube 
root ; l he 9th root is the cube root of the cube root ; etc. 

To Extract the Square Root.— Point off the given number into 
periods of two places each, beginning: with units. If there are decimals, 
point these off likewise, beginning at the decimal point, and supplying 
as many ciphers as may be needed. Find the greatest number whose 
square is less than the first left-hand period, and place it as the first 
figure in the quotient. Subtract its square from the left-hand period, 
and to the remainder annex the two figures of the second period for 
a dividend. Double the first figure of the quotient for a partial divisor; 
find how many times the latter is contained in the dividend exclusive 
of the right-hand figure, and set the figure representing that number of 
times as the second figure in the quotient, and annex it to the right of 
the partial divisor, forming the complete divisor. Multiply this divisor by 
the second figure in the quotient and subtract the product from the divi- 
dend. To the remainder bring down the next period and proceed as before, 
in each case doubling the figures in the root alread}* found to cbtain the 
trial divisor. Should the product of the second figure in the root by the 
completed divisor be greater than the dividend, erase the second figure both 
from the quotient and from the divisor, and substitute the next smaller 
figure, or one small enough to make the product of the second figure by the 
divisor less than or equal to the dividend. 

3.1415926536|1.77245 + 

271274 
1 189 
34712515 
| 2429 
3542 18692 
7084 
35444 ! 160865 
1141776 



fir 



To extract the square root of a fraction, extract the root of numerator 

/i 2 
and denominator separately. A / - = -, or first convert the fraction into a 



imal 'f/|= V- 



decimal, {/ = i/.4444 + = .6G66 + . 



To Extract the Cube Root.— Point off the number into periods of 
3 figures each, beginning at the right hand, or unit's place. Point off deci- 
mals in periods of 3 figures from the decimal point. Find the greatest cube 
that does not exceed the left-hand period ; write its root as the first figure 
in the required root Subtract the cube from the left-hand period, and to 
the remainder bring down the next period for a dividend. 

Square the first figure of the root ; multiply bv 300, and divide the product 
into the dividend for a trial divisor ; write the quotient after the first figure 
Of I lie root as a trial second figure. 

( lomplete the divisor by adding to 3C0 times the square of the first figure, 
30 times the product of the first by the second figure, and the square ofthe 
second figure. Multiply this divisor by the second figure: subtract the 
product from the remainder. (Should the product be greater than the 
remainder, the last figure of the root and the complete divisor are too large ; 



CUBE ROOT. 



substitute for the last figure the next smaller number, and correct the trial 
divisor accordingly.) 

To the remainder bring down the next period, and proceed as before to 
find the third figure of the root— that is, square the two figures of the root 
already found; multiply by 300 for a trial divisor, etc. 

If at any time the trial divisor is less than the dividend, bring down an- 
other period of 3 figures, and place in the root and proceed. 

The cube root of a number will contain as many figures as there are 
periods of 3 in the number. 

Shorter Methods of Extracting the Cube Root.— 1. From 
Wentworth's Algebra: 



1,881,365,963,625 |12345 



22 



881 



72* 



300 
60 

-I 364 

[ 641 153365 

= 43200 

3 = 10801 

3 2 = 91 

44289 (■ 132867 







1089J 20498963 


123 2 


— 


45387001 


123 x 


4 = 


14760 1 




4* = 


16] 
4553476 }- 18213904 



6J_ 2285059625 

300 x 1234 2 = 456826 
30 x 1234 x 5 = 185100 
5 2 = 25 



457011925 



2285059625 



After the first two figures of the root are found the next trial divisor is 
found by bringing down the sum of the 60 and 4 obtained in completing the 
preceding divisor , then adding the three lines connected by the brace, and 
annexing two ciphers. This method shortens the work in long examples, as 
is seen in the case of the last two trial divisors, saving the labor of squaring 
123 and 1234. A further shortening of the work is made by obtaining the 
last two figures of the root by division, the divisor employed being three 
times the square of the part of the root already found; thus, after finding 
the first three figures: 

3 x 1232 = 45387|20498963|45.1 + 

~181548 

234416 

226935 

74813 

The error due to the remainder is not sufficient to change the fifth figure of 
the root. 
2. By Prof. H. A. Wood (Stevens Indicator, July, 1890): 

I. Having separated the number into periods of three figures each, count- 
ing from the right, divide by the square of the nearest root of the first 
period, or first two periods ; the nearest root is the trial root. 

II. To the quotient obtained add twice the trial root, and divide by 3. 
This gives the root, or first approximation. 

III. By using the first approximate root as a new trial root, and proceed- 
ing as before, a nearer approximation is obtained, which process may be 
repeated until the root has been extracted, or the approximation carried as 
far as desired. 



10 ARITHMETIC. 

Example.— Required the cube root of 20. The nearest cube to 20 is 3 
3 2 = 9) 20.0 
2.2 
6_ 
3)8.1 
2.7 IstT. R. 
2.7» = 7.29) 20.000 
2.743 
5.4 
3)8.143 
2.714, 1st ap. cube root. 
2.7142 = 7.365796)2 0.0000000 
2.7152534 
5.428 



3) 8.1432534 
2.7144178 2d ap. cube root. 

Remark. — In the example it will be observed that the second term, or 
first two figures of the root, were obtained by using for trial root the root of 
the first period. Using, in like manner, these two terms for trial root, we 
obtained four terms of the root ; and these four terms for trial root gave 
seven figures of the root correct. In that example the last figure should be 
7. Should we take these eight figures for trial root we should obtain at least 
fifteen figures of the root correct. 

To Extract a Higher Root than tlie Cube.— The fourth root is 
the square root of the square root ; the sixth root is the cube root of the 
square root or the square root of the cube root. Other roots are most con- 
veniently found by the use of logarithms. 

ALLIGATION 

shows the value of a mixture of different ingredients when the quantity 
and value of each is known. 

Let the ingredients be a, b, c, d, etc. , and their respective values per unit 
w, x, y, z, etc. 

A = the sum of the quantities = a + & + c-f d, etc. 

P = mean value or price per unit of A. 

AP = aw -f bx + cy -f- dz, etc. 

_ aw -\- bx -f cy -\- dz 

- — • 

PERMUTATION 

shows in how many positions any number of things may be arranged in a 
row; thus, the letters a, b, c may be arranged in six positions, viz. abc, acb, 
cab, cba, bac, bca. 

Rule.— Multiply together all the numbers used in counting the things; thus, 
permutations of 1,2, and 3 = 1x2x3 = 6. In how many positions can 9 
things in a row be placed ? 

1X2X3X4X5X6X7X8X9 = 362880. 
COMBINATION 

shows how many arrangements of a few things may be made out of a 
greater number. Rule : Set down that figure which indicates the greater 
number, and after it a series of figures diminishing by 1, until as many are 
set down as the number of the few things to be taken in each combination. 
Then beginning under the last one set down said number of few things ; 
then going backward set down a series diminishing by 1 until arriving under 
the first of the upper numbers. Multiply together all the upper numbers to 
form one product, and all the lower numbers to form another; divide the 
upper product by the lower one. 



GEOMETRICAL PROGRESSION. 11 

How many combinations of 9 things can be made, taking 3 in each com- 
bination ? 



1X2X3 



ARITHMETICAL. PROGRESSION, 

in a series of numbers, is a progressive increase or decrease in each succes- 
sive number by the addition or subtraction of tbe same amount at each step, 
as 1, 2, 3, 4, 5, etc., or 15, 12, 9, 6, etc. The numbers are called terms, and the 
equal increase or decrease the difference. Examples in arithmetical pro- 
gression may be solved by the following formulae : 

Let a = first term, I = last term, d = common difference, n — number of 
terms, s = sum of the terms: 

I = + (n - l)d, = -\d±^2ds + (a-\dY 



2s 

~ n ' 




s (11 — l)d 
~n + 2 


= g ?i[2a + (n - 


i)d], 


l~\-a Z 2 _ ft 2 
~ 2 + 2d ' 


n 
= d + «) 5 . 




= ^ n l- 21 - ( n - !)<*]• 


= I — (n - l)d, 




s (n - l)d 

~ n ~ 2 ' 


-\*W-( l + 


\dY -2ds, 
2 ' 


11 


I — a 
= n — l' 




2(s - an) 
"~ n(n — 1)' 


Z 2 - a 2 




2(wZ - s) 


~ 2s - I — a 




— ?i(n. — 1) 


I — a , ' 


d - 2a ± |/(2a - cZ) 2 + 8cZs 


*— +. 1 ' 


2d 


2s 


2l-\-d ± 4/(21 + d) 2 - 8cZs 



GEOMETRICAL. PROGRESSION, 

in a series of numbers, is a progressive increase or decrease in each suc- 
cessive number by the same multiplier or divisor at each step, as 1, 2, 4, 8, 
16. etc., or 243, 81, 27, 9, etc. The common multiplier is called the ratio. 

Let a = first term, I = last term, r = ratio or constant multiplier, n = 
number of terms, in = any term, as 1st, 2d, etc., s = sum of the terms: 

I = ar «-., = "+"■-■>* . =fr=i>^, 

r r n — 1 

log I = log a + (n — 1) log r, l(s - l) n ~ 1 - a(s - a) n ~ 1 = 0. 

m = ar m lm log m = log a -f (m - 1) log r. 

s = a ^' n ~ *> = rl — a ■ n ~\/] n - " ~\Z a n i r n _ z 

VZ - 1/a r n - '' n x 






ARITHMETIC 






(r - l).s 
r n - 1 ' 


log a 


= log I - (n - 1) log r 


s — a 




lo „ r _ log * - log a 

?i — 1 



- f r + g "" = 0. 
a a 

. l og * - l og a , j 
logr ' 

log Z — log a 



■ s-l ' 



l +; 



- z - 



"log( 



a) - log (s - Z) 



+ 1, 



log [a + (?' - l)s] - log a 

log r 
log Z - log [Ir - (r - l)s] 



log i 



+ 1- 



Population of the United States. 

(A problem in geometrical progression.,) 

Increase in 10 Annual Increase, 



STear. 


Population. 


Years, per 


cent. 


per cei 


1860 


31,443,321 








1870 


39,818,449* 


26.63 




2.39 


1880 


50.155,783 


25.96 




2.33 


1890 


62,622,250 


24.86 




2.25 


1895 


Est. 69,733,000 







Est. 2.174 


1900 


" 77,652,000 


Est. 24.0 




" 2.174 



Estimated Population in Each Tear from 1860 to 1899. 
(Based on the above rates of increase, in even thousands.) 



I860.... 


31,443 


1870.... 


39,818 


1880.... 


50,156 


1890.... 


62.622 


1861 .... 


32,195 


1871.... 


40,748 


1881.... 


51,281 


I 1891.... 


63.984 


1862.... 


32,964 


1872.... 


41,699 


1882 ... 


52,433 


1892... 


65.375 


1863.... 


33,752 


1873. .. 


42,673 


1883.... 


53.610 


1893.... 


66,797 


1864.... 


34,558 


1874... 


43,670 


1884 ... 


54,813 


1894... 


68,249 


1865.... 


35,384 


1875.... 


44,690 


1885.... 


56.043 


1895.... 


69.733 


1806.... 


36.229 


1876.... 


45.373 


1886 ... 


57,301 


1 1896... 


71,219 


1867.... 


37,095 


1877 .. 


46,800 


1887.... 


58.588 


1897.... 


72.799 


1868.... 


37,981 


1878 ... 


47,893 


1888.... 


59,903 


1898 ... 


74,382 


1869.... 


38,889 


1879 .... 


49,011 


1889... 


61,247 


1899.... 


75,999 



The above table has been calculated by logarithms, as follows : 



logr 



: log Z - log a -T- (n - 1), 



log m — log a + (»"• — 1) log r 



Pop. 1870 . 
" I860.. 



39.si8449 1og = 
31,443321 log = 



7.6000S41 
7.4975288 



diff. = .1025553 
1 = 10, diff. ■+- 10 = .01025553 
add log for 1860 7.497528S 



:10gZ 

: log a 



: log r, 

: log a 



log for 1861 = 
add again 



log for 1862 7.51803986 No. = 32,964 . 



7.50778433 No. 
.01025553 



Compound interest is a form of geometrical progression; the ratio 
being 1 plus the percentage. 



* Corrected by addition of 1.260.078, estimated error of the census of 1870, 
Census Bulletin' No. 1G, Dee. 12. 1890. 



DISCOUNT. 13 

INTEREST AND DISCOUNT. 

Interest is money paid for the use of money for a given time; the fac 

tors are : 

p, the sum loaned, or the principal: 
t, the time iu years; 
?•, the rate of interest; 

i, the amount of interest for the given rate and time; 
a = p + i = the amount of the principal with interest 
at the end of the time. 
Formulae : 

i = interest = principal X time X rate per cent = i — *-—• 



- amount = principal + intere 

lOOi 

= rate = — — ; 
pt 

100; _ ptr 

tr ~ a 100 ; 

lOOi 

pr ' 



100' 
ph\ 
~ 100' 



= principal = -^- = a - 
-. time = 



If the rate is expressed decimally as a per cent,— thus, 6 per cent = .06,— 
the formulas become 

i = prt; a — p(\ -4- rf): r = — ; t = — • ; p — -- = — — - 

Rules for finding Interest.— Multiply the principal by the rate 
per annum divided by 100, and by the time in years and fractions of a year. 
T , " , ' principal x rate per annum 

If the time is given in days, interest = ~~o"^ 77^ — • 

In banks interest is sometimes calculated on the basis of 360 days to a 
year, or 12 months of 30 days each. 
Short rules for interest at 6 per cent, when 360 days are taken as 1 year: 
Multiply the principal by number of days and divide by 6000. 
Multiply the principal by number of months and divide by 200. 
The interest of 1 dollar for one month is y% cent. 

Interest of 100 Dollars for Different Times and Rates. 



Time. 2% 


Z% 


4% 


h% 


6% 


8% 


10* 


1 year $2.00 


$3.00 


$4.00 


$5.00 


$6.00 


$8.00 


$10.00 


1 month .16§ 


.25 


.83* 


.41§ 


.50 


.66f 


.83i 


1 day = 3^ year .0055| 


.0083J 


.01 11 J 


.013Sf 


.0166§ 


.0222| 


.0277| 


1 day = 3 i 5 year .005179 


.008219 


.010959 


.013699 


.016438 


.0219178 


.0273973 



Discount i^ interest deducted for payment of money before it is due. 

True discount is the difference between the amount of a debt pay- 
able at a future date without interest and its present worth. The present 
worth is that sum which put at interest at the legal rate will amount to the 
debt when it is due. 

To find the present worth of an amount due at future date, divide the 
amount by the amount of $1 placed at interest for the given time. The dis- 
count equals the amount minus the present worth. 

What discount should be allowed on $103 paid six months before it is due, 
interest being 6 per cent per annum ? 

■ = $100 present worth, discount = 3.00. 

1 + 1 X .06 X - 

Bank discount is the amount deducted by a bank as interest on 
money loaned on promissory notes. It is interest calculated not on the act- 
ual sum loaned, but on the gross amount of the note, from which the dis- 
count is deducted in advance. It is also calculated on the basis of 360 days 
in the year, and for 3 (in some banks 4) days more than the time specified in 
the note. These are called days of grace, and the note is not payable till 
the last of these days. 



14 



ARITHMETIC. 



What discount will be deducted by a bank in discounting a note for $103 
payable 6 months hence ? Six months = 182 days, add 3 days grace = 185 
. ,103 X 185 

days -6uTio— = $3 - 1 ' 6 - 

Compound Interest.— In compound interest the interest is added to 
the principal at the end of each year, (or shorter period if agreed upon). 

Letp = the principal, r = the rate expressed decimally, n = no of years, 
and a the amount : 



: amount = p (1 + r) n ; r = rate 



s/-p- 



p = principal, = , no of years = ?i, 



log a — log p 
'' log (1 + r) 



Compound Interest Table. 

(Value of one dollar at compound interest, compounded yearly, at 
3, 4, 5, and 6 per cent, from 1 to 50 years.) 



1 


Z% 


4% 


b% 


G% 


03 


S% 


4% 


5* 


6£ 


£ 










s 










i 


1.03 


1.04 


1.05 


1.06 


16 


1.6047 


1.8730 


2.1829 


2.5403 


2 


1.0609 


1.0816 


1.1025 


1.1236 


17 


1.6528 


1.9479 


2.2920 


2.6928 


3 


1.0927 


1.1249 


1.1576 


1.1910 


18 


1.7024 


2.0258 


2.4066 


2.8543 


4 


1.1255 


1.1699 


1.2155 


1.2625 


19 


1.7535 


2.1068 


2 5269 


3.0256 


5 


1.1593 


1.2166 


1.2763 


1.3382 


20 


1.8061 


2.1911 


2.6533 


3.2071 


6 


1.1941 


1.2653 


1.3401 


1.4185 


21 


1.8603 


2.2787 


2.7859 


3.3995 


7 


1.2299 


1.3159 


1.4071 


1.5036 


22 


1.9161 


2.3699 


2.9252 


3.6035 


8 


1.2668 


1.3686 


1.4774 


1.5938 


23 


1.9736 


2.4647 


3.0715 


3.8197 


9 


1.3048 


1.4233 


1.5513 


1.6895 


24 


2.0328 


2.5633 


3.2251 


4 0487 


10 


1.3439 


1.4802 


1.6289 


1.7908 


25 


2.0937 


2.6658 


3.3864 


4.2919 


11 


1.3842 


1.5394 


1.7103 


1.8983 


30 


2.4272 


3.2434 


4.3219 


5 7435 


12 


1.4258 


1.6010 


1.7058 


2.0122 


35 


2.8138 


3.9160 


5.5166 


7.6861 


13 


1.4685 


1.6651 


1.8856 


2.1329 


40 


3.2620 


4.8009 


7 0100 


10.2858 


14 


1.5126 


1.7317 


1.9799 


2.2609 


45 


3.7815 


5.8410 


8.9850 


13.7646 


15 


1.5580 


1.8009 


2.0789 


2.3965 


50 


4.3838 


7.1064 


11.6792 


18.4190 



At compound interest at 3 per cent money will double itself in 23J4 years, 
at 4 per cent in 17% years, at 5 per cent in 14.2 j r ears, and at per cent in 
11.9 years. 

EQUATION OF PAYMENTS. 

By equation of payments we find the equivalent or average time in which 
one payment should be made to cancel a number of obligations due at dif- 
ferent dates; also the number of days upon which to calculate interest or 
discount upon a gross sum which is composed of several smaller sums pay- 
able at different dates. 

Rule.— Multiply each item by the time of its maturity in days from a 
fixed date, taken as a standard, and divide the sum of the products by the 
sum of the items: the result is the average time in days from the standard 
date. 

A owes B $100 due in 30 days. $200 due in 60 days, and $300 due in 90 clays. 
In how many days may the whole be paid in one sum of $600 ? 

100 x 30 + 200 x 60 + 300 x 90 = 42,000 ; 42,000 h- 600 = 70 days, am. 

A owes B $100, $200, and $300, which amounts are overdue respectively 30, 
60. and 90 days. If lie now pays the whole amount. $600, how many days* 
interest should he pay on that sum ? Ans. 70 days. 






ANNUITIES. 



15 



PARTIAL. PAYMENTS. 

To compute interest on notes and bonds when partial payments have been 
made: 

United States Rule.— Find the amount of the principal to the time 
of the first payment, and, subtracting the payment from it, find the amount 
of the remainder as a new principal to the time of the next payment. 

If the payment is less than the interest, find the amount of the principal 
to the time when the sum of the payments equals or exceeds the interest 
due, and subtract the sum of the payments from this amount. 

Proceed in this manner till the time of settlement. 

Note.— The principles upon which the preceding rule is founded are: 

1st. That payments must be applied first to discharge accrued interest, 
and then the remainder, if any, toward the discharge of the principal. 

2d. That only unpaid principal can draw interest. 

Mercantile Method.— When partial payments are made on short 
notes or interest accounts, business men commonly employ the following 
method : 

Find the amount of the whole debt to the time of settlement ; also find 
the amount of each payment from the time it was made to the time of set- 
tlement. Subtract the amount of payments from the amount of the debt; 
the remainder will be the balance due. 

ANNUITIES. 

An Annuity is a fixed sum of money paid yearly, or at other equal times 
agreed upon. The values of annuities are calculated by the principles of 
compound interest. 

1. Let i denote interest on $1 for a year, then at the end of a year the 
amount will be 1 + i. At the end of n years it will be (1 + i) n . 

2. The sum which in n years will amount to 1 is or (l + i)~ n , or the 

(1 -f i) n 
present value of 1 due in n years. 

3. The amount of an annuity of 1 in any number of years n is 

4. The present value of an annuity of 1 for any number of years n is 
I_ ■(!. + <)-» 



q + «) n -i 



5. The annuity which 1 will purchase for any number of years n is 



6. The annuity which would amount to 1 in n years is 



(l+i) w -l 



Amounts, Present Values, etc., at 5% Interest. 



Years 


(1) 


(2) 


(3) 


(4) 


(5) 


(6) 




(1 + 9" 


a + ir n 


(1 + t)» - 1 


l-d+i)- 1 * 


i 


i 








* 


t 


l-(l-fz)-" 


(l + i)»--l 


1 


1.05 


.952381 


1. 


.952381 


1.05 


1. 


2 


1.1025 


.907029 


2.05 


1.859410 


.537805 


.487805 


3 


1.157625 


.863838 


3.1525 


2.723248 


.367209 


.317209 


4 


1.215506 


.822702 


4.310125 


3.545951 


.282012 


.232012 


5 


1.276282 


.783526 


5.525631 


4.329477 


.230975 


.180975 


6 


1.340096 


.746215 


6.801913 


5.075692 


.197017 


.147018 


7 


1.407100 


.710681 


8.142008 


5.786373 


.172820 


.122820 


8 


1.477455 


.676839 


9.549109 


6.463213 


.154722 


.104722 


9 


1.551328 


.644609 


11.026564 


7.107822 


.140690 


.090690 


10 


1.628895 


.613913 


12.577893 


7.721735 


.129505 


.079505 



16 



ARITHMETIC. 







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WEIGHTS AND MEASURES. 



17 



TABLES FOR CALCULATING SINKING-FUNDS AND 
PRESENT VALUES. 

Engineers and others connected with municipal work and industrial enter- 
prises often find it necessary to calculate payments to sinking-funds which 
will provide a sum of money sufficient to pay off a bond issue or other debt 
at the end of a given period, or to determine the present value of certain 
annual charges. The accompanying tables were computed by Mr. John W. 
Hill, of Cincinnati, Eng'g News, Jan. 25, 1894. 

Table I (opposite page) shows the annual sum at various rates of interest 
required to net $1000 in from 2 to 50 years, and Table II shows the present 
value at various rates of interest of an annual charge of $1000 for from 5 to 
50 years, at five-year intervals and for 100 years. 



Table II. 



Capitalization of Annuity of SI 000 for 
from 5 to 100 Years. 



Rate of Interest, per cent. 



23^ 



3^ 



VA 


5 


5^ 


4,389.91 


4,329.45 


4,268.09 


7,912.67 


7,721.73 


7,537.54 


10,739.42 


10,379.53 


10,037.48 


13,007.88 


12,462.13 


11,950.26 


14,828.12 


14,093.86 


13,413.82 


16,288.77 


15,372.36 


14,533.63 


17,460.89 


16,374.36 


15,390.48 


18,401.49 


17,159.01 


16.044.92 


19,156.24 


17,773.99 


16,547.65 


19,761 93 


18,255.86 


16,931.97 


21,949.21 


19,847.90 


18,095.83 



■20,930 
23,145 
2c, 103 

"",368. 
36,614 



4,579 
8,530 
11,937 

14,877, 
17,413, 

19,600. 



24,518 
25,729 
31,598 



,514.92 4,451.68 
,316.45 8,110.74 
,517.23 11,118.06 
,218.is!l3,590.21 

,481.28,15,621.93 

,891.85,17,291.86 
,000.43 18,664.37 
,354.83 19,792.65 

,495.23 20,719.89 
,455.21 21,482.08 
,655.36,24,504.96 



WEIGHTS AND MEASURES. 
Long Measure.— Measures of Length. 

12 inches = 1 foot. 

3 feet = 1 yard. 

5> yards, or 16£ feet = 1 rod, pole, or perch. 

40 poles, or 220 yards = 1 furlong. 

8 furlongs, or 1760 yards, or 5280 feet = 1 mile. 
3 miles = league. 

Additional measures of length in occasional use : 1000 mils = 1 inch; 
4 inches = 1 hand ; 9 inches = 1 span ; 2| feet = 1 military pace ; 2 yards = 
1 fathom. 

Old Land Measure.— 7.92 inches = 1 link; 100 links, or 66 feet, or 4 
poles = 1 chain; 10 chains = 1 furlong; 8 furlongs = 1 mile; 10 square chains 
= 1 acre. 

Nautical Measure. 
6080.26 feet, or 1.15156 stat- 



ute miles 
3 nautical miles 
60 nautical miles, or 6 

statute miles 
360 degrees 



Y = 1 nautical mile, or knot.* 

~ 1 league. 
i = .1 degree (at the equator). 

= circumference of the earth at the equator. 



* The British Admiralty takes the round figure of 6080 ft. which is the 
length of the " measured mile" used in trials of vessels. The value varies 
from 6080.26 to 6088.44 ft. according to different measures of the earth's di- 
ameter. There is a difference of opinion among writers as to the use of the 
word " knot " to mean length or a distance— some holding that it should be 



18 ARITHMETIC. 



Square Measure.— Measures of Surface. 



144 square inches, or 183.35 circular 
inches 



: 1 square foot. 



9 square feet = l square yard. 

30^ square yards, or 272J square feet = 1 square rod, pole, or perch. 

40 square poles = 1 rood. 

4 roods, or 10 sq. chains, or 160 sq. ) 

poles, or 4840 sq. yards, or 43560 V = 1 acre, 

sq. feet, ) 
640 acres = 1 square mile. 

An acre equals a square whose side is 208.71 feet. 

A circular inch is the area of a circle 1 inch in diameter = 0.7854 square 
inch. 

1 square inch = 1.2732 circular inches. 

A circular mil is the area of a circle 1 mil, or .001 inch in diameter. 
1000 2 or 1,000,000 circular mils = 1 circular inch. 

1 square inch = 1,273,239 circular mils. 

The mil, and circular mil are used in electrical calculations involving 
the diameter and area of wires. 

Solid or Cubic Measure.— Measures of Volume. 

1728 cubic inches = 1 cubic foot. 

27 cubic feet = 1 cubic yard. 

1 cord of wood = a pile, 4x4x8 feet = 128 cubic feet. 

1 perch of masonry = 16| X 1| X 1 foot = 24f cubic feet. 





Liquid Measure. 


4 gills 




= 1 pint. 


2 pints 




= 1 quart. 


4 quarts 




- 1 o-Allon i U - S - 231 cubic inches - 

- l gallon -j Eng 277 2U cubic inches# 


31J gallons 




= 1 barrel. 


42 gallons 




= 1 tierce. 


2 barrels, or 


63 gallons 


— 1 hogshead. 


84 gallons, or 


2 tierces 


= 1 puncheon. 


2 hogsheads 


or 126 gallons 


= 1 pipe or butt. 



' 2 pipes, or 3 puncheons = 1 tun. 

The U. S. gallon contains 231 cubic iuches; 7.4805 gallons = 1 cubic foot. 
A cylinder 7 in. diam. and 6 in. high contains 1 gallon, very nearly, or 230.9 
cubic inches. The British Imperial gallon contains 277.274 cubic inches 
= 1.20032 U. S. gallon. 

The Miner's Inch.— (Western U. S. for measuring flow of a stream 
of water). 

The term Miner's Inch is more or less indefinite, for the reason that Cali- 
fornia water companies do not all use the same head above the centre of 
the aperture, and the inch varies from 1.36 to 1.73 cubic feet per minute 
each; but the most common measurement is through an aperture 2 inches 
high and whatever length is required, and through a plank 1£ inches thick. 
The lower edge of the aperture should be 2 inches above the bottom of the 
measuring-box, and the plank 5 inches high above the aperture, thus mak- 
ing a 6-inch head above the centre of the stream. Each square inch of this 
opening represents a miner's inch, which is equal to a flow of lb cubic feet 
per minute. 

Apothecaries' Fluid Measure. 
60 minims = 1 fluid drachm. 

8 drachms, or 437* grains, or 1.732 cubic inches = 1 fluid ounce. 

Dry Measure, U. S. 

2 pints = 1 quart. 
8 quarts = 1 peck. 
4 pecks = 1 bushel. 

used only to denote a rate of speed. The length between knots on the log 
line is T £<j of a nautical mile or 50.7 ft. when a half -minute glass is used; so 
that a speed of 10 knots is equal to 10 nautical miles per hour. 



WEIGHTS AND MEASURES. 19 

The standard U. S. bushel is the Winchester bushel, which is in cylinder 
form, 18£ inches diameter and 8 inches deep, and contains 2150.42 cubic 
inches. 

A struck bushel contains 2150.42 cubic inches — 1.2445 cu. ft. ; 1 cubic foot 
= 0.80356 struck bushel. A heaped bushel is a cylinder 18J inches diam- 
eter and 8 inches deep, with a heaped cone not less than 6 inches high. 
It is equal to 1£ struck bushels. 

The British Imperial bushel is based on the Imperial gallon, and contains 
8 such gallons, or 2218.192 cubic inches = 1.2837 cubic feet. The English 
quarter = 8 Imperial bushels. 

Capacity of a cylinder in U. S. gallons = square of diameter, in inches x 
height in inches X .0034. (Accurate within 1 part in 100,000.) 

Capacity of a cylinder in U. S. bushels = square of diameter in inches X 
height in inches X .0003652. 

Shipping Measure. 

Register Ton.— For register tonnage or for measurement of the entire 
internal capacity of a vessel : 

100 cubic feet = 1 register ton. 

This number is arbitrarily assumed to facilitate computation. 
Shipping Ton. — For the measurement of cargo : 

( 1 U. S. shipping ton. 
40 cubic feet = < 31.16 Imp. bushels. 
( 32.143 U. S. " 
( 1 British shipping ton. 
42 cubic feet = ■{ 32.719 Imp. bushels. 
(33.75 U.S. 
Carpenter' 1 s Rule.— Weight a vessel will cany = length of keel X breadth 
at main beam x depth of hold in feet h-95 (the cubic feet allowed for a ton). 
The result will be the tonnage. For a double-decker instead of the depth 
of the hold take half the breadth of the beam. 

Measures of Weight.— Avoirdupois, or Commercial 
Weight. 

16 drachms, or 437.5 grains = 1 ounce, oz. 
16 ounces, or 7000 grains = 1 pound, lb. 
28 pounds = 1 quarter, qr. 

4 quarters = 1 hundredweight, cwt. = 112 lbs. 

20 hundred weight = 1 ton of 2240 pounds, or long ton. 

2000 pounds = 1 net, or short ton. 

2204.6 pounds — 1 metric ton. 

1 stone = 14 pounds ; 1 quintal = 100 pounds. 

Troy Weight. 

24 grains = 1 pennyweight, dwt. 

20 pennyweights = 1 ounce, oz. = 480 grains. 

12 ounces = 1 pound, lb. = 5760 grains. 

Troy weight is used for weighing gold and silver. The grain is the same 
in Avoirdupois, Troy, and Apothecaries' weights. A carat, used in weighing 
diamonds = 3.168 grains = .205 gramme. 

Apothecaries' Weight. 

20 grains = 1 scruple, 3 
3 scruples = 1 drachm, 3 = 60 grains. 
8 drachms = 1 ounce, § = 480 grains. 

12 ounces = 1 pound, lb. = 5760 grains. 

To determine whether a balance has unequal arms. — 

Afrer weighing an article and obtaining equilibrium, transpose the article 
and the weights. If the balance is true, it will remain in equilibrium ; if 
untrue, the nan suspended from the loneer arm will descend. 

To weigh correctly on an incorrect balance.— First, by 
substitution. Put the article to be weighed in one pan of the balance and 



20 ARITHMETIC. 

counterpoise it by any convenient heavy articles placed on the other pan. 
Remove the article to be weighed and substitute for it standard weights 
until equipoise is again established. The amount of these weights is the 
weight of the article. 

Second, by transposition. Determine the apparent weight of the article 
as usual, then its apparent weight after transposing the article and the 
weights. If the difference is small, add half the difference to the smaller 
of the apparent weights to obtain the true weight. If the difference is 2 
per cent the error of this method is 1 part in 10,000. For larger differences, 
or to obtain a perfectly accurate result, multiply the two apparent weights 
together and extract the square root of the product. 

Circular Measure. 

60 seconds, " = 1 minute, '. 
60 minutes, ' = 1 degree, °. 
90 degrees = 1 quadrant. 
360 ■' = circumference. 

Time. 

60 seconds = 1 minute. 
60 minutes = 1 hour. 
24 hours = 1 day. 
7 days = 1 week. 
365 days, 5 hours, 48 minutes, 48 seconds = 1 year. 

By the Gregorian Calendar every year whose number is divisible by 4 is a 
leap year, and contains 366 days, the other years containing 365 days, ex- 
cept rhat the centesimal years are leap years only when the number of the 
year is divisible by 400. 

The comparative values of mean solar and sidereal time are shown by the 
following relations according to Bessel : 

365.24292 mean solar days = 366.24222 sidereal days, whence 
1 mean solar day = 1.00273791 sidereal days; 
1 sidereal day = 99726957 mean solar day; 
24 hours mean solar time = 24 h 3 m 56 8 .555 sidereal time; 
24 hours sidereal time — 23 h 56 m 4 3 .091 mean solar time, 

whence 1 mean solar day is 3 m 55 s .91 longer than a sidereal day, reckoned in 
mean solar time. 

BOARD AND TIMBER MEASURE. 

Board Measure. 

In board measure boards are assumed to be one inch in thickness. To 
obtain the number of feet board measure (B. M.) of a board or stick of 
square timber, multiply together the length in feet, the .breadth in feet, and 
the thickness in inches. 

To compute the measure or surface in square feet.— When 
all dimensions are in feet, multiply the length by the breadth, and the pro- 
duct will give the surface required. 

When either of the dimensions are in inches, multiply as above and divide 
the product by 12. 

When all dimensions are in inches, multiply as before and divide product 
by 144. 

Timber Measure. 

To compute the volume of round timber.— When all dimen- 
sions are in feet, multiply the length by one quarter of the product of the 
mean girth and diameter, and the product will give the measurement in 
cubic feet. When length is given in feet and girth and diameter in inches, 
divide the product by 144 ; when all the dimensions are in inches, divide by 
1728. 

To compute the volume of square timber.— When all dimen- 
sions are in feet, multiply together the length, breadth, and depth; the 
product will be the volume in cubic feet. When one dimension is given in 
inches, divide by 12; when two dimensions are in inches, divide by 144; w T hen 
all three dimensions are in inches, divide by 1728. 



WEIGHTS AND MEASURES. 



21 



Contents in Feet of Joists, Scantling, and Timber, 

Length in Feet. 



Feet Board Measure. 



2X4 


8 


9 


11 


12 


13 


15 


16 


17 


19 


20 


2X6 


12 


14 


16 


18 


20 


22 


24 


26 


28 


30 


2X8- 


16 


19 


21 


24 


27 


29 


32 


35 


37 


40 


2 X 10 


20 


23 


27 


30 


33 


37 


40 


43 


47 


50 


2 X 12 


24 


28 


32 


36 


40 


44 


48 


52 


56 


60 


2 X 14 


28 


33 


37 


42 


47 


51 


56 


61 


65 


70 


3X8 


24 


28 


32 


36 


40 


44 


48 


52 


56 


60 


;•{ x io 


30 


35 


40 


45 


50 


55 


60 


65 


70 


75 


3 X 12 


36 


42 


48 


54 


60 


66 


72 


78 


84 


90 


3X14 


42 


49 


56 


63 


70 


77 


84 


91 


98 


105 


4X4 


16 


19 


21 


24 


27 


29 


32 


35 


37 


40 


4X6 


24 


23 


32 


36 


40 


44 


48 


52 


56 


60 


4X8 


32 


37 


43 


48 


53 


59 


64 


69 


75 


80 


4 X 10 


40 


47 


53 


60 


67 


73 


80 


87 


93 


100 


4 X 12 


48 


56 


64 


72 


80 


88 


96 


104 


112 


120 


4 X 14 


56 


65 


75 


84 


93 


103 


112 


121 


131 


140 


6X6 


36 


42 


48 


54 


60 


66 


72 


78 


84 


90 


6X8 


48 


56 


64 


72 


80 


88 


96 


104 


112 


120 


6 X 10 


60 


70 


80 


90 


100 


110 


120 


130 


140 


150 


6 X 12 


72 


84 


96 


108 


120 


132 


144 


156 


168 


180 


6X 14 


84 


98 


112 


126 


140 


154 


168 


182 


196 


210 


8X8 


64 


75 


85 


96 


107 


117 


128 


139 


149 


160 


8 X 10 


80 


93 


107 


120 


133 


147 


160 


173 


187 


200 


8 X 12 


96 


112 


128 


144 


160 


176 


192 


208 


224 


240 


8X 14 


112 


131 


149 


168 


187 


205 


224 


243 


261 


280 


10 X 10 


100 


117 


133 


150 


167 


183 


200 


217 


233 


250 


10 X 12 


120 


140 


160 


180 


200 


220 


240 


260 


280 


300 


10 X 14 


140 


163 


187 


210 


233 


257 


280 


303 


327 


350 


12 X 12 


144 


168 


192 


216 


240 


264 


288 


312 


336 


3(50 


12 X 14 


168 


196 


224 


252 


280 


308 


336 


364 


392 


420 


14 X 14 


196 


229 


261 


294 


327 


359 


392 


425 


457 


490 



FRENCH OR METRIC MEASCJRES„ 

The metric unit of length is the metre = 39.37 inches. 
The metric unit of weight is the gram = 15.432 grains. 

The following prefixes are used for subdivisions and multiples; Milli = 1055, 
Centi = Tt hj, Deci = r V, Deca = 10, Hecto = 100, Kilo = 1000, Myria = 10,000. 

FRENCH AND BRITISH (AN© AMERICAN) 
EQUIVALENT MEASURES. 

Measures of Length. 

French. British and U. S. 

1 metre = 39.37 inches, or 3.28083 feet, or 1.09361 yards. 

.3048 metre = 1 foot. 

1 centimetre = .3937 inch. 
2.54 centimetres = 1 inch. 

1 milinietre = .03937 inch, or 1/25 inch, nearly. 
2.54 millimetres = 1 inch. 

1 kilometre = 1093.61 yards, or 0.62137 mile. 



22 ARITHMETIC. 

Measures of Surface. 

French. British. 

1 snnarp metre - J 10 -'' 64 s q uare f eet, 

1 squaie metre _ ^ ] 196 square yards# 

.836 square metre = 1 square yard. 

.0929 square metre = 1 square foot. 

1 square centimetre = .155 square inch. 
6.452 square centimetres = 1 square inch. 

1 squaie millimetre = .00155 square inch. 
645.2 square millimetres = 1 square inch. 
1 centiare = 1 sq. metre = 10 764 square feet. 

1 are = 1 sq. decametre = 1076.41 " " 

1 hectare = 100 ares = 107641 " " = 2.4711 acres. 

1 sq. kilometre = .386109 sq. miles = 247.11 " 

1 sq. myriametre =38.6109" " 

Of Volume. 

French. British and U. S. 

1 pnhir metre - i 35 - 314 cubic feet ' 

1 cubic metre _ -J 1<gog cuWc yardg 

.7645 cubic metre = 1 cubic yard. 

.02832 cubic metre = 1 cubic foot. 

1 cubic decimetre -}«;& *£&£&«- 

28.32 cubic decimetres = 1 cubic foot. 
1 cubic centimetre = .061 cubic inch. 

16.387 cubic centimetres = 1 cubic inch. 
1 cubic centimetre = 1 millilitre = .061 cubic inch. 
1 centilitre = = .610 " 

1 decilitre = = 6.102 " 

1 litre = 1 cubic decimetre = 61.023 " " = 1.05671 quarts, IT. S. 

1 hectolitre or decistere = 3.314 cubic feet = 2.8375 bushels, " 

1 stere, kilolitre, or cubic metre = 1.308 cubic yards = 28.37 bushels, " 

Of Capacity. 

French. British and U. S. 

f 61.0-23 cubic inches, 

1 life (= 1 cubic decimetre) = \ : ~ b * g*^ 

1.2.202 pounds of water at 62° F. 
28.317 litres = 1 cubic foot. 

4.543 litres = 1 gallon (British). 

3.785 litres = 1 gallon (American). 

Of Weight. 

French. British and U. S. 

1 gramme = 15.432 grains. 

.0648 gramme = 1 grain. 

28.35 gramme = 1 ounce avoirdupois. 

1 kilogramme = 2.2046 pounds. 

.4536 kilogramme = 1 pound. 

1 tonne or metric ton = I ^ ^on °f 2240 pounds, 
1000 kilogrammes = ] ^4 Grounds. 

1.010 metric tons = ] t t f 9240 nol]nds 

1016 kilogrammes = } l ton ot .240 pounds. 

Mr. O. H. Titmann, in Bulletin No. 9 of the IT. S. Coast and Geodetic Sur- 
vey, discusses the work of various authorities who have compared the yard 
and the metre, and by referring all the observations to a common standard 
has succeeded in reconciling the discrepancies within very narrow limits. 
The following are his results for the number of inches in a metre according 
to the comparisons of the authorities named: 

1817. Hassler 39.36994 inches. 

1818. Kater 39.36990 " 

1835. Baily 39.36973 " 

1866. Clarke 39.36970 " 

1885. Comstock 39.36984 " 

The mean of these is 39.36982 " 



METRIC WEIGHTS AKD MEASURES. 23 

METRIC CONVERSION TABLES. 

The following tables, with the subjoined memoranda, were published in 
1890 by the United States Coast and Geodetic Survey, office of standard 
weights and measures, T. C. Mendenhall, Superintendent. 

Tables for Converting U. S. Weiglits and Measures- 
Customary to Metric 





Inches to Milli- 
metres. 


Feet to Metres. 


Yards to Metres. 


Miles to Kilo- 
metres. 


1 = 

2 = 

3 = 

4 = 

5 = 

6 = 

7 = 

8 = 

9 = 


25.4000 
50.8001 
76.2001 
101.6002 
127.0002 

152.4003 
177.8003 
203.2004 
228.6004 


0.304801 
0.609601 
0.914402 
1.219202 
1.524003 

1.828804 
2.133604 
2.438405 
2.743205 


0.914402 
1.828804 
2.743205 
3.657607 
4.572009 

5.486411 
6.400813 
7.315215 
8.229616 


1.60935 
3.21869 
4.82804 
6.43739 
8.04674 

9.65608 
11.26543 
12.87478 
14.48412 



SQUARE. 





Square Inches to 
Square Centi- 
metres. 


Square Feet to 
Square Deci- 
metres. 


Square Yards to 
Square Metres. 


Acres to 
Hectares. 


1 = 


6.452 


9.290 


0.836 


0.4047 


2 = 


12.903 


18.581 


1.672 


0.8094 


3 = 


19.355 


27.871 


2.508 


1.2141 


4 = 


25.807 


37.161 


3.344 


1.6187 


5 = 


32.258 


46.452 


4.181 


2.0234 


6 = 


38.710 


55.742 


5.017 


2.4281 


7 = ■ 


45.161 


65.032 


5.853 


2.8328 


8 = 


51.613 


74.323 


6.689 


3.2375 


9 = 


58.065 


83.613 


7.525 


3.6422 





Cubic Inches to 
Cubic Centi- 
metres. 


Cubic Feet to 


Cubic Yai*ds to 


Bushels to 




Cubic Metres. 


Cubic Metres. 


Hectolitres. 


1 = 


16.387 


0.02832 


0.765 


0.35242 


2 = 


32.774 


0.05663 


1.529 


0.70485 


3 = 


49.161 


0.08495 


2.294 


1.05727 


4 = 


65.549 


0.11327 


3.058 


1.40969 


5 = 


81.936 


0.14158 


. 3.82.3 


1.76211 


fi = 


98.323 


0.16990 


4.587 


2.11454 


7 = 


114.710 


0.19822 


5.352 


2.46696 


8 = 


131.097 


0.22654 


6.116 


2.81938 


9 = 


147.484 


0.25485 


6.881 


3.17181 



u 



ARITHMETIC. 
CAPACITY. 





Fluid Drachms 










to Millilitres or 


Fluid Ounces to 


Quarts to Litres. 


Gallons to Litres. 




Cubic Centi- 


Millilitres. 








metres. 








1 = 


3.70 


29.57 


0.-94636 


3.78544 


2 = 


7.39 


59.15 


1.89272 


7.57088 


3 = 


11.09 


88.72 


2.83908 


11.35632 


4 = 


14.79 


118.30 


3.78544 


15.14176 


5 = 


18.48 


147.87 


4.73180 


18.92720 


6 = 


22.18 


177.44 


5.67816 


22.71264 


7 = 


25.88 


207.02 


6.62452 


26.49808 


8 = 


29.57 


236.59 


7.57088 


30.28352 


9 = 


33.28 


266.16 


8.51724 


34.06896 





Grains to Milli- 
grammes. 


Avoirdupois 
Ounces to 
Grammes. 


Avoirdupois 
Pounds to Kilo- 
grammes. 


Troy Ounces to 
Grammes. 


1 = 

3 = 

4 = 

5 = 

6 = 

7 = 

8 = 

9 = 


64.7989 
129.5978 
194.3968 
259.1957 
323.9946 

388.7935 
453.5924 
518.3914 
583.1903 


28.3495 
56.6991 
85.0486 
113.3981 
141.7476 

170.0972 
198.4467 
226.7962 
255.1457 


0.45359 
0.90719 
1.36078 
1.81137 
2.26796 

2.72156 
3.17515 

3.62874 
4.08233 


31.10348 
62.20696 
93.31044 
124.41392 
155.51740 

186.62089 
217.72437 
248.82785 
279.93133 



1 chain = 20.1169 metres. 

1 square mile = 259 hectares. 

1 fathom = 1.829 metres. 

1 nautical mile = 1853.27 metres. 

1 foot = 0.304801 metre. 

1 avoir, pound = 453.5924277 gram. 

15432.35639 grains = 1 kilogramme. 



Tables for Converting IT. S. Weights and Measures- 
Metric to Customary. 







LINEAR. 






Metres to 
Inches. 


Metres to 
Feet. 


Metres to 
Yards. 


Kilometres to 
Miles. 


1 = 

2 = 

3 = 

4 = 

5 = 

6 = 

7 = 

8 = 

9 = 


39.3700 
78.7400 
118.1100 
157.4800 
196.8500 

236.2200 
275.5900 
314.9600 
354.3300 


3.28083 
6.56167 
9.84250 
13.12333 
16 ; 40417 

19.68500 
22.96583 
26.24667 
29.52750 


1.093611 
2.187222 
3.280883 
4.374444 
5.468056 

6.561667 
7.655278 

8.748889 
9.842500 


0.62137 
1.24274 

1 86411 
2.48548 
3.10685 

3.72822 
4.34959 
4.97096 
5.59233 



METRIC CONVERSION TABLES. 
SQUARE. 



25 





Square Centi- 
metres to 
Square Inches. 


Square Metres 
to Square Feet. 


Square Metres 
to Square Yards. 


Hectares to 
Acres. 


1 = 


0.1550 


10.764 




1.196 


2.471 


2 = 


0.3100 


21.528 




2.392 


4.942 


3 = 


0.4650 


32.292 




3.588 


7.413 


4 = 


o: 6<>oo 


43.055 




4.784 


9.884 


5 = 


0.7750 


53.819 




5.980 


12.355 


6 = 


0.9300 


64.583 




7.176 


14.826 


7 = 


1.0850 


75.347 




8.372 


17.297 


8 = 


1.2400 


86.111 




9.568 


19.768 


9 = 


1.3950 


96.874 




10.764 


22.239 


CUBIC. 




Cubic Centi- 
metres to Cubic 
Inches. 


Cubic Deci- 
metres to Cubic 
Inches. 


Cubic Metres to 
Cubic Feet. 


Cubic Metres to 
Cubic Yards. 


1 = 


0.0610 


61.023 




35.314 


1.308 


2 = 


0.1220 


122.047 




70.629 


2.616 


3 = 


0.1831 


183.070 




105.943 


S.924 


4 = 


0.2441 


244.093 




141.258 


5.232 


5 = 


0.3051 


305.117 




176.572 


6.540 


6 = 


0.3661 


366.140 




211.887 


7.848 


7 = 


0.4272 


427.163 




247.201 


9.156 


8 = 


0.4882 


488.187 




282.516 


10.464 


9 = 


0.5492 


549.210 




317.830 


11.771 


CAPACITY. 




Millilitres or 
Cubic Centi- 
litres to Fluid 
Drachms. 


Centilitres 
to Fluid 
Ounces. 


Litres to 
Quarts. 


Dekalitres 

to 

Gallons. 


Hektolitres 

to 

Bushels. 


1 = 


0.27 


0.338 


1.0567 




2.6417 


2.8375 


2 = 


0.54 


0.676 


2.1134 




5.2834 


5.6750 


3 = 


0.81 


1.014 


3.1700 




7.9251 


8.5125 


4 = 


1.08 


1.352 


4.2267 




10.5668 


11.3500 


5 - 


1.35 


1.691 


5.2834 




13.2085 


14.1875 


6 = 


1.62 


2.029 


6.3401 




15.8502 


17.0250 


7 = 


1.89 


2.368 


7.3968 




18.4919 


19.8625 


8 = 


2.16 


2.706 


8.4534 




21.1336 


22.7000 


9 = 


2.43 


3.043 


9.5101 




23.7753 


25 5375 



26 



AKITHMETIC. 
WEIGHT. 





Milligrammes 
to Grains. 


Kilogrammes 
to Grains. 


Hectogrammes 
(100 grammes) 
to Ounces Av. 


Kilogrammes 
to Pounds 
Avoirdupois. 


1 = 


0.01543 


15432.36 


3.5274 


2.20462 


2 = 


0.03086 


30864.71 


7.0548 


4.40924 


3 = 


0.04630 


46297.07 


10.5822 


6.61386 


4 = 


0.06173 


61729.43 


14.1096 


8.81849 


5 = 


0.07716 


77161.78 


17.6370 


11.02311 


6 = 


0.09259 


92594.14 


21.1644 


13.22773 


7 — 


0.10803 


108026.49 


24.6918 


15.43235 


8 = 


0.12346 


123458.85 


28.2192 


17.63697 


9 = 


0.13889 


138891.21 


31.7466 


19.84159 



WEIGHT— (Continued). 





Quintals to 
Pounds Av. 


Milliers or Tonnes to 
Pounds Av. 


Grammes to Ounces, 
Troy. 


1 = 

2 = 

3 = 

4 = 

5 = 

6 = 

7 = 

8 = 

9 = 


220.46 
440.92 
661.38 
881.84 
1102.30 

1322.76 
1543.22 
1763.68 
1984.14 


2204.6 
4409.2 
6613.8 
8818.4 
11023.0 

13227.6 
15432.2 
17636.8 
19841.4 


0.03215 
0.06430 
0.09645 
0.12860 
0.16075 

0.19290 
0.22505 
0.25721 
0.28936 

1 



The only authorized material standard of customary length is the 
Troughton scale belonging to this office, whose length at 59°. 62 Fahr. con- 
forms to the British standard. The yard in use in the United States is there- 
fore equal to the British yard. 

The only authorized material standard of customary weight is the Troy 
pound of the mint. It is of brass of unknown density, and therefore not 
suitable for a standard of mass. It was derived from the British standard 
Troy pound of 1758 by direct comparison. The British Avoirdupois pound 
was also derived from the latter, and contains 7000 grains Troy. 

The grain Troy is therefore the same as the grain Avoirdupois, and the 
pound Avoirdupois in use in the United States is equal to the British pound 
Avoirdupois. 

The metric system was legalized in the United States in 1866. 

By the concurrent action of the principal governments of the world an 
International Bureau of Weights and Measures has been established near 
Paris. 

The International Standard Metre is derived from the Metre des Archives, 
and its length is defined by the distance between two lines at 0° Centigrade, 
on a platinum-iridium bar deposited at the International Bureau. 

The International Standard Kilogramme is a mass of platinum-iridium 
deposited at the same place, and its weight in vacuo is the same as that of 
the Kilogramme des Archives. 

Copies of these international standards are deposited in the office of 
standard weights and measures of the U. S. Coast and Geodetic Survey. 

The litre is equal to a cubic decimetre of water, and it is measured by the 
quantity of distilled water which, at its maximum density, will counterpoise 
the standard kilogramme in a vacuum; the volume of such a quantity of 
water being, as nearly as has been ascertained, equal to a cubic decimetre, 



WEIGHTS AND MEASURES — COMPOUND UNITS. 27 

COMPOUND UNITS. 
Measures of Pressure and Weight. 



1 lb. per square inch. 



' 144 lbs. per square foot. 

2.0355 ins. of mercury at 32° F. 
2.0416 " " " " 62° F. 

2.309 ft. of water at 62° F. 
27.71 ins. " " " 62° F. 



f 2116.3 lbs. per square foot. 
I 33.947 ft. of water at 62° F. 
1 atmosphere (14.7 lbs. per sq. in.). = -j 30 ins. of mercury at 62° F. 

| 29.922 ins. of mercury at 32° F. 
1.760 millimetres of mercury at 32° F. 

{.0361 lb. per square inch. 
5.196 lbs. " " foot. 
.0736 in. of mercury at 62° F. 

., . , „ . . nnn tt, J 5.2021 lbs. per square foot. 

1 inch of water at 32° F. = -J 036]25 lb £ ^ „ inch 

( .433 lb. per square inch. 
1 foot of water at 62° F. =■{ 62.355 lbs. " " foot. 

( .883 in. of mercury at 62° F. 
f .49 lb. per square inch. 

• • i. * *■ «oo I? J 70.56 lbs. " " foot. 

1 inch of mercury at 62° F. = \ im ft of water at 62 o F- 

[13.98 ins. " : ' " 62° F. 
W eight of One Cubic Foot of Pure Water. 

At 32° F. (freezing-point) 62.418 lbs. 

" 39.1° F. (maximum density) 62.425 " 

" 62° F. (standard temperature) 62.355 " 

" 212° F. (boiling-point, under 1 atmosphere) 59.76 " 

American gallon = 231 cubic ins. of water at 62° F. = 8.3356 lbs. 

British " =277.274" " " " " " = 10 lbs. 

Measures of Work, Poorer, and. Duty. 

Work.— The sustained exertion of pressure through space. 

Unit of work.— One foot-pound, i.e., a pressure of one pound exerted 
through a space of one foot. 

Horse-power.- The rate of work. Unit of horse-power = 33,000 ft.- 
lbs. per minute, or 550 ft.-lbs. per second = 1,980,000 ft.-lbs. per hour. 

Heat unit = heat required to raise 1 lb. of water 1° F. (from 39° to 40°). 
33000 

Horse-power expressed in heat units = ~^?r = 42.416 heat units per min- 
ute = .707 heat unit per second = 2545 heat units per hour. 

iVii f f , i „ Q „ tt T > ™„ h™,. f 1,980.000 ft.-lbs. per lb. of fuel. 
1 lb. of fuel per H. I\ per hour= } £ m heat unitg ^ „ 

1,000,000 ft.-lbs. per lb. of fuel = 1.98 lbs. of fuel per H. P. per hour. 
5280 22 
Velocity.— Feet per second = ^x = 15 * mi,es per hour. 

Gross tons per mile = ^—z = — lbs. per yard (single rail.) 

French and British Equivalents of Weight and Press- 
ure per Unit of Area. 

French. British. 

1 gramme per square millimetre ""= 1.422 lbs. per square inch. 

1 kilogramme per square " =1422.32 " " " 

1 " " " centimetre = 14.223 " " " " 

1.0335 kilogrammes per square centimetre I _ ..„ >< u « ,. 
(1 atmosphere) j 
0.70308 kilogramme per square centimetre = 1 lb. per square inch. 



28 ARITHMETIC. 

WIRE AND SHEET-METAL. GAUGES COMPARED. 



"o 




Ot3 & 

C C M 


^E. 05 




British Imperial 
Standard 


dard 
or 

Plate 
Steel. 

dard 
1893.) 





a a 


'Sees 
CO 




Roebling'i 

Washbu 

& Moen 

Gauge 


° 05 03 
£ 


Wire Gauge. 
(Legal Standard 
in Great Britain 
since 
March 1, 1884.) 


U. S. Stan. 
Gauge f 
Sheet and '. 
Iron and S 
(Legal Stan 
since July 1, 


it 

es 




inch. 


inch. 


inch. 


inch. 


inch. 


millira. 


inch. 




0000000 






.49 




.500 


12.7 


.5 


7/0 


000000 






.46 




.464 


11.78 


.469 


6/0 


00000 






.43 


.45 


.432 


10.97 


.438 


5/0 


0000 


.454 


.46 


.393 


.40 


.4 


10.16 


.406 


4/0 


000 


.425 


.40964 


.362 


.36 


.372 


9.45 


.375 


3/0 


00 


.38 


.3648 


.331 


.33 


.348 


8.84 


.344 


2/0 





.34 


.32486 


.307 


.305 


.324 


8.23 


.313 





1 


.3 . 


.2893 


.283 


.285 


.3 


7.62 


.281 


1 


2 


.284 


.25763 


.263 


.265 


.276 


7.01 


.266 


2 


3 


.259 


.22942 


.244 


.245 


.252 


6.4 


.25 


3 


4 


.238 


.20431 


.225 


.225 


.232 


5.89 


.231 


4 


5 


.22 


.18194 


.207 


.205 


.212 


5.38 


.219 


5 


6 


.203 


.16202 


.192 


.19 


.192 


4.88 


.203 


6 


7 


.18 


.144-28 


.177 


.175 


.176 


4.47 


.188 


7 


8 


.165 


.12849 


.162 


.16 


.16 


4.06 


.172 


8 


9 


.148 


.11443 


.148 


.145 


.144 


3-66 


.156 


9 


10 


.134 


.10189 


.135 


.13 


.128 


3-26 


.141 


10 


11 


.12 


.09074 


.12 


.1175 


.116 


2.95 


.125 


11 


12 


.109 


.08081 


.105 


.105 


.104 


2-64 


.109 


12 


13 


.095 


.07196 


.092 


.0925 


.092 


2-34 


.094 


13 


14 


.083 


.06408 


.08 


.08 


.08 


2-03 


.078 


14 


15 


.072 


.05707 


.072 


.07 


.072 


1.83 


.07 


15 


16 


.065 


.05082 


.063 


.061 


.064 


1.63 


.0625 


16 


17 


.058 


.04526 


.054 


.0525 


056 


1.42 


.0563 


17 


18 


.049 


.0403 


.047 


.045 


.048 


1.22 


.05 


18 


19 


.042 


.03589 


.041 


.04 


.04 


1.01 


.0438 


19 


20 


.035 


.03196 


.035 


.035 


.036 


.91 


.0375 


20 


21 


.032 


.02846 


.032 


.031 


.032 


.81 


.0344 


21 


22 


.028 


.02535 


.028 


.028 


.028 


.71 


.0313 


22 


23 


.025 


.02257 


.025 


.025 


.024 


.61 


.0281 


23 


24 


.022 


.0201 


.023 


.0225 


.022 


.56 


.025 


24 


25 


.02 


.0179 


.02 


.02 


.02 


.51 


.0219 


25 


26 


.018 


.01594 


.018 


.018 


.018 


.45 


.0188 


26 


27 


.016 


.01419 


.017 


.017 


.0164 


.42 


.0172 


27 


28 


.014 


.01264 


.016 


.016 


.0148 


.38 


.0156 


28 


29 


.013 


.01126 


.015 


.015 


.0136 


.35 


.0141 


29 


30 


.012 


.01002 


.014 


.014 


.0124 


.31 


.0125 


30 


31 


.01 


.00893 


.0135 


.013 


.0116 


.29 


.0109 


31 


32 


.009 


.00795 


.013 


.012 


.0108 


.27 


.0101 


32 


33 


.008 


.00708 


.011 


.011 


.01 


.25 


.0094 


33 


34 


.007 


.0063 


.01 


.01 


.0092 


.23 


.0086 


34 


35 


.005 


.00561 


.0095 


.0095 


.0084 


.21 


.0078 


35 


36 


004 


.005 


.009 


.009 


.0076 


.19 


.007 


36 


37 




.00445 


.0085 


.0085 


.0068 


.17 


.0066 


37 


38 




.00396 


.008 


.008 


.006 


.15 


.0063 


38 


39 




.0g353 


.0075 


.0075 


.0052 


.13 




39 


40 




.00314 


.007 


.007 


.0048 


.12 




40 


41 










.0044 


.11 




41 


42 










.004 


.10 




42 


43 










.0036 


.09 




43 


44 










.0032 


.08 




44 


45 










.0028 


.07 




45 


46 










.0024 


.06 




46 


47 










.002 


.05 




47 


48 










.0016 


.04 




48 


49 










.0012 


.03 




49 


50 










.001 


.025 




50 



WIRE GAUGE TABLES. 



29 



EDISON, OR CIRCULAR Mil. GAUGE, FOR ELEC- 
TRICAL WIRES. 



Gauge 
Num- 
ber. 


Circular 

Mils. 


Diam- 
eter 
in Mils. 


3 


3,000 


54.78 


5 


5,000 


70.72 


8 


8,000 


89.45 


12 


12,000 


109.55 


15 


15,000 


122.48 


20 


20,000 


141.43 


25 


25,000 


158.12 


30 


30,000 


173.21 


35 


35,000 


187.09 


40 


40,000 


200.00 


45 


45,000 


212.14 


50 


50,000 


223.61 


55 


55,000 


234.53 


60 


60,000 


244.95 


65 


65,000 


254.96 



Gauge 
Num- 
ber. 


Circular 

Mils. 


Diam- 
eter 
in Mils. 


70 


70,000 


264.58 


75 


75,000 


273.87 


80 


80,000 


282.85 


85 


85,000 


291.55 


90 


90,000 


300.00 


95 


95,000 


308.23 


100 


100,000 


316.23 


110 


110,000 


331.67 


120 


120,000 


346.42 


130 


130,000 


360.56 


140 


140,000 


374.17 


150 


150,000 


387.30 


160 


160.000 


400.00 


170 


170,000 


412.32 


180 


180,000 


424.27 



Gauge 
Num- 
ber. 


Circular 
Mils. 


Diam- 
eter 
in Mils. 


190 


190,000 


435.89 


200 


200,000 


447.22 


220 


220,000 


469.05 


240 


240.000 


489.90 


260 


260,000 


509.91 


280 


280,000 


529.16 


300 


300,000 


547.73 


320 


320,000 


565.69 


340 


340,000 


583.10 


360 


360,000 


600.00 



TWIST DRILL AND STEEL WIRE GAUGE. 

(Morse Twist Drill and Machine Co.) 



No. 


Size. 




inch. 


1 


.2280 


2 


.2210 


3 


.2130 


4 


.2090 


5 


.2055 


6 


.2040 


7 


.2010 


8 


.1990 


9 


.1960 


10 


.1935 


11 


.1910 


12 


.1890 


13 


.1850 


14 


.1820 


15 


.1800 



No. 


Size. 


No. 


Size. 




inch. 




inch. 


16 


.1770 


31 


.1200 


17 


.1730 


32 


.1160 


18 


.1695 


33 


.1130 


19 


.1660 


34 


.1110 


20 


.1610 


35 


.1100 


21 


.1590 


36 


.1065 


22 


.1570 


37 


.1040 


23 


.1540 


38 


.1015 


24 


.1520 


39 


.0995 


25 


.1495 


40 


.0980 


26 


.1470 


41 


.0960 


27 


.1440 


42 


.0985 


28 


.1405 


43 


.0890 


29 


.1360 


44 


.0860 


30 


.1285 


45 


.0820 



No. 


Size. 




inch. 


46 


.0810 


47 


.0785 


48 


.0760 


49 


.0730 


50 


.0700 


51 


.0670 


52 


.0635 


53 


.0595 


54 


.0550 


55 


.0520 


56 


.0465 


57 


.0430 


58 


.0420 


59 


.0410 


60 


.0400 



STEEL MUSIC-WIRE GAUGE. 

(Washburn & Moen Mfg. Co.) 



No. 


Size. 


No. 


Size. 


• No. 


Size. 


No. 


Size. 




inch. 




inch. 




inch. 




inch. 


12 


.0295 


17 


.0378 


21 


.0461 


25 


.0585 


13 


.0311 


18 


.0395 


22 


.0481 


26 


.0626 


14 


.0325 


19 


.0414 


23 


.0506 


27 


.0663 


15 


.0313 


20 


.043 


24 


.0547 


28 


.0719 


16 


.0359 















30 ARITHMETIC. 

THE EDISON OR CIRCULAR OTIL WIRE GAUGE. 

(For table of copper wires by this'gauge, giving weights, electrical resist- 
ances, etc., see Copper Wire.) 

Mr. C. J. Field {Stevens Indicator, July, 1887) thus describes the origin of 
the Edison gauge: 

The Edison company experienced inconvenience and loss by not having a 
wide enough range nor sufficient number of sizes in the existing gauges. 
This was felt more particularly in the central-station work in making- 
electrical determinations for the street system. They were compelled to 
make use of two of the existing gauges at least, thereby introducing a 
complication that was liable to lead to mistakes by the contractors and 
linemen. 

In the incandescent system an even distribution throughout the entire 
system and a uniform pressure at the point of delivery are obtained by cal- 
culating for a given maximum percentage of loss from the potential as 
delivered from the dynamo. In carrying this out, on account of lack of 
regular sizes, it was often necessary to use larger sizes than the occasion 
demanded, and even to assume new sizes for large underground conductors. 
It was also found that nearly all manufacturers based their calculation for. 
the conductivity of their wire on a variety of units, and that not one used 
the latest unit as adopted by the British Association and determined from 
Dr. Matthiessen's experiments ; and as this was the unit employed in the 
manufacture of the Edison lamps, there was a further reason for construct- 
ing a new gauge. The engineering department of the Edison company, 
knowing the requirements, have designed a gauge that has the widest 
range obtainable and a large number of sizes which increase in a regular 
and uniform manner. The basis of the graduation is the sectional area, and 
the number of the wire corresponds. A wire of 100,000 circular mils area is 
No. 100 ; a wire of one half the size will be No. 50 ; twice the size No. 200. 

In the older gauges, as the number increased the size decreased. With 
this gauge, however, the number increases with the wire, and the number 
multiplied by 1000 will yive the circular -nils. 

The weight per mil-foot, 0.00000302705 pounds, agrees with a specific 
gravity of 8.889, which is the latest figure given for copper. The ampere 
capacity which is given was deduced from experiments made in the com- 
pany's laboratory, and is based on a rise of temperature of 50° F. in the wire. 

In 1893 Mr. Field writes, concerning gauges in use by electrical engineers: 

The B. and S. gauge seems to be in general use for the smaller sizes, up 
to 100,000 c. m., and in some cases a little larger. From between one and 
two hundred thousand circular mils upwards, the Edison gauge or its 
equivalent is practically in use, and there is a general tendency to designate 
all sizes above this in circular mils, specifying a wire as 200,000, 400,000, 500,- 
000, or 1,000,000 c. m. 

In the electrical business there is a large use of copper ware and rod and 
other materials of these large sizes, and in ordering them, speaking of them, 
specifying, and in every other use, the general method is to simply specify 
the circular milage. I think it is going to be the only system in the future 
for the designation of wires, and the attaining of it means practically the 
adoption of the Edison gauge or the method and basis of this gauge as the 
correct one for wire sizes. 

THE U. S. STANDARD GAUGE FOR SHEET AND 
PLATE IRON AND STEEL, 1893. 

The Committee on Coinage, Weights, and Measures of the House of 
Representatives in 1893, in introducing the bill establishing the new sheet 
and plate gauge, made a report from which we take the following : 

The purpose of this bill is to establish an authoritative standard gauge for 
the measurement of sheet and plate iron. 

There is in this country no uniform or standard gauge, and the same 
numbers in different gauges represent different thicknesses of sheets or 
plates. This has given rise to much misunderstanding and friction between 
employers and workmen and mistakes and fraud between dealers and con- 
sumers. 

The practice of describing the different thicknesses of sheet and plate 
iron by gauge numbers has been so long established and become so uni- 
versal both here and in Great Britain that it is not deemed advisable to 
change this mode of designation; but these descriptive gauge numbers 



GAUGE FOR SHEET AND 


PLATE 


IRON 


AND 


STEEL. 31 


IT. S. STANDARD 


GAUGE 


FOR 


SHEET AND PLATE 




IRON AND 


STEEL., 1893. 




o 
II 

S3 
; is 


.5 a> g o 
x a. 2 a 

Js ° ^ a 


Approximate 
Thickness iu 

Decimal 

Parts of an 

Inch. 


III 

£ 5 a S 

Oo' H .S 

f £ 1 


Weight per 
Square Foot 

in Ounces 
Avoirdupois. 


Weight per 

Square Foot 

in Pounds 

Avoirdupois. 


«"£ a 

A 9 & 
.L r -=- 


S3 -g 2 

ft! 

03.2 


Weight per 
Square Meter 

in Pounds 
Avoirdupois. 


0000000 


1-2 


0.5 


12.7 


320 


20. 


9.072 


97.65 


215.28 


000000 


15-32 


0.46S75 


11.90625 


300 


18.75 


8.505 


91.55 


201.82 


00000 


7-16 


0.4375 


11.1125 


280 


17.50 


7.938 


85.44 


188.37 


0000 


13-32 


0.40625 


10.31875 


260 


16.25 


7.371 


79.33 


174.91 


000 


3-8 


0.375 


9.525 


240 


15. 


6.804 


73.24 


161.46 


oa 


11-32 


0.34375 


8.73125 


220 


13.75 


6.237 


67.13 


148.00 





5-16 


0.3125 


7.9375 


200 


12.50 


5.67 


61.03 


134.55 


1 


9-32 


0.28125 


7.14375 


180 


11.25 


5.103 


54.93 


121.09 


2 


17-64 


0.265625 


6.746875 


170 


10.625 


4.819 


51.88 


114.37 


3 


1-4 


0.25 


6.35 


160 


10. 


4.536 


48.82 


107.64 


4 


15-64 


0.234375 


5.953125 


150 


9.375 


4.252 


45.77 


100.91 


5 


7-32 


0.21875 


5.55625 


140 


8.75 


3.969 


42.72 


94.18 


6 


13-64 


0.203125 


5.159375 


130 


8.125 


3.685 


39.67 


87.45 


7 


3-16 


0.1875 


4.7625 


120 


7.5 


3.402 


36.62 


80.72 


8 


11-64 


0.171875 


4.365625 


110 


6.875 


3.118 


33.57 


74.00 


9 


5-32 


0.15625 


3.96875 


100 


6.25 


2.835 


30.52 


67.27 


10 


9-64 


0.140625 


3.571875 


90 


5.625 


2.552 


27.46 


60.55 


11 


1-8 


0.125 


3.175 


80 


5. 


2.268 


24.41 


53.82 


12 


7-64 


0.109375 


2.778125 


70 


4.375 


1.984 


21.36 


47.09 


13 


3-32 


0.09375 


2.38125 


60 


3.75 


1.701 


18.31 


40.36 


14 


5-64 


0.078125 


1.984375 


50 


3.125 


1.417 


15.26 


33.64 


15 


9-128 


0.0703125 


1.7859375 


45 


2.8125 


1.276 


13.73 


30.27 


16 


1-16 


0.0625 


1.5875 


40 


2.5 


1.134 


12.21 


26.91 


17 


9-160 


0.05625 


1.42875 


36 


2.25 


1.021 


10.99 


24.22 


18 


1-20 


0.05 


1.27 


32 


2. 


0.9072 


9.765 


21.53 


19 


7-160 


0.04375 


1.11125 


28 


1.75 


0.7938 


8.544 


18.84 


20 


3-80 


0.0375 


0.9525 


24 


1.50 


0.6804 


7.324 


16.15 


21 


11-320 


0.034375 


0.873125 


22 


1.375 


0.6237 


6.713 


14.80 


22 


1-32 


0.03125 


0.793750 


20 


1.25 


0.567 


6.103 


13 46 


23 


9-320 


0.028125 


0.714375 


18 


1.125 


0.5103 


5.493 


12.11 


24 


1-40 


0.025 


0.635 


16 


1. 


0.4536 


4.882 


10.76 


25 


7-320 


0.021875 


0.555625 


14 


0.875 


0.3969 


4.272 


9.42 


26 


3-160 


0.01875 


0.47625 


12 


0.75 


0.3402 


3.662 


8.07 


27 


11-640 


0.0171875 


0.4365625 


11 


0.6875 


0.3119 


3.357 


7.40 


28 


1-64 


0.015625 


0.396875 


10 


0.625 


0.2835 


3.052 


6.73 


29 


9-640 


0.0140625 


0.3571875 


9 


0.5625 


0.2551 


2.746 


6.05 


30 


1--80 


0.0125 


0.3175 


8 


0.5 


0.2268 


2.441 


5.38 


31 


7-640 


0.0109375 


0.2778125 


7 


0.4375 


0.1984 


2.136 


4.71 


32 


13-1280 


0.01015625 


0.25796875 


^ 


0.40625 


0.1843 


1.983 


4.37 


33 


3-320 


0.009375 


0.238125 


6 


0.375 


0.1701 


1.831 


4.04 


34 


11-1280 


00859375 


0.21828125 


5^ 


0.34375 


0.1559 


1.678 


3 70 


35 


5-640 


0.0078125 


0.1984375 


5 


0.3125 


0.1417 


1.526 


3.36 


36 


9-1280 


0.00703125 


0.17859375 


$A 


0.28125 


).1270 


1.373 


3.03 


37 


17-2560 


0.006640625 


0.168671875 


m 


0.265625 


0.1205 


1.297 


2.87 


38 


1-160 


0.00625 


0.15875 


4 


0.25 


0.1134 


1.221 


2.69 





















32 MATHEMATICS. 

ought to have the same meaning and significance at all times and under all 
circumstances. 

To accomplish this and furnish a legal guide in the collection of govern- 
ment duties, the United States should establish a legal standard gauge. 
None of the existing gauge-tables or scales exactly meet the requirements 
of accuracy and convenience, nor rest on a systematic basis; but the one 
submitted by your committee is believed to fully meet these requirements. 

It is based on the fact that a cubic foot of iron weighs 480 pounds. This 
is the same basis on which the Imperial gauge of Great Britain rests, and 
also the New Birmingham and Amalgamated Association gauges. 

A sheet of iron 1 foot square and 1 inch thick weighs 40 pounds, or 640 
ounces, and 1 ounce in weight should be 1/640 inch thick. The scale has been 
arranged so that each descriptive number represents a certain number of 
ounces in weight, and an equal number of six hundred and fortieths of an 
inch in thickness, and the weights, and hence the thicknesses, have been 
arranged in a regular series of gradations. A micrometer for measuring 
the thickness of sheets and plates can be constructed to indicate six hun- 
dred and fortieths of an inch as easily as one thousandths, and thus the 
measurement of a sheet of iron will give the thickness in six hundred and 
fortieths of an inch and in weight in ounces at the same time. 

It is probable that the adoption of this gauge will gradually lead to the 
abandonment of the numbers and to the use of the number of ounces in 
weight per square foot as the descriptive terms of the different thicknesses 
of sheet and plate iron. It will become as easy to order a 20-ounce sheet as 
a No. 22, or a 10 ounce as a No. 28; and this will cause a more general and 
intelligent comprehension of just what is being contracted for, and the 
opportunity for mistake or fraud growing out of an uncertainty of designa- 
tion will be removed. 

A natural consequence also will be the substitution of such weight desig- 
nation for the arbitrary methods now in vogue of describing tin and terne 
plates as IC, IX, IXX, DC, DX, etc. 

The law establishing the new gauge enacts as follows : 

That for the purpose of securing uniformity, the following is established 
as the only standard gauge for sheet and plate iron and steel in the United 
States of America, namely : 

And on and after July 1, 1893, the same and no other shall be used in 
determining duties and taxes levied by the United States of America on 
sheet and plate iron and steel. 

Sec. 2. That the Secretary of the Treasury is authorized and required to 
prepare suitable standards in accordance herewith. 

Sec. 3. That in the practical use and application of the standard gauge 
hereby established a variation of 2\£ per cent either way may be allowed. 



33 



ALGEBRA. 

Addition.— Add a and 6. Ans. a-\-b. Add a, b, and — c. Ans. a-\-b — c. 

Add 2a and — 3a. Ans. — a. Add 2ab, — Sab, — c, — 3c. Ans. — ab — 4c. 

Subtraction. — Subtract a from b. Ans. 6 — a. Subtract — a from — 6. 
Ans. — b + «• 

Subtract 6 -f- c from a. Ans. a — & — c. Subtract 3a 2 6- 9c from 4a 2 6 + c. 
Ans. a 2 b -f- 10c. Rule: Change the signs of the subtrahend and proceed as 
in addition. 

Multiplication.— Multiply a by b. Ans. ab. Multiply ab by a 4- 6. 
Ans. a 2 b + a6 2 . 

Multiply a 4- 6 by a + 6. Ans. (a 4- &)(a 4- b) = a 2 4- 2a6 + 6 2 . 

Multiply — a by — 6. Ans. ab. Multiply - a by b. Ans. — a&. Like 
signs give plus, unlike signs minus. 

Powers of numbers, — The product of two or more powers of any 
number is the number with an exponent equal to the sum of the powers: 
a 2 x a 3 = a 5 ; a 2 6 2 xa6 = a 3 6 3 ; - lab x 2ac = - 14 a 2 6c. 

To multiply a polynomial by a monomial, multiply each term of the poly- 
nomial by the monomial and add the partial products: (6a — 36) x 3c = 18ac 
-36c. 

To multiply two polynomials, multiply each term of one factor by each 
term of the other and add the partial products: (5a — 66) x (3a — 46) = 
15a 2 - 38a6 + 246 2 . 

The square of the sum of two numbers = sum of their squares 4- twice 
their product. 

The square of the difference of two numbers = the sum of their squares 
— twice their product. 

The product of the sum and difference of two numbers = the difference 
of their squares: 

(a 4- 6) 2 = a 2 4- 2a6 4- 6 2 ; (a - 6) 2 = a - 2a6 4- 6 2 ; 
(a 4- 6) x (a- 6) = a 2 -& 2 . 

The square of half the sums of two quantities is equal to their product plus 

( a _j_ \ 2 / a - 6 \ 2 

— - — ) = a6 4- ( — T~ ) 

The square of the sum of two quantities is equal to four times their prod- 
ucts, plus the square of their difference: (a 4 6j 2 — 4a6 4- (a — 6) 2 

The sum of the squares of two quantities equals twice their product, plus 
the square of their difference: a 2 4- 6 2 = 2a6 + (a — 6j 2 . 

The square of a trinomial = the square of each term + twice the product 
of each term by each of the terms that follow it: (a -\-b -\-c)' 2 = a 2 4"6 2 4" 
c 2 4- 2ab 4- 2ac + 26c ; (a - 6 - c) 2 = a 2 + 6 2 4- c 2 - 2a6 - 2ac + 26c. 

The square of (any number 4- J^>) — square of the number + the number 
4- 14; = the number X (the number + 1) 4- M; 
(a + ^) 2 = a 2 4-«4-M, =a(a 4-D + M- (4^) 2 =4 2 + 4 + M=4 x 5 + 14 = 2 0^. 

The product of any number 4- }/k by an .Y other number 4- V& = product of 
the numbers + half their sum 4- 14. (a 4- V>) X 64- H) = ab -\- J^(a+ 6)4- J4. 
41/3 X 6^ = 4 X 6 4- 1/ 2 (4 4- 6) + 14 = 24 + 5 4- U = 29J4- 

Square, cube, 4tn power, etc., of a binomial a 4 6. 

(a 4- 6) 2 = a 2 + 2a6 4- 6 2 ; (a + 6) 3 = a 3 + 3a 2 6 4 3a6 2 + b 3 ; 
(a + 6) 4 = a 4 4- 4a 3 6 + 6a 2 6 2 + 4ab 3 + 6*. 

In each case the number of terms is one greater than the exponent of 
the power to winch the binomial is raised. 

2. In the first term the exponent of a is the same as the exponent of the 
power to which the binomial is raised, aud it decreases by 1 in each succeed- 
ing term. 

3. 6 appears in the second term with the exponent 1, and its exponent 
increases by 1 in each succeeding term. 

4. The coefficient of the first term is 1. 

5. The coefficient of the second term is the exponent of the power to 
which the binomial is raised. 

6. The coefficient of each succeeding term is found from the next pre- 
ceding term by multiplying its coefficient by the exponent of a, and divid- 
ing the product by a number greater by 1 than the exponent of 6. (See 
Binomial Theorem, below.) 



34 ALGEBRA. 

Parentheses.— When a parenthesis is preceded by a plus sign it may be 
removed without changing the value of the expression: a -f- b 4- (a + b) — 
2a + 2b. When a parenthesis is preceded by a minus sign it may be removed 
if we change the signs of all the terms within the parenthesis: 1 — (a — b 
— c) = 1 — a -4- b + c. When a parenthesis is within a parenthesis remove 
the inner one first: a - \b 



-(d-e) }.] = a- [6_ Jc-d + e}] 



= a -■ [b - c +'d - e] = a - b + c - d + e. 

A multiplication sign, X, has the effect of a parenthesis, in that the oper- 
ation indicated by it must be performed before the operations of addition 
or subtraction, a -f 6 X a + b = a -4- ab -4- b; while (a -\- b) X (a -\- b) — 
a 2 + 2ab + 6 2 , and (a -f 6) X a + 6 = a 2 -f ab + 6. 

Division.— -The quotient is positive when the dividend and divisor 
have Jike signs, and negative when they have unlike signs: abc -*■■& = ac; 
abc -. b = — ac. 

To divide a monomial by a monomial, write the dividend over the divisor 
with a line between them. If the expressions have common factors, remove 
the common factors: 

a?bx ax a 4 a 3 1 _ 2 

a 2 bx-r-aby= — r — = — ; — = a; — = — = a 
aby y a 3 a 5 a 2 

To divide a polynomial by a monomial, divide each term of the polynomial 
by the monomial: (8ab — Y2ac) -=- 4a = 2b — 3c. 

To divide a polynomial by a polynomial, arrange both dividend and divi- 
sor in the order of the ascending or descending powers of some common 
letter, and keep this arrangement throughout the operation. 

Divide the first term of the dividend by the first term of the divisor, and 
write the result as the first term of the quotient. 

Multiply all the terms of the divisor by the first term of the quotient and 
subtract the product from the dividend. If there be a remainder, consider 
it as a new dividend and proceed as before: (a 2 — ft 2 ) -*- (a -f- b). 
a 2 - o 2 | a + ft. 
cfl + ab\ a_-_b. 

- ab - 6 2 . 

- ab - V *. 

The difference of two equal odd powers of any two numbers is divisible 
by their difference and also by their sum: 

(a 3 - 6 3 ) ^ ( a _ 5) = a 2 + ab _|_ 52 . (a 3 _ 53) -h (a + &) = a 2 - a& + 6 2 . 

The difference of two equal even powers of two numbers is divisible by 
their difference and also by their sum: (a 2 — 6 2 ) -h (a — b) — a + b. 

The sum of two equal even powers of two numbers is not divisible by 
either the difference or the sum of the numbers; but when the exponent 
of each of the two equal powers is composed of an odd and an even factor, 
the sum of the given power is divisible by the sum of the powers expressed 
by the even factor. Thus x & -j- y 6 is not divisible by x + y or byx — y, but is 
divisible by x 2 -\- y 2 . 

Simple equations. — An equation is a statement of equality between 
two expressions; as, a -\- b = c -{- d. 

A simple equation, or equation of the first degree, is one which contains 
only the first power of the unknown quantity. If equal changes be made 
(by addition, subtraction, multiplication, or division) in both sides of an 
equation, the results will be equal. 

Any term may be changed from one side of an equation to another, pro- 
vided its sign be changed: a -f- b = c + d; a — c -\- d — b. To solve an 
equation having one unknown quantity, transpose all the terms involving 
the unknown quantity to one side of the equation, and all the other terms 
to the other side; combine like terms,.and divide both sides by the coefficient 
of the unknown quantity. 

Solve 8a; - 29 = 26 - 3x. 8x + Sx = 29 + 26; 11a; = 55; x = 5, ans. 

Simple algebraic problems containing one unknown quantity are solved 
by making x = the unknown quantity, and stating the conditions of the 
problem in the form of an algebraic equation, and then solving the equa- 
tion. What two numbers are those whose sum is 48 and difference 14 v Let 
x — the smaller number, x + 14 the greater, x -j- x -4- 14 = 48. 2a; = 34, x 
— 17; x -\- 14 = 31, ans. 

Find a number whose treble exceeds 50 as much as its double falls short 
of 40. Let x = the number. 3a; - 50 = 40 - 2a;; 5a; = 90; x - 18, ans. Prov^ 
ing, 54 - 50 = 40 - 36, • 



ALGEBRA. 35 

liquations containing two unknown quantities.— If one 

equation contains two unknown quantities, x and y, an indefinite u umber of 
pairs of values of x and y may be found that will satisfy the equation, but if 
a second equation be given only one pair of values can be found that will 
satisfy both equations. Simultaneous equations, or those that may be satis- 
fied by the same values of the unknown quantities, are solved by combining 
the equations so as to obtain a single equation containing only one unknown 
quantity. This process is called elimination. 

Elimination by addition or subtraction. — Multiply the equation by 
such numbers as will make the coefficients of one of the unknown quanti- 
ties equal in the resulting equation. Add or subtract the resulting equa- 
tions according as they have unlike or like signs. 

bolve j 4x - 5y = 3. Subtract: 4x - 5y = 3 lly = .11; y = 1/ 

Substituting value of y in first equation, 2x + 3 = 7: x = 2. 

Elimination by substitution.— From one of the equations obtain the 
value of one of the unknown quantities in terms of the other. Substitu- 
tute for this unknown quantity its value in the other equation and reduce 
the resulting equations. 

'. j 2x + 3y = 8. (1). From (1) we find x = — ^—^ . 
Solve i „ , „ r /()1 2 

t 3a; -f 7y = 7. (2). 

Substitute this value in (2): 3^ - : i— ) + 7y — 7; = 24 - 9y + Uy = 14, 

whence y — — 2. Substitute this value in (1): 2x — 6 = 8; x = 7. 

Elimination by comparison. — From each equation obtain the value of 
one of the unknown quantities in terms of the other. Form an equation 
from these equal values, and reduce this equation. 



I 3a; - 4y = 7. 

Equating these values of x, ~^- — - — — — ; 19^ = — 19; y = : — 1. 

Substitute this value of y in (1): 2a; + 9 = 11; x = 1. 

If three simultaneous equations are given containing three unknown 
quantities, one of the unknown quantities must be eliminated between two 
pairs of the equations: then a second between the two resulting equations. 

Quadratic equations. — A quadratic equation contains the square 
of the unknown quantity, but no higher power. A pure quadratic contains 
the square only; an affected quadratic both the square and the first power. 

To solve a pure quadratic, collect the unknown quantities on one side, 
and the known quantities on the other; divide by the coefficient of the un- 
known quantity and extract the square root of each side of the resulting 
equation. 

Solve 3a; 2 - 15 = 0. 3a; 2 = 15; a; 2 = 5; x = V5 

A root like ^5, winch is indicated, but which can be found only approxi- 
mately, is called a surd. 

Solve 3a; 2 + 15 = 0. 3a; 2 = - 15; a; 2 = - 5; x = V- 5. 

The square root of — 5 cannot be found even approximately, for the square 
of any number positive or negative is positive; therefore a root which is in- 
dicated, but cannot be found even approximately, is called imaginary. 

To solve an affected quadratic. — 1. Convert the equation into the form 
a 2 * 2 ± 2abx = c, multiplying or dividing the equation if necessary, so as 
to make the coefficient of x 2 a square number. 

2. Complete the square of the first member of the equation, so as to con- 
vert it to the form of a^x 2 ± 2abx -f- 6 2 , which is the square of the binomial 
ax ± b, as follows: add to each side of the equation the square of the quo- 
tient obtained by dividing the second term by twice the square root of the 
first term. 

3, Extract the square root of each side of the resulting equation. 

Solve 3.x 2 — 4x = 32. To make the coefficient of a; 2 a square number, 
multiply by 3: 9a* 2 - 12a; = 96; 12a; -f- (2 x 3a;) = 2; 2 2 = 4. 
Complete the square: 9a; 2 - 12a; -\- 4 — 100. Extract the root: 3a; — 2 = ± 



36 ALGEBRA. 

10, whence x - 4 or - 2 2/3. The square root of 100 is either -j- 10 of - 10, 
since tlie square of — 10 as well as -j- 10 2 = 100. 

Problems involving quadratic equations have apparently two solutions, as 
a quadratic has two roots. Sometimes both will be true solutions, but gen- 
erally one only will be a solution and the other be inconsistent with the 
conditions of the problem. 

The sum of the squares of two consecutive positive numbers is 481. Find 
the numbers. 

Let x = one number, x + 1 the other, re 2 + (x + l) 2 = 481. 2a; 2 -f 2x + 1 
= 481. 

x 2 + x — 240. Completing the square, a; 2 + x-\- 0.25 = 240.25. Extracting 
the root we obtain a* -j- 0.5 = ± 15.5; x = 15 or — 16. 

The positive root gives for ihe numbers 15 and 16. The negative root — 
16 is inconsistent with the conditions of the problem. 

Quadratic equations containing two unknown quantities require different 
methods for their ^plution, according to the form of the equations. For 
these methods reference must be made to works on algebra. 

Theory of exponents.— ya when n is a positive integer is one of n 

equal factors of a. y a m means a is to be raised to the with power and the 
nth root extracted. 

{ya) '"means that the nth root of a is to be taken and the result 
raised to the mtli power. 

n — h - m m 

y a m = \ya ) = a n . When the exponent is a fraction, the numera- 
tor indicates a power, and the denominator a root, a? — y a = a 3 ; as = 

|/a3 = a 1 " 5 - 

To extract the root of a quantity raised to an indicated power, divide 
the exponent by the index of the required root; as, 

ty~a™ = a "Z "' 4/a 6 = J = a 2 . 

Subtracting 1 from the exponent of a is equivalent to dividing by a : 

a 2 -i = a J = a; a* - 1 = a = - = 1 ; a - 1 = a - 1 = - ; a- 1 -* = a ~' 2 = ~ 2 

A number with a negative exponent denotes the reciprocal of the number 
with the corresponding positive exponent. • ' 

A factor under the radical sign whose root can be taken may, by having 
the root taken, be removed from under the radical sign: 

y/~(tfb = -f/o 2 x |/6 = a y'b. 
A factor outside the radical sign may be raised to the corresponding 
power and plact-d under it: 

\/\ = \/% - v^l = 1 ** \/i = 1 ^ 

Binomial Theorem.— To obtain any power, as the nth, of an ex- 
pression of the form x + a 

, , sn n , «-_i . n (n - !) aM ~ 2 ^.. 1 n (n - 

(a + x) n = a n -f na x -\ — x* -\ g g 

The following laws hold for any term in the expansion of (a -j- x) n . 

The exponent of x is less by one than the number of terms. 

The exponent of a is n minus the exponent of x. 

The last factor of the numerator is greater by one than the exponent of a. 

The last factor of the denominator is the same as the exponent of x. 

In the rth term the exponent of x will be r — 1. 

The exponent of a will be n — (r — 1). or n — r -4- 1. 

The last factor of the numerator will be n — r -\- 2. 

The last factor of the denominator will be = r — 1. 

Hence the rth term = "° l "^^ ~ 8) ' ' '^'iT^' + ^ a»-r+l x r-l 



GEOMETRICAL PROBLEMS. 



3? 



GEOMETRICAL PROBLEMS. 




f 



-hr+- 








Y~ 






Fig 


5. 




c 






D 


1 

1 






1 
1 



1 . To bisect a straight line, 
or an arc of a circle (Fig. l).— 
From the ends A, B. as centres, de- 
scribe arcs intersecting at C and D, 
and draw a line through C and D 
which will bisect the line at E or the 
arc at F. 



2. To draw a perpendicular 
to a straight line, or a radial 
line to a circular arc— Same as 
in Problem 1. C D is perpendicular to 
the line A B, and also radial to the arc. 

3. To draw a perpendicular 
to a straight line from a given 
point in that line (Fig. a).— With 
any radius, from the given point A in 
the line B C, cut the line at B and C. 
With a longer radius describe arcs 
from B and C, cutting each other at 
D, and draw the perpendicular D ^4. 

4. From the end A of a gievn 
line -1 J> to erect a perpendic- 
ular A JE (Fig. 3).— From any centre 
F, above A D, describe a circle passing 
through the given point A, and cut- 
ting the given line at D. Draw D F 
and produce it to cut the circle at E, 
and draw the perpendicular A E. 

Second Method (Fig. 4).— From the 
given point A set off a distance A E 
equal to three parts, by any scale ; 
and on the centres A and E, with radii 
of four and five parts respectively, 
describe arcs intersecting at C. Draw 
the perpendicular A C. 

Note.— This method is most useful 
oh very large scales, where straight 
edges are inapplicable. Any multiples 
of the numbers 3, 4, 5 may be taken 
with the same effect as 6, 8, 10, or 9, 
12, 15. 



5 . To draw a perpendicular 
to a straight line from any 
point without it (Fig. 5.)— From 
the point A y with a sufficient radius 
cut the given line at F and G, and 
from these points describe arcs cut- 
ting at E. Draw the perpendicular 
A E. 



6. To draw a straight line 
parallel to a given line, at a 
given distance apart (Fig. 6). — 

From the centres A, B, in the given 
liue, with the given distance as radius, 
describe arcs C, D, aud draw the par- 
allel lines C D touching the arcs. 



38 



GEOMETRICAL PROBLEMS. 
Q 











7. To divide a straight line 
Into a number of equal parts 

(Fig. 7).— To divide the line A B into, 
say, five parts, draw the line A C at 
an angle from .4; set off five equal 
parts; draw B 5 and draw parallels to 
it from the other points of division in 
A C. These parallels divide A B as 
required. 

No'iE.— By a similar process a line 
may be divided into a number of un- 
equal parts; setting off divisions on 
A C, proportional by a scale to the re- 
quired divisions, and drawing parallel 
cutting A B. The triangles ^411, A22, 
A&i, etc., are similar triangles. 



8. Upon a straight line to 
draw an angle equal to a 
giveu angle (tig K).— Let A be lite 
given aiijrie and b G the line. From 
the point A with any radius describe 
the arc D E. From F with the same 
radius describe 1 H. Set off the arc 
/ H equal to D E, and draw F H. The 
angle Fis equal to A, as required. 



9. To draw angles of 60° 

and. 30° (Fig. 9).— From F, with 
any radius F I, describe an arc I J?; 
and from I, with the same radius, cut 
the arc at H and draw F H to form 
the required angle IF H. Draw the 
perpendicular H Kto the base line to 
form the angle of 30° F H K. 



10. To draw an angle of 45° 

(Fig. 10).— Set off the distance F I; 
draw the perpendicular 1 H equal to 
IF, and join HF to form the angle at 
F. The angle at H is also 45°. 



11. To bisect an angle (Fig. 
11).— Let A C Bbe the angle; with G 
as a centre draw an arc cutting the 
sides at A, B. From A and B as 
centres, describe arcs cutting each 
other at D. Draw C D, dividing the 
angle into two equal parts. 



12. Through two given 
points to describe an arc of 
a circle with a given radius 

(Fig. 12).— From the points A and B 
as centres, with the given radius, de- 
scribe arcs cutting at C; and from 
C with the same radius describe an 
arc A B. 



GEOMETRICAL PROBLEMS. 



39 





■V 




13. To find the centre of a 
circle or of an arc of a circle 

(Fig. 13). — Select three points, A, B, 
C, in the circumference, Avell apart; 
with the same radius describe arcs 
from these three points, cutting each 
other, and draw the two lines, D E, 
F G, through their intersections. The 
point 0, where they cut, is the centre 
of the circle or arc. 

To describe a circle passing 
through three given points. 
— Len A, B, C be the given points, and 
proceed as in last problem to find the 
centre 0, from which the circle may 
be described. 

14. To describe an arc of 
a circle passing through 
three given points when 
the centre is not available 
(Fig. 14). —From the extreme points 
A, B, as centres, describe arcs A H, 
B G. Through the third point 
draw A E, B F, cutting the arcs. 
Divide A F and B E into any num- 
ber of equal parts, and set off a 
series of equal parts of the same 
length on the upper portions of the 
arcs beyond the points E F. Draw 
straight lines, B L, B M, etc., to 
the divisions in A F, and A I, A K, 
etc., to the divisions in E G. The 
successive intersections N, 0, etc., 
of these lines are points in the 
circle required between the given 
points A and C. which may be 
drawn in ; similarly the remaining 
part of the curve B C may be 
described. (See also Problem 54.) 

15. To draw a tangent to 
a circle from a given point 
in the circumference (Fig. 15). 
—Through the given point A, draw the 
radial line A C, and a perpendicular 
to it, F G, which is the tangent re- 
quired. 



16. To draw tangents to a 
circle from a point without 

it (Fig. 16).— From A, with the radius 
A C, describe an arc B CD, and from 
C. with a radius equal to the diameter 
of the circle, cut the arc at B D. Join 
B C, CD, cutting the circle at E F y 
and draw A E, A F, the tangents. 

Note.— When a tangent is already 
drawn, the exact point of contact may 
be found by drawing a perpendicular 
to it from the centre. 



17. Between two inclined lines to draw a series of cir- 
cles touching these lines and touching each other (Fig. 1?). 
-Bisect the inclination of the given lines A B, C D, by the line N 0. From 
v point P in this line draw the perpendicular P B to the line A B, and 



40 



GEOMETRICAL PROBLEMS. 
A 




on P describe the circle B D, touching 
the lines and cutting: the centre line 
at E. From J? draw EF perpendicular 
to the centre line, cutting A B at F, 
and from F describe an arc E G, cut- 
ting A B at G. Draw G H parallel to 
B P, giving H, the centre of the next 
circle, to be described with the radius 
H E, and so on for the next circle IN. 
Inversely, the largest circle may be 
described first, and the smaller ones 
in succession. This problem is of fre- 
quent use in scroll-work. 

18. Between two inclined 
lines to draw a circular seg- 
ment tangent to the lines and 
passing through a point F 
on the line^ C which bisects 
the angle of the lines (Fig. 18). 
— Through .Fdraw D A at right angles 
to F G ; bisect the angles A and D, as 
in Problem 11, by lines cutting at C, 
and from Cwith radius C Fdva,w the 
arc H F G required. 

19. To draw a circular arc 
that will he tangent to two 
given lines A B and C D in- 
clined to one another, one 
tangential point E being 
given (Fig. 19).— Draw the centre 
line G F. From E draw E F at right 
to angles A B ; then F is the centre 
of the circle required. 

20. To describe a circular 
arc joining two circles, and 
touching one of them at a 
given point (Fig. 20). — To join the 
circles A B, F G, by an arc touching 
one of them at F, draw the radius E F, 
and produce it both ways. Set off F H 
equal to the radius A C of the other 
circle; join C H and bisect it with the 
perpendicular LI, cutting E F at I. 
On the centre I, with radius IF, de- 
scribe the arc F A as required. 



21. To draw a circle with a 
given radius It that will be 
tangent to two given circles 

A and Ji (Fig. 21).— From centre 
of circle A with radius equal B plus 
radius of A, and from centre of B with 
radius equal to R + radius of B, draw 
two arcs cutting each ot her in C, which 
will be the centre of the circle re- 
quired. 

22. To construct an equi- 
lateral triangle, the sides 
being given (Fig. 22).— On the ends 
of out- side, A, B, with A B as radius, 
describe arcs cutting at C, and draw 
AC,CB, 



GEOMETRICAL PROBLEMS. 



41 




23. To construct a triangle 
of unequal sides (Fig. 23).— On 
either end of the base A D, with the 
side B as radius, describe an arc; 
and with the side C as radius, on the 
other end of the base as a centre, cut 
the arc at E. Join A E, D E. 




24. To construct a square 
on a given straight line A B 

(Fig. 24). — At ..4 erect a perpendicular 
A C, as in Problem 4. Lay off A D 
equal to A B ; from D and B as centres 
with radius equal A B, describe arcs 
cutting each other in E. Join D E and 
BE. 




25. To construct a rect- 
angle with given base M F 

and height E H (Fig. 25).— On the 
base E Fdvaw the perpendiculars EH, 
F G equal to the height, and join Q H. 






26. To describe a circle 
about a triangle (Fig. 26).— 
Bisect two sides A B, A C of the tri- 
angle at E F, and from these points 
draw perpendiculars cutting at K. On 
the centre K, with the radius K A, 
draw the circle ABC. 



27. To inscribe a circle in 
a triangle (Fig. 27).— Bisect two of 
the angles A, C, of the triangle by lines 
cutting at D ; from D draw a per- 
pendicular D Eto any side, and with 
D E as radius describe a circle. 

When the triangle is equilateral, 
draw a perpendicular from one of the 
angles to the opposite side, and from 
the. side set off one third of the per- 
pendicular. 

28. To describe a circle 
about a square, and to in- 
scribe a square in a circle (Fig. 
28). — To describe the circle, draw the 
diagonals A B, C D of the square, cut- 
ting at E. On the centre E, with the 
radius A E, describe the circle. 

To inscribe the square.— 
Draw the two diameters, A B, CD, at 
right angles, and join the points A, B, 
C D, to form the square. 

Note.— In the same way a circle may 
be described about a rectangle. 



42 



GEOMETRICAL PROBLEMS. 






29* To inscribe a circle in a 
square (Fig. 29).— To inscribe the 
circle, draw the diagonals A B, C D 
of the square, cutting at E; draw the 
perpendicular E F to one side, and 
with the radius E F describe the 
circle. 



30. To describe a square 
about a circle (Fig. 30).— Draw two 
diameters A B, CD at right angles. 
With the radius of the circle and A, B, 
C and D as centres, draw the four 
half circles which cross one another 
in the corners of the square. 



31. To inscribe a pentagon 
in a circle (Fig. 31).— Draw diam- 
eters AC, B D at right angles, cutting 
at o. Bisect A o at E. and from E, 
with radius E B, cut A C at F ; from 
B, with radius B F, cut the circumfer- 
ence at G, H, and with the same radius 
step round the circle to Zand K\ join 
the points so found to form the penta- 
gon. 



32. To construct a penta- 
gon on a given line A & (Fig. 
32).— From B erect a perpendicular 
B G half the length of A B; join A C 
and prolong it to D, making CD — B C. 
Then B D is the radius of the circle 
circumscribing the pentagon. From 
A and B as centres, with B D as radius, 
draw arcs cutting each other in O, 
which is the centre of the circle. 

33. To construct a hexagon 
upon a given straight line 

(Fig. 33).— From A and B, the ends of 
the given line, with radius A B, de- 
scribe arcs cutting at g ; from g, with 
the radius g A, describe a circle ; with 
the same radius set off the arcs A G, 
G F, and B D, D E. Join the points so 
found to form the hexagon. The side 
of a hexagon = radius of its circum- 
scribed circle. 

34. To inscribe a hexagon 
in a circle (Fig. 34).— Draw a diam- 
eter A CB. From A and B as centres, 
with the radius of the circle A C, cut 
the circumference at D, E, F, G, and 
draw A D,D E, etc., to form the hexa- 
gon. The radius of the circle is equal 
to the side of the hexagon ; therefore 
the points D, E, etc., may also be 
found by stepping the radius six 
times round the circle. The angle 
between the diameter and the sides of 
a hexagon and also the exterior angle 
between a side and an adjacent side 
prolonged is 60 degrees; therefore a 
hexagon may conveniently be drawn 
by the use of a 60-degree triangle. 



L 



GEOMETRICAL PROBLEMS. 



43 





35. To describe a hexagon 
about a circle (Fig. 35).— Draw a 
diameter A D B, and with the radius 
A D, on the centre A, cut the circum- 
ference at C ; join A C, and bisect it 
with the radius D E ; through E draw 
FG, parallel to A C, cutting the diam- 
eter at F, and with the radius D ^de- 
scribe the circumscribing circled if. 
Within this circle describe a hexagon 
by the preceding problem. A more 
convenient method is by use of a 60- 
degree triangle. Four of the sides 
make angles of 60 degrees with the 
diameter, and the other two are par- 
allel to the diameter. 

36. To describe an octagon 
on a given straight line (Fig. 

36).— Produce the given line A B both 
ways, and draw perpendiculars A E, 
B F; bisect the external angles A and 
B by the lines A H, B C, which make 
equal to A B. Draw C D and H G par- 
allel to A E, and equal to A B ; from 
the centres G, D, with the radius A B, 
cut the perpendiculars at E, F, and 
draw S F to complete the octagon. 

37. To convert a square 
into an octagon (Fig. 37).— Draw 
the diagonals of the square cutting at 
e ; from the corners A, B, C, D, with 
ieas radius, describe arcs cutting 
the sides at gn, fk, hm, and ol, and 
join the points so found to form the 
octagon. Adjacent sides of an octa- 
gon make an angle of 135 degrees. 



38. To inscribe an octagon 
in a circle (Fig. 38).— Draw two 
diameters, A C, B D at right angles; 
bisect the arcs A B, B C, etc., at ef, 
etc., and join A e, e B, etc., to form 
the octagon. 



39. To describe an octagon 
about a circle (Fig. 39).— Describe 
a square about the given circle A B ; 
draw perpendiculars h k, etc., to the 
diagonals, touching the circle to form 
the octagoD. 



40. To describe a polygon of any number of sides upon 
a given straight line (Fig. 40).— Produce the given line A B, and on A, 



44 



GEOMETRICAL PROBLEMS. 






with the radius A B, describe a semi- 
circle; divide the semi-circumference 
into as many equal parts as there are 
to be sides in the polygon— say, in this 
example, five sides. Draw lines from 
A through the divisional points D, 6, 
and c, omitting one point a ; and on 
the centres B, D, with the radius AB, 
cut A b at E and A c at F. Draw D E, 
E F, F B to complete the polygon. 

41. To inscribe a circle 
w it lain a polygon (Figs. 41, 42).— 
When the polygon has an even number 
of sides (Fig. 41), bisect two opposite 
sides at A audi?; draw AB. and bisect 
it at C by a diagonal D E, and with 
the radius C A describe the circle. 

When the number of sides is odd 
(Fig. 42), bisect two of the sides at A 
and B, and draw lines A E, B D to the 
opposite angles, intersecting at C ; 
from C, with the radius C A, describe 
the circle. 



42. To describe a circle 
without a polygon (Figs. 41. 42). 
— Find the centre (J as before, and with 
the radius C D describe the circle. 




43. To inscribe a polygon 
of any number of sides with- 
in a circle (Fig. 43).— Draw the 
diameter A B and through the centre 
E draw the perpendicular EC, cutting 
the circle at F. Divide E F into four 
equal parts, and set off three parts 
equal to those from F to G. Divide 
the diameter A B into as many equal 
parts as the polygon is to have sides ; 
and from C draw C D, through the 
second point of division, cutting the 
circle at D. Then A D is equal to one 
side of the polygon, and by stepping 
round the circumference with the 
length A D the polygon may be com- 
pleted. 



TABLE OF POLYGONAL ANGLES. 



Number 


Angle 


Number 


Angle 


Number 


Angle 


of Sides. 


at Centre. 


of Sides. 


at Centre. 


of Sides. 


at Centre. 


No. 


Degrees. 


No. 


Degrees. 


No. 


Degrees. 


3 


120 


9 


40 


15 


24 


4 


90 


10 


36 


16 


22£ 


5 


72 


11 


32 T 8 T 


17 


21 & 


6 


60 


i 12 


30 


18 


20 


7 


51f 


i 13 


2?A 


19 


19 


8 


45 


!- H 


25f 


20 


18 



GEOMETRICAL PROBLEMS. 



45 




In this table the angle at the centre is found by dividing 360 degrees, the 
number of degrees in a circle, by the number of sides in the polygon; and 
by setting off round the centre of the circle a succession of angles by means 
of the protractor, equal to the angle in the table due to a given number of 
sides, the radii so drawn will divide the circumference into the same number 
of parts. 

44. To describe an ellipse 
when the length and breadth 
are given (Fig. 44).— A B, transverse 
axis; C D, conjugate axis; F G, foci. 
The sum of the distances from C to 
i^and G, also the sum of the distances 
from F and G to any other point in 
the curve, is equal to the transverse 
axis. From the centre C, with A E as 
radius, cut the axis A B at i^and G, 
the foci ; fix a couple of pins into the 
axis at F and G, and loop on a thread 
or cord upon them equal in length to 
the axis A B, so as when stretched to 
reach to the extremity C of the con- 
jugate axis, as shown in dot-lining. 
Place a pencil inside the cord as at H, 
and guiding the pencil in this way, 
keeping the cord equally in tension, 
carry the pencil round the pins F, G, 
and so describe the ellipse. 

Note.— This method is employed in 
setting off elliptical garden - plots, 

2d Method (Fig. 45). — Along the 
straight edge of a slip of stiff paper 
mark off a distance a c equal to A C, 
half the transverse axis; and from the 
same point a distance a b equal to 
C D, half the conjugate axis. Place 
the slip so as to bring the point b on 
the line A B of the transverse axis, 
and the point c on the line D E ; and 
set off on the drawing the position of 
the point a. Shifting the slip so that 
the point b travels on the transverse 
axis, and the point c on the conjugate 
axis, any number of points in the 
curve may be found, through which 
the curve may be traced. 

3d Method (Fig. 46).— The action of 
the preceding method may be em- 
bodied so as to afford the means of 
describing a large curve continuously 
by means of a bar m fc, with steel 
points m, 1, fc, riveted into brass slides 
adjusted to the length of the semi- 
axis and fixed with set-screws. A 
rectangular cross E G, with guiding- 
slots is placed, coinciding with the 
two axes of the ellipse A C and B H. 
By sliding the points k, I in the slots, 
and carrying round the point m, the 
curve may be continuously described. 
A pen or pencil may be fixed at m. 

4th Method (Fig. 47).— Bisect the 
transverse axis at C. and through C 
draw the perpendicular D E, making 
C D and C E each equal to half the 
conjugate axis. From D or E, with 
the radius A C, cut the transverse 
axis at F, F', for the foci. Divide 
A C into a number of parts at the 





46 



GEOMETRICAL PROBLEMS. 





G 






;'' 


D "N 




' x > i\\ 


aV h !' ' ! 


c | \\ •] 






avd^'i 


' /"• 'A 


X^QL/ 


\j^r*/ 


&x!?"~~ 


E \i / 








If" 



B 



Fig. 48. 



points 1, 2, 3, etc. With the radius A I on F and .F 1 ' as centres, describe 
arcs, and with the radius B I on the same centres cut these arcs as shown. 
Repeat the operation for the other 
divisions of the transverse axis. The 
series of intersections thus made are 
points in the curve, through which the 
curve may be traced. 

5th Method (Fig. 48).— On the two 
axes A B, D E as diameters, on cent re 
C, describe circles; from a number of 
points a, 6, etc., in the circumference 
A FB, draw radii cutting the inner 
circle at a', b', etc. From a, b, etc., 
draw perpendiculars to AB; and from 
a', b', etc., draw parallels to A B, cut- 
ting the respective perpendiculars at 
n, o. etc. The intersections are points 
in the curve, through which the curve 
may be traced. 

6th Method (Fig. 49). — When the 
transverse and conjugate diameters 
are given, A B, C D, draw the tangent 
EF parallel to AB. Produce CD, 
and on the centre G with the radius 
of half A B, describe a semicircle 
HDK; from the centre G draw any 
number of straight lines to the points 
E, r, etc., in the line E F, cutting the 
circumference at I, m, n, etc.; from 
the centre O of the ellipse draw 
straight lines to the points E, r, etc. ; 
and from the points I, m, n, etc., draw 
parallels to G C, cutting the lines E, 
O r, etc., at L, M, N, etc. These are 
points in the circumference of the 
ellipse, and the curve may be traced 
through them. Points in the other 
half of the ellipse are formed by ex- 
tending the intersecting lines as indi- 
cated in the figure. 

45. To describe an ellipse 
approximately by means of 
circular arcs.— First.— With arcs 
of two radii (Fig. 50).— Find the differ- 
ence of the two axes, and set it off 
from the centre O to a and c on O A 
and O C ; draw ac, and set off half 
a c to d ; draw d i parallel to a c; set 
off O e equal to O d; join e i, and draw 
the parallels e m, d m. From m, with 
radius m C, describe an arc through 
C ; and from i describe an arc through 
D; from d and e describe arcs through 
A and B. The four arcs form the 
ellipse approximately. 

Note.— This method does not apply 
satisfactorily when the conjugate axis 
is less than two thirds of the trans- 
verse axis. 

2d Method (by Carl G. Barth, 
Fig. 51). —In Fig. 51 a b is the major 
and c d the minor axis of the ellipse 
to be approximated. Lay off b e equal 
to the semi-minor axis c O, and use a e 
as radius for the arc at each extremity 
of the minor axis. Bisect e o sit f and 
lay off e g equal to ef, and use g b as 
radius for the arc at each extremity 
of the major axis. 




Fig. 49. 





^~~c 


\ 

c \ 




I /h 


a 





'V\ J 


V" 


\ 


/ 


V 






m 





B 



D 
Fig. 50. 




Fig. 51. 



GEOMETRICAL PROBLEMS. 



47 



£^~? 


X^K"""! 


s'\ 






Fig 


J 

i 

) 



The method is not considered applicable for cases in which the minor 
axis is less than two thirds of the major. 

, 3d Method : With arcs of three radii 

(Fig. 52).— On the transverse axis A B 

draw the rectangle B G on the height 

0(7; to the diagonal A C draw the 

perpendicular G H D\ set off O K 

,^\ \^ equal to O C, and describe a semi- 

IjjK. y\ \ / X sy circle on A K, and produce OCtoL; 

// v, V N / ,Jp\ \j set off O If equal to C L, and from D 

- ~^— A— , — !- — i- '-'—i — _-J describe an arc with radius D M ; from 

A, with radius O L, cut this arc at a. 
Thus the five centres D, a, b, H, H' 
are found, from which the arcs are 
described to form the ellipse. 

Note. — This process works well for 
nearly all proportions of ellipses. It 
is employed in striking out vaults and 
stone bridges. 

4th Method (by F. R. Honey, Figs. 53 and 54).— Three radii are employed. 
With the shortest radius describe the two arcs which pass through the ver- 
tices of the major axis, with the longest the two arcs which pass through 
the vertices of the minor axis, and with the third radius the four arcs which 
connect the former. 
A. simple method of determining the radii of curvature is illustrated in 

Fig. 53. Draw the straight 
lines a /and a c, forming any 
angle at a. With a as a centre, 
and with radii a b and ac, re- 
spectively, equal to the semi- 
ininor and semi-major axes, 
draw the arcs b e and c d. Join 
ed, and through b and c r 
speetively draw b g and c / 
parallel to e d, intersecting a c 
at g, and af at/; af is the 
radius of curvature at the ver- 
tex of the minor axis; and a g 
the radius of curvature at the 
vertex of the major axis. 

Lay off d h (Fig. 53) equal to one eighth of b d. Join e h, and draw c k and 
o l parallel to e h Take a k for the longest radius (=B),al for the shortest 
ramus (= r) and the arithmetical mean, or one half the sum of the semi-axes, 
as'follows- US (= p) ' and e,n P lo y these radii for the eight-centred oval 

Let a b and c d (Fig. 54) 
be the major and minor 
axes. Lay. off a e equal 
to r, and af equal to p; 
also lay off c g equal to B, 
and c h equal to p. With 
g as a, centre and g h as a 
radius, draw the arc h k; 
with the centre e and 
radius e f draw the arc fk, 
intersecting/ifc at k. Draw 
the line g k and produce it, 
making g I equal to B. 
Draw k e and produce it, 
making k m equal to p. 
With the centre g and 
radius g c (= B) draw the 
arc c I ; with the centre k 
and radius k I (= p) draw 
the arc I m, and with the 
centre e and radius e m 
(= r) draw the arc m a- 

The remainder of the 
work is symmetrical with 
respect to the axes. 





48 



GEOMETRICAL PROBLEMS. 



K 




i 


L 






A 




E 




G 


nL. 


F 




v 


71/ 


o 


\ 


V 




o 




\ 




o 




\ % 


D 


B 
b 




'a c 



Fig. 55. 



' 46. The Parabola. —A parabola 
(D A C, Fig. 55) is a curve such that 
every point in the curve is equally- 
distant from the directrix KL and the 
focus F. The focus lies in the axis 
A B drawn from the vertex or head of 
the curve A, so as to divide the figure 
into two equal parts. The vertex A ' 
is equidistant from the directrix and 
the focus, or A e — A F. Any line 
parallel to the axis is a diameter. A 
straight line, as E G or DC, drawn 
across the figure at right angles to the 
axis is a double ordinate, and either 
half of it is an ordinate. The ordinate 
to the axis E F G, drawn through the 
focus, is called the parameter of the 
axis. A segment of the axis, reckoned 
from the vertex, is an abscissa of the 
axis, and it is an abscissa of the ordi- 
nate drawn from the base of the ab- 
scissa. Thus, A B is an abscissa of 
the ordinate B C. 
Abscissae of a parabola are as the squares of their ordinates. 
To describe a parabola when an abscissa and its ordi- 
nate are given (Fig. 55).— Bisect the given ordinate B Cat a. draw A a, 
and then a b perpendicular to it, meeting the axis at b. Set off A e. A F, 
each equal to B b; and draw K e L perpendicular to the axis. Then K L is 
the directrix and F is the focus. Through F and any number of points, o, o, 
etc., in the axis, draw doubJe ordinates, n o n, etc , and from the centre F, 
with the radii Fe, o e, etc., cut the respective ordinates at E, G, n, n, etc. 
The curve may be traced through these points as shown. 



2d Method : By means of a square 
and a cord (Fig. 56).— Place a straight- 
edge to the directrix EN, and apply 
to it a square LEG. Fasten to the 
end G one end of a thread or cord 
equal in length to the edge E G, and 
attach the other end to the focus F ; 
slide the square along the straight- 
edge, holding the cord taut against the 
eiige of the square by a pencil D, by 
which the curve is described. 





; b a B a b c d 
Fig. 57. 



Zd Method : When the height and 
the base are given (Fig. 57). — Let A B 
be the given axis, and CD a double 
ordinate or b»se; to describe a para- 
bola of which the vertex is at A. 
Through A draw EF parallel to CD, 
and through C and D draw C E and 
D F parallel to the axis. Divide B C 
and B D into any number of equal 
parts, say five, at a, b, etc., and divide 
C E and DF'iuto the same number of 
parts. Through the points a, 6, c, d in 
the base CD on each side of the axis 
draw perpendiculars, and through 
a,b,c, din C E and D F draw lines to 
the vertex A, cutting the perpendicu- 
lars at e, /, g, h. These are points in 
the parabola, and the curve C AD may 
be traced as shown, passing through 
them. 



GEOMETRICAL PROBLEMS. 



49 




Fig. 59. 



47. The Hyperbola (Fig. 58).— A hyperbola is a plane curve, such 
that the difference of the distances from any point of it to two fixed points 
is equal to a given distance. The fixed 
points are called the foci. 

To construct a hyperbola. 
—Let F' and F be the foci, and F' F 
the distance between them. Take a 
ruler longer than the distance F' F, 
and fasten one of its extremities at the 
focus F'. At the other extremity, H, 
attach a thread of such a length that 
the length of the ruler shall exceed 
the length of the thread by a given 
distance A B. Attach the other ex- 
tremity of the thread at the focus F. 

Press a pencil, P, against the ruler, 
and keep the thread constantly tense, 
while the ruler is turned around F' as 
a centre. The point of the pencil will 
describe one branch of the curve. 

2d Method: By points (Fig. 59).— 
From the focus F' lay off a distance 
F' N equal to the transverse axis, or 
distance between the two branches of 
the curve, and take any other distance, 
as F'H, greater than F'N. 

With F" as a centre and F'H as a 
radius describe the arc of a circle. 
Then with .Fas a centre and N H as a 
radius describe an arc intersecting 
the arc before described at p and q. 
These will be points of the hyperbola, for F' q — Fq is equal to the trans- 
verse axis A B. 

If, with F as a centre and F' H as a radius, an arc be described, and a 
second arc be described with F' as a centre and N H as a radius, two points 
in the other branch of the curve will be determined. Hence, by changing 
the centres, each pair of radii will determine two points in each branch. 

The Equilateral Hyperbola.— The transverse axis of a hyperbola 
is the distance, on a line joining the foci, between the two branches of the 
curve. The conjugate axis is a line perpendicular to the transverse axis, 
drawn from its centre, and of such a length that the diagonal of the rect- 
angle of the transverse and conjugate axes is equal to the distance between 
the foci. The diagonals of this rectangle, indefinitely prolonged, are the 
asymptotes of the hyperbola, lines which the curve continually approaches, 
but touches only at an infinite distance. If these asymptotes are perpen- 
dicular to each other, the hyperbola is called a rectangular or equilateral 
hyperbola. It is a property of this hyperbola that if the asymptotes are 
taken as axes of a rectangular system of coordinates (see Analytical Geom- 
etry), the product of the abscissa and ordinate of any point in the curve is 
equal to the product of the abscissa and ordinate of any other point ; or, if 
p is the ordinate of any point and v its abscissa, and p x and v } are the ordi- 
nate and abscissa of any other point, pv—pi v x ; or pv = a constant. 

48. The Cycloid 
(Fig. 60).— If a circle Ad 
be rolled along a straight 
line ^4 6, any point of the 
circumference as A will 
describe a curve, which is 
called a cycloid. The circle 
is called the generating 
circle, and A the generat- 
ing point. 

To draw a cycloid. 
— Divide the circumference 
of the generating circle into an even number of equal parts, as A 1, 12, etc., 
and set off these distances on the base. Through the points 1, 2, 3, etc., on 
the circle draw horizontal lines, and on them set off distances la — Al, 
26 = A2, 'ic = AS, etc. The points A, a, b, c, etc., will be points in the cycloid, 
through which draw the curve. 




50 



GEOMETRICAL PROBLEMS. 




49. The Epicycloid (Fig. 61) is 
generated by a point D in one circle 
D C rolling upon the circumference of 
another circle A C B, instead of on a 
flat surface or line; the former being 
the generating circle, and the latter 
the fundamental circle. The generat- 
ing circle is shown in four positions, in 
which the generating point is succes- 
sively marked D, D', D", £>'". A D'" B 
is the epicycloid. 



50. The Hypocycloid (Fig. 62) 

is generated by a point in the gener- 
ating circle rolling on the inside of ihe 
fundamental circle. 

When the generating circle = radius 
of the other circle, the hypocycloid 
becomes a straight line. 



51. The Tractrix or 

Schiele's anti-friction curve 

(Fig. 63).— R is the radius of the shaft, 
C, 1,2, etc., the axis. From O set off 
on R a small distance, oa; with radius 
R and centre a cut the axis at 1, join 
a 1, and set off a like small distance 
ab; from b with radius R cut axis at 
2, join b 2, and so on, thus finding 
points o, a, 6, c, d, etc., through which 
the curve is to be drawn. 
Fig. 63. 
52. The Spiral.— The spiral is a curve described by a point which 

moves along a straight line according to any given law, the line at the same 

time having a uniform angular motion. The line is called the radius vector. 
If the radius vector increases directly 
as the measuring angle, the spires, 
or parts described in each revolution, 
thus gradually increasing their dis- 
tance from each other, the curve is 
known as the spiral of Archimedes 
(Fig. 64). 

This curve is commonly used for 
cams. To, describe it draw the radius 
vector in several different directions 
around the centre, with equal angles 

between them; set off the distances 1, 2, 3, 4, etc., corresponding to the scale 

upon which the curve is drawn, as shown in Fig. 64. 
In the common spiral (Fig. 64) the pitch is uniform; that is, the spires are 

equidistant. Such a spiral is made by rolling up a belt of uniform thickness. 



To construct a spiral with 
four centres (Fig. 65).— Given the 
pitch of the spiral, construct a square 
about the centre, with the sum of the 
four sides equal to the pitch. Prolong 
the sides in one direction as shown; 
the corners are the centres for each 
arc of the external angles, forming a 
quadrant of a spire. 





Fig. 65. 



GEOMETRICAL PROBLEMS. 



51 




Fig. 66. 



53. To find the diameter of a circle into which a certain 
number of rings will fit on its inside (Fig. 66).— For instance, 
what is the diameter of a circle into which twelve ^-h>ch rings will fit. as 
per sketch ? Assume that we have found the diameter of the required 

circle, and have drawn the rings inside 
of it. Join the centres of the rings 
by straight lines, as shown : we then 
obtain a regular polygon with 12 
sides, each side being equal to the di- 
ameter of a given ring. We have now 
to find the diameter of a circle cir- 
cumscribed about this polygon, and 
add the diameter of one ring to it; the 
sum will be the diameter of the circle 
into which the rings will fit. Through 
the centres A and D of two adjacent 
rings draw the radii CA and CD; 
since the polygon has twelve sides the 
angle A C D = 30° and A C B = 15°. 
One half of the side A D is equal to 
A B. We now give the following pro- 
portion : The sine of the angle AC B 
is to AB as 1 is to the required ra- 
dius. From this we get the following 
rule : Divide A B by the sine of the angle A C B ; the quotient will be the 
radius of the circumscribed circle ; add to the corresponding diameter the 
diameter of one rine: ; the sum will be the required diameter FG. 

54. To describe an arc of a circle which is too large to 
be drawn by a beam compass, by means of points in the 
arc, radius being given.— Suppose the radius is 20 feet and it is 
desired to obtain five points in an arc whose half chord is 4 feet. Draw a 
line equal to the half chord, full size, or on a smaller scale if more con- 
venient, and erect a perpendicular at one end, thus making rectangular 
axes of coordinates. Erect perpendiculars at points 1, 2, 3, and 4 feet from 
the first perpendicular. Find values of y in the formula of the circle. 
a* 2 + 2/ 2 = B 2 by substituting for x the values 0, 1, 2, 3, and 4, etc., and for H 2 
the_square of _the r adiu s, or 400. The values will he y — V R 2 — x 2 — ^400, 
^399, ^396, V 391, V 384; = 20, 19.975, 19.90, 19.774, 19.596. 
Subtract the smallest, 

or 19.596, leaving 0.404, 0.379, 0.304, 0.178, feet. 

Lay off these distances on the five perpendiculars, as ordinates from the 
half chord, and the positions of five points on the arc will be found. 
Through these the curve may be 
drawn. (See also Problem 14.) 

55. The Catenary is the curve 
assumed by a, perfectly flexible chord 
when its ends are fastened at two 
points, the weight of a unit length 
being constant. 
The equation of the catenary is 







Fig. 67. 



in which e is the 

base of the Naperian system of log- 
arithms. 
To plot the catenary.— Let o 

(Fig. 67) be the origin of coordinates. 
Assigning to a any value as 3, the 
equation becomes 



= ?le3 



(* + .-.-). 



To find the lowest point of the curve. 

q/ o -o\ o 

Vxxtx = 0\.-.y=ile fe J = |(1 y- 1) = 3. 



52 



GEOMETRICAL PROBLEMS. 



Then put x = 1 ; 






(1.396 + 0.717) = 3.17. 



-(1.948 4- 0.513) = l 



Put x = 3, 4, 5, etc., etc., and find the corresponding values of y. For 
each value of y we obtain two symmetrical points, as for example p and p 1 . 
In this way, by making a successively equal to 2, 3, 4, 5, 6, 7, and 8, the 
curves of Fig. 68 were plotted. 

In each case the distance from the origin to the lowest point of the curve 
is equal to a ; for putting x = o, the general equation reduces to y — a. 

For values of a — 6, 7, and 8 the catenary closely approaches the parabola. 
For derivation of the equation of the catenary see Bowser's Analytic 
Mechanics. For comparison of the catenary with the parabola, see article 
by F. It. Honey, Amer. Machinist, Feb. 1, 1894. 
56. The Involute is a name given to the curve which is formed by 
the end of a string which is unwound 
from a cylinder and kept taut ; con- 
sequently the string as it is unwound 
will always lie in the direction of a 
tangent to the cylinder. To describe 
the involute of any given circle, Fig. 
68, take any point A on its circum- 
ference, draw a diameter A B, and 
from B draw B b perpendicular to AB. 
Make Bb equal in length to half the 
circumference of the circle. Divide 
Bb and the semi-circumference into 
the same number of equal parts, 
say six. From each point of division 
1, 2, 3, etc., on the circumference draw 
lines to the centre C of the circle. 
Then draw 1 a perpendicular to C 1 ; 
2a, 2 perpendicular to C2; and so on. 
Make 1 a equal to b />, ; 2 q a equal 
to b 6 2 ; 3 a 3 equal to. ft b 3 ; and so on. 
by a curve ; this curve will be the 




Fig. 68. 



, etc., 



Join the 'points A, a x \ > 
required involute. 

57. Method of plotting angles without using a protrac- 
tor. --The radius of a circle whose circumference is 360 is 57.3 (more ac- 
curately 57.296). Striking a semicircle with a radius 57.3 by any scale, 
spacers set to 10 by the same scale will divide the arc into 18 spaces of 10° 
each, and intermediates can be measured indirectly at the rate of 1 by scale 
for each 1°, or interpolated by eye according to the degree of accuracy 
required. The following table shows the chords to the above-mentioned 
radius, for every 10 degrees from 0° up to 110°. By means of one of these, 



Angle, 



10°. 
20°. 



40° 
50° 



Chord. 

, 0.999 

. 9.988 

. 19.899 

. 29.658 

. 39.192 

. 48.429 



Angle. 

60° . . . 

70°... 

80°... 

90°... 
100°... 
110°... 



Chord. 

57.296 
. 65.727 
. 73.658 
. 81.029 

. 87.782 



a 10° point is fixed upon the paper next less than the required angle, and 
the remainder is laid off at the rate of 1 by scale for each degree. 



GEOMETRICAL PROPOSITIONS. 53 



GEOMETRICAL PROPOSITIONS. 

In a right-angled triangle the square on the hypothenuse is equal to the 
sum of the squares on the other two sides. 

If a triangle is equilateral, it is equiangular, and vice versa. 

If a straight line from the vertex of an isosceles triangle bisects the base, 
it bisects the vertical angle and is perpendicular to the base. 

If one side of a triangle is produced, the exterior angle is equal to the sum 
of the two interior and opposite angles. 

If two triangles are mutually equiangular, they are similar and their 
corresponding sides are proportional. 

If the sides of a polygon are produced in the same order, the sum of the 
exterior angles equals four right angles. 

In a quadrilateral, the sum of the interior angles equals four right angles. 

In a parallelogram, the opposite sides are equal ; the opposite angles 
are equal; it is bisected by its diagonal; and its diagonals bisect each 
other. 

If three points are not in the same straight line, a circle may be passed 
through them. 

If two arcs are intercepted on the same circle, they are proportional to 
the corresponding angles at the centre. 

If two arcs are similar, they are proportional to their radii. 

The areas of two circles are proportiona"l to the squares of their radii. 

If a radius is perpendicular to a chord, it bisects the chord and it bisects 
the arc subtended by the chord. 

A straight line tangent to a circle meets it in only one point, and it is 
perpendicular to the radius drawn to that point. 

If from a point without a circle tangents are drawn to touch the circle, 
there are but two; they are equal, and they make equal angles with the 
chord joining the tangent points. 

Jf two lines are parallel chords or a tangent and parallel chord, they 
intercept equal arcs of a circle. 

If an angle at the circumference of a circle, between two chords, is sub- 
tended by the same arc as an angle at the centre, between two radii, the 
angle at the circumference is equal to half the angle at the centre. 

If a triangle is inscribed in a semicircle, it is right-angled. 

If an angle is formed by a tangent and chord, it is measured by one half 
of the arc intercepted by the chord; that is, it is equal to half the angle at 
the centre subtended by the chord. 

If two chords intersect each other in a circle, the rectangle of the seg- 
ments of the one equals the rectangle of the segments of the other. 

And if one chord is a diameter and the other perpendicular to it, the 
rectangle of the segments of the diameter is equal to the square on half the 
other chord, and the half chord is a mean proportional between the seg- 
ments of the diameter. 



54 MENSURATION. 



MENSURATION. 

PLANE SURFACES. 

Quadrilateral.— A four-sided figure. 

Parallelogram.— A quadrilateral with opposite sides parallel. 

Varieties.— Square : four sides equal, all angles right angles. Rectangle: 
opposite sides equal, all angles right angles. Rhombus: four sides equal, 
opposite angles equal, angles not right angles. Rhomboid: opposite sides 
equal, opposite angles equal, angles not right angles. 

Trapezium.— A quadrilateral with unequal sides. 

Trapezoid. — A quadrilateral with only one pai* of opposite sides 
parallel. 

Diagonal of a square = |/2x side' 2 = 1.4142 x side. 

Diagonal of a rectangle = ^/product of two adjacent sides. 

Area of any parallelogram = base x altitude. 

Area of rhombus or rhomboid = product of two adjacent sides 
X sine of angle included between them. 

Area of a trapezium = half the product of the diagonal by the sum 
of the perpendiculars let fall on it from opposite angles. 

Area of a trapezoid = product of half the sum of the two parallel 
sides by the perpendicular distance between them. 

To find the area of any quadrilateral figure.— Divide ihe 
quadrilateral into two triangles; the sum of the areas of the triangles is the 
area. 

Or, multiply half the product of the two diagonals by the sine of the angle 
at their intersection. 

To find the area of a quadrilateral inscribed in a circle. 
—From halt' the sum of the four sides subtract each side severally; multi- 
ply the four remainders together; the square root of the product is the area. 

Triangle.— A three-sided plane figure. 

Varieties.— Right-angled, having one right angle; obtuse-angled, having 
one obtuse angle ; isosceles, having two equal angles and two equal sidet.; 
equilateral, having three equal sides and equal angles. 

The sum of the three angles of every triangle — 180°. 

The two acute angles of a right-angled triangle are complements of each 
other. 

Hypothenuse of a right-angled triangle, the side opposite the right angle. 

= |/sum of the squares of the other two sides. 
To find the area of a triangle : 

Rule 1. Multiply the base by half the altitude. 

Rule 2.- Multiply half the product of two sides by the sine of the included 
angle. 

Rule 3. From half the sum of the three sides subtract each side severally ; 
multiply together the half sum and the three remainders, and extract the 
square root of the product. 

The area of an equilateral triangle is equal to one fourth the square of one 

of its sides multiplied by the square root of 3, = ~ , a being the side; or 

4 
a 2 X .433013. 

Hypothenuse and one side of right-angled triangle given, to find other side, 
Required side = ^hyp 2 — given side 2 . 

If the two sides are equal, side = hyp -f- 1.4142; or hyp X .7071. 

Area of a triangle given, to find base: Base = twice area -5- perpendicular 
height. 

Area of a triangle given, to find height: Height = twice area -*- base. 

Two sides and base given, to find perpendicular height (in a triangle in 
which both of the angles at the base are acute). 

Rule.— As the base is to the sum of the sides, so is the difference of the 
sides to the difference of the divisions of the base made by drawing the per- 
pendicular. Half this difference being added to or subtracted from half 
the base will give the two divisions thereof. As each side and its opposite 



PLAtfE SURFACES. 



55 



division of the base constitutes a right-angled triangle, the perpendicular is 
ascertained by the rule perpendicular = Vhyp 2 — base 2 . 

Polygon. — A plane figure having three or more sides. Regular or 
irregular, according as the sides or angles are equal or unequal. Polygons 
are named from the number of their sides and angles. 

To find the area of an Irregular polygon.— Draw diagonals 
dividing the polygon into triangles, and find the sum of the areas of these 
triangles. 

To find the area of a regular polygon : 

Rule.— Multiply the length of a side by the perpendicular distance to the 
centre; multiply the product by the number of sides, and divide it by 2. 
Or, multiply half the perimeter by the perpendicular let fall from the centre 
on one of the sides. 

The perpendicular from the centre is equal to half of one of the sides of 
the polygon multiplied by the cotangent of the angle subtended by the half 
side. 

The angle at the centre = 360° divided by the number of sides. 







TABLE OF REGULAR POLYGONS. 












Radius of Cir- 
















cumscribed 


t_j 










8 




Ci 


•cle. 


.c-* 


°1§ 




< 




bo 








5 II 






a ^ 




>> 


11 


£-• 




CO CD 


T3 


a 


£2 

CD 5 


02 


o 


cd 

02 


g ll 


II 


cw&2 

Si" 






O 


a 


e3 


530 


-3 


18 


g^O 


to 


11 a3 
C-r-> 


£ 


fc 


< 


Ph 


02 


H 


M 


< 


< 


3 


Triangle 


.4330127 


2. 


.5773 


.2887 


1.732 


120° 


60° 


4 


Square 


1. 


1.414 


.7071 


.5 


1.4142 


90 


90 


5 


Pentagon 


1.7204774 


1.238 


.8506 


.6882 


1.1756 


72 


108 


6 


Hexagon 


2 5980762 


1.156 


1. 


.866 


1. 


60 


120 


7 


Heptagon 


3.6339124 


1.11 


1.1524 


1.0383 


.8677 


5126' 


128 4-7 


8 


Octagon 


4.8284271 


1.083 


1.3066 


1.2071 


.7653 


45 


135 


9 


Nonagon 


6.1818242 


1.064 


1.4619 


1.3737 


.684 


40 


140 


10 


Decagon 


7.6942088 


1.051 


1.618 


1.5388 


.618 


36 


144 


11 


Undecagon 


9.3656399 


1.042 


1.7747 


1.7028 


.5634 


32 43' 


147 3-11 


12 


Dodecagon 


11.1961524 


1.037 


1.9319 


1.866 


.5176 


30 


150 



To find the area of a regular polygon, when the length 
of a side only is given : 

Rule. — Multiply the square of the side by the multiplier opposite to the 
name of the polygon in the table. 

To find the area of an ir- 
regular figure (Fig. 69).— Draw or- 
di nates across its breadth at equal 
distances apart, the first and the last 
ordinate each being one half space 
from the ends of the figure. Find the 
average breadth by adding together 
the lengths of these lines included be- 
tween the boundaries of the figure, 
and divide by the number of the lines 
added ; multiply this mean breadth by 
the length. The greater the number 
of lines the nearer the approximation. 



2 3 



~J 



- -Length 

Fig. 69. 
In a, figure of very ^regular outline, as an indicator-diagram from a high- 



speed steam-engine, mean lines may be substituted for the actual lines of the 
figure, being so traced as to intersect the undulations, so that the total area 
of the spaces cut off may be compensated by that of the extra spaces in- 
closed. 



56 MENSURATlOtf. 

2d Method: The Trapezoidal Rule. — Divide the figure into any suffi- 
cient number of equal parts; acid half the sum of the two end ordinates to 
the sum of all the other ordinates; divide by the number of spaces (that is, 
one less than the number of ordinates) to obtain the mean ordinate, and 
multiply this by the length to obtain the area. 

3d Method : Simpson's Rule.— Divide the length of the figure into any 
even number of equal parts, at the common distance D apart, and draw or- 
dinates through the points of division to touch the boundary lines. Add 
together the first and last ordinates and call the sum A ; add together the 
even ordinates and call the sum B; add together the odd ordinates, except 
the first and last, and call the sum C. Then, 



area of the figure = 



4th Method : Durand's Rule.— Add together 4/10 the sum of the first and 
last ordinates, 1 1/10 the sum of the second and the next to the last (or the 
penultimates), and the sum of all the intermediate ordinates. Multiply the 
sum thus gained by the common distance between the ordinates to obtain 
the area, or divide this sum by the number of spaces to obtain the mean or- 
dinate. 

Prof. Durand describes the method of obtaining his rule in Engineering 
News, Jan. 18, 1894. He claims that it is more accurate than Simpson's rule, 
and practically as simple as the trapezoidal rule. He thus describes its ap- 
plication for approximate integration of differential equations. Any defi- 
nite integral may be represented graphically by an area. Thus, let 



-/' 



u dx 



be an integral in which u is some function of x, either known or admitting of 
computation Or measurement. Any curve plotted with x as abscissa and u 
as ordinate will then represent the variation of u with x, and the area be- 
tween such curve and the axis Xwill represent the integral in question, no 
matter how simple or complex may be the real nature of the function u. 

Substituting in the rule as above given the word " volume " for "area " 
and the word " section " for " ordinate," it becomes applicable to the deter- 
mination of volumes from equidistant sections as well as of areas from 
equidistant ordinates. 

Having approximately obtained an area by the trapezoidal rule, the area 
by Durand's rule may be found by adding algebraically to the sum of the 
ordinates used in the trapezoidal rule (that is, half the sum of the end ordi- 
nates + sum of the other ordinates) 1/10 of (sum of penultimates — sum of 
first and last) and multiplying by the common distance between the other 
ordinates. 

5th Method.— Draw the figure on cross-section paper. Count the number 
of squares that are entirely included within the boundary; then estimate 
the fractional parts of squares that are cut by the boundary, add together 
these fractions, and add the sum to the number of whole squares. The 
result is the area in units of the dimensions of the squares. The finer the 
ruling of the cross-section paper the more accurate the result. 

6th Method.— Use a planimeter. 

7th Method.— -With a chemical balance, sensitive to one milligram, draw 
the figure on paper of uniform thickness and cut it out carefully; weigh the 
piece cut out, and compare its weight with the weight per square inch of the 
paper as tested by weighing a piece of rectangular shape. 



THE CIRCLE. 



57 



THE CIRCLE. 

Circumference = diameter x 3.1416, nearly; more accurately, 3.14159265359. 
22 355 

Approximations, — = 3.143; — — = 3.1415929. 

The ratio of circum. to diam. is represented by the symbol 



Multiples of 7T. 

1tt = 3.14159265359 

2tt = 6.28318530718 
3tt= 9.42477796077 
4tt = 12.56637001436 
577 = 15.70796326795 
for =18.84955592154 
7ir = 21.99114857513 
8n = 25.13274122872 
9tt = 28.27433388231 
Ratio of diam. to circumference = 



(called Pi). 
Multiples of n -. 

r = .7853982 



Reciprocal of -n = 1 .2732 
Multiples of -. 

— = .31831 

n 

— = .63662 

IT 

— = .95493 
-=1.27324 

— = 1.59155 

— = 1.90986 



x 2 = 1.5707963 
x 3 = 2.3561945 
x 4 = 3.1415927 
x 5 = 3.9269908 
x 6 = 4.7123890 
x 7 = 5.4977871 
x 8 = 6.2831853 
x 9 = 7.0685835 
■eciprocal of n = 0.3183099. 

0.2617! 



— = 2.22817 

- = 2.54648 



10 _ 
12 



= 1.570796 
= 1.047197 



12 



- = 0.101321 

7T 2 

Vn = 1.772453 
<y/l = 0.564189 

Log it = 0.4971498? 



Diam. in ins. = 13.5405 ^area in sq. ft. 

Area in sq. ft. = (diam. in inches) 2 x .0054542. 

D = diameter, B = radius, C = circumference, 



C=7T.D;=277#; = -g-; = 2*Vk; = 3.545'V^. ; 



A=D*x .7854 



. . _CB 

' ~ 2 ' 


= 4£ 2 x .785 


7T 


= 0.31831C; 


*~b 


= 0.159155C; 



W A -, 



8VA; 



■i-Vi 



= 0.564189 ^A. 



Areas of circles are to each other as the squares of their diameters. 
To find the length of an arc of a circle : 

rule 1. As 360 is to the number of degrees in ihe arc, so is the circum- 
ference of the circle to the length of the arc. 

rule 2. Multiply the diameter of the circle by the number of degrees in 
the arc, and this product by 0,0087266, 



58 MENSUKATIOK. 

Relations of Arc, Chord, Chord of Half the Arc, 
Versed Sine, etc. 

Let R — radius, D — diameter, Arc = length of arc, 
Cd = chord of the arc, ch — chord of half the arc, 
V — versed sine, D — V = diam. minus ver. sin., 

8ch -Cd, ,' Vcdy+W* x 10F2 
Arc = (very nearly), = — iwdt + ZZV* ^ ' nearl y- 

2ch x 10V . _ . 

ArC = WD^27V + 2ch ' neai ' ly - 

Chord of the arc = 2 V c h*-V*; = ^D* - (D-2V)*; = 8ch - 3 Arc. 

= 2VRi-(R-V)*; =2V(D-V) x V. 
Chord of half the arc, ch - - v ™ 2 ->- A ™ • -VTHTTr. _ 



Diameter 




Versed sine = ^— ; = ~(D - V D*- Cd*) 

l(D + VD* - Cd 2 ), if F is greater than radius. 



-V. 



«».-^. 



Half the chord of the arc is a mean proportional between the versed sine 
and diameter minus versed sine : 



\cd = Vv x {D- V). 



Length of a Circular Arc— Huyghens's Approximation. 

Let C represent the length of the chord of the arc and c the length of the 

chord of half the arc; the length of the arc 

r 8c- C 

L = ~S~' 

Professor Williamson shows that when the arc subtends an angle of 30°, the 

radius being 100,000 feet (nearly 19 miles), the error by this formula is about 

two inches, or 1/600000 part of the radius. When the length of the arc is 

equal to the radius, i.e., when it subtends an angle of 57°. 3, the error is less 

than 1/7680 part of the radius. Therefore, if the radius is 100.000 feet, the 

error is less than ' = 13 feet. The error increases rapidly with the 

increase of the angle subtended. 

In the measurement of an arc which is described with a short radius the 
error is so small that it may be neglected. Describing an arc with a radius 
of 12 inches subtending an angle of 30°, the error is 1/50000 of an inch. For 
57°. 3 the error is less than 0".0015. 

In order to measure an arc when it subtends a large angle, bisect it and 
measure each half as before— in this case making B = length of the chord of 
half the arc, and b — length of the chord of one fourth the arc ; then 
T 166 - 2B 

L = 8~ " 

Relation ot the Circle to its Equal, Inscribed, and Cir- 
cumscribed Squares. 
Diameter of circle x .88623 ( _ - , - , 

Circumference of circle x .28209 f - side ot equal squaie. 
Circumference of circle x 1.1281 = perimeter of equal square, 



THE ELLIPSE. 59 

Diameter of circle x .7071 ) 

Circumference of circle x .22508 >• = side of inscribed square. 
Area of circle x .90031-f- diameter ) 

Area of circle x 1.2732 = area of circumscribed square. 

Area of circle x .63662 = area of inscribed square. 

Side of square x 1.4142 = diam. of circumscribed circle. 

" " x 4.4428 = circum. " " " 

" " x 1.1284 = diam. of equal circle. 

" " x 3.5449 = circum. " " 

Perimeter of square x 0.88623 = " " " 

Square inches x 1.2732 = circular inches. 

Sectors and Segments.- To find the area of a sector of a circle. 
Rule 1. Multiply the arc of the sector by half its radius. 
Rule 2. As 360 is to the number of degrees in the arc, so is the area of 
the circle to the area of the sector. 

Rule 3. Multiply the number of degrees in the arc by the square of the 
radius and by .008727. 

To find the area of a segment of a circle: Find the area of the sector 
which has the same arc, and also the area of the triangle formed by the 
chord of the segment and the radii of the sector. 

Then take the sum of these areas, if the segment is greater than a semi- 
circle, but take their difference if it is less. 

Another Method: Area of segment =-- (arc — sin A) in which A is the 

central angle, R the radius, and arc the length of arc to radius 1. 

To find the area of a segment of a circle when its chord and height or 
versed sine only are given. First find radius, as follows : 

1 rsquare of half the chord , . ...,~] 
radius = - [_ 5^ + height J. 

2. Find the angle subtended by the arc, as follows: ^ = sine 

radius 
of half the angle. Take the corresponding angle from a table of sines, and 
double it to get the angle of the arc. 

3. Find area of the sector of which the segment is a part ; 

„ . , degrees of arc 

area of sector = area of circle x -— . 

ooU 

4. Subtract area of triangle under the segment: 

~ vl «,^«o — height of segment). 

The remainder is the area of the segment. 

When the chord, arc, and diameter are given, to find the area. From the 
length of the arc subtract the length of the chord. Multiply the remainder 
by the radius or one-half diameter; to the product add the chord multiplied 
by the height, and divide the sum by 2. 

Another rule: Multiply the chord by the height and this product by .6834 
plus one tenth of the square of the height divided by the radius. 

To find the chord: From the diameter subtract the height; multiply the 
remainder by four times the height and extract the square root. 

When the chords of the arc and of half the arc and the versed sine are 
given: To the chord of the arc add four thirds of the chord of half the arc; 
multiply the sum by the versed sine and the product by .40426 (approximate). 

Circular Ring. — To find the area of a ring included between the cir- 
cumferences of two concentric circles: Take the difference between the areas 
of the two circles; or, subtract the square of the less radius from the square 
of the greater, and multiply their difference by 3.14159. 

The area of the greater circle is equal to irR*; 
and the area of the smaller, nr 2 . 

Their difference, or the area of the ring, is 7r(i? 2 - r 2 ). 

The Ellipse.— Area of an ellipse = product of its semi-axes x 3.14159 
= product of its axes x .785398. 



1416^ + 



Tlie Ellipse.— Circumference (approximate) = 3.1416 V — ^ — , D and d 

being the two axes. 

Trautwine gives the following as more accurate: When the longer axis D 
is not more than five times the length of the shorter axis, d, 



60 MENSURATION". 







Circumfei 


ence 


= 3.1416 ^ 




When D is 


more than 5d, the divisor 8 


.8 is 


divisors : 














D 
d 


6, 


7, 8, 


9, 


10, 


12, 


14, 


Divisor = 


9 


9.2, 9.3, 


9.35, 


9.4, 


9.5, 


9.G, 



_ ( D- d) 2 
8.8 
is to be replaced by the following 

16, 18, 20, 30, 40, 50. 

9.68, 9.75, 9.8, 9.92, 9.98, 10. 

/ w 2 n* n 6 \ 
Reuleaux gives : Circumference = n (a + b)yi -f T+ fi4+or fi +--« )» in 

which n = — r— , , a and b being the semi-axes. 
a 4-6 

Area of a segment of an ellipse the base of which is parallel to one of 
the axes of the ellipse. Divide the height of the segment by the axis of 
which it is part, and find the area of a circular segment, in a table of circu- 
lar segments, of which the height is equal to the quotient; multiply the area 
thus found by the product of the two axes of the ellipse. 

Cycloid.— A curve generated by the rolling of a circle on a plane. 
Length of a cycloidal curve = 4 X diameter of the generating circle. 
Length of the base = circumference of the generating circle. 
Area of a cycloid = 3 X area of generating circle. 

Helix (Screw).— A line generated by the progressive rotation of a 
point arouud an axis and equidistant from its centre. 

Length of a helix.— To the square of the circumference described by the 
generating'-point add the square of the distance advanced in one revolution, 
and take the square root of their sum multiplied by the number of revolu- 
tions of the generating point. Or, 

y(c 2 + h 2 )n = length, n being number of revolutions. 

Spirals.— Lines generated by the progressive rotation of a point around 
a fixed axis, with a constantly increasing distance from the axis. 

A plane spiral is when the point rotates in one plane. 

A conical spiral is when the point rotates around an axis at a progressing 
distance from its centre, and advancing in the direction of theaxis, as around 
a cone. 

Length of a plane spiral line. — When the distance between the coils is 
uniform. 

Rule.— Add together the greater and less diameters; divide their sum by 
2; multiply the quotient by 3.1416, and again by the number of revolutions. 
Or, take the mean of the length of the greater and less circumferences and 
multiply it by the number of revolutions. Or, 

length = mi — - — , d and d' being the inner and outer diameters. 

Length of a conical spiral line.— Add together the greater and less diam- 
eters; divide their sum by 2 and multiply the quotient by 3.1416. To the 
square of the product of this circumference and the number of revolutions 
of the spiral add the square of the height of its axis and take the square 
root of the sum. 



Or, length = |/(™^^) 2 + ft 2 . 

SOLID BODIES. 

The Prism.— To find the surface of a right prism : Multiply the perim- 
eter of the base by the altitude for the convex surface. To this add the 
areas of the two ends when the entire surface is required. 

Volume of a prism = area of its base X its altitude. 

The pyramid.— Convex surface of a regular pyramid = perimeter of 
its base X half the slant height. To this add area of the base if the whole 
surface is required. 

Volume of a pyramid = area of base X one third of the altitude. 



SOLID BODIES. 61 

To find the surface of a frustum of a regular pyramid : Multiply half the 
slant height by the sum of the perimeters of the two bases for the convex 
surface. To this add the areas of the two bases when the entire surface is 
required. 

To find the volume of a frustum of a pyramid : Add together the areas of 
the two bases and a mean proportional between them, and multiply the 
sum by one third of the altitude. (Mean proportional between two numbers 
= square root of their product.) 

Wedge.— 'A wedge is a solid bounded by five planes, viz.: a rectangular 
base, two trapezoids, or two rectangles, meeting in an edge, and two tri- 
angular ends. The altitude is the perpendicular drawn from an}' point in 
the edge to the plane of the base. 

To find the volume of a wedge : Add the length of the edge to twice the 
length of the base, and multiply the sum by one sixth of the product of the 
height of the wedge and the breadth of the base. 

Rectangular prismoid.— A rectangular prismoid is a solid bounded 
by six planes, of which the two bases are rectangles, having their corre- 
sponding sides parallel, and the four upright sides of the solids are trape- 
zoids. 

To find the volume of a rectangular prismoid: Add together the areas of 
the two bases and four times the area of a parallel section equally distant 
from the bases, and multiply the sum by one sixth of the altitude. 

Cylinder.— Convex surface of a cylinder = perimeter of base X altitude. 
To this add the areas of the two ends when the entire surface is required. 
Volume of a cylinder = area of base X altitude. 

Cone.— Convex surface of a cone = circumference of base X half the slant 
side. To this add the area of the base when the entire surface is required. 

Volume of a cone = area of base X k altitude. 

To find the surface of a frustum of a cone : Multiply half the side by the 
sum of the circumferences of the two bases for the convex surface; to this 
add the areas of the two bases when the entire surface is required. 

To find the volume of a, frustum of a cone : Add together the areas of the 
two bases and a mean proportional between them, and multiply the sum 
by one third of the altitude. 

Spliere.— To find the surface of a sphere : "Multiply the diameter by the 
ciicumference of a great circle; or, multiply the square of the diameter by 
3.14159. 

Surface of sphere = 4 x area of its great circle. 

" " " = convex surface of its circumscribing cylinder. 

Surfaces of spheres are to each other as the squares of their diameters. 

To find the volume of a sphere : Multiply the surface by one third of the 
radius; or, multiply the cube of the diameter by 1/Qn; that is, by 0.5236. 

Value of |rr to 10 decimal places = .5235987756. 

The volume of a sphere = 2/3 the volume of its circumscribing cylinder. 

Volumes of spheres are to each other as the cubes of their diameters. 

Spherical triangle.— To find the area of a spherical triangle : Com- 
pute the surface of the quadrantal triangle, or one eighth of the surface of 
the sphere. From the sum of the three angles subtract two right angles; 
divide the remainder by 90, and multiply the quotient by the area of the 
quadrantal triangle. 

Spherical polygon. —To find the area of a spherical polygon : Com- 
pute the surface of the quadrantal triangle. From the sum of all the angles 
subtract the product of two right angles by the number of sides less two; 
divide the remainder by 90 and multiply the quotient by the area of the 
quadrantal triangle. 

The prismoid.— The prismoid is a solid having parallel end areas, and 
may be composed of any combination of prisms, cylinders, wedges, pyra- 
mids, or cones or frustums of the same, whose bases and apices lie in the 
end areas. 

Inasmuch as cylinders and cones are but special forms of prisms and 
pyramids, and w T arped surface solids may be divided into elementary forms 
of them, and since frustums may also be subdivided into the elementary 
forms, it is sufficient to say that all prismoids may be decomposed into 
prisms, wedges, and pyramids. If a formula can be found which is equally 
applicable to all of these forms, then it will apply to any combination of 
them. Such a formula is called 



62 MENSURATION. 

The Prismoidal Formula. 

Let A = area of the base of a prism, wedge, or pyramid; 
A], A 2 , Am — the two end and the middle areas of a prismoid, or of any of 
its elementary solids; 
h = altitude of the prismoid or elementary solid; 

V = its volume; 

V = J ^U 1 +4Am^A 2 ) i 

For a prism A x , Am and A 2 are equal, = A ; V — - X 6A = hA. 

b 

For a wedge with parallel ends, A 2 = 0, Am — -A 1 ; V — ^(A. x + 2A X ) — — • 

For a cone or pyramid, A 2 = 0, ^l»i = '-zA\\ V = 7.(^*1 +^i) = ir- 

The prismoidal formula is a rigid formula for all prismoids. The only 
approximation involved in its use is in the assumption that the given solid 
may be generated by a right line .moving over the boundaries of the end 
areas. 

The area of the middle section is never the mean of the two end areas if 
the prismoid contains any pyramids or cones among its elementary forms. 
When the three sections are similar in form the dimensions of the middle 
area are always the means of the corresponding end dimensions. This fact 
often enables the dimensions, and hence the area of the middle section, to 
be computed from the end areas. 

Polyedrons.— A polyedron is a solid bounded by plane polygons. A 
regular polyedron is one whose sides are all equal regular polygons. 

To find the surface of a regular polyedron..— Multiply the area of one of 
the faces by the number of faces ; or, multiply the square of one of the 
edges by the surface of a similar solid whose edge is unity. 

A Table of the Regular Polyedrons whose Edges are Unity. 

Names. No. of Faces. Surface. Volume. 

Tetraedron 4 1.7320508 0.1178513 

Hexaedron. 6 6.0000000 1 .0000000 

Octaedron 8 3.4641016 0.4714045 

Dodecaedron 12 20.6457288 7.6631189 

Icosaedron 20 8.6602540 2.1816950 

To find the volume of a regular polyedron.— Multiply the 
surface by one third of the perpendicular let fall from the centre on one of 
the faces ; or, multiply the cube of one of the edges by the solidity of a 
similar polyedron whose edge is unity. 

Solid of revolution.— The volume of any solid of revolution is 
equal to the product of the area of its generating surface by the length of 
the path of the centre of gravity of that surface. 

The convex surface of any solid of revolution is equal to the product of 
the perimeter of its generating surface by the length of path of its centre 
of gravity. 

Cylindrical ring.— Let d = outer diameter ; d' — inner diameter ; 
1 1 1 

■= (d — d') = thickness = t ; - n P* = sectional area ; - (d -\- d') = mean diam- 
eter = M ; nt = circumference of section ; nM — mean circumference of 
ring; surface = n t X n M; = - ^ (d 2 - d' 2 ); = 9.86965 1 M; = 2.46741 (cZ 2 -d' 2 ); 

volume = InP Mn; = 2AQ74WM. 
4 

Spherical zone.— Surface of a spherical zone or segment of a sphere 
= its altiiude x the circumference of a great circle of the sphere. A great 
circle is one whose plane passes through the centre of the sphere. 

Volume of a zone of a sphere. — To the sum of the squnres of the radii 
of the ends add one third of the square of the height ; multiply the sum 
by the height and by 1.5708. 

Spherical segment.— Volume of a spherical segment with one base.— 



SOLID BODIES. 63 

Multiply half the height of the segment by the area of the base, and the 
cube of the height by .5236 and add the two products. Or, from three times 
the diameter of the sphere subtract twice the height of the segment; multi- 
ply the difference by the square of the height and by .5236. Or, to three 
times the square of the radius of the base of the segment add the square of 
its height, and multiply the sum by the height and by .5236. 

Spheroid or ellipsoid.— When the revolution of the spheroid is about 
the transverse diameter it is prolate, and when about the conjugate it is 
oblate. 

Convex surface of a segment of a spheroid.— Square the diameters of the 
spheroid, and take the square root of half their sum ; then, as the diameter 
from which the segment is cut is to this root so is the height of the 
segment to the proportionate height of the segment to the mean diameter. 
Multiply the product of the other diameter and 3.1416 by the proportionate 
height. 

Convex surface of a frustum or zone of a spheroid.— Proceed as by 
previous rule for the surface of a segment, and obtain the proportionate 
height of the frustum. Multiply the product of the diameter parallel to the 
base of the frustum and 3.1416 by the proportionate height of the frustum. 

Volume of a spheroid is equal to the product of the square of the revolving 
axis by the fixed axis and by .5236. The volume of a spheroid is two thirds 
of that of the circumscribing cylinder. 

Volume of a segment of a spheroid.— 1. When the base is parallel to the 
revolving axis, multiply the difference between three times the fixed axis 
and twice the height of the segment, by the square of the height and by 
.5236. Multiply the product by the square of the revolving axis, and divide 
by the square of the fixed axis. 

2. When the base is perpendicular to the revolving axis, multiply the 
difference between three times the revolving axis and twice the height of 
the segment by the square of the height and by .5236. Multiply the 
product by the length of the fixed axis, and divide by the length of the 
revolving axis. 

Volume of the middle frustum of a spheroid.— 1. When the ends are 
circular, or parallel to the revolving axis : To twice the square of the 
middle diameter add the square of the diameter of one end ; multiply the 
sum by the length of the frustum and by .2618. 

2. AVhen the ends are elliptical, or perpendicular to the revolving axis: 
To twice the product of the transverse and conjugate diameters of the 
middle section add the product of the transverse and conjugate diameters 
of one end ; multiply the sum by the length of the frustum and by .2618. 

Spindles.— Figures generated by the revolution of a plane area, when 
the curve is revolved about a chord perpendicular to its axis, or about its 
double ordinate. They are designated by the name of the arc or curve 
from which they are generated, as Circular, Elliptic, Parabolic, etc., etc. 

Convex surface of a circular spindle, zone, or segment of it — Rule: Mul- 
tiply the length by the radius of the revolving arc; multiply this arc by the 
central distance, or distance between the centre of the spindle and centre 
of the revolving arc ; subtract this product from the former, double the 
remainder, and multiply it by 3.1416 

Volume of a circxdar spindle. — Multiply the central distance by half the 
area of the revolving segment; subtract the product from one third of the 
cube of half the length, and multiply the remainder by 12 5664. 

Volume of frustum or zone of a circxdar spiudle.— From the square of 
half the length of the whole spindle take one third of the square of half the 
length of. the frustum, and multiply the remainder by the said half length 
of the frustum ; multiply the central distance by the revolving area which 
generates the frustum ; subtract this product from the former, and multi- 
ply the remainder by 6.2832. 

Volume of a segment of a circxdar sj)indle. — Subtract the length of the 
segment from the half length of the spindle ; double the remainder and 
ascertain the volume of a middle frustum of this length ; subtract the 
result from the volume of the whole spindle and halve the remainder. 

Volnxne of a cycloidal spindle = five eighths of the volume of the circum- 
scribing cylinder. — Multiply the product of the square of twice the diameter 
of the generating circle and 3.927 by its circumference, and divide this pro- 
duct by 8. 

Parabolic conoid.— Volume of a parabolic conoid (generated by the 
revolution of a parabola on its axis).— Multiply the area of the base by half, 
the height. 



64 MEISSUKATION". 

Or multiply the square of the diameter of the base by the height and by 
.3927. 

Volume of a frustum of a parabolic conoid.— Multiply half the sum of 
the areas of the two ends by the height. 

Volume of a parabolic spindle (generated by the revolution of a parabola 
on its base).— Multiply the square of the middle diameter by the length 
and by .4189. 

The volume of a parabolic spindle is to that of a cylinder of the same 
height and diameter as 8 to 15. 

Volume of the middle frustum of a parabolic spindle.— Add together 
8 times the square of the maximum diameter, 3 times the square of the end 
diameter, and 4 times the product of the diameters. Multiply the sum by 
the length of the frustum and by .05236. 

This rule is applicable for calculating the content of casks of parabolic 
form. 

Casks. — To find the volume of a cash of any form. — Add together 39 
times the square of the bung diameter, 25 times the square of the head 
diameter, and 26 times the product of the diameters. Multiply the sum by 
the length, and divide by 31,773 for the content in Imperial gallons, or by 
26,470 for U. S. gallons. 

This rule was framed by Dr. Hutton, on the supposition that the middle 
third of the length of the cask was a frustum of a parabolic spindle, and 
each outer third was a frustum of a cone. 

To find the idlage of a cask, the quantity of liquor in it when it is not full. 
1. For a lying cask : Divide the number of wet or dry inches by the bung 
diameter in inches. If the quotient is less than .5, deduct from it one 
fourth part of what it wants of .5. If it exceeds .5, add to it one fourth part 
of the excess above .5. Multiply the remainder or the sum by the whole 
content of the cask. The product is the quantity of liquor in the cask, in 
gallons, when the dividend is ivet inches; or the empty space, if dry inches. 

2. For a standing cask : Divide the number of wet or dry inches by the 
length of the cask. If the quotient exceeds .5, add to it one tenth of its 
excess above .5; if less than .5, subtract from it one tenth of what it wants 
of .5. Multiply the sum or the remainder by the whole content of the cask. 
The product is the quantity of liquor in the cask, when the dividend is wet 
inches; or the empty space, if dry inches. 

Volume of cask (approximate) U. S. gallons = square of mean diam. 
X length in inches X .0034. Mean diam. — half the sum of the bung and 
head diams. 

Volume of an irregular solid.— Suppose it divided into parts, 
resembling prisms or other bodies measurable by preceding rules. Find 
the content of each part; the sum of the contents is the cubic contents of 
the solid. 

The content of a small part is found nearly by multiplying half the sum 
of the areas of each end by the perpendicular distance between them. 

The contents of small irregular solids may sometimes be found by im- 
mersing them under water in a prismatic or cylindrical vessel, and observ- 
ing the amount by which the level of the water descends when the solid is 
withdrawn. The sectional area of the vessel being multiplied by the descent 
of the level gives the cubic contents. 

Or, weigh the solid in air and in water; the difference is the weight of 
water it displaces. Divide the weight in pounds by 62.4 to obtain volume in 
cubic feet, or multiply it by 27.? to obtain the volume in cubic inches. 

When the solid is very large and a great degree of accuracy is not 
requisite, measure its length, breadth, and depth in several < hTerent 
places, and take the mean of the measurement for each dimension, and 
multiply the three means together. 

When the surface of the solid is very extensive it is better to divide it 
into triangles, to find the area of each triangle, and to multiply it by the 
mean depth of the triangle for the contents of each triangular portion; the 
contents of the triangular sections are to be added together. 

The mean depth of a triangular section is obtained by measuring the 
depth at each angle, adding together the. three measurements, and taking 
one third of the sum. 



PLANE TRIGONOMETRY. 



65 



PLANE TRIGONOMETRY. 



Trigonometrical Functions. 

Every triangle has six parts — three angles and three sides. When any 
three of these parts are given, provided one of them is a side, the other 
parts may be determined. By the solution of a triangle is meant the deter- 
mination of the unknown parts of a triangle when certain parts are given. 

The complement of an angle or are is what remains after subtracting the 
angle or arc from 90°. 

In general, if we represent any arc by A, its complement is 90° — A. 
Hence the complement of an arc that exceeds 90° is negative. 
- Since the two acute angles of a right-angled triangle are together equal to 
a right angle, each of them is the complement of the other. 

The supplement of an angle or arc is what remains after subtracting the 
angle or arc from 180°. If A is an arc its supplement is 180° — A. The sup- 
plement of an arc that exceeds 180° is negative. 

The sum of the three angles of a triangle is equal to 180°. Either angle is 
the supplement of the other two. In a right-angled triangle, the right angle 
being equal to 90°, each of the acute angles is the complement of the other. 

In all right-angled triangles having the same acute angle, the sides have 
to each other the same ratio. These ratios have received special names, as 
follows: 

If A is one of the acute angles, a the opposite side, b the adjacent side, 
and c the hypothenuse. 

The sine of the angle A is the quotient of the opposite side divided by the 
a 
hypothenuse. Sin. A = -.■ 

The tangent of the angle A is the quotient of the opposite side divided by 
a * 

the adjacent side. Tang. A = t- 

The secant of the angle A is the quotient of the hypothenuse divided by 
c 
the adjacent side. Sec. A = t' 

The cosine, cotangent, and cosecant of an angle are respec- 
tively the sine, -tangent, and secant of the complement of that angle. The 
terms sine, cosine, etc., are called trigonometrical functions. 

In a circle whose radius is unity, the sine of .an arc, or of the angle at the 
centre measured by that arc, is the perpendicular let fall from one extrem- 
ity of the arc upon the diameter passing through the other extremity. 

The tangent of an arc is the line which touches the circle at one extrem- 
ity of the arc, and is limited by the diameter {produced) passing through 
the other extremity. 

- The secant of an arc is that part of the produced diameter which is 
intercepted betireen the centre and the tangent. 

The versed, sine of an arc is that part of the diameter intercepted 
between the extremity of the arc and the foot of the sine. 
• In a circle whose radius is not unity, the trigonometric functions of an arc 
I will be equal to the lines here defined, divided by the radius of the circle. 

If. I C A (Fig. 70) is an angle in the first quadrant, and C F— radius, 

mu . f+u . FG _ CG KF 

The sine of the angle — 



Tang. 



I A 
z Rad.' 

Cosec. = i 



"Rad." Rad, 

Secant =R^uT Cot - = 
CL GA 

Rad." Versin ' = Rad/ 
If radius is 1, then Rad. in the denominator is 
omitted, and sine = F G, etc. 

The sine of an arc = half the chord of twice the 
arc. 

The sine of the supplement of the arc is the same 
as that of the arc itself. Sine of arc B D F = F G = 
sin arc -FM. 



I 




66 



PLANE TRIGONOMETRY. 



The tangent of the supplement is equal to the tangent of the arc, but with 
a contrary sign. Tang. B D F — B M. 

The secant of the supplement is equal to the secant of the arc, but with a 
contrary sign. Sec. B D F — CM. 

Signs of the functions in the four quadrants.— If we 
divide a circle iuto four quadrants by a vertical and a horizontal diame- 
ter, the upper right-hand quadrant is called the first, the upper left the sec- 
ond, the lower left the third, and the lower right the fourth. The signs of 
the functions in the four quadrants are as follows: 

First quad. Second quad. Third quad. Fourth quad. 
Sine and cosecant, + + — — 

Cosine and secant, -j- — — 4- 

Tangent and cotangent, + — 4- — 

The values of the functions are as follows for the angles specified: 



Angle 

Sine 

Cosine 

Tangent 

Cotangent . 

Secant 

Cosecant . . . 

"Versed sine 






„ 








• 




„ 














30 


45 


60 


90 


120 


135 


150 


180 


270 . 





1 


1 


v 3 


1 


Vs 


1 


1 







2 


V~i 


2 


2 


V2 


2 




1 


V3 


1 


1 





1 


1 


V3 







2 


Vg 


2 




2 


V2 


2 







Vs 


1 


v 3 

1 


00 


-*s 


-1 


1 

Vs 





c, 


ao 


V'6 




f3 





~vl 


- 1 


- Vs 


cc 





1 

GO 


vi 

2 


V% 
V~2 


2 
2 
V-3 


00 

1 


2 


~V2 


2 

13 
2 


-1 

<x 


■X 

-1 





2- ^3 


V2-1 


1 




3 


V2+1 


2 +|/3 








2 


V : 2 


2 




2 


V2 


2 







TRIGONOMETRICAL FORMULiE. 

The following relations are deduced from the properties of similar tri- 
angles (Radius = 1): 

cos A : sin A :: 1 : tan A, whence tan A = 
sin A : cos A :: 1 : cot A, " cotan A = 



cos .4 : 1 



: 1 : sec A, 



sec A -- 



: 1 : cosec A, " cosec A - 



cos 


A' 


cos 

sin 


A 


1 




cos 


A 


1 

sin 


~A % 



tan A : 1 



: 1 : cot A 



tan A - 



cot A' 



The sum of the square of the sine of an arc and the square of its cosine 
equals unitv. Sin 2 A + cos 2 A = 1. 

Formulae for the functions of the sum and difference ot 
two angles : 

Let the two angles be denoted by A and B, their sum A 4- B = C, and their 
difference A — B by D. 



gin {A 4- B) = sin AcosB -\- cos 4 sin J5; 



(1) 



TRIGONOMETRICAL FORMULA. 



67 



cos A + B = cos A cos B - sin A sin B; . . . 

sin {A- B) = sin A cos 5 — cos A sin 5; . . .' 

cos U. — B) - cos J. cos B 4- sin .4 sin B. . . . 
From these four formulae by addition and subtraction we obtain 

sin (A + B) + sin (A - B) = 2 sin ^4 cos 5; . . . 

sin (J. + i?) - sin (A — B) = 2 cos A sin B; . . . 

cosC4 4- #) + cos(J. - B) - 2 cos ^ cos B; . . . 

cos (A - B) - cos (A + B) = 2 sin Asm B. . . . 



(2) 
(3) 

(4) 

(5) 
(6) 
(7) 

(8) 



1 



If we put A + S = (7, and A - B = D, then A = -{C + D) and B = -(C ■ 
D), and we have 

sin C+ sin D = 2 sin \{C + D) cos |(C - D); . . . . (< 



sin C - sin D = 2 cos -(C4- D) sin -(C - D); 



1 



(10) 
(ID 



cos D - cos a = 2 sin 1«7 + Z>) sin \{C - D). 



(12) 



Equation (9) may be enunciated thus: The sum of the sines of any two 
angles is equal to twice the sine of half the sum of the angles multiplied by 
the cosine of half their difference. These formulae enable us to transform 
a sum or difference into a product. 

The sum of the sines of two angles is to their difference as the tangent of 
half the sum. of those angles is to the tangent of half their difference. 



sin A + sin B = * ** \u + B)oo*^A - B) = t ^jA + B) 
sin A- sin B 2 C08jfA + B)sin±{A -B) tan\(A - B) 



(13) 



The sum of the cosines of two angles is to their difference as the cotan- 
gent of half the sum of those angles is to the tangent of half their difference. 



cos A + c 



2 cos \{A + B) cos \(A - B) cot \{A + B) 



cos B - cos A a gin 1 {A + B) gin l u _ B) tftn 1 {A _ B - 



(14) 



The sine of the sum of two angles is to the sine of their difference as the 
sum of the tangents of those angles is to the difference of the tangents. 

sin ( A + B) _ tan A + tan B 

sin {A — B) tan A — tan B" 1 ' ' 



(15) 



sin (A + B) 
cos A cos B 
sin (A - B) 
cos A cos B 
cos (A + B) 
cos A cos i? 
cos (A - B) 



— tan A + tan £; 
= tan A - tan B; 
= 1 — tan ^4 tan i?; 
= 1 + fan A tan i?; 



tan (A + B) = 
tan (.4 - B) = 
cot C4 -f- B) = 
cot U - 5) = 






tan ^ 4- tan ff 
1 - tan A tan 5' 

tan A - tan i? _ 
1 4- tan A tan B' 
cot ^4 cot B — 1 . 

cot-B-f cot4 ' 
cot ,4 cot ff 4- 1 

cot B — cot A ' 



68 PLANE TRIGONOMETRY. 

Solution of Plane Right-angled Triangles. 

Let A and B be the two acute angles and C the right angle, and a, b, and 
c the sides opposite these angles, respectively, then we have 



1. 


sin A = cos B — - ; 
c 


3. tan A = cot B = T ; 
6 


2. 


cos A = sin B = -; 


4. cot A = tan 5 = - . 



1. In any plane right-angled triangle the sine of either of the acute angles 
is equal to the quotient of the opposite leg divided by the hypothenuse. 

2. The cosine of either of the acute angles is equal to the quotient of the 
adjacent leg divided by the hypothenuse. 

3. The tangent of either of the acute angles is equal to the quotient of the 
opposite leg divided by the adjacent leg. 

4. The cotangent of either of the acute angles is equal to the quotient of 
the adjacent leg divided by the opposite leg. 

5. The square of the hypothenuse equals the sum of the squares of the 
other two sides. 

Solution of Oblique-angled Triangles. 

The following propositions are proved in works on plane trigonometry. In 
any plane triangle — 

Theorem 1. The sines of the angles are proportional to the opposite sides. 

Theorem 2. The sum of any two sides is to their difference as the tangent 
of half the sum of the opposite angles is to the tangent of half their differ- 
ence. 

Theorem 3. If from any angle of a triangle a perpendicular be drawn to 
the opposite side or base, the whole base will be to the sum of the other two 
sides as the difference of those two sides is to the difference of the segments 
of the base. 

Case I. Given two angles and a side, to find the third angle and the other 
two sides. 1. The third angle = 180° — sum of the two angles. 2. The sides 
may be found by the following proportion : 

The sine of the angle opposite the given side is to the sine of the angle op- 
posite the required side as the given side is to the required side. 

Case II. Given two sides and an angle opposite one of them, to find the 
third side and the remaining angles. 

The side opposite the given angle is to the side opposite the required angle 
as the sine of the given angle is to the sine of the required angle. 

The third angle is found by subtracting the sum of the other two. from 180°, 
and the third side is found as in Case I. 

Case III. Given two sides and the included angle, to find the third side and 
the remaining angles. 

The sum of the required angles is found by subtracting the given angle 
from 180°. The difference of the required angles is then found by Theorem 
II. Half the difference added to half the sum gives the greater angle, and 
half the difference subtracted from half the sum gives the less angle. The 
third side is then found by Theorem I. 

Another method : 

Given the sides c, 6, and the included angled, to find the remaining side n 
and the remaining angles B and C. 

From either of the unknown angles, as B, draw a perpendicular B e to the 
opposite side. 

Then 

Ae = c cos A, i? e = c sin .4, e C -— b — Ae, Be h- e C — tan C. 

Or, in other words, solve Be, Ae and B e C as right-angled triangles. 

Case IV. Given the three sides, to find the angles. 

Let fall a perpendicular upon the longest side from the opposite angle, 
dividing the given triangle into two right-angled triangles. The two seg- 
ments of the base may be found by Theorem III. There will then be given 
the hypothenuse and one side of a right-angled triangle, to find the angles. 

For areas of triangles, see Mensuration. 



ANALYTICAL GEOMETRY. 69 



ANALYTICAL GEOMETRY. 

Analytical geometry is that branch of Mathematics which has for 
its object the determination of the forms and magnitudes of geometrical 
magnitudes by means of analysis. 
Or di nates and abscissas.— In analytical geometry two intersecting 
Y lines YY', XX' are used as coordinate axes, 

XX' being the axis of abscissas or axis of X, 
and YY' the axis of ordinates or axis of Y. 
A. the intersection, is called the origin of co- 
ordinates. The distance of any point P from 
the axis of Y measured parallel to the axis of 
X is called the abscissa of the point, as AD or 
CP, Fig. 71. Its distance from the axis of X. 
measured parallel to the axis of Y, is called 
the ordinate, as AC or PD. The abscissa and 
ordinate taken together are called the coor- 
dinates of the point P. The angle of intersec- 
tion is usually taken as a right angle, in which 
case the axes of Xand Y are called rectangu- 
lar coordinates. 

The abscissa of a point is designated by the letter x and the ordinate by y. 
^ The equations of a point are the equations which express the distances of 
thp point from the axis. Thus x = a, y = b are the equations of the point P. 
Equations referred to rectangular coordinates.— The equa- 
tion uf a line expresses the relation which exists between the coordinates of 
every puint of the line. 

Equation of a straight line, y = ax ± b, in which a is the tangent of the 
angle the liue makes with the axis of X, and b the distance above A in which 
the line cuts the axis of Y. 

Every equation of the first degree between two variables is the equation of 
a straight line, as Ay -j- Bx 4- C = 0, which can be reduced to the form y = 
ax ± b. 
Equation of the distance between two points: 









P 




/c 


/ 


7 


/a 




D 




Y' 








Fig. 


1. 







D = \\x" - a;') 2 + W - V') 2 , 
in which x'y', x"y" are the coordinates of the two points. 
Equation of a line passing through a given point: 

y — ?j' — a(x — x'), 

in which x'y' are the coordinates of the given point, a, the tangent of the 
angle the line makes with the axis of x, being undetermined, since any num- 
ber of lines may be drawn through a given point. 
Equation of a line passing through two given points: 

y-y' = y x „ ~ v , {x - x'). 

Equation of a line parallel to a given line and through a given point: 

y - y 1 - a(x - x'). 

Equation of an angle V included between two given lines: 

. yT a' — a 

tang V— , , 

& 1 + a'a 

in which a and a' are the tangents of the angles the lines make with the 
axis of abscissas. 
If the lines are at right angles to each other tang V = a>, and 

1 -f a'a = 0. 
Equation of an intersection of two lines, whose equations are 
y = ax + &, and y = a'x -f- b\ 

b — b' , ab' — a'b 

x = • , and y = — — . 

a — a' a — a' 



70 ANALYTICAL GEOMETRY. 

Equation of a perpendicular from a given point to a given line: 

y — y' — (x — x'). 

Equation of the length of the perpendicular P: 

p _ y' - ax' - b 

Vl X o« 

"The circle.— Equation of a circle, the origin of coordinates being at the 
centre, and radius = R : 

a: 2 -f ?/ 2 = iJ 2 . 

If the origin is at the left extremity of the diameter, on the axis of X: 

2/ 2 = 2Rx - a; 2 . 

If the origin is at any point, and the coordinates of the centre are a-'//' : 

(x - a;') 2 + (y- yV = R 2 . 

Equation of a tangent to a circle, the coordinates of the point of tangency 
being x"y" and the origin at the centre, 

yy" + xx" - .R 2 . 

The ellipse. —Equation of an ellipse, referred to rectangular coordi- 
nates with axis at the centre: 

A i y i 4. 52^2 _ A 1 B 2 % 

in which A is half the transverse axis and B half the conjugate axis. 

Equation of the ellipse when the origin is at the vertex of the transverse 
axis : 

?/ 2 = ^Ax - a,- 2 ). 

The eccentricity of an ellipse is the distance from the centre to either 
focus, divided by the semi-transverse axis, or 



Va* - # 2 

e = A— 

The parameter of an ellipse is the double ordinate passing through the 
focus. It is a third proportional to the transverse axis and its conjugate, or 



Any ordinate of a circle circumscribing an ellipse is to the corresponding 
ordinate of the ellipse as the semi-transverse axis to the semi-conjugate. 
Any ordinate of a circle inscribed in an ellipse is to the corresponding ordi- 
nate of the ellipse as the semi-conjugate axis to the semi-transverse. 

Equation of the tangent to an ellipse, origin of axes at the centre : 

• A*yy" + B^xx" = A^B*, 

y"x" being the coordinates of the point of tangency. 

Equation of the normal, passing through the point of tangency, and per- 
pendicular to the tangent: 

y - y" xx -Efrfa - x )• 

The normal bisects the angle of the two lines drawn from the point of 
tangency to the foci. 

The lines drawn from the foci make equal angles with the tangrent. 

Tlae parabola. —Equation of the parabola referred to rectangular 
coordinates, the origin being at the vertex of its axis, y" 1 — 2px, in which 2p 
is the parameter or double ordinate through the focus. 



ANALYTICAL GEOMETRY. 71 

The parameter is a third proportional to any abscissa and its corresponding 
ordinate, or 

x :y ::y :2p. 
Equation of the tangent: 

yy" = p(x + x"), 

y''x" being coordinates of the point of tangency. 
Equation of the normal: 

y - y" xx — —(x — x"). 

The sub-normal, or projection of the normal on the axis, is constant, and 
equal to half the parameter. 

The tangent at any point makes equal angles with the axis and with the 
line drawn from the point of tangency to the focus. 

The hyperbola.— Equation of the hyperbola referred to rectangular 
coordinates, origin at the centre: 

A 2 y 2 - B 2 x 2 = - AW 2 , 

in which A is the semi-transverse axis and B the semi-conjugate axis. 
Equation when the origin is at the vertex of the transverse axis: 

Conjugate and equilateral hyperbolas. — If on the conjugate 

axis, as a transverse, and a focal distance equal to VA 2 -+- B 2 , we construct 
the two branches of a hyperbola, the two hyperbolas thus constructed are 
called conjugate hyperbolas. If the transverse and conjugate axes are 
equal, the hyperbolas are called equilateral, in which case y 2 — x 2 — — A 2 
when A is the transverse axis, and x 2 — y 2 = — B 2 when B is the trans- 
verse axis. 

The parameter of the transverse axis is a third proportional to the trans- 
verse axis and its conjugate. 

2 A : 2B : : 2B : parameter. 

The tangent to a hyperbola bisects the angle of the two lines drawn from 
the point of tangency to the foci. 

The asymptotes of a hyperbola are the diagonals of the rectangle 
described on the axes, indefinitely produced in both directions. 

In an equilateral hyperbola the asymptotes make equal angles with the 
transverse axis, and are at right angles to each other. 

The asymptotes continually approach the hyperbola, and become tangent 
to it at an infinite distance from the centre. 

Conic sections.— Every equation of the second degree between two 
variables will represent either a circle, an ellipse, a parabola or a hyperbola. 
These curves are those which are obtained by intersecting the surface of a 
cone by planes, and for this reason they are called conic sections. 

Logarithmic curve.— A logarithmic curve is one in which one of the 
coordinates of any point is the logarithm of the other. 

The coordinate axis to which the lines denoting the logarithms are parallel 
is called the axis of logarithms., and the other the axis of numbers. If y is 
the axis of logarithms and x the axis of numbers, the equation of the curve 
is y = log x. 

If the base of a system of logarithms is a, we have a v = x, in which y is the 
logarithm of x. 

Each system of logarithms will give a different logarithmic curve. If y — 
0, x - 1. Hence every logarithmic curve will intersect the axis of numbers 
at a distance from the origin equal to 1, 



72 DIFFERENTIAL CALCULUS. 



DIFFERENTIAL CALCULUS. 

The differential of a variable quantity is the difference between any two 
of its consecutive values; hence it is indefinitely small. It is expressed by 
writing d before the quantity, as dx, which is read differential of x. 

The term -,- is called the differential coefficient of y regarded as a func- 
tion of x. 

The differential of a function is equal to its differential coefficient mul- 
tiplied by the differential of the independent variable; thus, -i-dx = dy. 

The limit of a variable quantity is that value to which it continually 
approaches, so as at last to differ from it by less than any assignable quan- 
tity. 

The differential coefficient is the limit of the ratio of the increment of the 
independent variable to the increment of the function. 

The differential of a constant quantity is equal to 0. 

The differential of a product of a constant by a variable is equal to the 
constant multiplied by the differential of the variable. 

If u — Av, du — Adv. 

In any curve whose equation is y=f(x), the differential coefficient 

rr' = tan a: hence, the rate of increase of the function, or the ascension of 
dx 

the curve at any point, is equal to the tangent of the angle which the tangent 
line makes with the angle of abscissas. 
All the operations of the Differential Calculus comprise but two objects: 

1. To find the rate of change in a function when it passes from one state 
of value to another, consecutive with it. 

2. To find the actual change in the function : The rate of change is the 
differential coefficient, and the actual change the function. 

Differentials of algebraic functions.— The differential of the 
sum or difference of any number of functions, dependent on the same 
variable, is equal to the sum or difference of their differentials .taken sepa- 
rately : 

If u — y -f- z — w, du — dy + dz — div. 

The differential of a product of two functions dependent on the same 
variable is equal to the sum of the products of each by the differential of 
the other : 

, . . , , , d(uv) du . dv 

d(uv) = vdu + udv. = . 

v J , : uv u v 

The differential of the product of any number of functions is equal to the 
sum of the products which arise by multiplying the differential of each 
function by the product of all the others: 

d(uts) = tsdu -f- usdt 4- utds. 
The differential of a fraction equals the denominator into the differential 
of the numerator minus the numerator into the differential of the denom- 
inator, divided by the square of the denominator : 



-G)= 



If the denominator is constant, dv — 0, and dt = — — = — > 

-u 2 v 

If the numerator is constant, du — 0, and dt xx — 

The differential of the square root of a quantity is equal to the differen- 
tial of the quantity divided by twice the square root of the quantity: 

If v == tt% or v — Vu, dv = -; = -u~ ?du. 

2 Yu ^ 



DIFFERENTIAL CALCULUS. 73 

The differential of any power of a function is equal to the exponent multi- 
plied by the function raised to a power less one, multiplied by the differen- 
tial of the function, d(ii n ) = nu n ~ 1 du. 

Formulas for differentiating algebraic functions. 

/x\ _ ydx - xdy 



1. d(a) = 0. 

2. d (ax) — adx. 
3.d(x-\-y) = dx + dy. 

4. d (x — y) = dx — dy. 

5. d (xy) = xdy + ydx. 



6. d | 

7. d (x m ) = mx m ' 

dx 

s. div*) = -^; 

9.d\x 7 = - - 



dx. 



To find the differential of the form u = (a -\~ bx n ) '■ 

Multiply the exponent of the parenthesis into the exponent of the varia- 
ble within the parenthesis, into the coefficient of the variable, into the bi- 
nomial raised to a power less 1, into the variable within the parenthesis 
raised to a power less 1, into the differential of the variable. 

du = d(a + bx n ) m = mnb(a + bx n ) m " l x n ~ 1 dx. 

To find the rate of change for a given value of the variable : 
Find the differential coefficient, and substitute the value of the variable in 
the second member of the equation. 

Example.— If tc is the side of a cube and u its volume, u = x 3 , -=- : '— 3ic 2 . 

dx 
Hence the rate of change in the volume is three times the square of the 
ige. If the edge is denoted by 1, the rate of change is 3: 

Application. The coefficient of expansion by heat of the volume of a body 
is three times the linear coefficient of expansion. Thus if the side of a cube 
expands .001 inch, its volume expands .003 cubic inch. 1.001 3 = 1.003003001. 

A partial differential coefficient is the differential coefficient of 
a function of two or more variables under the supposition that only one of 
them has changed its value. 

A partial differential is the differential of a function of two or more vari- 
ables under the supposition that only one of them has changed its value. 

The total differential of a function of any number of variables is equal to 
the sum of the partial differentials. 

If u — f(xy), the partial differentials are -rdx, ~rdy. 

If u = x" 2 + V 3 — z, die — —dx + -—dy + -z-dz; = 2xdx-\- Zy^dy — dz. 
ax ay dz 

Integrals.— An integral is a functional expression derived from a 
differential. Integration is the operation of finding the primitive function 
from the differential function. It is indicated by the sign /, which is read 
u the integral of." Thus / 2xdx = x* : read, the integral of 2xdx equals x 2 . 

To integrate an expression of the form mx m ~~ 1 dx or x m dx, add 1 to the 
exponent of the variable, atid divide by the new exponent and by the differ- 
ential of the variable: f 3x 2 dx = x 3 . (Applicable in all cases except when 



- 1. For lx dx see formula 2 page 78.) 



The integral of the product of a constant by the differential of a vari- 
able is equal to the constant multiplied by the integral of the differential: 

fax m dx — afx m dx = a — — , x m + a - 
J m 4- 1 

The integral of the algebraic sum of any number of differentials is equal to 
the algebraic sum of their integrals: 

du = 2ax*dx - bydy - z*dz; fdu = %ix 3 - ^y* - -*. 

Since the differential of a constant is 0, a constant connected with a vari- 
able by the sign + or — disappears in the differentiation ; thus d(a + x m ) = 
ix m = mx m ~ dx. Hence in integrating a differential expression we must 



74 DIFFERENTIAL CALCULUS. 

annex to the integral obtained a constant represented by C to compensate 
for the term which may have been lost in differentiation. Thus if we have 
dy = adx\ fdy = afdx. Integrating, 

y = ax ± C. 

The constant 0, which is added to the first integral, must have such a 
value as to render the functional equation true for every possible value that 
may be attributed to the variable. Hence, after having found the first 
integral equation and added the constant C, if we then make the variable 
equal to zero, the value which the function assumes will be the true value 
of C. 

An indefinite integral is the first integral obtained before the value of the 
constant C is determined. 

A particular integral is the integral after the value of C has been found. 

A definite integral is the integral corresponding to a given value of the 
variable. 

Integration between limits. — Having found the indefinite inte- 
gral and the particular integral, the next step is to find the definite integral, 
and then the definite integral between given limits of the variable. 

The integral of a function, taken between two limits, indicated by given 
values of x, is equal to the difference of the definite integrals correspond- 
ing to those limits. The expression 



I dy = a I dx 



is read: Integral of the differential of y, taken between the limits x' and x' , 
the least limit, or the limit corresponding to the subtractive integral, being 
placed below. 

Integrate du = 9x'*dx between the limits x = 1 and x = 3, u being equal to 
81 when x = 0. fdu = f9x'*dx = 3x 3 + C; C = 81 when x = 0, then 



t/x -■ 



du = 3(3) 3 + 81, minus 3(1) 3 + 81 = 



Integration of particular forms. 

To integrate a differential of the form du — (a + bx n ) m x n ' 1 dx. 

1. If there is a constant factor, place it without the sign of the integral, 
and omit the power of the variable without the parenthesis and the differ- 
ential; 

2. Augment the exponent of the parenthesis by 1, and then divide this 
quantity, with the exponent so increased, by the exponent of the paren- 
thesis, into the exponent of the variable within the parenthesis, into the co- 
efficient of the variable. Whence 



fdu = ^ + ^ m+1 = C. 
J (m + \)nb 



The differential of an arc is the hypothenuse of a right-angle triangle ol 
which the base is dx and the perpendicular dy. 

If z is an arc, dz = Vdx* + dy* z =f Vdx^-^-dy 1 . 

Quadrature of a plane figure. 

The differential of the area of a plane surf ace is equal to the ordinate into 
the differential of the abscissa. 

ds — ydx. 

To apply the principle enunciated in the last equation, in finding the area 
of any particular plane surface : 

Find the value of y in terms of x, from the equation of the bounding line; 
substitute this value in the differential equation, and then integrate between 
the required limits of x. 

Area of tlie parabola.— Find the area of any portion of the com- 
mon parabola whose equation is 

2/2 — 2px; whence y = ^2px. 



DIFFERENTIAL CALCULUS. 75 

Substituting this value of y in the differential equation ds — ydx gives 

r r , ,- r x 2 i/%p 3 

I ds= I y->pxdx = y-2p I x^dx = — - — xs -f C; 

2/i/2px x x 2 
or, s = r 3 = 3 *y+C. 

Tf we estimate the area from the principal vertex, x = 0. y = 0, and C — 0; 
and denoting the particular integral by s', s' — - xy. 

That is, the area of any portion of the parabola, estimated from the ver- 
tex, is equal to % of the rectangle of the abscissa and ordinate of the extreme 
point. The curve is therefore qnadrable. 

Quadrature of surfaces of revolution. —The differential of a 
surface of revolution is equal to the circumference of a circle perpendicular 
to the axis into the differential of the arc of the meridian curve. 



ds = 2iry^/dx 2 + dy*; 

in which y is the radius of a circle of the bounding surface in a plane per- 
pendicular to the axis of revolution, and x is the abscissa, or distance of the 
plane from the origin of coordinate axes. 
Therefore, to find the volume of any surface of revolution: 
Find the value of y and dy from the equation of the meridian curve in 
terms of x and dx, then substitute these values in the differential equation, 
and integrate between the proper limits of x. 
By application of this rule we may find : 

The curved surface of a cylinder equals the product of the circumference 
of the base into the altitude*. 

The convex surface of a cone equals the product of the circumference of 
the base into half the slant height. 

The surface of a sphere is equal to the area of four great circles, or equal 
to the curved surface of the circumscribing cylinder. 

Cubature of volumes of revolution.— A volume of revolution 
is a volume generated by the revolution of a plane figure about a fixed line 
I called the axis. 

I If we denote the volume by V, dV = ny 2 dx. 

I The area of a circle described by any ordinate y is try 7 ", hence the differ- 
I ential of a volume of revolution is equal to the area of a circle perpendicular 
I to the axis into the differential of the axis. 

I The differential of a volume generated by the revolution of a plane figure 
J about the axis of Y is irx 2 dy. 

I To find the value of F"for any given volume of revolution : 
I Find the value of y 2 in terms of x from the equation of the meridian 
I curve, substitute this value in the differential equation, and then integrate 
I between the required limits of x. 
I By application of this rule we may find : 

t The volume of a cylinder is equal to the area of the base multiplied by the 
I altitude. 

The volume of a cone is equal to the area of the base into one third the 
: altitude. 

The volume of a prolate spheroid and of an oblate spheroid (formed by 
the revolution of an ellipse around its transverse and its conjugate axis re- 
spectively) are each equal to two thirds of the circumscribing cylinder. 
If the axes are equal, the spheroid becomes a sphere and its volume = 

^jt.R 2 x D — - irD 3 ; B being radius and D diameter. 
6 o 

The volume of a paraboloid is equal to half the cylinder having the same 
base and altitude. 

The volume of a pyramid equals the area of the base multiplied by one 
third the altitude. 

Second, third* etc., diiferentials.— The differential coefficient 
being a function of the independent variable, it may be differentiated, and 
we thus obtain the second differential coefficient: 

Dividing by dx, we have for the second differential coeffi- 



,/du\ c 
d \dx) = i 



76 DIFFERENTIAL CALCULUS. 

cient — j, which is read: second differential of u divided by the square of 

the differential of x (or dx squared). 
d 3 u 
The third differential coefficient — -= is read: third differential of u divided 
ax 6 
by dx cubed. 
The differentials of the different orders are obtained by multiplying the 

differential coefficients by the corresponding powers of dx: thus — dx 3 = 

third differential of u. 

Sign of the first differential coefficient.— If we have a curve 
whose equation is y = fx, referred to rectangular coordinates, the curve 

will recede from the axis of X when -r~ is positive, and approach the 

axis when it is negative, when the curve lies within the first angle of the 
coordinate axes. For all angles and every relation of y and x the curve 
will recede from the axis of X when the ordinate and first differential co- 
efficient have the same sign, and approach it when they have different 
signs. If the tangent of the curve becomes parallel to the axis of X at any 

point -p = 0. If the tangent becomes perpendicular to the axis of X at any 

. . dy 
point — -co. 

Sign of the second differential coefficient.— The second dif- 
ferential coefficient has the same sign as the ordinate when the curve is 
convex toward the axis of abscissa and a contrary sign when it is concave. 

Maclaurin's Theorem.— For developing into a series any function 
of a single variable as u — A-\- Bx + Cx 2 -f- Dx 3 -J- Ex*, etc., in which A, B, 
C, etc., are independent of x: 

u = (u) +(-r) x + r-*\z-*) x + 1 o q V^—W « 3 + etc. 
x = q \dx/ x = 1 .2^-dx 2/ x = 1 • 2 . 3^dx 3 / x = 

In applying the formula, omit the expressions x = 0, although the coeffi- 
cients are always found under this hypothesis. 
Examples : 

(a + x) m = a m -\- ma m 

«i l 111 _ n Ini — 9\ ... .. 

' 6 x 3 + etc. 

1 _ J x . , 

a -\-x ~ a a 2 a 3 a 4 ' a » + i ' 

Taylor's Theorem.— For developing into a series any function of the 
sum oV difference of two independent variables, as u' — fix '± y): 

, du , dHi y* , d 3 u y 3 , '., 

in which u is what u' becomes when y = 0, — is what -, becomes when 

dx dx 

y = 0, etc. 

Maxima and minima.— To find the maximum or minimum value 
of a function of a single variable: 

1. Find the first differential coefficient of the function, place it equal to 0, 
and determine the roots of the equation. 

2. Find the second differential coefficient, and substitute each real root, 
in succession, for the variable in the second member of the equation. Each 
root which gives a negative result will correspond to a maximum value of 
the function, and each which gives a positive result will correspond to a 
minimum value. 

Example.— To find the value of x which will render the function y a 
maximum or minimum in the equation of the circle, y 2 -f- x 2 = B' 2 ; 

dy x . . x . . . 

■f- = ; making = gives x = 0, 

dx y y 




DIFFERENTIAL CALCULUS. 77 

d 2 y x 2 4- v^ 

The second differential coefficient is: — - = -^~- 

dx y 3 

d 2 y 1 

When x = U, y = R; hence -— - = — — , which being negative, y is a maxi- 
mum for R positive. 

In applying the rule to practical examples we first find an expression for 
the function which is to be made a maximum or minimum. 

2. If in such expression a constant quantity is found as a factor, it may 
be omitted in the operation ; for the product will be a maximum or a mini- 
mum when the variable factor is a maximum or a minimum. 

3. Any value of the independent variable which renders a function a max- 
imum or a minimum will render any power or root of that funciiou a 
maximum or minimum; hence we may square both members of an equa- 
tion to free it of radicals before differentiating. 

By these rules we may find: 

The maximum rectangle which can be inscribed in a triangle is one whose 
altitude is half the altitude of the triangle. 

The altitude of the maximum cylinder which can be inscribed in a cone is 
one third the altitude of the cone. 

The surface of a cylindrical vessel of a given volume, open at the top, is a 
minimum when the altitude equals half the diameter. 

The altitude of a cylinder inscribed in a sphere when its convex surface is 
a maximum is r |/2. r = radius. 

The altitude of a cylinder inscribed in a sphere when the volume is a 
2r 
maximum is ~7. ' 

Differential of an exponential function. 

If u - a x (1) 

then du = da x = a x k dx, (2) 

in which A; is a constant dependent on a. 
l 

The relation between a and 7c is a k = e; whence a — e k , ..... (3) 

in which e = 2.7182818 . . . the base of the Naperian system of logarithms. 
Logarithms.— The logarithms in the Naperian system are denoted by 
Z, Nap. log or hyperbolic log, hyp. log, or log e ; and in the common system 
always by log. 

k — Nap. log a, log a — k log e (4) 

The common logarithm of e, = log 2.7182818 . . . = .4342945 . .- . , is called 
the modulus of the common system, and is denoted by M. Hence, if we have 
the Naperian logar thm of a number we can find the common logarithm of 
the same number by multiplying by the modulus. Reciprocally, Nap. 
log - com. log x 2 3025851. 

If in equation (4) we make a — 10, we have 

1 = k log e, or - = log e = M. . 

That is, the modulus of the common system is equal to 1, divided by the 

Naperian logarithm of the common base. 
From equation (2) we have 

du da x , , 

— = = kdx. 

u a x 

If we make a = 10, the base of the common system, x = log u, and 

,,, , , du 1 du 

d(log u) = dx — — x - = — x M . 
u k u 

That is, the differential of a common logarithm of a quantity is equal to the 
diffei-ential of the quantity divided by the quantity, into the modulus. 
If we make a — e, the base of the Naperian system, x becomes the Nape- 



78 DLFFEBENTIAL CALCULUS. 

rian logarithm of u, and k becomes 1 (see equation (3)); hence M = 1, and 

d(Nap. log u) = dx = — ; = — . 
a x u 

That is, the differential of a Naperian logarithm of a quantity is equal to the 
differential of the quantity divided by the quantity; and in the Naperian 
system the modulus is 1. 

Since k is the Naperian logarithm of a, du = a x I a dx. That is, the 
differential of a function of the form a x is equal to the function, into the 
Naperian logarithm of the base a, into the differential of the exponent. 

It* we have a differential in a fractional form, in which the numerator is 
the differential of the denominator, the integral is the Naperian logarithm 
of the denominator. Integrals of fractional differentials of other forms are 
given below: 

Differential forms which have known integrals; ex- 
ponential functions. (I — Nap. log.) 



/■ 



a x I a dx — a x -f- C; 
J ~ = J dxx~ 1 = lx + C; 

I (xy x ~ 1 dy + y x ly x dx) = y* + C; 

/dx , 

,. = Kx + \/x* ± a 2 ) 4- C; 

4/a; 2 ± a 2 

dx , / , \ 

l{x ± a + |/x 2 ± 2ax) + C; 



/ 



2adx 
2 -x* 



/' 2 adx 
a;2- tt 2 



j/rr 2 ± 2ax 

/ 2adx _ / i/a* + x* - a\ 
#|/a 2 + x* \|/aa+ls» + a/ 

/ 2adx -i( a ~ \/<* r ~^\ 

/x ~ 2 dx /l -(- |/F 

V x + x~ 2 ~ ~ \ « 



Circular functions.— Let 2 denote an arc in the first quadrant, y its 
sine, a; its cosine, v its versed sine, and t its tangent; and the following nota- 
tion be employed to designate an arc by any one of its functions, viz., 

sin -1 y denotes an arc of which y is the sine 
cos a; " " " " " re is the cosine, 

tan -1 t " " " " " t is the tangent 



DIFFERENTIAL CALCULUS. 



(read "arc whose sine is y," etc.), — we have the following differential forms 
which have known integrals (r = radius): 



cos z dz — sin z 4- C; 



f 

/— dx -\ \ n 

— • = cos x x +- C\ 
|/l - X 2 

f 



sin zdz — cos z-\- C\ 

Q" 1 y + C; 



-. ver-sin 1 v + C; 



\/%v - v 2 

/rdy - -i 

— — =sin l y -f- C; 
y,. 2 _ ^2 

/ — = cos x+ C; 

J |A- 2 - x 2 



I sin z dz = ver-sin z -\- C\ 

— — = tan z + C; 
cos 2 z 

/ rd v 
tfitrv + jfl = versin _1 v + C > 

/Mi* =tan- 1 ^0; 

J 4/a 2 - w 2 a 

-7 = cos" 1 - 4- C; 

4/a 2 - W 2 » 



e/ j/ita 



/*/2aw — u 2 

/ adu 
a 2 4 « 2 



= ver-sin ~" * — \- C: 
a 

: tan x - 4- G. 



The cycloid.— If a circle he rolled along a straight line, any point of 
the circumference, as P, will describe a curve which is called a cycloid. The 
circle is called the generating circle, and _Pthe generating point. 

The transcendeutal equation of the cycloid is 



and the differential equation is dx - 



1 y - \/*ry 

ydx 
\/'Zry - y 2 . 



The area of the cycloid is equal to three times the area of the generating 
circle. 

The surface described by the arc of a cycloid when revolved about its base 
is equal to 64 thirds of the generating circle. 

The volume of the solid generated bv revolving a cycloid about its base is 
equal to five eighths of the circumscribing cylinder. 

Integral calculus. — In the integral calculus we have to return from 
the differential to the function from which it was derived A number of 
differential expressions are given above, each of which has a known in- 
tegral corresponding to it, and which being differentiated, will produce the ' 
given differential. 

In all classes of functions any differential expression may be integrated 
when it is reduced to one of the known forms; and the operations of the 
integral calculus consist mainly in making such transformations of given 
differential expressions as shall reduce them to equivalent ones w r hose in- 
tegrals are known. 

For methods of making these transformations reference must be made to 
<he text-books on differential and integral calculus. 



80 



MATHEMATICAL TABLES. 
RECIPROCALS OF NUMBERS. 



No. 


Recipro- 
cal. 


No. 


Recipro- 
cal. 


No. 


Recipro- 
cal. 


No. 


Recipro- 
cal. 


No. 


Recipro- 
cal. 


1 


.10000000 


64 


.01562500 


127 


,00787402 


190 


.00526316 


253 


.00395257 


2 


.50000000 


5 


.0153S461 


8 


.00781250 


1 


.00523560 


4 


.00393701 


3 


.33333333 


6 


.01515151 


9 


; 00775194 


2 


.00520833 


5 


.00392157 


4 


.25000000 


7 


.01492537 


130 


; 00769231 


3 


.00518135 


6 


.00390625 


5 


.20000000 


8 


.01470588 


1 


.00763359 


4 


.00515464 


7 


.00389105 


6 


.16666667 


9 


.01449275 


2 


:. 00757576 


5 


.00512820 


8 


.00387597 


7 


.14285714 


70 


.01428571 


3 


:. 00751 880' 


6 


.00510204 


9 


.00386100 


8 


. 12500000 


1 


.01408451 


4 


.00746269 


7 


.00507614 


260 


.00384615 


9 


.11111111 


2 


.01388889 


5 


.00740741 


8 


.00505051! 


1 


.00383142 


10 


.10000000 


3 


.01369863 


6 


.00735294 


9 


.00502513 


2 


.00381679 


11 


.09090909 


4 


.01351351 


7 


.00729927 


200 


.00500000 


3 


.00380228 


12 


.08333333 


5 


.01333333 


8 


.00724638 


1 


.00497512 


4 


.00378788 


13 


.07692308 





.01315789 


9 


.00719424 


2 


.00495049 


5 


.00377358 


14 


.07142857 


7 


.01298701 


140 


.00714286 


3 


.00492611 


6 


.00375940 


15 


.06666667 


8 


.01282051 


1 


.00709220 


4 


.00490196 


7 


.00374532 


10 


.06250000 


9 


.01265823 


2 


.00704225 


5 


.00487805 


8 


.00373134 


17 


. 05582353 


80 


.01250000 


3 


.00699301 


6 


.00485437 


9 


.00371747 


18 


. 05555556 


1 


.01234568 


4 


.00694444 


7 


.00483092 


271 


.00370370 


19 


.05263158 


2 


.01219512 


5 


.00689655 


■ 8 


.00480769 


1 


.00369004 


20 


.05000000 


3 


.01204819 


6 


.00684931 


9 


.00478469 


2 


.00367647 


1 


.04761905 


4 


.01190476 


7 


.00680272 


210 


.00476190 


3 


.00366300 


2 


.04545455 


5 


.01176471 


8 


.00675676 


11 


.00473934 


4 


.00364963 


3 


.04347826 


6 


.01162791 


9 


.00671141 


12 


.00471698 


5 


.00363636 


4 


.04166667 


7 


.01149425 


150 


.00666667 


13 


.00469484 


6 


.00362319 


5 


.04000000 


8 


.01136364 


1 


.00662252 


14 


.00467290 


7 


.00361011 


6 


.03846154 


9 


.01123595 


2 


.00657895 


15 


.00465116 


8 


.00359712 


7 


.03703704 


90 


.01111111 


3 


.00653595 


16 


.00462963 


f 


.00358423 


8 


.03571429 


1 


.01098901 


4 


.00649351 


17 


.00460829 


28l 


.00357143 


9 


.03448276 


2 


.01086956 


5 


.00645161 


18 


.00458716 


1 


.00355872 


30 


.03333:333 


3 


.01075269 


6 


.00641026 


19 


.00456621 


2 


.00354610 


1 


.03225806 


4 


.01063830 




.00636943 


220 


.00454545 


3 


.00353357 


2 


.03125000 


5 


.01052632 


8 


.00632911 


1 


.00452489 


4 


.00352113 


3 


.03030303 


6 


.01041667 


9 


.00628931 


2 


.00450450 


5 


.00350877 


4 


.02941176 


7 


.01030928 


160 


.00625000 


3 


.00448430 


6 


.00349850 


5 


.02857143 


8 


.01020408 


1 


.00621118 


4 


.00446429 


7 


.00348432 


6 


.02777778 


9 


.01010101 


2 


.00617284 


5 


.00444444 


8 


.00347222 


7 


.02702703 


100 


.01000000 


3 


.00613497 


6 


.00442478 


9 


.00346021 


8 


.02631579 


1 


.00990099 


4 


.00609756 


7 


.00440529 


290 


.00344828 


9 


.02564103 


2 


.00980392 


5 


.00606061 


8 


.00438596 


1 


.00343643 


40 


.02500000 


3 


.00970874 


6 


.00602410 


9 


.00436681 


2 


.00342466 


1 


.02439024 


4 


.00961538 


7 


.00598802 


230 


.00434783 


3 


.00341297 


2 


.02380952 


5 


.00952381 


8 


.00595238 


1 


.00432900 


4 


.00340136 


3 


.02325581 


6 


.00943396 


9 


.00591716 


2 


.00431034 


5 


.00338983 


4 


.02272727 


7 


.00934579 


170 


.00588235 


3 


.00429184 


6 


.00337838 


5 


.02222222 


8 


.00925926 


1 


.00584795 


4 


.00427350 


7 


.00336700 


6 


.02173913 


9 


.00917431 


2 


.00581395 


5 


.00425532 


8 


.00335570 


7 


.02127660 


110 


.00909091 


3 


.00578035 


6 


.00423729 


9 


.00334448 


8 


.02083333 


11 


.00900901 


4 


.00574713 


7 


.00421941 


300 


.00333333 


9 


.02040816 


12 


.00892857 


5 


.00571429 


e 


.0042016S 


1 


.00332226 


50 


.02000000 


13 


.00884956 


G 


.00568182 


9 


.00418410 


a 


.00331126 


1 


.01960784 


14 


.00877193 


7 


.00564972 


240 


.00416667 


3 


.00330033 


2 


.01923077 


15 


.00869565 


8 


.00561798 


1 


.00414938 


4 


.00328947 


3 


.01886792 


16 


.00862069 


9 


.00558659 


2 


.00413223 


5 


.00327869 


4 


.01851852 


17 


.00854701 


180 


.00555556 


3 


.00411523 


6 


.00326797 


5 


.01818182 


18 


.00847458 


1 


.00552486 


4 


.00409836- 


7 


.00325733 


6 


.01785714 


19 


.00840336 


2 


.00549451 


5 


.00408163 


8 


.00324675 


7 


.01754386 


120 


.00833333 


3 


.00546448 


6 


.00406504 


9 


.00323625 


8 


.01724138 


1 


.00826446 


4 


.00543478 


7 


.00404858 


oil 


.00322581 


9 


.01694915 


2 


.00819672 


5 


.005405)0 


8 


.00403226 


11 


.00321543 


60 


.01666667 


3 


.00813008 


6 


.00537634 


9 


.00401606 


12 


.00320513 


1 


.01639344 


4 


.00806452 


7 


.00534759 


250 


.00400000 


13 


.00319489 


2 


.01612903 


5 


.00800000 


8 


.00531914 


1 


.00398406 


14 


.00318471 


3 


.01587302 


6 


.00793651 


9 


.00529100 


2 


.00390825 


15 


.00317460 



RECIPROCALS OP NUMBERS. 



81 



No 


Recipro- 
cal. 


I 


Recipro- 
cal. 


No. 
446 


Recipro- 
cal. 


No. 


Recipro- 
cal. 


No. 


Recipro- 
cal. 


316 


.00316456 


381 


.00262467 


.00224215 


511 


.00195695 


576 


.00173611 


17 


.00315457 


2 


.00261780 


7 


.00223714 


12 


.00195312 


7 


.0017 3310 


18 


.00314465 


3 


.00261097 


8 


.00223214 


13 


.00194932 


8 


.00173010 


19 


.00313480 


4 


.00260417 


9 


.00222717 


14 


.00194552 


9 


.00172712 


3:20 


.00312500 


5 


.00259740 


450 


.00222222 


15 


.00194175 


580 


.00172414 


1 


.00311526 


6 


.00259067 


1 


.00221729 


16 


.00193798 


1 


.00172117 


2 


.00310559 


7 


.00258398 


2 


.00221239 


17 


.C0193424 


2 


.00171821 


3 


.00309597 


8 


.00257732 


3 


.00220751 


18 


.00193050 


3 


.00171527 


4 


.00308642 


9 


.00257069 


4 


.00220264 


19 


.00192678 


4 


.90171233 


5 


.00307692 


390 


.00256410 


5 


.00219780 


520 


.00192308 


5 


.00170940 


6 


.00306748 


1 


.00255754 


6 


.00219298 


1 


.00191939 


6 


.00170648 


7 


.00305S10 


2 


.00255102 


7 


.00218818 


2 


.00191571 


7 


.00170358 


8 


.00304878 


3 


.00254453 


8 


.00218341 


3 


.00191205 


8 


.00170068 


9 


.00303951 


4 


.00253807 


9 


.00217865 


4 


.00190840 


9 


.00169779 


330 


.00303030 


5 


.00253165 


460 


.00217391 


5 


.00190476 


590 


.00169491 


1 


.00302115 


6 


.00252525 


1 


.00216920 


6 


.00190114 


1 


.00169205 


2 


.00301205 


7 


.00251889 


2 


.00216450 


7 


.00189753 


2 


.00168919 


3 


.00300300 


8 


.00251256 


3 


.90215983 


8 


.00189394 


3 


.00168634 


4 


.00299401 


9 


.00250627 


4 


.00215517 


9 


.00189036 


4 


.00168350 


5 


.00298507 


400 


.00250000 


5 


.00215054 


530 


.00188679 


5 


.00168067 


6 


.00297619 


1 


.00249377 


6 


.00214592 


1 


.00188324 


6 


.00167785 


7 


.00296736 


2 


.00248756 


7 


.00214133 


2 


.00187970 


7 


.00167504 


8 


.00295858 


3 


.00248139 


8 


.00213675 


3 


.00187617 


8 


.00167224 


9 


.00294985 


4 


.00247525 


9 


.00213220 


4 


.00187266 


9 


.00166945 


340 


.00294118 


5 


.00246914 


470 


.00212766 


5 


.00186916 


600 


.00166667 


1 


.00293255 


6 


.00246305 


1 


.00212314 


6 


.00186567 


1 


.00166389 


2 


.00292398 


7 


.00245700 


2 


.00211864 


7 


.CO 186220 


2 


.00166113 


3 


.00291545 


8 


.00245098 


3 


.00211416 


8 


.00185874 


3 


.00165837 


4 


.00290698 


9 


.00244499 


4 


.00210970 


9 


.00185528 


3 


.00165563 


5 


.00289855 


410 


.00243902 


5 


.00210526 


540 


.00185185 


5 


.00165289 


6 


.00289017 


11 


.00243309 


6 


.00210084 


1 


.C0184S43 


6 


.00165016 


7 


.00288184 


12 


.00242718 


7 


.00209644 


2 


.00184502 


7 


.00164745 


8 


.00287356 


13 


.00242131 


8 


.00209205 


3 


.00184162 


8 


.00164474 


9 


.00286533 


14 


.00241546 


9 


.00208768 


4 


.00183823 


9 


.00164204 


350 


.00285714 


15 


.00240964 


480 


.00208333 


5 


.00183486 


610 


.00163934 


1 


.00284900 


16 


.00240385 


1 


.00207900 


6 


.00183150 


11 


.00163666 


2 


.00284091 


17 


.00239808 


2 


.00207469 


7 


.00182815 


12 


.90163399 


3 


.00283286 


18 


.00239234 


3 


.00207039 


8 


.00182482 


13 


.00163132 


4 


.00282486 


19 


.00238663 


4 


.00206612 


9 


.00182149 


14 


.00162866 


5 


.00281690 


420 


.00238095 


5 


.00206186 


£50 


.00181818 


15 


.00162602 


6 


.00280899 


1 


.00237530 


6 


.00205761 


1 


.00181488 


16 


.00162338 


7 


.00280112 


2 


.00236967 


7 


.00205339 


2 


.00181159 


17 


.00162075 


8 


.90279330 


3 


.00236407 


8 


.00204918 


3 


.00180832 


18 


.00161812 


9 


.00278551 


4 


.00235849 


9 


.00204499 


4 


.00180505 


19 


.00161551 


360 


.00277778 


5 


.00235294 


490 


.00204082 


5 


.00180180 


620 


.00161290 


1 


.00277008 


G 


.00234742 


1 


.00203666 


6 


.00179856 


1 


.00161031 


2 


.00276243 


7 


.00234192 


2 


.00203252 


7 


.00179533 


2 


.00160772 


3 


.00275482 


8 


.00233645 


3 


.00202840 


8 


.00179211 


3 


.00160514 


4 


.00274725 


9 


.00233100 


4 


.00202429 


9 


.00178891 


3 


.00160256 


5 


.00273973 


430 


.00232558 


5 


.00202020 


560 


.00178571 


5 


.00160000 


6 


.00273224 


1 


.00232019 


6 


.00201613 


1 


.00178253 


6 


.00159744 


7 


.00272480 


2 


.00231481 


7 


.00201207 


2 


.00177936 


7 


.00159490 


8 


.00271739 


3 


.00230947 


8 


.00200803 


3 


.00177620 


8 


.00159236 


9 


.00271003 


4 


.00230415 


9 


.00200401 


4 


.00177305 


E 


.00158982 


370 


.00270270 


5 


.00229885 


500 


.00200000 


5 


.00176991 


630 


.00158730 


1 


.00269542 


6 


.00229358 


1 


.00199601 


6 


.00176678 


1 


.00158479 


2 


.00268817 


7 


.00228833 


2 


.00199203 


7 


.00176367 


2 


.00158228 


3 


.00268096 


8 


.00228310 


3 


.00198807 


8 


.00176056 


3 


.00157978 


4 


.00267380 


9 


.00227790 


4 


.00198413 


9 


.00175747 


4 


.00157729 


5 


.00266667 


440 


.00227273 


5 


.00198020 


570 


.00175439 


5 


.00157480 


6 


.00265957 


1 


.00226757 


6 


.00197628 


1 


.00175131 


6 


.00157233 


7 


.00265252 


2 


.00226244 


7 


.00197239 


2 


.00174825 


7 


.00156986 


8 


.00264550 


3 


.00225734 


& 


.001968o0 


3 


.00174520 


8 


.00156740 


9 


.00263852 


4 


.00225225 


9 


.00196464 


4 


.00174216 


9 


.00156494 


380 


,00263158 


5 


.00224719 


510 


.00196078 


51 .00173913 


640 


.00156250 



S2 



MATHEMATICAL TABLES. 



No. 


Recipro- 
cal. 


No . 


Recipro- 
cal. 


No. 


Recipro- 
cal. 


No. 


Recipro- 
cal. 


No. 


Recipro- 
cal. 


641 


.00156006 


706 


.00141643 


771 


.00129702 


836 


.00119617 


901 


.00110988 


2 


.00155763 


7 


.00141443 


2 


.00129534 


7 


.00119474 


2 


.00110865 


3 


.001555-21 


8 


.00141243 


3 


.00129366! 


8 


.00119332 


3 


.00110742 


4 


.00155279 


9 


.00141044 


4 


.00129199, 


9 


.00119189 


4 


.00110619 


5 


.00155039 


710 


.00140845 


5 


.00129032 


840 


.00119048 


5 


.00110497 


6 


.00154799 


11 


.00140647 


6 


.00128S66 


1 


.00118906 


6 


.00110375 


7 


.00154559 


12 


.00140449 


7 


.00128700' 


2 


.00118765 


7 


.00110254 


8 


.001543-2! 


13 


.00140252 


8 


.00128535) 


3 


.00118624 


8 


.00110132 


9 


.00154083 


14 


.00140056 


9 


.00128370 


4 


.00118483 


9 


.00110011 


650 


.00153846 


15 


.00139860 


780 


.00128205 


5 


.00118343 


910 


.00109890 


1 


.00153610 


16 


.00139665 


1 


.00128041 


6 


.00118203 


11 


.00109769 


2 


.00153374 


17 


.00139470 


2 


.00127877 


7 


.00118064 


12 


.00109649 


3 


.00153140 


18 


.00)39276 


3 


.00127714 


8 


.00117924 


13 


.00109529 


4 


.00152905 


19 


.00139082 


4 


.00127551 


9 


.00117786 


14 


.00109409 


5 


.00152672 


720 


.00138889 


5 


.00127388 


850 


.00117647 


15 


.00109290 


6 


.00152439 


1 


.00138696 


6 


.00127226 


1 


.00117509 


16 


.00109170 


7 


.00152-207 


2 


.00138504 


7 


.00127065 


2 


.00117371 


17 


.00109051 


8 


.00151975 


3 


.00138313 


8 


.00126904 


3 


00117233 


18 


.0010893-2 


9 


.00151745 


4 


.00138121 


9 


.00126743 


4 


.00117096 


19 


.00108814 


660 


.00151515 


5 


.00137931 


790 


.00126582 


5 


.00116959 


920 


.00108696 


1 


.00151286 


6 


.00137741 


1 


.00126422 


6 


.00116822 


1 


.00108578 


2 


.00151057 


7 


.00137552 


2 


.00126263; 


7 


.00116686 


2 


: 00 108 160 


3 


.00150830 


8 


.00137363 


3 


.00126103 


8 


.00116550 


3 


.00108342 


4 


.00150602 


9 


.00137174 


4 


.00125945 


9 


.00116414 


4 


.00108225 


5 


.00150376 


73c 


.00136986 


5 


.00125786 


860 


.00116279 


5 


.00108108 


6 


.00150150 


1 


.00136799 


6 


.00125628 


1 


.00116144 


6 


.00107991 


7 


.001499-25 


2 


.00136612 


7 


.00125470 


2 


.00116009 


7 


.00107875 


8 


.00149701 


3 


.00136426 


8 


.00125313 


3 


.00115875 


8 


.00107759 


8 


.00149477 


4 


.00136240 


9 


.00125156 


4 


.00115741 


9 


.00107643 


67C 


.00149254 


5 


.00136054 


800 


.00125000 


5 


.00115607 


930 


.00107527 


1 


.00149031 


j 


.00135870 


1 


.00124844 


6 


.00115473 


1 


.00107411 


S 


.00148809 




.00135685 


2 


.00124688 


7 


.00115340 


2 


.00107296 




.00148588 


£ 


.00135501 


3 


.00124533 


8 


.00115207 


3 


.00107181 


1 


.00148368 


£ 


.00135318 


4 


.00124378 


9 


.00115075 


4 


.60107066 




.00148148 


741 


.00135135 


5 


.00124224 


870 


.00114942 


5 


.00106952 


I 


.00147929 


1 


.00134953 


6 


.00124069 


1 


.00114811 


6 


.00106838 




.00147710 


£ 


.00134771 


7 


.00123916 


2 


.00114679 




.00106724 


£ 


.00147493 


' 


.00134589 


8 


.00123762 


3 


.00114547 


8 


.00106610 


1 


.00147275 




.00134409 


9 


.00123609 


4 


.00114416 


9 


.00106496 


68( 


) .00147059 




.00134228 


810 


.00123457 


5 


.00114286 


940 


.00106383 


] 


.00146843 


f 


.00134048 


11 


.00123305 


6 


.00114155 


1 


.00106270 




J .00146628 


r 


.00133869 


12 


.00123153 


7 


.00114025 




.00106157 




} .00146413 


t 


.00133690 


13 


.00123001 


8 


.00113895 


3 


.00106044 




1 .00146199 


J 


.00133511 


14 


.00122850 


9 


.00113766 


4 


.00105932 




> .00145985 


75( 


.00133333 


15 


.00122699 


880 


.00113636 


5 


.00105820 


i 


5 .00145773 


1 


.00133156 


16 


.00122549 


1 


.00113507 


6 


.00105708 




• .00145560 




.00132979 


17 


.00122399 


2 


.00113379 




.00105597 


J 


1 .00145319 




.0013-2802 


18 


.00122249 


3 


.00113250 


8 


.00105485 


( 


) .00145137 


4 


.00132626 


19 


.00122100 


4 


.00113122 


9 


.00105374 


69( 


) .00144927 




.00132450 


820 


.00121951 


5 


.00112991 


950 


.00105263 


] 


.00144718 


i 


.00132275 


1 


.00121803 


6 


.00112867 


1 


.00105152 




> .00144509 




.00132100 


2 


.00121654 


7 


.00112740 


2 


.00105042 




5 .00144300 


£ 


.00131926 


3 


.00121507 


8 


.00112613 


3 


.00104932 


' 


.00144092 


i 


.00131752 


4 


.00121359 


9 


.00112486 


4 


.00104822 


j 


.00143885 


7GC 


.00131579 


5 


.00121212 


S90 


.00112360 


5 


.00104712 


i 


.00143678 


1 


.00131406 


6 


.90121065 


1 


.00112233 


6 


.00104602 




.00143472 




.00131234 


7 


.001-20919 


2 


.00112108 




.00104493 


8 


.00143266 


2 


.00131062 


8 


.00120773 


3 


.00111982 


8 


.00104384 


£ 


.00143061 


4 


.00130890 


9 


.00120627 


4 


.00111857 


9 


.00104275 


700 


.00142857 


5 


.00130719 


. 830 


.00120482 


5 


.00111732 


960 


.00104167 


1 


.00142653 


1 


.00130548 


1 


.00120337 


6 


.00111607 


1 


.00104058 


2 


.00142450 


7 


.00130378 




.00120192 


7 


.00111483 


2 


.00103950 


3 


.00142247 


8 


.00130208 


3 


.00120048 


8 


.00111359 


3 


.00103842 


4 


.00142045 


9 


.00130039 


4 


.00119904 


9 


.00111235 


4 


.00103734 


ft 


.nni41844 


770 


.00129870 


5 


.00119760 


900 


.00111111 


5 


.00103627 



RECIPROCALS OF NUMBERS. 



83 



No. 


Recipro- 
cal. 


No. 


Recipro- 
cal. 


No. 


Recipro- 
cal. 


No. 


Recipro- 
cal. 


No. 


Recipro- 
cal. 


966 


.00103520 


1031 


.000969932 


1096 


.000912409 


116! '.000861328 


1226 


.000815661 


7 


,00103413 


2 


.000968992 


7 


.000911577 


2'' .000860585 


7 


.000814996 


8 


.00103306 


3 


.000988054 


8 


.000910747 


3 .000859845 


8 


.000814332 


9 


.00103199 


4 


.000967118 


9 


.000909918 


41.000859106 


9 


.000813670 


970 


.00103093 


5 


.000966184 


1100 


.000909091 


5 '.000858369 


12 


.000813008 


1 


.00102987 


6 


.000965251 


1 


.000908265 


6 .000857633 


1 


.000812348 


2 


.00102881 


7 


.000964320 


2 


000907441 


7 1.000856898 


2 


.000811688 


3 


.00102775 


8 


.000963391 


3 


.000906618 


81.000856164 


3 


.000811030 


4 


.C01 02669 


9 


.000962464 


4 


.000905797 


9 


.000855432 


4 


.000810373 


5 


.00102564 


1040 


.000961538 


5 


.000904977 


1170 


.000854701 


5 


.000809717 


6 


.00102459 


1 


.000960615 


6 


.000904159 


1 


.000853971 


6 


.000809061 


7 


.00102354 


2 


.000959693 


7 


.000903342 


2 


.000853242 


7 


.000808407 


8 


.00102250 


3 


.000958774 


8 


.000902527 


3 


.000852515 


8 


.000807754 


9 


.00102145 


4 


.000957854 


9 


.000901713 


4 


.000851789 


9 


.000807102 


980 


.00102041 


5 


.000956938 


1110 


.000900901 


5 


.090851064 


1240 


.000806452 


I 


.00101937 


6 


.000956023 


11 


000900090 


6 


.000850340 


1 


.000805802 


2 


.00101833 


7 


.000955110 


12 


.000899281 


7 


.000849618 


2 


.000805153 


3 


.00101729 


8 


.000954198 


13 


.000898473 


8 


.000848896 


3 


.000804505 


4 


.00101626 


9 


.000953289 


14 


.000897666 


9 


.000848176 


•4 


.000803858 


5 


.00101523 


1050 


.000952381 


15 


.000896861 


1180 


.000847457 


5 


.000803213 


6 


.00101420 


1 


.000951475 


16 


.000896057 


1 


.000846740 


6 


.000802568 


7 


.00101317 


2 


.000950570 


17 


.000895255 


2 


.000846024 


7 


.000801925 


8 


.00101215 


3 


.000949668 


18 


.000894454 


3 


.000S45308 


8 


.000801282 


9 


.00101112 


4 


.000948767 


19 


.000893655 


4 


.000844595 


9 


.000800640 


990 


.00101010 


5 


.000947867 


1120 


.000892857 


5 


.000843882 


1250 


.000800000 


1 


.00100908 


6 


.000946970 


1 


.000892061 


6 


.000843170 


1 


.000799360 


2 


.00100806 


7 


.000946074 


2 


.000891266 


7 


.000842460 


2 


.000798722 


3 


.00100705 


8 


.000945180 


3 


.000890472 


8 


.000841751 


3 


.000798085 


4 


.00100604 


9 


.000944287 


4 


.000889680 


9 


.000841043 


4 


.000797448 


5 


.00100502 


1060 


.000943396 


5 


.000888889 


1190 


.000840336 


5 


000796813 


6 


.00100102 


1 


.000942507 


6 


.000888099 


1 


.000839631 


6 


.000796178 


7 


.00100301 


2 


.000941620 


7 


.000887311 


2 


.000838926 


7 


000795545 


8 


.00100200 


3 


.000940734 


8 


.000886525 


3 


.000838222 


8 


.000794913 


9 


.00100100' 


4 


.000939850 


9 


.000885740 


4 


000837521 


9 


.000794281 


1000 


.00100000 


5 


.000938967 


1130 


.000884956 


5 


.000836820 




.000793651 


1 


.000999001 


6 


000938086 


1 


.000884173 


6 


.000836120 


1 


.000793021 


2 


.000998004 


7 


.000937207 


2 


.000883392 


7 


.000835422 


2 


.000792393 


3 


.000997009 


8 


.000936330 


3 


.000882612 


8 


.000834724 


3 


.000791766 


4 


.000996016 


9 


.000935454 


4 


.000881834 


9 


.000834028 


4 


.000791139 


5 


.000995025 


1070 


.000934579 


5 


.000881057 


1200 


.600833333 


5 


.000790514 


6 


.000994036 


1 


.000933707 


e 


.000880282 


1 


000832639 


6 


.000789889 


7 


.000993049 


2 


.000932836 


7 


.000879508 


2 


.000831947 


7 


.000789266 


8 


.000992063 


3 


.000931966 


8 


.000878735 


3 


.000831255 


8 


.000788643 


9 


.000991080 


4 


.000931099 


9 


.000877963 


4 


.000830565 


9 


.000788022 


1010 


.000990099 


5 


.000930233 


1140 


.000877193 


5 


.000829875 


1270 


.000787402 


11 


.000989120 


6 


.000929368 


1 


.000876424 


6 


.000829187 


1 


.000786782 


12 


.000988142 


7 


.000928505 


2 


.000875657 


7 


.000828500 


2 


.000786163 


13 


.000987167 


8 


.000927644 


I 


.000874891 


8 


.000827815 


3 


.000785546 


14 


.000986193 


£ 


.000926784 


4 


.000874126 


9 


.000827130 


4 


.000784929 


15 


.000985222 


1080 


.000925926 


5 


.000873362 


1210 


.000826446 


5 


.000784314 


16 


.000984252 


1 


.000925069 


6 


.000872600 


11 


.000825764 


6 


.000783699 


17 


.000983284 


2 


.000924214 


7 


.000871840 


12 


.000825082 


7 


.000783085 


18 


.000982318 


3 


.000923361 


8 


.000871080 


13 


.000824402 


8 


.000782473 


19 


.000981354 


4 


.000922509 


9 


090870322 


14 


.000823723 


9 


.000781861 


1020 


.000980392 


5 


.000921659 


1150 


.000869565 


15 


.000823045 


1280 


.000781250 


1 


.000979432 


6 


.000920810 


1 


.000868810 


16 


.000822368 


1 


.000780640 


2 


.000978474 


7 


.000919963 


2 


.000868056 


17 


.000821693 


2 


.000780031 


3 


.000977517 


8 


.000919118 


3 


.000867303 


18 


000821018 


3 


.000779423 


4 


.000976562 


9 


000918274 


4 


.000866551 


19 


.000820344 


4 


.000778816 


5 


.000975610 


1090 


.000917431 


5 


.000865801 


1220 


.000819672 


5 


.000778210 


6 


.000974659 


1 


.000916590 


6 


.000865052 


1 


.000819001 


6 


.000777605 


7 


.000973710 


2 


.000915751 


7 


.000864304 


2 


.000818331 


7 


.000777001 


8 


.000972763 


3 


.000914913 


,8 


.000863558 


3 


.000817661 


8 


000776397 


9 


.000971817 


4 


.000914077 


9 


.000862813 


4 


.000816993 


9 


.000775795 


1030 


.000970874 


5 


.000913242 


1160 


.000862009 


5 


.000816326 


! 


.000775194 



84 



MATHEMATICAL TABLES. 



No. 


Recipro- 
cal. 


No. 


Recipro- 
cal. 


No. 


Recipro- 
cal. 


No. 


Recipro- 
cal. 


No. 


Recipro- 
cal. 


1291 


.000774593 


1356 


.000737463 


1421 '.000703730 


1486 


.000672948 


1551 


.000644745 


2 


.000773994 


7 


000736920 


2 .000703235 


7 


.000672495 


2 


.000644330 


3 


.000773395 


8 


.000736377 


3 .000702741 


8 


.000672043 


3 


.000643915 


4 


.000772797 


9 


.000735835 


4 .900702247 


9 


.000671592 


4 


.000643501 


5 


.000772. '01 


1360 


.000735294 


5 .000701754 


1490 


.000671141 


5 


.000643087 


6 


.000771605 


1 


.000734754 


6 .000701262 


1 


.000670691 


6 


.000642673 


7 


.000771010 


2 


.000734214 


7 .000700771 


2 


.000670241 


7 


.000642261 


8 


.000770416 


3 


.000733676 


8 .0U07 00280 


3 


.000669792 


8 


.000641848 


9 


.000769823 


4 


.000733138 


9 .000699790 


4 


.000669344 


9 


.000641437 


1300 


.000769231 


5 


.00073 .'601 


1430 .000699301 


5 


.000668896 


1560 


.000641026 


1 


.000768639 


6 


.000732064 


1. 000698812 


6 


.000668449 


1 


.000640615 


2 


.000768049 


7 


.000731529 


2 .000698324 


7 


.000668003 


2 


.000640205 


3 


.000767459 


8 


.000730994 


3 .000097837 


8 


.000667557 


3 


.000639795 


4 


.000766871 


9 


.000730460 


4 .000697350 


9 


.000667111 


4 


.000639386 


5 


.000766283 


1370 


.000729927 


5 .000696864 


1500 .000666667 


5 


.000638978 


6 


.000765697 


1 


.000729395 


6 '.000696379 


1 


.000666223 


6 


.000638570 


7 


.000765111 


2 


.000728863 


7;. 000695894 


2 


.000665779 


7 


.000638162 


8 


.000764526 


3 


.000728332 


8 ! . 000695410 


3 


.000655336 


8 


.000637755 


9 


.000763942 


4 


.000727802 


9 .000694927 


4 


.000664894 


9 


.000637319 


1310 


.000763359 


5 


.000727273 


1440 .000694444 


5 


.000664452 


1570 


.000636943 


11 


.000762776 


6 


.000726744 


1 .0006 


6 


.000664011 


1 


.000636537 


12 


.000762195; 


7 


.000726216: 


2 .000693481 


7 


.000663570 


2 


.000636132 


13 


.000761615 


8 


.000725689 


3 .000693001 


8 


.000663130 


3 


.000635728 


14 


.000761035: 


9 


.000725163 


4 .000692521 


9 


.000662691 


4 


.000635324 


15 


.000760456 


1380 


.000724638 


5 .000692041 


1510 


.000662252 


5 


.000634921 


16 


.000759878 


1 


.000724113 


6.000691563 


11 


.000661813 


6 


.000634518 


17 


.000759301 


2 


.000723589 


71.000691085 


12 


.000661376 


7 


.000634115 


18 


.000758725; 


3 


.000723066 


8 . 0006 90608 


13 


.000660939 


8 


.000633714 


19 


.000758150 


4 


.000722543 


9,-000690131 


14 


.000660502 


9 


.000633312 


1320 


.000757576 


5 


.000722022 


14501.000689655 


15 


.000660066 


1580 


.000632911 


1 


.000757002 


6 


.000721501 


1 .000689180 


16 


.000659631 


1 


.000632511 


2 


.000756430 


7 


.000720980 


2 .000688705 


17 


.000659196 


2 


.000632111 


3 


.000755858 


8 


.000720461 


3 .000688231 


18 


.000658761 


3 


.000631712 


4 


.000755287 


9 


.000719942 


4 


.000687758 


19 


.000658328 


4 


.000631313 


5 


.000754717 


1390 


.000719424 


5 


.000687285 


1520 


.000657895 


5 


.000630915 


6 


.000754148 


1 


.000718907 


6 


.000686813 


1 


.000657462 


6 


.000630517 


7 


.000753579 


2 


000718391 


7 


.000686341 


2 


.000657030 


7 


.000630120 


8 


.000753012 


3 


.000717875 


8 


.000685871 


3 


.000656598 


8 


.000629723 


9 


.000752445 


4 


.000717360 


9 


.000685401 


4 


.000656168 


9!. 000629327 




.000751880 


5 


.000716846 


1460 


.000684932 


5 


.000655738 


1590 


.000628931 


1 


.000751315 


6 


.000716332 


1 


.000684463 


6 


.000655308 


] 


.000628536 


2 


.000750750 


7 


.000715820 


2 


.000683994 


7 


.000654879 


2 


.000628141 


3 


.000750187 


8 


.000715308 


3 .000683527 


8 


.000654450 


3 


.000627746 


4 


.000749625 


9 


.000714796 


4 .000683060 


9 


.000654022 


4 


.000627353 


5 


.000749064 


1400 


.000714286 


5 .000682594 


1530 


.000653595 


5 


.000626959 


6 


.000748503 


1 


.000713776 


61.000682128 


1 


.000653168 


6 


.000626566 


7 


.000747943 


2 


.000713267 


7 .000681663 


2 


.000652742 


7 


.000626174 


8 


.0007473S4 


3 


.000712758 


81 .000681199 


3 


.000652316 


8 


.000625782 


9 


. 000746826 


4 


.000712251 


9; .000680735 


4 


.000651890 


9 


.000625391 




.0007462691 


5 


.000711744 


1470|. 000680272 


5 


.000651466! 


1600 


.000625000 


1 


.000745712' 


6 


.000711238 


1 .000679810 


6 


.000651042; 


2 


.000624219 


2 


.000745156; 


7 


.000710732 


2i .000679348 


7 


.0006506181 


4 


.000623441 


3 


.009744602 


8 


.000710227 


3 . 000678887 


8 


.000650195' 


6 


.000622665 


4 


.000744048 


9 


.000709723 


4;. 000678426 


9 


.000649773 


8 


.000621890 


5 


.000743494! 


1410 


.000709220 


5 .000677966 


1540 


. 000649351 : 


1610 


.000621118 


6 


.000742942 


11 


.000708717 


6 .000677507 


1 


.000648929 


2 


.000620347 


7 


.000742390 


12 


.000708215 


7 .000677048 


2 


. 000648508 ! 


4 


.000619578 


8 


.000741840 


13 


.000707714 


81.000676590 


3 


.000648088 


6 


.000618812 


9 


.000741290 


14 


.000707214 


91.000676132 


4 


.000647668 


8 


.000618047 


1350 


. 000740741 ; 


15 


.000706714 


1480, 000675676 


5 


.000647249 


1620 


.000617284 


1 


.000740192 


16 


.000706215 


1. 00' »ii75219 


6 


.000646830 


2 


.000616523 


2 


000739645 


17 


.000705716 


2 .000674 "64 


7 


.000646412 


4 


.000615763 


3 


.000739098' 


18 


.000705219 


3 .000674309 


8 


. 0001545995 


6 


.000615006 


4 


.000738552 


19 


.000704722 


4.000673s:,4 


9 


.000645578 


8 


.000614250 


5 


.0007380071 


1420 


.000704225 


5 


.000673*01 


1550 


.000615161 


1630 


.000613497 






RECIPROCALS OF NUMBERS. 



85 



No. 


Recipro- 
cal. 


No. 


Recipro- 
cal. 


No. 


Recipro- 
cal. 


No. 


Recipro- 
cal. 


No. 


Recipro- 
cal. 


1632 


.000612745 


1706 


.000586166 


1780 


.000561798 


1854 


.000539374 


1928 


.000518672 


4 


.000611995 


8 


.000585480 


2 


.000561167 


6 


.000538793 


1930 


.000518135 


6 


.000611247 


1710 


.000584795 


4 


.000560538 


8 


.000538213 


2 


.000517599 


8 


.000610500 


12 


.000584112 


6 


.000559910 


1860 


.000537634 


4 


.000517063 


1640 


.000609756 


14 


.000583430 


8 


.000559284 


2 


.000537057 


6 


.000516528 


2 


.000609013 


16 


.000582750 


17 90 


.000558659 


4 


.000536480 


8 


.000515996 


4 


.000608272 


18 


.000582072 


2 


.000558035 


6 


.000535905 


1940 


.000515464 


6 


.000607533 


1720 


.000581395 


4 .000557413 


8 


.000535332 


2 


.000514933 


8 


.000606796 


2 


.000580720 


6:. 000556793 


1870 


.000534759 


4 


.000514403 


165U 


.000606061 


4 


.000580046 


8 


.000556174 


2 


.000534188 


6 


.000513874 


2 


.000605327 


6 


.000579374 


18 00 


.000555556 


4 


.000533618 


8 


.000513347 


4 


.000604595 


8 


.000578704 


2 


.000554939 


6 


.000533049 


1950 


.000512820 


6 


.000603865 


1730 


.000578035 


4 


.000554324 


8 


.000532481 


2 


.000512295 


8 


.000603136 


2 


.000577367 


6 


.000553710 


1880 


.000531915 


4 


.000511770 


161 


.000602410 


4 


.000576701 


8 


. 000553097 


2 


.000531350 


6 


.000511247 


2 


.000601685 


6 


.000576037 


18101.000552486 


4 


.000530785 


8 


.000510725 


4 


.000600962 


8 


.000575374 


12 .000551876 


6 


.000530222 


1960 


.000510204. 


6 


.000600240 


1740 


.000574713 


14 


.000551268 


8 


.000529661 


2 


.000509684 


8 


.000599520 


2 


.000574053! 


16 


.000550661 


1890 


.000529100 


4 


.000509165 


1670 


.000598802 


4 


.000573394 


18 


.000550055 


2 


.000528541 


6 


.000508647 


2 


.000598086 


6 


.000572737 


1820 


.000549451 


4 


.000527983 


8 


.000508130 


4 


.000597371 


8 


.000572082 


2 


.000548848 


6 


.000527426 


1970 


.000507614 


6 


.000596658 


1750 


.000571429 


4 


.000548246 


8 


.000526870 


2 


.000507099 


8 


.000595947 


2 


.000570776 


6 


.000547645 


1900 


.000526316 


4 


.600506585 


ig-;o 


.000595238 


4 


.000570125! 


8 


.000547046 


2 


.000525762 


6 


.000506073 


' 2 


.000594530 


6 


.000569476' 


1830 


.000546448 


4 


.0005^5210 


8 


.000505561 


4 


.000593824 


8 


.000568828 


2 


.000545851 


6 


.000524659 


1980 


.000505051 


6 


.000593120 


1760 


.000568182 


4 .000545253 


8 


.000524109 


2 


.000504541 


8 


.000592417 


2 


.000567537 


61.000544662, 


1910 


.000523560 


4 


.000504032 




.000591716 


4 


.000566893 


8 


.000544069 


12 


.000523012 


6 


.000503524 


2 


.000591017 


6 


.000566251 


1840 


.000543478 


14 


.000522466 


8 


.000503018 


4 


.000590319 


8 


.000565611 


2 


.000542888 


16 


.000521920 


1990 


.000502513 


6 


.000589622 


1770 


.000564972 


4 


.000542299 


18 


.000521376 


2 


.000502008 


8 


.000588928 


2 


.000564334 


6 


.000541711 


1920 


.000520833 


4 


.000501504 


1700 


.000588235 


4 


.000563698] 


8 


.000541125 


2 


.000520291 


6 


.000501002 


2 


.000587544 


6 


.000563063 


1850 .000540540 


4 


.000519750 


8 


.000500501 


4 


.000586854 


8 


.000562430! 


2. 000539957 


6 


.000519211 


2000 


000500000 



"Use of reciprocals,,— Reciprocals may be conveniently used to facili- 
tate computations in long division. Instead of dividing as usual, multiply 
the dividend by the reciprocal of the divisor. The method is especially 
useful when many different dividends are required to be divided by the 
same divisor. In this case find the reciprocal of the divisor, and make a . 
small table of its multiples up to 9 times, and use this as a multiplication- i 
table instead of actually performing the multiplication in each case. 

Example.— 9871 and several other numbers are to be divided by 1638. The 
reciprocal of 1638 is .000610500. 
Multiples of the 

reciprocal : 

1. .0006105 

2. .0012210 

3. .0018315 

4. .0024420 

5. .0030525 

6. .0036630 

7. .0042735 

8. .0048840 

9. .0054945 
10. .0061050 



The table of multiples is made by continuous addition 
of 6105. The tenth line is written to check the accuracy 
of the addition, but it is not afterwards used. 
Operation: 

Dividend 9871 
Take from table 1 . . 



.0006105 
0.042735 
00.48840 
005.4945 

6.0262455 



Quotient. . 

Correct quotient by direct division 6.0262515 

The result will generally be correct to as many figures as there are signifi- 
cant figures in the reciprocal, less one, and the error of the next figure will in 
general not exceed one. In the above example the reciprocal has six sig- 
nificant figures, 610500, and the result is correct to five places of figures. 



86 



MATHEMATICAL TABLES. 



SQUARES, CUBES, SQUARE ROOTS AND CUBE 
ROOTS OF NUMBERS FROM .1 TO 1600. 



No. 


Square. 


Cube. 


Sq. 
Root. 


Cube 
Root. 


No. 


Square. 


Cube. 


Sq. 
Root. 


Cube 
Root. 


.1 


.01 


.001 


.3162 


.4642 


3.1 


9.61 


29.791 


1.761 


1.458 


.15 


.0225 


.0034 


.3873 


.5313 


.2 


10.24 


32.768 


1.789 


1.474 


.2 


.04 


.008 


.4472 


.5848 


.3 


10.89 


35.937 


1.817 


1.489 


.25 


.0625 


.0156 


.500 


.6300 


.4 


11.56 


39.304 


1.844 


1.504 


.3 


.09 


027 


.5477 


.6694 


.5 


12.25 


42.875 


1.871 


1.518 


.35 


.1225 


.0429 


.5916 


.7047 


.6 


12.96 


46.656 


1.897 


1.533 


.4 


.16 


.064 


.6325 


.7368 


7 


13.69 


50.653 


1.924 


1.547 


.45 


.2025 


.0911 


.6708 


.7663 


!8 


14.44 


54.872 


1.949 


1.560 


.5 


.25 


.125 


.7071 


.7937 


.9 


15.21 


59.319 


1.975 


1.574 


.55 


.3025 


.1664 


.7416 


.8193 


4. 


16. 


64. 


2. 


1.5874 


.6 


.36 


.216 


.7746 


.8434 


.1 


16.81 


68.921 


2.025 


1.601 


.65 


.4225 


.2746 


.8062 


.8062 


.2 


17.64 


74.088 


2.049 


1.613 


.7 


.49 


.343 


.8367 


.8879 


.3 


18.49 


79.507 


2.074 


1.626 


.75 


.5625 


.4219 


.8600 


.9086 


.4 


19.36 


85.184 


2.098 


1.639 


.8 


.64 


.512 


.8944 


.9283 


.5 


20.25 


91.125 


2.121 


1.651 


.85 


.7225 


.6141 


.9219 


.9473 


.6 


21.16 


97.336 


2.145 


1.663 


.9 


.81 


.729 


.9487 


.9655 


7 


22.09 


103.823 


2.168 


1.675 


.95 


.9025 


.8574 


.9747 


.9830 


'.8 


23.04 


110.592 


2.191 


1.687 


1 


1. 


1. 


1. 


1. 


.9 


24.01 


117.649 


2.214 


1.698 


1.05 


1.1025 


1.158 


1.025 


1.016 


5. 


25. 


125. 


2.2361 


1.7100 


1.1 


1.21 


1.331 


1.049 


1.032 


.1 


26.01 


132.651 


2.258 


1.721 


1.15 


1.3225 


1.521 


1.072 


1.048 


.2 


27.04 


140.608 


2.280 


1.732 


1.2 


1.44 


1.728 


1.095 


1.063 


.3 


28.09 


148.877 


2.302 


1.744 


1.25 


1.5625 


1.953 


1.118 


1.077 


.4 


29.16 


157.464 


2.324 


1.754 


1.3 


1.69 


2.197 


1.140 


1.091 


.5 


30.25 


166.375 


2.345 


1.765 


1.35 


1.8225 


2.460 


1.162 


1.105 


.6 


31.36 


175.616 


2.366 


1.776 


1.4 


1.96 


2.744 


1 . 183 


1.119 


7 


32.49 


185.193 


2.387 


1.786 


1.45 


2.1025 


3.049 


1.204 


1.132 


'.8 


33.64 


195.112 


2.408 


1.797 


1.5 


2.25 


3.375 


1.2247 


1 . 1447 


.9 


34.81 


205.379 


2.429 


1.807 


1.55 


2.4025 


3.724 


1.245 


1.157 


6. 


36. 


216. 


2.4495 


1.8171 


1.6 


2.56 


4.096 


1.265 


1.170 


.1 


37.21 


226.981 


2.470 


1.827 


1.05 


2.7225 


4.492 


1.285 


1.182 


.2 


38.44 


238.328 


2.490 


1.837 


1.7 


2.89 


4.913 


1.304 


1.193 


.S 


39.69 


250.047 


2.510 


1.847 


1.75 


3.0625 


5.359 


1.323 


1.205 


A 


40.96 


262.144 


2.530 


1.857 


1.8 


3.24 


5.832 


1.342 


1.216 


.5 


42.25 


274.625 


2.550 


1.866 


1.85 


3.4225 


6.332 


1.360 


1.228 


.6 


43.56 


287.496 


2.569 


1.S76 


1.9 


3.61 


6.859 


1.378 


1.239 


7 


44.89 


300.763 


2.588 


1.885 


1.95 


3.8025 


7.415 


1.396 


1.249 


!8 


46.24 


314.432 


2.608 


1.895 


2. 


4. 


8. 


1.4142 


1.2599 


.9 


47.61 


328.509 


2.627 


1.904 


.1 


4.41 


9.261 


1.449 


1.281 


7. 


49. 


343. 


2.6458 


1.9129 


.2 


4.84 


10.648 


1.483 


1.301 


.1 


50.41 


357.911 


2.665 


1.922 


.3 


5.29 


12.167 


1.517 


1.320 


.2 


51.84 


373.248 


2.683 


1.931 


A 


5.76 


13.824 


1.549 


1.339 


.3 


53.29 


389.017 


2.702 


1.940 


.5 


6.25 


15.625 


1.581 


1.357 


.4 


54.76 


405.224 


2.720 


1.949 


.6 


6.76 


17.576 


1.612 


1.375 


.5 


56.25 


421.875 


2.739 


1.957 


7 


7.29 


19.683 


1.643 


1.392 


.6 


57.76 


438.976 


2.757 


1.966 


'.8 


7.84 


21.952 


1.673 


1.409 


.7 


59.29 


456.533 


2.775 


1.975 


.9 


8.41 


24.389 


1.703 


1.426 


.8 


60.84 


474.552 


2.793 


1.983 


3. 


9. 


27. 


1.7321 


1.4422 


.9 


62.41 


493.039 


2.811 


1.992 



SQUARES, CUBES, SQUARE AKD CUBE ROOTS. 8? 



No. 


Square. 


Cube. 


Sq. 
Root. 


Cube 
Root. 


No. 


Square. 


Cube. 


Sq. 
Root. 


Cube 
Root. 


8. 


64. 


512. 


2.8284 


2. 


45 


2025 


91125 


6.7082 


3.5569 


.1 


65.61 


531.441 


2.846 


2.008 


46 


2116 


97336 


6.7823 


3.5830 


.2 ' 


67.24 


551.368 


2.864 


2.017 


47 


2209 


103823 


6.8557 


3.6088 


.3 


68.89 


571.787 


2.881 


2 025 


48 


2304 


110592 


6.9282 


3.6342 


.4 


70.56 


592.704 


2.898 


2.033 


49 


2401 


117649 


7. 


3.6593 


.5 


72.25 


614.125 


2.915 


2.041 


50 


2500 


125000 


7.0711 


3.6840 


.6 


73.96 


636.056 


2.933 


2.049 


51 


2601 


132651 


7.1414 


3.7084 


.7 


75 69 


658.503 


2.950 


2.057 


52 


2704 


140608 


7.2111 


3.7325 


.8 


77^44 


681.472 


2.966 


2.065 


53 


2809 


148877 


7.2801 


3.7563 


.9 


79.21 


704.969 


2.983 


2.072 


54 


2916 


157464 


7.3485 


3.7798 


9. 


81. 


729. 


3. 


2.0801 


55 


3025 


166375 


7.4162 


3.8030 


.1 


82.81 


753.571 


3.017 


2.088 


56 


3136 


175616 


7.4833 


3.825^) 


.52 


84.64 


778.688 


3.033 


2.095 


57 


3249 


185193 


7.5498 


3.84S5 


.3 


86.49 


804.357 


3.050 


2.103 


58 


3364 


195112 


7.6158 


3.8709 


.4 


88.36 


830.581 


3.066 


2.110 


59 


3481 


205379 


7.6811 


3.8930 


.5 


90.25 


857.375 


3.082 


2.118 


60 


3600 


216000 


7.7460 


3.9149 


.6 


92.16 


884.736 


3.098 


2.125 


61 


3721 


226981 


7.8102 


3.9365 


.7 


94.09 


912.673 


3.114 


2 133 


62 


3844 


238328 


7.8740 


3.9579 


.8 


96.04 


941.192 


3.130 


2.140 


63 


3969 


250047 


7.9373 


3.9791 


.9 


98.01 


970.299 


3.146 


2.147 


64 


4096 


262144 


8. 


4. 


10 


100 


1000 


3.1623 


2.1544 


65 


4225 


274625 


8.0623 


4.0207 


11 


121 


1331 


3.3166 


2.2240 


66 


4356 


287496 


8.1240 


4.0412 


12 


144 


1728 


3.4641 


2.2894 


67 


4189 


300763 


8.1854 


4.0615 


13 


169 


2197 


3.6056 


2.3513 


68 


4624 


314432 


8.2462 


4.0S17 


14 


196 


2744 


3.7417 


2.4101 


69 


4761 


328509 


8.3066 


4.1016 


15 


225 


3375 


3.8730 


2.4662 


70 


4900 


343000 


8.3666 


4.1213 


16 


256 


4096 


4. 


2.5198 


71 


5041 


357911 


8.4261 


4.1408 


17 


289 


4913 


4.1231 


2.5713 


72 


5184 


373248 


8.4853 


4.1602 


18 


324 


5832 


4.2426 


2.6207 


73 


5329 


389017 


8.5440 


4.1793 


19 


361 


6859 


4.3589 


2.6684 


74 


5476 


405224 


8.6023 


4.1983 


20 


400 


8000 


4.4721 


2.7144 


75 


5625 


421875 


8.6603 


4.2172 


21 


441 


9261 


4.5826 


2 7589 


76 


5776 


438976 


8.7178 


4.2358 


22 


484 


10648 


4.6904 


2.8020 


77 


5929 


456533 


8.7750 


4.2543 


23 


529 


12167 


4.7958 


2.8439 


78 


6084 


474552 


8.8318 


4.2727 


24 


576 


13824 


4.8990 


2.8845 


79 


6241 


493039 


8.8882 


4.2908 


25 


625 


15625 


5. 


2.9240 


80 


6400 


512000 


8.9443 


4.3089 


26 


676 . 


17576 


5.0990 


2.9625 


81 


6561 


531441 


9. 


4.3267 


27 


729 


19683 


5.1962 


3. 


82 


6724 


551368 


9.0554 


4.3445 


28 


784 


21952 


5.2915 


3 0366 


83 


6889 


571787 


9.1104 


4.3621 


29 


841 


24389 


5.3852 


3.0723 


84 


7056 


592704 


9.1652 


4.3795 


30 


900 


27000 


5.4772 


3.1072 


85 


7225 


614125 


9.2195 


4.3968 


31 


961 


29791 


5.5678 


3.1414 


86 


7396 


636056 


9.2736 


4.4140 


32 


1024 


32768 


5.6569 


3.1748 


87 


7569 


658503 


9 3276 


4.4310 


33 


1089 


35937 


5.7446 


3.2075 


88 


7744 


OKI 472 


9.3808 


4.4480 


34 


1156 


39304 


5.8310 


3.2396 


89 


7921 


704969 


9.4340 


4.4647 


35 


1225 


42875 


5.9161 


3.2711 


90 


8100 


729000 


.9.4868 


4.4814 


36 


1296 


46656 


6. 


3.3019 


91 


8281 


753571 


9.5394 


4.4979 


37 


1369 


50653 


6.0828 


3.3322 


92 


8464 


778688 


9.5917 


4.5144 


38 


1444 


54872 


6.1644 


3.3620 


93 


8649 


804357 


9 6437 


4.5307 


39 


1521 


59319 


6.2450 


3.3912 


94 


8836 


830584 


9.6954 


4.5468 


40 


1600 


64000 


6.3246 


3 4200 


95 


9025 


857375 


9 7468 


4.5629 


41 


1681 


68921 


6.4031 


3.4482 


96 


9216 


884736 


9.7980 


4.5789 


42 


1764 . 


74088 


6.4807 


3.4760 


97 


9409 


912673 


9.8489 


4.5947 


43 


1849 


79507 


6.5574 


3.5034 


98 


9604 


941192 


9.8995 


4.6104 


44 


1936 


85184 


6.6332 


3.5303 


99 


9801 


970299 


9.9499 


4.6261 



MATHEMATICAL TABLES. 



No. 


Square. 


Cube. 


Sq. 
Root. 


Cube 
Root. 


No. 


Square. 


Cube. 


Sq. 
Root. 


Cube 
Root. 


100 


10000 


1000000 


10. 


4.6416 


155 


240.25 


3723875 


12.4499 


5.3717 


101 


10201 


1030301 


10.0199 


4.6570 


156 


24336 


3796416 


12.49UI 


5.3832 


10:2 


10404 


1061208 


10.0995 


4.6723 


157 


24649 


3869893 


I2.53IN 


5.3947 


103 


10609 


1092727 


10.1489 


4.6875 


158 


24964 


3944312 


12.569b 


5.4061 


104 


10816 


1124864 


10.1980 


4.7027 


159 


25281 


4019679 


12.6095 


5.4175 


105 


11025 


1157625 


10.2470 


4.7177 


160 


25600 


4096000 


12.6491 


5.4288 


106 


112 6 


1191016 


10.2956 


4.7326 


161 


25921 


4173281 


12.6886 


5.4401 


107 


11449 


1225043 


10.3141 


4.7475 


162 


26244 


4251528 


12.7279 


5 4514 


108 


11664 


1259712 


10.3923 


4.7622 


163 


26569 


4330747 


12.7671 


5.4626 


109 


11881 


1295029 


10.4403 


4.7769 


164 


26896 


4410944 


12.8062 


5.4737 


110 


12100 


1331000 


10.4881 


4.7914 


165 


27225 


4492125 


12.8452 


5.4848 


111 


12321 


1367631 


10.5357 


4.8059 


166 


27556 


4574296 


12.8841 


5.4959 


112 


12544 


1404928 


10.5S30 


4; 8203 


167 


27889 


4657463 


12.9228 


5.5069 


113 


12769 


1442897 


10.6301 


4.8346 


168 


28224 


4741632 


12.9615 


5.5178 


114 


12996 


1481544 


10.6771 


4:8488 


169 


28561 


4826809 


13.0000 


5.5288 


115 


13225 


1520875 


10.7238 


4.8629 


170 


28900 


4913000 


13.0384 


5.5397 


110 


13456 


1560896 


10.7703 


4.8770 


171 


29241 


5000211 


13.0767 


5.5505 


117 


13689 


1601613 


10.816? 


4.8910 


172 


29584 


5088448 


13.1149 


5.5613 


118 


13924 


1613032 


10.K62K 


4.9049 


173 


29929 


5177717 


13.1529 


5.5721 


119 


14161 


1685159 


10.9087 


4.9187 


174 


30276 


5268024 


13.1909 


5.5828 


120 


14400 


1728000 


10.9545 


4.9324 


175 


30625 


5359375 


13.2288 


5.5934 


121 


14641 


1771561 


11.0000 


4.9461 


176 


30976 


5451776 


13.2665 


5.6041 


122 


14884 


1815848 


11.0454 


4.9597 


177 


31329 


5545233 


13.3041 


5.6147 


123 


15129 


1860867 


11.0905 


4.9732 


178 


31684 


5639752 


13.3417 


5.6252 


124 


15376 


1906624 


11.1355 


4.9866 


179 


32041 


5735339 


13.3791 


5.6357 


125 


15625 


1953125 


11.1803 


5,0000 


180 


32400 


5832000 


13.4164 


5.6462 


126 


15876 


2000376 


11.2250 


5.0133 


181 


32761 


5929741 


13.4536 


5.6567 


127 


16129 


2048383 


11.2694 


5.0265 


182 


33124 


602S568 


13.4907 


5.6671 


128 


16384 


2097152 


11.3137 


5.0397 


183 


33489 


6128487 


13.5277 


5.6774 


129 


16641 


2146689 


11.3578 


5.0528 


184 


33856 


6229504 


13.5617 


5.6877 


130 


16900 


2197000 


11.4018 


5.0658 


185 


34225 


6331625 


13.6015 


5.6980 


131 


17161 


2248091 


11.4455 


5.0788 


186 


34596 


6434856 


13.6382 


5.7083 


- 


17424 


2299968 


11.4891 


5.0916 


187 


34969 


6539203 


13. oris 


5.7185 


133 


17689 


2352637 


11.5326 


5.1045 


188 


35344 


6644672 


13.7113 


5.7287 


134 


17956 


2406104 


11.5758 


5.1172 


189 


35721 


6751269 


13.7477 


5.7388 


135 


18225 


2460375 


11.6190 


5.1299 


190 


36100 


6859000 


13.7840 


5.7489 


136 


18496 


2515456 


11.6619 


5.1426 


191 


36481 


6967871 


13.8203 


5.7590 


137 


18769 


2571353 


11.7047 


5.1551 


192 


36864 


7077888 


13.8564 


5.7690 


138 


19044 


2628072 


11.7473 


5.1676 


193 


37249 


7189057 


13.8924 


5 7790 


139 


19321 


2685619 


11.7898 


5.1801 


194 


37636 


7301384 


13.9284 


5.7890 


140 


19600 


2744000 


11.8322 


5.1925 


195 


38025 


7414875 


13.9642 


5.7989 


141 


19881 


2803221 


11.8743 


5.2048 


196 


38416 


7529536 


14.0000 


5.8088 


112 


20164 


2863288 


11.9164 


5.2171 


197 


38809 


7645373 


14.0357 


5.8186 


143 


20449 


2924207 


11.9583 


5.2293 


198 


39204 


7762392 


14.0712 


5.8285 


144 


20736 


2985984 


12.0000 


5.2415 


199 


39601 


7880599 


14.1067 


5.8383 


145 


21025 


3048625 


12.0416 


5.2536 


200 


40000 


8000000 


14.1421 


5.8480 


146 


21316 


3112136 


12.0S30 


5.2656 


201 


40401 


8120601 


14.1774 


5.8578 


147 


21609 


3176523 


12.1244 


5.2776 


202 


40804 


8242408 


14.2127 


5.8675 


148 


21904 


3241792 


12.1655 


5.2896 


203 


41209 


8365127 


14.2478 


5.8771 


149 


22201 


3307949 


12.2066 


5.3015 


204 


41616 


8489664 


14.2829 


5.8868 


150 


22500 


3375000 


12.2474 


5.3133 


205 


42025 


8615125 


14.3178 


5.8964 


151 


22801 


3442951 


12.2882 


5.1251 


206 


42436 


8741816 


14.3527 


5 9059 


152 


23104 


3511803 


12.32SS 


5.3368 


207 


42849 


8869743 


14.3875 


5.9155 


153 


23409 


358 1 577 


12. 3693 


5.3485 


208 


43264 


8998912 


14.4222 


5.9250 


154 


23716 


3652264 


12 4097 


5.3601 


.209 


43681 


9129329 


14.4568 


5.9345 



SQUARES, CUBES, SQUARE AND CUBE ROOTS. 



89 



No. 


Square. 


2 10 


44100 


an 


44521 


212 


44944 


213 


45369 


214 


45796 


215 


46225 


216 


46656 


217 


47089 


218 


47524 


219 


47961 


250 


48400 


221 


48841 


222 


49284 


228 


49729 


224 


50176 


225 


50625 


226 


51076 


227 


51529 


228 


51984 


229 


52441 


280 


52900 


231 


53861 


282 


53824 


288 


54289 


284 


54756 


285 


55225 


236 


55696 


28? 


56169 


288 


56644 


289 


57121 


240 


57600 


241 


58081 


242 


58564 


248 


59049 


244 


59536 


245 


60025 


246 


60516 


247 


61009 


24 S 


61504 


249 


62001 


250 


62500 


251 


63001 


252 


63504 


25 5 


64009 


251 


64516 


255 


65025 


256 


65536 


257 


66049 


258 


66564 


259 


67081 


260 


67600 


21 i1 


68121 


282 


68644 


2(18 


69169 


264 


69696 



Cube. 


Sq. 
Root. 


Cube 
Root. 


9261000 


14.4914 


5.9439 


9393931 


14.5258 


5.9533 


9528128 


14.5602 


5.9627 


9663597 


14.5945 


5.9721 


9800344 


14.6287 


5,9814 


9938375 


14.6629 


5.9907 


10077696 


14.6969 


6.0000 


10218313 


14.7309 


6 0092 


10360232 


14.7648 


6.0185 


10503459 


14.7986 


6 0277 


10648000 


14.8324 


6.0368 


10793861 


14.8661 


6.0459 


10941048 


14.8997 


6.0550 


11089567 


: « •■ 


6.0641 


11239424 


14.9666 


6.0732 


11390625 


15.0000 


6.0822 


11543176 


15.0333 


6.0912 


11697083 


15.0665 


6.1002 


11852352 


15.099? 


6.1091 


12008989 


15.1327 


6.1180 


12167000 


15.1658 


6.1269 


12326391 


15.1987 


6.1358 


12487168 


15.2315 


6.1446 


12649337 


15.2643 


6.1534 


12812904 


15.2971 


6,1622 


12977875 


15.3297 


6.1710 


13144256 


15.3623 


6.179? 


13312053 


15 3948 


6.1885 


13481272 


15.4272 


6.1972 


13651919 


15.4596 


6.2058 


13824000 


15.4919 


6.2145 


13997521 


15.5242 


6.2231 


14172488 


15.5563 


6.2317 


14348907 


15.5885 


6.2403 


14526784 


15.6205 


6.2488 


14706125 


15.6525 


6.2573 


14886936 


15.6844 


6.2658 


15069223 


15.7162 


6.2743 


15252992 


15.7480 


6.2828 


15438249 


15.7797 


6.2912 


15625000 


15.8114 


6.2996 


15813251 


15.8430 


6.3080 


16003008 


15.8745 


6.3164 


16194277 


15.9060 


6.3247 


16387064 


15.9374 


6.3330 


16581375 


15.9687 


6.3413 


16777216 


16.0000 


6.3496 


16974593 


16.0312 


6.3579 


17173512- 


16.0624 


6.3661 


17373979 


16.0935 


6.3743 


17576000 


16.1245 


6 3825 


17779581 


16.1555 


6.390? 


17984728 


16.1864 


6.3988 


18191447 


16.2173 


6.4070 


18399744 


16.2481 


6.4151 



No. 


Square. 


Cube. 


265 


70225 


1S609625 


266 


70756 


18821096 


267 


71289 


19034163 


268 


71824 


19248832 


269 


72361 


19465109 


270 


72900 


19683000 


271 


73441 


19902511 


272 


73984 


20123648 


273 


74529 


20346417 


274 


75076 


20570824 


275. 


75625 


20796875 


276 


76176 


21024576 


277 


76729 


21253933 


278 


77284 


21484952 


279 


77841 


21717639 


280 


78400 


21952000 


281 


78961 


22188041 


282 


79524 


22425768 


283 


80089 


22665187 


284 


80656 


22906304 


285 


81225 


23149125 


286 


81796 


23393656 


287 


82369 


23639903 


288 


82944 


23887872 


289 


83521 


24137569 


290 


84100 


24389000 


291 


84681 


24642171 


292 


85264 


24897088 


293 


85849 


25153757 


294 


86436 


25412184 


295 


87025 


25672375 


296 


87616 


25934336 


297 


88209 


26198073 


298 


88804 


26463592 


299 


89401 


26730899 


300 


90000 


27000000 


301 


90601 


27270901 


302 


91204 


27543608 


303 


91809 


27818127 


304 


92416 


28094464 


305 


93025 


28372625 


305 


93636 


28652616 


307 


94249 


28934443 


308 


94864 


29218112 


309 


95481 


29503629 


310 


96100 


29791000 


311 


96721 


30080231 


312 


9?'344 


30371328 


313 


97969 


30664297 


314 


98596 


30959144 


315 


99225 


31255875 


316 


99856 


31554496 


317 


100489 


31855013 


318 


101124 


32157432 


319 


101761 


32461759 



Sq. 


Cube 


Root. 


Root. 


16.2788 


6.4232 


16.3095 


6.4312 


16.3401 


6.4393 


16.8707 


6.4473 


16.4012 


6.4553 


16 431? 


6.4633 


16.4621 


6.4713 


16.4924 


6.4792 


16.5227 


6.4872 


16.5529 


6.4951 


16.5831 


6.5030 


16.6132 


6.5108 


16.6433 


6.5187 


16.6733 


6.5265 


16.7033 


6.5343 


16.7332 


6.5421 


16.7631 


6.5499 


16.7929 


6.5577 


16.8226 


6.5654 


16.8523 


6.5731 


16.8819 


6.5808 


16.9115 


6.5885 


16.9411 


6.5962 


16.9706 


6.6039 


17.0000 


6.6115 


17.0294 


6.6191 


17.0587 


6.6267 


17.0880 


6.6343 


17.1172 


6.6419 


17.1464 


6.6494 


17.1756 


6.6569 


17.2047 


6.6644 


17.2337 


6.6719 


17.2627 


6.6794 


17.2916 


6.6869 


17.3205 


6.6943 


17.3494 


6.7018 


17.3781 


6.7092 


17.4069 


6.7166 


17 4356 


6.7240 


17.4642 


6.7313 


17.4929 


6.7387 


17.5214 


6.7460 


17.5499 


0.7533 


17.5784 


6.7606 


17.6068 


6.7679 


17.6352 


6.7752 


17.6635 


6.7824 


17.6918 


6.7897 


17.7200 


6.7969 


17.7482 


6.8041 


17.7764 


6.8113 


17.8045 


6.8185 


17.8326 


6.8256 


17.8606 


6.8328 



90 



MATHEMATICAL TABLES. 



No. 


Square. 


Cube. 


Sq. 

Root. 


Cube 
Root. 


No. 


Square. 


Cube. 


Sq. 

Root. 


Cube 
Root. 


320 


102400 


32768000 


17.8885 


6.8399 


375 


140625 


52734375 


19.3649 


7.2112 


321 


103041 


33076161 


17.9165 


6.8470 


376 


141376 


53157376 


19.3907 


7.2177 


322 


103684 


33386248 


17.9444 


6.8541 


377 


142129 


53582633 


19.4165 


7.2240 


323 


104329 


33698267 


17.9722 


6.8612 


37 S 


142884 


54010152 


19.4422 


7.2304 


324 


104976 


34012224 


18.0000 


6.8683 


379 


143641 


54439939 


19.4679 


7.2368 


325 


105625 


34328125 


18.0278 


6.8753 


380 


144400 


54872000 


19.4936 


7.2432 


326 


106276 


34645976 


18.0555 


6.8824 


381 


145161 


55306341 


19.5192 


7.2495 


327 


106929 


34965783 


18.0831 


6.8894 


3S2 


145924 


55742968 


19.5448 


7.2558 


328 


107584 


35287552 


18.1108 


6.8964 


383 


146689 


56181887 


16.5704 


7.2622 


329 


108241 


35611289 


18.1384 


6.9034 


384 


147456 


56623104 


19.5959 


7.2685 


330 


108900 


35937000 


18.1659 


6.9104 


385 


148225 


57066625 


19.6214 


7.2748 


331 


109561 


36264691 


18.1934 


6.9174 




148996 


57512456 


19.6469 


7.2811 


332 


110224 


36594368 


18.2209 


6.9244 




149769 


57960603 


19.6723 


7.2874 


333 


110889 


36926037 


18.2483 


6.9313 


3 


150544 


58411072 


19.6977 


7.2936 


334 


111556 


37259704 


18.2757 


6.9382 


389 


151321 


58863869 


19.7231 


7.2999 


335 


112225 


37595375 


18.3030 


6.9451 


390 


152100 


59319000 


19.7484 


7.3061 


336 


112896 


37933056 


18.3303 


6.9521 


391 


152881 


59776471 


19.7737 


7.3124 


337' 


113569 


38272753 


18.3576 


6.9589 


192 


153664 


60236288 


19.7990 


7.3186 


33S 


114244 


38614472 


18.3848 


6.9658 




154449 


60698457 


19.8242 


7.3248 


339 


114921 


38958219 


18.4120 


6.9727 


394 


155236 


61162984 


19.8494 


7.3310 


340 


115600 


39304000 


18.4391 


6.9795 


395 


156025 


61629875 


19.8746 


7.3372 


341 


116281 


39651821 


18.4662 


6.9864 


J9I 


156816 


62099136 


19.8997 


7.3434 


342 


116964 


40001688 


18.4932 


6 9932 


397 


157609 


62570773 


19.9249 


7.3496 


343 


117649 


40353607 


18.5203 


7.0000 


39S 


158404 


63044792 


19.9499 


7.3558 


344 


118336 


40707584 


18.5472 


7.0068 


399 


159201 


63521199 


19.9750 


7.3619 


315 


119025 


41063625 


18.5742 


7.0136 


400 


160000 


64000000 


20 0000 


7.3681 


346 


119716 


41421736 


18.6011 


7.0203 


401 


160801 


64481201 


20 0250 


7.3742 


347 


120409 


41781923 


18.6279 


7.0271 


402 


161604 


64964808 


20.0499 


7.3803 


348 


121104 


42144192 


18.6548 


7.0338 


403 


162409 


65450827 


20 0749 


7.3864 


349 


121801 


42508549 


18.6815 


7.0406 


404 


163216 


65939264 


20.0998 


7.3925 


350 


122500 


42875000 


18.7083 


7.0473 


405 


164025 


66430125 


20.1246 


7.3986 


351 


123201 


43243551 


18.7350 


7.0540 


406 


164836 


66923416 


20.1494 


7.4047 


352 


123904 


43614208 


18.7617 


7.0607 


407 


165649 


67419143 


20.1742 


7.4108 


353 


124609 


43986977 


1S.7S-3 


7.0674 


40S 


166464 


67917312 


20.1990 


7.4169 


354 


125316 


44361864 


18.8149 


7.0740 


409 


167281 


68417929 


20.2237 


7.4229 


355 


126025 


44738875 


18.8414 


7.0807 


410 


168100 


68921000 


20.2485 


7.4290 


3 6 


126736 


45118016 


18.8680 


7.0873 


411 


168921 


69426531 


20.2731 


7.4350 


357 


127449 


45499293 


18.8944 


7.0940 


412 


169744 


69934528 


20.2978 


7.4410 


358 


128164 


45882712 


18 9209 


7.1006 


413 


170569 


70444997 


20.3224 


7.4470 


359 


128881 


46268279 


18.9473 


7.1072 


414 


171396 


70957944 


20.3470 


7.4530 


360 


129600 


46656000 


18.9737 


7.1138 


415 


172225 


71473375 


20.3715 


7.4590 


361 


130321 


47045881 


19.0000 


7.1204 


416 


173056 


71991296 


20.3961 


7.4650 


362 


131044 


47437928 


19.0263 


7.1269 


417 


173889 


72511713 


20.4206 


7.4710 


363 


131769 


47832147 


17.0526 


7.1335 


418 


174724 


73034632 


20.4450 


7.4770 


364 


132496 


48228544 


19.0788 


7.1400 


419 


175561 


73560059 


20.4695 


7.4829 


365 


133225 


48627125 


19.1050 


7.1466 


420 


176400 


74088000 


20.4939 


7.4889 


366 


133956 


49027896 


19.1311 


7.1531 


421 


177241 


74618461 


20.5183 


7.4948 


367 


134689 


49430863 


19.1572 


7.1596 


422 


178084 


75151448 


20.5426 


7.5007 


368 


135424 


49836032 


19.1833 


7.1661 


423 


178929 


75686967 


20.5670 


7.5067 


309 


136161 


50243409 


19.2094 


7.1726 


424 


179776 


76225024 


20.5913 


7.5126 


370 


136900 


50653000 


19.2354 


7.1791 


425 


180625 


76765625 


20.6155 


7.5185 


371 


137641 


51064811 


19.2614 


7.1855 


426 


181476 


77308776 


20.6398 


7.5244 


372 


138384 


51478848 


19.2873 


7.1920 


427 


182329 


77854483 


20.6640 


7.5302 


S73 


139129 


51895117 


19.3132 


7.1984 


428 


183184 


78402752 


20 6882 


7.5361 


374 


139876 


52313624 


19.3391 


7.2048 


429 


184041 


78953589 


20.7123 


7.5420 



SQUARES, CUBES, SQUARE AKB CUBE ROOTS. 91 



No. 


Square. 


Cube. 


Sq. 
Root. 


Cube 
Root. 


No. 


Square. 


Cube. 


Sq. 
Root. 


Cube' 
Root. 


430 


184900 


79507000 


20.7364 


7.5478 


485 


235225 


114084125 


22.0227 


7.8568 


431 


185761 


80062991 


20.7605 


7.5537 


486 


236196 


114791256 


22.0454 


7.8622 


432 


186624 


80621568 


20.7846 


7.5595 


487 


237169 


115501303 


22.0681 


7.8676 


433 


1S7489 


SI 182737 


20.80S7 


7.5654 


488 


238144 


116214272 


22.0907 


7.8730 


434 


188356 


81746504 


20.8327 


7.5712 


489 


239121 


116930169 


22.1133 


7.8784 


435 


189225 


82312875 


20.8567 


7.5770 


490 


240100 


117649000 


22.1359 


7.8837 


436 


190096 


82881856 


20.8806 


7.5828 


491 


241081 


118370771 


22.1585 


7.8891 


437 


190969 


83453453 


20.9045 


7.5886 


492 


242064 


119095488 


22.1811 


7.8944 


438 


191844 


84027672 


20.9284 


7.5944 


492 


243049 


119823157 


22.2036 


7.8998 


439 


192721 


84604519 


20.9523 


7.6001 


494 


244036 


120553784 


22.2261 


7.9051 


440 


193600 


85184000 


20.9762 


7.6059 


495 


245025 


121287375 


22.2486 


7.9105 


441 


194481 


85766121 


21.0000 


7.6117 


496 


246016 


122023936 


22.2711 


7.9158 


442 


195364 


86350888 


21.0238 


7. 6174 


497 


247009 


122763473 


■J2.2!)85 


7.9211 


443 


196249 


86938307 


21.0476 


7.6232 


498 


248004 


123505992 


22.3159 


7.9264 


444 


197136 


87528384 


21.0713 


7.6289 


499 


249001 


124251499 


22 3383 


7.9317 


445 


198025 


88121120 


21.0950 


7.6346 


500 


250000 


125000000 


22.3607 


7.9370 


446 


198916 


88716536 


21.1187 


7.6403 


501 


251001 


125751501 


22.3830 


7.9423 


447 


199809 


89314623 


21.1424 


7.6460 


502 


252004 


126506008 


22.4054 


7.9476 


448 


200704 


89915392 


21.1660 


7.6517 


503 


253009 


127263527 


22.4277 


7.9528 


449 


201601 


90518849 


21.1896 


7.6574 


504 


254016 


128024064 


22.4499 


7.9581 


450 


202500 


91125000 


21.2132 


7.6631 


505 


255025 


128787625 


22.4722 


7.9634 


451 


203401 


91733851 


21.2368 


7.6688 


506 


256036 


129554216 


22.4944 


7.9686 


452 


204304 


92345408 


21.2603 


7.6744 


507 


257049 


130323843 


22.5167 


7.9739 


453 


205209 


92959677 


21.2838 


7.6800 


508 


258064 


131096512 


22.5389 


7.9791 


454 


206116 


93576664 


21.3073 


7.6857 


509 


259081 


131872229 


22.5610 


7.9843 


455 


207025 


94196375 


21.3307 


7.6914 


510 


260100 


132651000 


22.5832 


7.9896 


456 


207936 


94818816 


21.3542 


7.6970 


511 


261121 


133432831 


22.6053 


7.9948 


457 


208849 


95443993 


21.3776 


7.7026 


512 


262144 


134217728 


22.6274 


8.0000 


458 


209764 


96071912 


21 4009 


7.7082 


513 


263169 


135005697 


22.6195 


8.0052 


459 


210681 


96702579 


21.4243 


7.7138 


514 


264196 


135796744 


22.6716 


8.0104 


460 


211600 


97336000 


21.4476 


7.7194 


515 


265225 


136590875 


22.6936 


8.0156 


461 


212521 


97972181 


21.4709 


7.7250 


516 


266256 


137388096 


22.7156 


8.0208 


462 


213444 


98611128 


21.4942 


7.7306 


517 


267289 


138188413 


22.7376 


8.0260 


463 


214369 


99252847 


21.5174 


7.7362 


518 


26S324 


138991832 


22.7596 


8.0311 


464 


215296 


99897344 


21.5407 


7.7418 


519 


269361 


139798359 


22.7816 


8.0363 


465 


216225 


100544625 


21.5639 


7.7473 


520 


270400 


140608000 


22.8035 


8.0415 


466 


217156 


101194696 


21.5870 


7.7529 


521 


271441 


141420761 


22.8254 


8.0466 


467 


218089 


101847563 


21.6102 


7.7584 


522 


272484 


142236648 


22.8473 


8.0517 


468 


219024 


102503232 


21.6333 


7.7639 


523 


273529 143055667 


22.8692 


8.0569 


469 


219961 


103161709 


21.6564 


7.7695 


524 


274576 443877824 


22.8910 


8.0620 


470 


220900 


103823000 


21.6795 


7.7750 


525 


275625 ' 144703 125 


22.9129 


8.0671 


471 


221841 


104487111 


21.7025 


7.7805 


526 


276676 445531576 


22.9317 


8.0723 


472 


222784 


105154048 


21.7256 


8.7860 


527 


277729 1146363183 


22.1)0(15 


8.0774 


473 


223729 


105823817 


21.7486 


7.7915 


528 


278784 147 197952 


22.9783 


8.0825 


474 


224676 


106496424 


21.7715 


7.7970 


529 


279841 148035889 


23.0000 


8.0876 


475 


225625 


107171875 


21.7945 


7.8025 


530 


280900 148877000 


23.0217 


8.0927 


476 


226576 


107850176 


21.8174 


7.8079 


531 


281961 i 149721291 


23.0434 


8.0978 


477 


227529 


108531333 


21.S403 


7.8134 


532 


283024 150568768 


23.0651 


8.1028 


478 


228484 


109215352 


21.8632 


7.8188 


533 


284089 ! 151419437 


23.0868 


8.1079 


479 


229441 


109902239 


21.8861 


7.8243 


534 


285156 |152273304 


23.1084 


8.1130 


480 


230400 


110592000 


21.9089 


7.8297 


535 


286225 153130375 


23.130' 


8.1180 


481 


231361 


111284641 


21.9317 


7.8352 


536 


287296 1153990656 


23.1517 


8.1231 


482 


232324 


111980168 


21.9545 


7.8406 


537 


288369 1 1548541 53 


23.1733 


8.1281 


483 


233.'89 


112678587 


21.9773 


7.8460 


538 


289444 J155720872 


23.1948 


8.1332 


484 


234256 


113379904 


22.0000 


7.8514 


539 


290521 156590819 


23.2164 8.1382 



92 



MATHEMATICAL TABLES. 



No. 


Square. 


Cube. 


Sq. 
Root. 


Cube 
Root. 


No. 


Square. 


Cube. 


Sq. 
Root. 


Cube 
Root. 


540 


291600 


157464000 


23.2379 


8.1433 


595 


354025 


210644875 


24.3926 


8.4108 


541 


292681 


158340421 


>:, :;/)! 


8.1483 


596 


355216 


211708736 


24.4131 


8.4155 


542 


293764 


159220088 


23.2809 


8.1533 


597 


356409 


212776173 


24.4336 


8.4202 


543 


294849 


160103007 


23.3(124 


8.1583 


598 


357604 


213847192 


24.4540 


8.4249 


544 


295936 


160989184 


23.3238 


8.1633 


599 


358801 


214921799 


24.4745 


8.4296 


545 


297025 


161878625 


23.3452 


8.1683 


6C0 


360000 


216000000 


24.4949 


8 4343 


546 


298116 


162771336 


23.3666 


8.1733 


601 


361201 


217081801 


24.5153 


8.4390 


547 


299209 


163667323 


23.3880 


8.1783 


602 


362404 


218167208 


24.5357 


8.4437 


548 


300304 


164566592 


23.4094 


8.1833 


603 


363609 


219256227 


24.5561 


8.4484 


549 


301401 


165469149 


23.4307 


8.1882 


604 


364816 


220348864 


24.5764 


8.4530 


550 


302500 


166375000 


23.4521 


8.1932 


605 


366025 


221445125 


24.5967 


8.4577 


551 


30360! 


167284151 


23.4734 


8.1982 


606 


367236 


222545016 


21.6171 


8.4623 


552 


304704 


168196608 


23.4947 


8.2031 


607 


30S449 


223648543 


24.6374 


8.4670 


553 


305809 


169112377 


23.5160 


8.2081 


608 


369664 


224755712 


24.6577 


8.4716 


554 


306916 


170031464 


23.537.2 


8.2130 


609 


370881 


225866529 


24.6779 


8.4763 


555 


308025 


170953875 


23.5584 


8.2180 


610 


372100 


226981000 


24 6982 


8.4809 


556 


309136 


171879616 


23.571)7 


8.2229 


611 


373321 


228099131 


24.71S4 


8.4856 


557 


310249 


172808693 


23.6008 


8.2278 


612 


374544 


229220928 


24 7386 


8.4902 


558 


311364 


173741112 


23.6220 


8.2327 


613 


375769 


230346397 


24.7588 


8.4948 


559 


312481 


174676879 


23.6432 


8.2377 


614 


376996 


231475544 


24.7790 


8.4994 


560 


313600 


175616000 


23.6643 


8.2426 


615 


378225 


232608375 


24.7992 


8.5040 


561 


314721 


176558481 


!>:::.! 


8.2475 


616 


379456 


233744896 


24.8193 


8.5086 


562 


315844 


177504328 


23.7065 


8.2524 


617 


380689 


234885113 


24.8395 


8.5132 


563 


316969 


178453547 


23.7276 


8.2573 


618 


381924 


236029032 


24.8596 


8.5178 


564 


318096 


179406144 


23.7487 


8.2621 


619 


383161 


237176659 


24.8797 


8.5224 


565 


319225 


180362125 


23.7697 


8.2670 


620 


384400 


238328000 


24.8998 


8.5270 


566 


320356 


181321496 


23.7908 


8.2719 


621 


385641 


239483061 


24.9199 


8.5316 


567 


321489 


182284263 


23.8118 


8.2768 


622 


386884 


240641848 


24.9399 


8.5362 


568 


322624 


183250432 


23.8328 


8.2816 


623 


388129 


241804367 


24.9600 


8.5408 


569 


323761 


184220009 


23.8537 


8.2865 


624 


389376 


242970624 


24.9800 


8.5453 


570 


324900 


185193000 


23.8747 


8.2913 


625 


390625 


244140625 


25.0000 


8.5499 


571 


326041 


186169411 


23.8956 


8.2962 


626 


391876 


245314376 


25.0200 


8.5544 


572 


327184 


187149248 


23.9165 


8.3010 


627 


393129 


246491883 


25.0400 


8.5590 


573 


328329 


188132517 


23.9374 


8.3059 


628 


394384 


247673152 


25.0599 


8.5635 


574 


329476 


189119224 


23.9583 


8.3107 


629 


395641 


248858189 


25.0799 


8.5681 


575- 


330625 


190109375 


23.9792 


8.3155 


630 


396900 


250047000 


25.0998 


8.57'26 


576 


331776 


191102976 


24.0000 


8.3203 


631 


398161 


251239591 


25.1197 


8.5772 


577 


332929 


192100033 


24.0208 


8.3251 


632 


390424 


252435968 


25.1396 


8.5817 


.578 


334084 


193100552 


24.0416 


8.3300 


633 


400689 


253636137 


25.1595 


8.5862 


579 


335241 


194104539 


24.0624 


8.3348 


634 


401956 


254840104 


25.1794 


8.5907 


580 


336400 


195112000 


24.0832 


8.3396 


635 


403225 


256047875 


25.1992 


8.5952 


581 


337561 


196122941 


24.1039 


8.3443 


636 


404496 


257259456 


25.2190 


8.5997 


582 


338724 


197137368 


24.1247 


8.3491 


637 


405769 


258474853 




8.6043 


583 


339889 


198155287 


24.1454 


8.3539 


638 


407044 


259694072 


25.2587 


S.G0SS 


584 


341056 


199176704 


24.1661 


8.3587 


639 


408321 


260917119 


25.2784 


8.6132 


585 


342225 


200201625 


24.1868 


8.3634 


640 


409600 


262144000 


25.2982 


8.6177 


586 


343396 


201230056 


24.2074 


8.3682 


641 


410881 


263374721 


25.3180 


8.6222 


587 


344569 


202262003 


24.2281 


8.3730 


642 


412164 


264609288 


25.3377 


8.6267 


588 


345744 


203297472 


24.2487 


8.3777 


643 


413449 


265847707 


25.3574 


8.6312 


589 


346921 


204336469 


24.2693 


8.3825 


644 


414736 


267089984 


25.3772 


8.6357 


590 


348100 


205379000 


24.2899 


8.3872 


645 


416025 


268336125 


25.3969 


8.6401 


591 


349281 


206425071 


24 3105 


8.3919 


646 


417316 


269586136 


25.4165 


8 6446 


592 


350464 


207474688 


24.3311 


8.3967 


647 


418609 


270840023 


25.4362 


8.6490 


593 


351649 


208527857 


24.3516 


8.4014 


648 


419904 


27'2097792 


25.4558 


8.6535 


594 


352836 


209584584 


24.3721 


8.4061 


649 


421201 


273359449 


25.4755 


8.6579 



SQUARES, CUBES, SQUARE AKD CUBE ROOTS. 93 



-No. 


Square. 


Cube. 


Sq. 

Root. 


Cube 
Root. 


No. 


Square. 


Cube. 


Sq. 
Root. 


Cube 
Root. 


650 


422500 


274625000 


25.4951 


8.6624 


705 


497025 


350402625 


26.5518 


8.9001 


651 


423801 


275894451 


25.5147 


8.6668 


706 


498436 


351895816 


26.5707 


8.9043 


652 


425104 


277167808 


25. 53 1 3 


8.6713 


707 


499849 


353393243 


26.5895 


8.9085 


653 


426409 


278445077 


25.5539 


8.6757 


708 


501264 


154894912 


26.6083 


8.9127 


654 


427716 


279726264 


25.5734 


8.6801 


709 


502681 


356400829 


26.6271 


8.9169 


655 


429025 


281011375 


25.5930 


8.6845 


710 


504100 


357911000 


26.6458 


8.9211 


656 


430336 


282300416 


25.6125 


8.6890 


711 


505521 


359425431 


26.6646 


8.9253 


657 


431649 


283593393 


25.6320 


8.6934 


712 


506944 


360944128 


26.6833 


8.9295 


658 


432964 


284890312 


25.6515 


8 6978 


713 


508369 


362467097 


26.7021 


8.9337 


659 


434281 


286191179 


25.6710 


8.7022 


714 


509796 


363994344 


26.7208 


8.9378 


660 


435600 


287496000 


25.6905 


8.7066 


715 


511225 


365525875 


26.7395 


8.9420 


661 


436921 


288804781 


25.71)911 


8.7110 


716 


512656 


367061696 


-.6.7582 


8.9462 


662 


438244 


290117528 


25 7294 


8.7154 


717 


514089 


368601813 


26.7769 


8.9503 


603 


439569 


291434247 


25.74S8 


8.7198 


718 


515524 


370146232 


•.'6.7955 


8.9545 


664 


440896 


292754944 


25.7682 


8.7241 


719 


516961 


371694959 


26.8142 


8.9587 


665 


442225 


294079625 


25.7876 


8.7285 


720 


518400 


373248000 


26.8328 


8.9628 


666 


443556 


295408296 


25.8070 


8.7329 


721 


519841 


374805361 


26.S514 


8.9670 


667 


444889 


296740963 


25.S263 


8.7373 


722 


521284 


376367048 


26.8701 


8.9711 


668 


4462-24 


298077632 


25.8457 


8.7416 


723 


522729 


377933067 


26.8887 


8.9752 


669 


447561 


299418309 


25.8650 


8.7460 


724 


524176 


379503424 


26.9072 


8.9794 


670 


448900 


300763000 


25.8844 


8.7503 


725 


525625 


381078125 


26.9258 


8.9835 


671 


450241 


302111711 


25.9037 


8.7547 


726 


527076 


382657176 


26.9444 


8.9876 


672 


451584 


303464448 


25.9230 


8.7590 


727 


528529 


384240583 


26.9629 


8.9918 


673 


452929 


304821217 


25.9122 


8.7634 


728 


5299S4 


385828352 


26.9815 


8.9959 


674 


454276 


306182024 


25.9615 


8.7677 


729 


531441 


387420489 


27.0000 


9.0000 


675 


455625 


307546875 


25.9808 


8.7721 


730 


532900 


389017000 


27 0185 


9 0041 


676 


456976 


308915776 


26.0000 


8.7764 


731 


534361 


390617891 


27.0370 


9.0082 


677 


458329 


310288733 


26.0192 


8.7807 


732 


535824 


392223168 


27.0555 


9.0123 


678 


459684 


311665752 


26.0384 


8.7850 


733 


537289 


393832837 


27.0740 


9.0164 


679 


461041 


313046839 


26.0576 


8.7893 


734 


538756 


395446904 


27.0924 


9.0205 


6S0 


462400 


314432000 


26.0768 


8.7937 


735 


540225 


397065375 


27.1109 


9.0246 


681 


463761 


315821241 


. O'.itii; 


8.7980 


736 


541696 


398688256 


27.1293 


9.0287 


682 


465124 


317214568 


26.1151 


8.8023 


737 


543169 


400315553 


27.1477 


9.0328 


683 


466489 


318611987 


26.1343 


8.8066 


738 


544644 


401947272 


27.1662 


9.0369 


684 


467856 


320013504 


26.1534 


8.8109 


739 


546121 


403583419 


27.1846 


9.0410 


685 


469225 


321419125 


26.1725 


8.8152 


740 


547600 


405221000 


27.2029 


9.0450 


686 


470596 


322828856 


26.1916 


8.8194 


741 


549801 


406869021 


27.221o 


9.0491 


687 


471969 


324242703 


26.2107 


8.8237 


742 


550564 


408518488 


27.2397 


9.0532 


688 


473344 


325660672 


26.22!H 


8.8280 


743 


552049 


410172407 


27.2581 


9.0572 


689 


474721 


327082769 


26.2488 


8.8323 


744 


553536 


411830784 


27.2764 


9.0613 


600 


476100 


328509000 


26.2679 


8.8366 


745 


555025 


413493625 


27.2947 


9.0654 


691 


477481 


329939371 


26.2869 


8.8408 


746 


556516 


415160936 


27.3130 


9.0694 


692 


478864 


331373888 


26.3059 


8.8451 


747 


558009 


416832723 


27.3313 


9.0735 


693 


480249 


332812557 


.; :;• ' 


8.8493 


748 


559504 


418508992 


27.3496 


9.0775 


694 


481636 


334255384 


26.3439 


8.8536 


749 


561001 


420189749 


27.3679 


9.0816 


695 


483025 


335702375 


26.3629 


8.8578 


750 


562500 


421875000 


27.3861 


9.0856 


696 


484416 


337153536 


26.3818 


8.8621 


751 


564001 


423564751 


27.4044 


9.0896 


697 


485809 


338608873 


26.4008 


8.8663 


752 


565504 


425259008 


27.4226 


9.0937 


698 


487204 


340068392 


23.4197 


8.8706 


753 


567009 


426957777 


27.4408 


9.0977 


699 


488601 


341532099 


26.4386 


8.8748 


754 


568516 


428661064 


27.4591 


9.1017 


700 


490000 


343000000 


26.4575 


8.8790 


755 


570025 


430368875 


27.4773 


9.1057 


701 


491401 


344472101 


26.4764 


8.8833 756 


571536 


432081216 


27.4955 


9.1098 


702 


492804 


345948408 


26.4953 


8. 8875 1 757 


573049 


433798093 


27.5136 


9.1138 


r m 


494209 


347428927 


26.5141 


8.8917 1 758 


574564 


435519512 


27.5318 


9.1178 


:m 


4956 10 


318:)lo664 26.5330 


8.89591759 


576081 


437245479 


27.5500 


9.1218 













94 



MATHEMATICAL TABLES. 



No. 


Square. 


760 


577600 


761 


579121 


702 


580644 


763 


582169 


764 


583696 


765 


585225 


766 


586756 


767 


588289 


7'6S 


589824 


769 


591361 


770 


592900 


771 


594441 


77-2 


595984 


778 


597529 


774 


599076 


775 


600625 


776 


602176 


777 


603729 


778 


605284 


779 


606841 


780 


608400 


781 


609961 


782 


611524 


788 


613089 


784 


614656 


785 


616225 


786 


617796 


787 


619369 


788 


620944 


789 


622521 


790 


624100 


791 


625681 


792 


627264 


798 


628849 


794 


630436 


795 


632025 


796 


633616 


797 


635209 


798 


636804 


799 


638401 


800 


640000 


SOI 


641601 


802 


643204 


808 


644809 


804 


646416 


805 


648025 


80(1 


649636 


807 


651249 


SOS 


652S64 


809 


654481 


810 


656100 


811 


657721 


812 


659344 


81.-, 


660969 


814 


662596 



438976000 
440711081 
442450728 
444194947 
445943744 

447697125 

449455096 

451217663 

452! 

454756609 

456533000 
458314011 
460099648 
461889917 
463684824 

465484375 2' 
4672S8576 2' 
469097433 2' 
470910952 2' 
472729139 

474552000 
476379541 
478211768 
480048687 
481890304 



Sq. 
Root. 



48373662: 
485587656 
487443403 
489303872 
491169069 

493039000 
494913671 
496793088 
498677257 
500566184 

502459875 
504358336 
506261573 
508169592 
510082399 

512000000 
513922401 
515849608 
517781627 
519718464 

521660125 
523606616 
525557943 
527514112 
529475129 28.4429 

53144100o'28.4605 
533411731 28.4781 
5358S7828 28.4956 
537867797 28.5132 
539353144 28.5807 



5681 
5862 
6043 
6225 
.6405 

.6586 
.676' 



17.7489 
7669 

78 M) 
8029 
8209 



0285 
9464 

'7.96 18 



Cube 
Root. 



28.0000 

28.0179' 
28.0357 
28.0585 
28.0713 
28. 0891 j 

28.1069, 

28.1247 
28.1425 
28.1603 
28.1780 

2S.1957 
28.2135 

28.2812 
28.2489 



28.2843 
28.3019 



3373 

28.3549 



28.407 
28.4253 



9.1258 
9.1298 
9.1338 



9.1537 
9.1577 
9.1617 

9.1657 

9.1696 
9.1736 
9.1775 
9.1815 

9.1855 
9.1894 
9.1933 
9.1973 
9.2012 

9.2052 

9.2091 
9.2130 

9^2209 

9.2248 
9.2287 
9 2326 
9.2365 
9.2404 

9.2443 

9.2482 
9.2521 
9.2560 



9.267 
9.2716 
9.2754 
9.2793 

9.2 

9.2870 



9.3025 
9.3063 
9.3102 
9.3140 
9.3179 

9.3217 

9.3255 
9.3294 
9.3332 

9.8370 



No. Square, 



815 664225 

816 665856 

817 i 667489 

818 669121 
819' 670761 

820 ' 672400 
821 1 674041 
822 : 675684 



678976 

680625 
682276 
683929 

685584 
687241 



700569 
702244 
703921 

705600 
707281 
708964 
710649 
712336 

714025 
715716 
717409 
719104 
720801 

722500 
724201 
725904 

727609 



731025 
732736 

781119 
736164 



739600 
741321 



751689 
753424 
755161 



Sq. Cube 
Root. Root. 



541343375 
543338496 
545338513 
547343432 
549353259 

551368000 
553387661 
555412248 
557441767 
559476224 

561515625 
563559976 
565609283 
567663552 
569722789 

571787000 
573856191 
575930368 
578009537 
580093704 

582182875 

584277056 

586376253 

5884804' 

590589719' 

592704000 
594823321 
5969476S8 
599077107 
601211584 



28.5482 
28.5657 
28.5832 
28.6007 
28.6182 

28.6356 
28.6531 
28.6705 



28.7402 
28.7576 
28.7750 
28.79$ 

28.8097 
28.8271 
28.8444 
28.8617 
28.8791 



28.9137 
28.9310 
28.9482 
28.9655 



29.0000 
29.0175 
29.0345 
29.0517 



603351125 29. ( 
605495736 29.0861 
607645423 29.1O 
6098001 92 j 29. 1204 
611960049 29. 1376 



614125000 
616295051 
618470208 
620650477 
622835864 

625026375 
627222016 
629422793 
631628712 



.1548 

.1719 

29.1890 

29.2062 

29.2233 

29.2404 

29.2575 

2746 

29.2916 



636056000 
638277381 
640503928 
642735647 
644972544 

647214625 
649461896 
651714363 
653972032 
656234909 



29.3258 
29 342* 
29.359* 
29.3769 
29.3939 

4109 

29.427' 

.4449 

29.4618 

29.47* 



SQUARES, CUBES, SQUARE AND CUBE ROOTS. 



95 



No. 


Square. 


Cube. 


Sq. 
Root. 


Cube 
Root. 


No. 


Square. 


Cube. 


Sq. 
Root. 


Cube 
Root. 


S70 


756900 


658503000 


29.4958 


9.5464 


925 


855625 


791453125 


30.4138 


9.7435 


871 


758641 


660776311 


29.5127 


9.5501 


926 


857 170 


794022776 730.4302 


9.7470 


87:2 


760384 


663054848 


29.5296 


9.5537 


927 


859329 


796597983 30.4467 


9.7505 


878 


762129 


665338617 


29.5466 


9.5574 


928 


861184 


7'99 178752 30.4631 


9 7540 


874 


763876 


667627624 


29.5635 


9.5610 


929 


863041 


801765089 30.4795 


9.7575 


875 


765625 


669921875 


29.5804 


9.5647 


930 


864900 


804357000 30.4959 


9.7610 


876 


767376 


672221376 


29.5973 


9.5683 


931 


866761 


806954491 [30. 5123 


9.7645 


877 


769129 


674526133 


29.6142 


9.5719 


932 


868624 


809557568,30.5287 


9.76S0 


878 


7708S4 


676836152 


29.6311 


9.5756 


933 


870489 


812166237 30.5450 


9.7715 


879 


772641 


679151439 


29.6479 


9.5792 


934 


872356 


814780504 30.5614 


9.7750 


880 


774400 


681472000 


29.6648 


9.5828 


935 


874225 


817400375 30.5778 


9.7785 


881 


776161 


683797841 


29.6816 


9.5865 


936 


876096 


820025856 30.5941 


9.7819 


882 


777924 


686128968 


29.6985 


9.5901 


937 


877969 


822656953 30.6105 


9.7S54 


88.3 


779689 


688465387 


29.7153 


9.5937 


938 


879844 


825293672:30.6268 


9.7889 


884 


781456 


690807104 


29.7321 


9.5973 


939 


881721 


827936019 30.6431 


9.7924 


885 


783225 


693154125 


29.7489 


9.6010 


940 


883600 


830584000 


30.6594 


9.7959 


886 


784996 


695506456 


29.7658 


9.6046 


941 J 885481 


833237621 


30.6757 


9.7993 


SS7 


786769 


697864103 


29.7825 


9.6082 


942; 887364 


835896888 


30.6920 


9.8028 


888 


788544 


700227072 


29.7993 


9.6118 


943 i 889249 


838561807 


30.7083 


9.8063 


889 


790321 


702595369 


29.8161 


9.6154 


944 891136 


841232384 


30.7246 


9.8097 


890 


792100 


704969000 


29.8329 


9.6190 


945' 893025 


84390S625 


30.7409 


9.8132 


891 


■93881 


707347971 


29.8496 


9.6226 


946! 894916 


846590536 


30.7571 


9.8167 


89-2 


795664 


709732288 


29.8664 


9.6262 


947 896809 


849278123 


30.7734 


9.8201 


8! 13 


797449 


712121957 


29.SS31 


9.6298 


948 898704 


851971892 


30.7S1MI 


9.8236 


894 


799236 


714516984 


29.8998 


9.6334 


949 900601 


854670349 


30.S058 


9.8270 


895 


801025 


716917375 


29.9166 


9.6370 


950 902500 


857375000 


30.8221 


9.8305 


890 


802816 


719323136 


29.9333 


9.6406 


951 904401 


860085351 j 30. 8383 


9.8339 


897 


804609 


721734273 


29.9500 


9.6442 


952 906304 


862801408 30.8545 


9.8374 


898 


806404 


724150792 


29 . 9666 


9.6477 


953 908209 


865523177 


30.8707 


9.8408 


899 


808201 


726572699 


29.9833 


9 6513 


954 910116 


868250664 


30.8869 


9.8443 


900 


810000 


729000000 


30 0000 


9.6549 


955 


912025 


870983875 


30.9031 


9.8477 


901 


811801 


731432701 


30.0167 


9.6585 


956 


913936 


87'3722816 


30.9192 


9.8511 


902 


813604 


733870S08 


30.0333 


9.6620 


957 


915849 


876467493 30.9354 


9.8546 


903 


815409 


736314327 


30.0500 


9.6656 


958 


917764 


879217912 


30.9516 


9.8580 


901 


817216 


738763264 


30.0066 


9.6692 


959 


919681 


881974079 


30.9677 


9.8614 


905 


819025 


741217625 


30.0832 


9.6727 


960 


921600 


884736000 


30.9839 


9.8648 


900 


820836 


743677416 


30.0998 


9.8763 


961 


923521 


887503681 


31.0000 


9.8683 


907 


822649 


746142643 30.1164 


9.6799 


962 


925444 


890277128 


31.0161 


9.8717 


908 


824464 


748613312 


30.1330 


9.6S34 


963 


927369 


893056347 


31.0322 


9.8751 


909 


826281 


751089429 


30.1496 


9.6870 


964 


929296 


895841344 


31.0483 


9.8785 


910 


828100 


753571000 


30.1662 


9.6905 


965 


931225 


898632125 


31.0644 


9.8819 


911 


829921 


756058031 


30.1828 


9.6941 


966 


933156 


901428696 


31.0805 


9.8854 


912 


831744 


758550528 


30.1993 


9.6976 


967 


935089 


904231063 


31.0966 


9.8888 


913 


833569 


761048497 


30.2159 


9.7012 


968 


937024 


907039232 


31.1127 


9.8922 


914 


835396 


763551944 


30.2324 


9.7'047 


969 


938961 


909853209 


31.1288 


9.8956 


915 


837225 


766060875 


30.2490 


9.7082 


970 


940900 


912673000 


31.1448 


9.8990 


916 


839056 


768575296 


30.2655 


9.7118 


971 


942841 


915498611 


31.1609 


9.9024 


91? 


840889 


771095213 


30.2820 


9.7153 


972 


9447S4 


918330048 


31.1769 


9.9058 


918 


842724 


773620632 


30.2985 


9.7188 


973 


946729 


921167317 


31.1929 


9.9092 


919 


844561 


776151559 


30.3150 


9.7224 


974 


948676 


924010424 


31.2090 


9.9126 


920 


846400 


778688000 


30.3315 


9.7259 


975 


950625 


926859375 


31.2250 


9.9160 


921 


848241 


781229961 


30,3480 


9.7294 


976 


952576 


929714176 


31.2410 


9.9194 


922 


850084 


783777448 




9.7329 


977 


954529 


932574833 


31.2570 


9 9227 


923 


851929 


786330467 


30.3809 


9.7364| 


978 


956484 


935141352 


31.2730 


9.9261 


924 


853776 


788889024 


30.3974 


9.7400! 


979 


958441 


938313739 


31.2890 


9.9295 



96 



MATHEMATICAL TABLES. 



No. 


Square. 


Cube. 


Sq. 
Root. 


Cube. 
Root. 


No. 


Square. 


Cube. 


Sq. 
Root. 


Cube 
Root. 


980 


960400 


941192000 


31.3050 


9.9329 


1035 


1071225 


1108717875 


32.1714 


10.1153 


981 


962361 


944076141 


31.32(i! 


9.9363 


1036 


1073296 


1111934656 


32.1870 


10.1186 


982 


964324 


946966168 


31.3369 


9.9396 


1037 


1075369 


1115157653 32.202.') 


10.1218 


983 


966289 


949862087 


31.3528 


9.9430 


1038 


1077444 


1118386872 32.2180 


10.1251 


984 


968256 


952763904 


31.3688 


9.9464 


1039 


1079521 


1121622319 


32.2335 


10.1283 


985 


970225 


955671625 


31.3847 


9.9497 


1040 


1081600 


1124864000 


32.2490 


10.1316 


980 


972196 


958585256 


31.4006 


9.9531 


1041 


1083681 


1128111921 




10.1348 


1M 


974169 


961504803 


31.4166 


9.9565 


1042 


1085764 


1131366088 


32.2800 


10.1381 


988 


976144 


964430272 


31 . 4325 


9.9598 


1043 


1087S49 


1134626507 


32.2955 


10.1413 


989 


978121 


967361669 


31.4484 


9.9632 


1044 


1089936 


1137893184 


32.3110 


10.1446 


990 


980100 


970299000 


31.4643 


9.9666 


1045 


1092025 


1141166125 


32.3265 


10.1478 


991 


982081 


973242271 


31.4802 


9.9699 


1046 


1094116 


1144445336 


32.3419 


10.1510. 


99',' 


984064 


976191488 


31.4960 


9.9733 


1047 


1096209 


1147730823 


32.3574 


10.1543 


993 


986049 


979146657 


31.5119 


9.9766 


104S 


1098304 


1151022592 


32.3728 


10.1575 


994 


988036 


982107784 


31.5278 


9.9800 


1049 


1100401 


1154320649 


32.3883 


10.1607 


995 


990025 


985074875 


31.5436 


9.9833 


1050 


1102500 


1157625000 


32.4037 


10.1640 


!)9(i 


992016 


988047936 


31.5595 


9.9866 


1051 


1104601 


1160935651 


32.4191 


10.1672 


997 


994009 


991026973 


31.5753 


9.9900 


1052 


1106704 


1164252608 


32.4345 


10.1704 


998 


996004 


994011992 


31.5911 


9 9933 


1053 


1108809 


1167575877 


32.4500 


10.1736 


999 


998001 


997002999 


31.6070 


9.9967 


1054 


1110916 


1170905464 


32.4654 


10.1769 


1000 


1000000 


1000000000 


31.6228 


10.0000 


1055 


1113025 


1174241375 


32.4808 


10.1801 


1001 


1002001 


1003003001 


31.6386 


10.0033 


1056 


1115136 


1177583616 


32.4962 


10.1833 




1004004 


1006012008 


31.6544 


10.0067 


1057 


1117249 


1180932193 


32.5115 


10.1865 


1003 


1006009 


1009027027 


31.6702 


10.0100 


1058 


1119364 


1184287112 


32.5269 


10.1897 


1004 


1008016 


1012048064 


31.6860 


10.0133 


1059 


1121481 


1187648379 


32.5423 


10.1929 


1005 


1010025 


1015075125 


31.7017 


10.0166 


1060 


1123600 


1191016000 


32.5576 


10.1961 


H 


1012036 


1018108216 


31.7175 


10.0200 


1061 


1125721 


1194389981 


32.5730 


10.1993 


li 


1014049 


1021147343 


31.7333 


10 0233 


1062 


1127844 


1197770328 


32.5883 


10.2025 


1008 


1016064 


1024192512 


31.7490 


10.0266 


1063 


1129969 


1201157047 


32.6036 


10.2057 


1009 


1018081 


1027243729 


31.7648 


10.0299 


1064 


1132096 


1204550144 


32.6190 


10.2089 


1010 


1020100 


1030301000 


31.7805 


10.0332 


1065 


1134225 


1207949625 


32.6343 


10.2121 


1011 


1022121 


1033364331 


31.7962 


10.0365 


1066 


1136356 


1211355496 


32.6497 


10.2153 


1012 


1024144 


1036433728 


31.8119 


10.0398 


1067 


1138489 


1214767763 


32.6650 


10.2185 


1013 


1026169 


1039509197 


31.8277 


10.0431 


1068 


1140624 


1218186432 


32.6803 


10.2217 


1014 


1028196 


1042590744 


31.8434 


10.0465 


1069 


1142761 


1221611509 


32.6956 


10.2249 


1015 


1030225 


1045678375 


31.8591 


10.0498 


1070 


114490Q 
1147041 


1225043000 


32.7109 


10.2281 


1016 


1032251; 


1048772096 


31.8748 


10.0531 


1071 


1228480911 


32.7261 


10.2313 


1017 


1034289 


1051871913 


31.8904 


10.0563 


1072 


1149184 


1231925248 


32.7414 


10.2345 


1018 


1036324 


1054977832 


31.9061 


10.0596 


1073 


1151329 


1235376017 


32.7567 


10.2376 


1019 


1038361 


1058089859 


31.9218 


10.0629 


1074 


1153476 


1238833224 


32.7719 


10.2408 


1020 


1040400 


1061208000 


31.9374 


10.0662 


1075 


1155625 


1242296875 


32.7872 


10.2440 


1021 


1042441 


1064332261 


31.9531 


10.0695 


1076 


1157776 


1245766976 


32.8024 


10.2472 




1044484 


1067462648 


31.9687 


10.0728 


1077 


1159929 


1249243533 


32.8177 


10.2503 




1046529 


1070599167 


31.9844 


10.0761 


1078 


1162084 


1252726552 


32.8329 


10.2535 


1024 


1048576 


1073741824 


32.0000 


10.0794 


1079 


1164241 


1256216039 


32.8481 


10.2567 


1025 


1050625 


1076890625 


32.0156 


10.0826 


1080 


1166400 


1259712000 


32.8634 


10.2599 




1052676 


1080045576 


32.0312 


10.0859 


1081 


1168561 


1263214441 


32.8786 


10.2630 




1054729 


1083206683 


32.0468 


10.0892 


1082 


1170724 


1266723368 


32.8938 


10.2662 




1056784 


1086373952 


32.0624 


10.0925 


10S3 


1172889 


1270238787 


32.9090 


10.2693 


1029 


1058841 


1089547389 


32.0780 


10.0957 


1084 


1175056 


1273760704 


32.9242 


10.2725 


1030 


1060900 


1092727000 


32.0936 


10.0990 


1085 


1177225 


1277289125 


32.9393 


10.2757 


103! 


1062961 


1095912791 


32.1092 


10.1023 


1086 


1179396 


1280824056 




10.2788 


1032 


1065024 


1099104768 


32.1248 


10.1055 


1087 


1181569 


1284365503 


32.9697 


10.2820 


1033 


1067089 


1102302937 


32.1403 


10.1088 


1088 


1183744 


1287913472 


32.9848 10.2851 


1034 


1069156 


1105507304 


32.1559 


10.1121 


1089 


1185921 


1291467969 


33.00001 


10.2883 



SQUABES, CUBES, SQUABE AKD CUBE BOOTS. 



97 



No. 


Square. 


Cube. 


Sq. 

Boot. 


Cube 
Boot. 


No. 


Square. 


Cube. 


Sq. 
Boot. 


Cube 
Boot. 


1090 


1188100 


1295029000 


33.0151 


10.2914 


1145 


1311025 


1501123625 


33.8378 


10.4617 


11)91 


1190281 


129S596571 


33.0303 


10.2946 


1146 


1313316 


1505060136 




10.4647 


• 


1192464 


1302170688 


33.0454 


10.2977 


1147 


1315609 


1509003523 


33.8674 


10.4678 


1093 


1194649 


1305751357 


33.0606 


10.3009 


1148 


1317904 


1512953792 


33.8821 


10.4708 


1094 


1196836 


1309338584 


33.0757 


10.3040 


1149 


1320201 


1516910949 


33.8969 


10.4739 


1095 


1199025 


1312932375 


33.0908 


10.3071 


1150 


1322500 


1520875000 


33.9116 


10.4769 


1096 


1201216 


1316532736 


33.1059 


10.3103 


1151 


1324801 


1524845951 


33.9264 


10.4799 


1097 


1203409 


1320139673 


33.1210 


10.3134 


1152 


1327104 


1528823808 


33.9411 


10.4830 


1098 


1205604 


1323753192 


33.1361 


10.3165 


1153 


1329409 


1532808577 


33 . 9559 


10.4860 


1099 


1207801 


1327373299 


33.1512 


10.3197 


1154 


1331716 


1536800264 


33.9706 


10.4890 


1100 


1210000 


1331000000 


33.1662 


10.3228 


1155 


1334025 


1540798875 


33.9853 


10.4921 


1101 


1212201 


1334633301 


33.1813 


10.3259 


1156 


1336336 


1544804416 


34.0000 


10.4951 


110:2 


1214404 


1338273208 


33.1964 


10.3290 


1157 


1338649 


1548816893 


34.0147 


10.4981 


1103 


1216609 


1341919727 


33.2114 


10.3322 


1158 


1340964 


1552836312 


34.0294 


10.5011 


1104 


1218816 


1345572864 


33.2264 


10.3353 


1159 


1343281 


1556862879 


34.0441 


10.5042 


1105 


1221025 


1349232625 


33.2415 


10.3384 


1160 


1345600 


1560896000 


34.0588 


10.5072 


1106 


1223236 


1352899016 


33.2566- 


10.3415 


1161 


1347921 


1564936281 


34.0735 


10.5102 


1107 


1225449 


1356572043 


33.2716 


10.3447 


1162 


1350244 


1568983528 


34.0881 


10.5132 


11 OS 


1227664 


1360251712 


33.2866 


10.3478 


1163 


1352569 


1573037747 


34.1028 


10.5162 


1109 


1229881 


1363938029 


33.3017 


10.3509 


1164 


1354896 


1577098944 


34.1174 


10.5192 


1110 


1232100 


1367631000 


33.3167 


10.3540 


1165 


1357225 


1581167125 


34.1321 


10.5223 


1111 


1234321 


1371330631 




10.3571 


1166 


1359556 


1585242296 


34.1467 


10.5253 


1112 


1236544 


1375036928 33.3407 


10.3602 


1167 


1361889 


1589324463 


34.1614 


10.5283 


1113 


1238769 


1378749897133.3617 


10.3633 


1168 


1364224 


1593413632 


34.1760 


10.5313 


in* 


1240996 


1382469544 33.3766 


10.3664 


1169 


1366561 


1597509809 


34.1906 


10.5343 


1115 


1243225 


1386195875 33.3916 


10 3695 


1170 


1368900 


1601613000 


34.2053 


10.5373 


1116 


1245456 


1389928896 33.4066 


10.3726 


1171 


1371241 


1605723211 


34.2199 


10.5403 


1117 


1247689 


13931568613 33.4215 


10.3757 


1172 


1373584 


1609840448 


34.2345 


10.5433 


HIS 


1249924 


1397415032 33.4365 


10.3788 


1173 


1375929 


1613964717 


34.2491 


10.5463 


1119 


1252161 


1401168159 33.4515 


10.3819 


1174 


1378276 


1618096024 


34.2637 


10.5493 


1120 


1254400 


1404928000 33.4664 


10.3850 


1175 


1380625 


1622234375 


34.2783 


10.5523 


1121 


1256641 


1403694561 33.4813 


10.3881 


1176 


1382976 


1626379776 


34.2929 


10.5553 


1 122 


1258884 


1412467848 33.4963 


10.3912 


1177 


1385329 


1630532233 


34.3074 


10.5583 


1123 


1261129 


1416247867 33.5112 


10.3943 


1178 


1387684 


1634691752 


34.3220 


10.5612 


1124 


1263376 


1420034624 33.5261 


10.3973 


1179 


1390041 


163S858339 


34.3366 


10.5642 


1125 


1265625 


1423828125 33.5410 


10.4004 


1180 


1392400 


1643032000 


34.3511 


10.5672 


1 126 


1267876 


1427628376 33.5559 


10.4035 


1181 


1394761 


1647212741 


34.3657 


10.5702 


1127 


1270129 


1431435383 33.5708 


10.4066 


1182 


1397124 


1651400568 


34.3802 


10.5732 




12723S4 


1435249152 33.5857 


10.4097 


1183 


1399489 


1655595487 


34.3948 


10 5762 


1129 


1274641 


1439069689 33.6006 


10.4127 


1184 


1401856 


1659797504 


34.4093 


10.5791 


1130 


1276900 


1442897000 33.6155 


10.4158 


1185 


1404225 


1664006625 


34.4238 


10.5821 


1131 


1279161 


1446731091 




10.4189 


1186 


1406596 


1668222856 


34.4384 


10.5851 


1132 


1281424 


1450571968 


33.6452 


10.4219 


1187 


1408969 


1672446203 


34.4529 


10.5881 




1283689 


1454419637 


33.6601 


10.4250 


1188 


1411344 


1676676672 


34.4674 


10.5910 


1134 


1285956 


1458274104 


33.6749 


10.4281 


1189 


1413721 


1680914269 


34.4819 


10.5940 


1135 


1288225 


1462135375 


33.689S 


10.4311 


1190 


1416100 


1685159000 


34.4964 


10.5970 


1136 


1290496 


1466003456 


33.7046 


10.4342 


1191 


1418481 


1689410871 


34.5109 


10.6000 


1137 


1292769 


1469878353 


33.7174 


10.4373 


1192 


1420864 


1693669888 


34.5254 


10.6029 


1138 


1295044 


1473760072 


33.7342 


10.4404 


1193 


1423249 


1697936057 


34.539S 


10.6059 


1139 


1297321 


1477648619 


33.7491 


10.4434 


1194 


1425636 


1702209384 


34.5543 


10.6088 


1140 


1299600 


1481544000 


33.7639 


10.4464 


1195 


1428025 


1706489875 


34.5688 


10.6118 


1141 


1301881 


1485446221 


33.7787 


10.4495 


1196 


1430416 


1710777536 


34.5832 


10.6148 


114.2 


1304164 


1489355288 


33.7935 


10.4525 


1197 


1432809 


1715072373 


34.5077 


10 6177 


1143 


1306449 


1493271207 


33. 8083! 10. 4556 


1198 


1435204 


1719374392 


34.6121 


10.6207 


1144 


1308736 


1497193984 


33.8231 


10.4586 


1199 


,437601 


1723683599 


34.6266 


10.6236 



98 



MATHEMATICAL TABLES. 



No. 


Square. 


Cube. 


Sq. 
Root, 


Cube 
Root. 


1200 


1440000 


1 
1728000000 34.6410 


10.6266 


1201 


1442401 


1732323601 34.6554 


10.6295 


1202 


1444804 


1736654408 34.6699 


10.6325 


1203 


1447209 


1740992427 34.6843 


10.6354 


1204 


1449616 


1745337664 34.6987 


10.6384 


1205 


1452025 


1749690125 


34.7131 


10.6413 


120(1 


1454436 


1754049816 


34.7275 


10.6443 


1207 


1456849 


1758416743 34.7419 


10.6472 


1208 


1459264 


1762790912 


34.7563 


10.6501 


1209 


1461681 


1767172329 


34.7707 


10.6530 


1210 


1464100 


1771561000 


34.7851 


10.6560 


1211 


1466521 


1775956931 


34.7994 


10.8590 


1212 


1468944 


1780360128 


34.8138 


10.6619 


1213 


1471369 


1784770597 


34.8281 


10.6648 


1214 


1473796 


1789188344 


34.8425 


10.6678 


1215 


1476225 


1793613375 


34.8569 


10.6707 


1216 


1478656 


1798045696 


34.8712 


10.6736 


1217 


1481089 


1802485313 


34.8855 


10.6765 


1218 


1483524 


1806932232 


34.8999 


10.6795 


1219 


1485961 


1811386459 


34.9142 


10.6824 


1220 


1488400 


1815848000 


34.9285 


10.6853 


1221 


1490841 


1820316861 


34.9428 


10.6882 


1222 


1493284 


1824793048 


31.9571 


10.6911 


1223 


1495729 


1829276567 


34.9714 


10.6940 


1224 


1498176 


1833767424 


34.9857 


10.6970 


1225 


1500625 


1838265625 


35.0000 


10.6999 


1226 


1503076 


1842771176 


35.0143 


10.7028 


1227 


1505529 


1847284083 


35.0286 


10.7057 


1228 


1507984 


1851804352 


35.0428 


10.7086 


1229 


1510441 


1856331989 


35.0571 


10.7115 


1230 


1512900 


1860867000 


35.0714 


10.7144 


1231 


1515361 


1S65409391 


35.0856 


10.7173 


1232 


1517824 


1869959168 


35.0999 


10.7202 


1233 


1520289 


1874516337 


35.1141 


10.7231 


1234 


1522756 


187 9080904 


35.1283 


10.7260 


1235 


1525225 


1883652875 


35.1426 


10.7289 


1236 


1527696 


1888232256 


35.1568 


10.7318 


1237 


1530169 


1892819053 


35.1710 


10.7347 


1238 


1532644 


1897413272 


35.1852 


10.7376 


1239 


1535121 


1902014919 


35.1994 


10.7405 


1240 


1537600 


1906624000 


35.2136 


10.7434 


1241 


1540081 


1911240521 


35.2278 


10.7463 


1242 


1542564 


1915864488 


35.2-120 


10.7491 


1243 


1545049 


1920495907 


35.2562 


10.7520 


1244 


154? 536 


1925134784 


35.2704 


10.7549 


1245 


1550025 


1929781125 


35.2846 


10.7578 


1216 


1552516 


1934434936 


35.29s? 


10.7607 


1247 


1555009 


1939096223 


35.3129 


10.7635 


121S 


1557504 


1943764992 


35.3270 


10.7664 


1249 


1560001 


1948441249 


35.3412 


10.7693 


1250 


1562500 


1953125000 


35.3553 


10.7722 


1251 


1505001 


1957816251 


35.3695 


10.7750 


1252 


1567504 


1962515008 


35.3836 


10.7779 


1253 


1570009 


1967221277 


35.3977 


10.7808 


125 4 


1572516 


1971935064 


35.4119 


10.7837 



No. 


Square. 


Cube. 


Sq. 
Root. 


Cube 
Root. 


1255 


1575025 


1976656375 35.4260 


10.7865 


1256 


1577536 


1981385216 35.4401 


10.7894 


1257 


1580049 


1986121593 35.4542 


10.7922 


1258 


1582564 


1900S05512 35.4683 


10.7951 


1259 


1585081 


1995616979 35.4824 


10.7980 


1260 


1587600 


2000376030.35.4965 


10.8008 


1201 


,1590121 


2005142581 35.5106 


10.8037 


1262 


1592644 


2009916728 35.5246 


10.8065 


1263 


1595169 


2014698447 35.5387 


10.8094 


1264 


1597696 


2019487744 35.5528 


10.8122 


1265 


1600225 


2024284625 35.5668 


10.8151 


1266 


1602756 


2029089096 


35.5809 


10.8179 


1267 


1605289 


2033901163 




10 8208 


1268 


1607824 


2038720S32 


35.6090 


10.8236 


1269 


1610361 


2043548109 


35.6230 


10.8265 


1270 


1612900 


2048383000 


35.6371 


10.8293 


1271 


1615441 


2053225511 


35.6511 


10.8322 


1272 


1617984 


2058075648 


35.6651 


10.8350 


1273 


1620529 


2062933417 


35.6791 


10 8378 


1274 


1623076 


2067798824 


35.6931 


10.8407 


1275 


1625625 


2072671875 


35.7071 


10.8435 


1276 


1628176 


2077552576 


35.7211 


10.8463 


1277 


1630729 


2082440933 


35.7351 


10.8492 


1278 


1633284 


2087336952 


35.7491 


10.8520 


1279 


1635841 


2092240639 


35.7631 


10.8548 


1280 


1638400 


2097152000 


35.7771 


10.8577 


12S1 


1640961 


2102071041 


35.7911 


10.8605 


1 282 


1643524 


2106997768 


35.8050 


10.8633 




1646089 


2111932187 


35.8190 


10.8661 


1284 


1648656 


2116874304 


35.8329 


10.8690 


1285 


1651225 


2121824125 


35.8469 


10.8718 




1653796 


2126781656 


35.8608 


10.8746 


12S7 


1656369 


2131746903 


35.8748 


10.8774 


12SS 


1658944 


2136719872 


35.8887 


10.8802 


3289 


1661521 


2141700569 


35.9026 


10.8831 


1290 


1664100 


2146689000 


35.9166 


10.8859 


1291 


1666681 


2151685171 


35.9305 


10.8887 


1202 


1669264 


21566890S8 


35.9444 


10.8915 


1293 


1671849 


2161700757 


35.9583 


10.8943 


1294 


1674136 


2166720184 


35.9722 


10.8971 


1295 


1677025 


2171747375 


35.9861 


10.8999 


1296 


1679616 


2176782336 


36.0000 


10.9027 


1207 


1682209 


2181825073 


36.0139 


10.9055 


12 '.is 


1684804 


2186875592 


36.0278 


10.9083 


1209 


1687401 


2191933899 


36.0416 


10.9111 


1300 


1690000 


2197000000 


36.0555 


10.9139 


1301 


1692601 


2202073901 


36.0694 


10.9167 


1302 


1605201 


2207155608 


36.0832 


10.9195 


1303 


1697809 


2212245127 


36.0971 


10.9223 


1304 


1700416 


2217342464 


36.1109 


10.9251 


1305 


1703025 


2222447625 


36.1248 


10.9279 


1306 


1705636 2227560616 


36.1386 


10.9307 


1307 


170824912232681443 


36.1525 


10.9335 


130S 


17108642237810112 


36.1663 


10.9363 


1300 


1713481 


2242946629 


36.1801 


10.9391 



SQUARES, CUBES, SQUARE AKD CUBE ROOTS. 09 



No. Square. 



1716100 2248091000 
17187212253243231 
1721344 225840332S 
1723969 2263571297 
1720596 2268747144 

i 
1729225 22' 
1731856 2279122496 
1734489 2284322013 
1737124 2289529432 
17397612294744759 

1742400 2299968000 

1745041 

1747684 

1750329 

1752971 

1755625 

17582' 

1760929 

1763584 

1766241 



1768900 
1771561 
1774224 

1776 



1782225 

1784 

1787 

1790244 

1792921 

1795600 

1798281 
1800964 
1803649 
1806336 

1809025 
1811716 
1814409 
1817104 
1819801 

1822500 
1825201 
1827904 
1830609 



231568526"; 
2320940224 

2326203125 
2331473976 
2336752783 
2342039552 
2347334289 

2352637000 

2357947 

2363266368 



2373927704 

2379270375 
2384621056 
2389979753 
2395346472 
2400721219 

2406104000 
2411494821 



1841449 
1844164 

1846881 

1849600 
1852321 
1855044 
1857769 
1860496 



i. 191 

36.20' 
36.2215 
36.2353 
36.2491 

2629 

36.276' 

2905 

36.3043 

36.3180 

36.3318 

6.3456 
30.3593 
36.3731 



242230060' 
2427715584 

2433138625 
2438569736 
2444008923 
2449456192 
2454911549 

2460375000 
2465846551 
2471326208 
2476813977 
2482309864 

2487813875 
2493326016 
2498846293 
2504374712 
250991127 

2515456000 

2521008881 

2526569928 

253213914 

2537716544 



Sq. 

Root. 



36.4005 
36.4143 
36.4280 
36.4417 
36.4555 

36.4692 
36.4829 
36.4966 
36.5103 
36.5240 

36.5377 

36.5513 

36.5650 

5787 



36.6197 
36.6333 
36.6469 



36.6742 
36.6879 
36.7015 
36.7151 
36.7287 

36.7423 

36.7560 



36.8103 

5.8239 



Cube 
Root, 



10.9418 
10.9446 
10.9474 
10.9502 
10.9530 

10.9557 
10.9585 
10.9613 
10.9640 
10.9668 

10.9696 

10.9721 
10.9752 
10.9779 
10.9807 

10.9834 
10.9862 
10.9890 
10.9917 
10.9945 

10.9972 
11.0000 
11.0028 
11.0055 
11.0083 

11.0110 

11.0138 
11.0165 
11 0193 
11.0220 

11.0247 
11.0275 
11.0302 
11.0330 
11.0357 

11.0384 
11.0412 
11.0439 
11.0466 
11.0494 

11.0521 
11.0548 
11.0575 
11.0603 
11.0630 

11.0657 
11.0684 
11.0712 
11.0739 
11.0766 



8782 11.0793 
36.8917 11.0820 
36.9053 11.0847 
36.9188.11.0875 
36.9324 11.0902 



No. 


Square. 


Cube. 


Sq. 
Root. 


Cube 
Root. 


1365 


1863225 


2543302125 


36.9459 


11.0929 


1300 


1865956 


2548895896 


36.9594 


11.0956 


1367 


1868689 


2554497863 


36.9730 


11.0983 


1368 


1871424 


2560108032 


36.9865 


11.1010 


1369 


1874161 


2565726409. 


37.0000 


11.1037 


1370 


1876900 


2571353000 


37.0135 


11.1064 


1371 


1879641 


2576987811 


37.0270 


11.1091 


1372 


1882384 


2582630848 


37.0405 


11.1118 


1373 


1885129 


2588282117 


37.0540 


11.1145 


1374 


1887876 


2593941624 


37.0675 


11. 1172 


1375 


1890625 


2599609375 


37.0810 


11.1199 


1376 


1893376 


2605285376 


37 0945 11.1226 


1377 


1896129 


2610969633 


37.1080 


11.1253 


137S 


1898884 


2616662152 


37.1214 


11.1280 


1379 


1901641 


2622362939 


37.1349 


11.1307 


1380 


1904400 


2628072000 


37.1484 


11.1334 


1381 


1907161 


2633789341 


37.1618 


11.1361 


1382 


1909924 


2639514968 


37.1753 


11.1387 


13S3 


1912689 


2645248887 


37.1887 


11.1414 


1384 


1915456 


2650991104 


37.2021 


11.1441 


1385 


1918225 


2656741625 


37.2156 


11.1468 


1386 


1920996 


2662500456 


37.2290 


11.1495 


1387 


1923769 


2668267603 


37.2424 


11.1522 


1388 


1926544 


2674043072 


37.2559 


11.1548 


1389 


1929321 


2679826869 


37.2693 


11.1575 


1390 


1932100 


2685619000 


37.2827 


11.1602 


1391 


1934881 


2691419471 


37.2961 


11.1629 


1392 


1937664 


2697228288 


37.3095 


11.1655 


1393 


1940449 


2703045457 


37.3229 


11.1682 


1394 


1943236 


2708870984 


37.3363 


11.1709 


1395 


1946025 


2714704875 


37.3497 


11.1736 




1948816 


2720547136 


37.3631 


11.1762 


1397 


1951609 


2726397773 


37.3765 


11.1789 


1398 


1954404 


2732256792 


37.3898 


11.1816 


1399 


1957201 


2738124199 


37.4032 


11.1842 


1400 


1960000 


2744000000 


37.4166 


11.1869 


1401 


1962801 


2749884201 


37.4299 


11.1896 


1402 


1905604 


2755776808 


37.4433 


11.1922 


1403 


1968409 


2761677827 


37.4566 


11.1949 


1404 


1971216 


2767587264 


37.4700 


11.1975 


1405 


1974025 


2773505125 


37.4833 


11.2002 


1406 


1976836 


2779431416 


37.4967 


11.2028 


1407 


1979649 


2785366143 


37.5100 


11.2055 


1408 


1982464 


2791309312 


37.5233 


11.2082 


1409 


1985281 


2797260929 


37.5366 


11.2108 


1410 


1988100 


2803221000 


37.5500 


11.2135 


1411 


1990921 


28091S9531 


37.5633 


11.2161 


1412 


1993744 


2815166528 


37.5766 


11.2188 


1413 


1996569 


2821151997 


37.5899 


11.2214 


1414 


1999396 


2827145944 


37.6032 


11.2240 


1415 


2002225 


2833148375 


37.6165 


11.2267 


1416 


2005056 


2839159296 


37.6298 


11 2293 


1417 


2007889 


2845178713 


37.6431 


11.2320 


1418 


2010724 


2851206632 


37.6563 11.2346 


1419 


2013561 


2857243059 


37.6696 


11.2373 



100 



MATHEMATICAL TABLES. 



No. 


Square 


Cube. 


Sq. 
Root. 


Cube 
Root. 


No. 


Square. 


Cube. 


Sq. 
Root. 


Cube 
Root. 


1420 


2016400 


2863288000 37.6829 


11.2399 


1475 


2175625 


3209046875 


38.4057 


11.3832 


1421 


2019241 


2869341461 37.6962 


11.2425 


1476 


2178576 


3215578176 


38.4187 


11.3858 




2022084 


2875403448 37.7094 


11.2452 


1477 


2181529 


3222118333 


38.4318 


11.3883 




2024929 


2881473967 37.7227 


11.2478 


1478 


2184484 


3228667352 


38.4448 


11.3909 


1424 


2027776 


2887553024.37.7359 


11.2505 


1479 


2187441 


3235225239 


38.4578 


11.3935 


1425 


2030625 


2893640625 37.7492 


11.2531 


1480 


2190400 


3241792000 


38.4708 


11.3960 




2033476 


2899736776 s 37. 7624 


11.2557 


1481 


2193361 


3248367641 


38.4838 


11.3986 




2036329 


2905841483 37.7757 


11.2583 


1482 


2196324 


3254952168 


38.4968 


11.4012 




2039184 


2911954752 37.7889 


11.2610 


1483 


2199289 


3261545587 


38.5097 


11.4037 


1429 


2042041 


2918076589-37.8021 


11.2636 


1484 


2202256 


3268147904 


38.5227 


11.4063 


1430 


2044900 


2924207000 37.8153 


11.2662 


1485 


2205225 


3274759125 


38.5357 


11.4089 


1431 


2047761 


2930345991 '37.8286 


11.2689 


1486 


2208196 


3281379256 


38.5487 


11.4114 




2050624 


2936493568 37.8418 

2942040:37 37.8550 


11.2715 


1487 


2211169 


3288008303 


38.5616 


11.4140 




2053489 


11.2741 


1488 


2214144 


3294646272 


38.5746 


11.4165 


1434 


2056356 


2948814504 37.8682 


11.2767 


1489 


2217121 


3301293169 


38.5876 


11.4191 


1435 


2059225 


2954987875 37.8814 


11.2793 


1490 


2220100 


3307949000 


38.6005 


11.4216 




2062096 


2961169856 37.8946 


11.2820 


1491 


2223081 


3314613771 


38.6135 


11.4242 




2064969 


2967360453 37.9078 


11.2846 


1492 


2220004 


3321287488 


38.6264 


11.4268 


143K 


2067844 


2973559672 37.9210 


11.2872 


1493 


2229049 


3327970157 


38.6394 


11.42(13 


1439 


2070721 


2979767519:87.9342 


11.2898 


1494 


2232036 


3634661784 


38.6523 


11.4319 


1440 


2073600 


2985984000 37.9473 


11.2924 


1495 


2235025 


3341362375 


38.6652 


11.4344 


1441 


2076481 


2992209121 37.9605 


11.2950 


1496 


2238016 


3348071936 


38.6782 


11.4370 




2079364 


2998442888 37.9737 


11.2977 


1497 


2241009 


3354790473 


38.6911 


11.4395 




2082249 


3004685307 37.9868 11.3003 


1498 


2244004 


3361517992 


38.7040 


11.4421 


1444 


2085136 


3010936384 38.0000,11.3029 


1499 


2247001 


3368254499 


38.7169 


11.4446 


1445 


2088025 


3017196125 


38.0132 


11.3055 


1500 


2250000 


3375000000 


38.7298 


11.4471 




2090916 


3023464536 


38.0263 


11.3081 


1501 


2253001 


3381754501 


38.7427 


11.4497 


1447 


2093809 


3029741623 


38.0395 


11.3107 


1502 


2256004 


3388518008 


38.7556 


11.4522 




2096704 


3036027392 


38.0526 


11.3133 


1503 


2259009 


3395290527 


38.7685 


11.4548 




2099601 


3042321849 


38.0657 


11.3159 


1504 


2262016 


3402072064 


38.7814 


11.4573 


1450 


2102500 


3048625000 


38.0789 


11.3185 


1505 


2265025 


3408862625 


38.7943 


11.4598 


1451 


2105401 


3054936851 


38.0920 


11.3211 


1506 


2268036 


3415662216 


38.8072 


11.4624 




2108304 


3061257408 


38.1051 


11.3237 


1507 


2271049 


3422470843 


38.8201 


11.4649 




2111209 


3067586677 


38.1182 


11.3263 


1508 


2274064 


3429288512 




11.4675 


1454 


2114116 


3073924664 


38.1314 


11.3289 


1509 


2277081 


3436115229 


38.8458 


11.4700 


1455 


2117025 


3080271375 


38.1445 


11.3315 


1510 


2280100 


3442951000 


38.8587 


11.4725 


1450 


2119936 


3086626816 


38. 1576; 11. 3341 


1511 


2283121 


3449795831 


38.8716 


11.4751 


1 457 


2122849 


3092990993 


38. 1707 111. 3367 


1512 


2286144 


3456649728 


38 8844 


11.4776 


1458 


2125764 


3099363912 


38.1838 11.3393 


1513 


2289169 


3463512697 


38.8973 


11.4801 


1459 


2128681 


3105745579 


38.1969 


11.3419 


1514 


2292196 


3470384744 


38.9102 


11.4826 


1 4 oo 


2131600 


3112136000 


38.2099 


11.3445 


1515 


2295225 


3477265875 


38.9230 


11.4852 


1401 


21 845.' 1 


3118535181 




11.3471 


1516 


2298256 


3484156096 


38.9358 


11.4877 


1462 


2137444 


3124943128 


38.2361 


11.3496 


1517 


2301289 


3491055413 


38.9487 


11.4902 




2140369 


3131359847 


38.2492 


11.3522 


1518 


2304324 


3597963832 


38.9615 


11.4927 


1464 


2143296 


3137785344 


38.2623 


11.3548 


1519 


2307361 


3504881359 


38.9744 


11.4953 


14(35 


2146225 


3144219625 


38.2753 


11.3574 


1520 


2310400 


3511808000 


38.9872 


11.4978 




2149156 


3150662696 


3S.2884 


11.3600 


1521 


28134 41 


3518743761 


39.0000 


11.5003 




2152089 


3157114563 


38.3014 


11.3626 


1522 


2316484 


3525688648 


39.0128 


11.5028 




2155024 


3163575232 


3^.81 15 


11.3652 


1523 


2319529 


3532642667 


39.0256 


11 5054 


1409 


2157961 


3170044709 


38.3275 


11.3677 


1524 


2322576 


3539605824 


39.0384 


11.5079 


1470 


2160900 


3176523000 


38.3406 


11.3703 


1525 


2325625 


3546578125 


39.0512 


11.5104 


1471 


2163841 


3183010111 


38.3536 


11.3729 


1526 


2328676 


3553559576 


39.0640 


11.5129 


1472 


2166784 


3189506048 


38.3667 


11.3755 


1527 


2881720 


3560550183 


39.0768 


11.5154 


1473 


21 OUT-.'!) 


3196010817 


8s . 8797 


11.3780 


1528 


2334784 


3567549952 


39.0896 


11.5179 


1474 2172076 


3202524424 


38.3927 


11.3806 


1529 


2337841 3574558889 39.1024 11.5204 



SQUARES, CUBES, SQUARE AHD CUBE ROOTS. 101 



No. 


Square. 


Cube. 


Sq. 
Root. 


Cube 
Root. 


No. 


Square. 


Cube. 


Sq. 
Root. 


Cube 
Root. 


1530 


2340900 


358157700( 


39.1152 


11.5230 


1565 


2449225 


3833037125 


39.5601 


11.6102 


1531 


2343961 


3588604291 


39.1280 


11.5255 


1566 


2452356 


3840389496 


39.5727 


11.6126 


1532 


2347024 


359564076* 


39.1408 


11.5280 


1567 


2455489 3847751263 


39.5854 


11.6151 


1533 


2350089 


360268643" 


39.1535 


11.5305 


1568 


2458624 3855123432 


39.5980 


11.6176 


1534 


2353156 


360974130- 


39.1663 


11.5330 


1569 


2461761; 3862503009 


39.6106 


11.6200 


1535 


2356225 


3616S0537c 


39.1791 


11.5355 


1570 


2464900 3869893000 


39.6232 


11.6225 


1536 


2359296 


3623878656 i 39. 1918 


11.5380 


1571 


2468041 3877292411 


39.6358 


11.6250 


1537 


2362369 


3630961 153' 39. 2046 


11.5405 


1572 


2471184 3884701248 


39.6485 


11.6274 


1538 


2365444 


363805287; 


39.2173 


11.5430 


1573 


24743293892119517 


39.6611 


11.6299 


1539 


2368521 


36451538H 


39.2301 


11.5455 


1574 


2477476 3899547224 


39.6737 


11.6324 


1540 


2371600 


365226400C 


39.2428 


11.5480 


1575 


2480625 '3906984375 


39.6863 


11.6348 


1541 


2374681 


3659383421 


39.2556 


11.5505 


1576 


2483776 3914430976 


39.6989 


11.6373 


1542 


2377761 


36665 1208* 


39.2683 


11.5530 


1577 


2486929 3921887033 


39.7115 


11.6398 


1543 


2380S49 


367365000" 


39.2810 


11.5555 


1578 


2490084 


3929352552 


39.7240 


11.6422 


1544 


2383936 


3680797184 


39.2938 


11.5580 


1579 


2493241 


3936827539 


39.7366 


11.6447 


1545 


2387025 


368795362c 


39.3065 


11.5605 


1580 


2496400 


3944312000 


39.7492 


11.6471 




2390116 


36951 19336 


39.3192 


11.5630 


1581 


2499561 


3651805941 


39.7618 


11.6496 


1547 


2393209 


370229432: 


39.3319 


11.5655 


1582 


2502724 


3959309368 


39.7744 


11.6520 


1548 


2396304 


370947859; 


39.3446 


11.5680 


1583 


2505889 


3966822287 


39.7869 


11.6545 


1549 


2399401 


371667214S 


39.3573 


11.5705 


1584 


2509056 


3974344704 


39.7995 


11.6570 


1550 


2402500 


372387500C 


39.3700 


11.5729 


1585 


2512225 


3981876625 


39.8121 


11.6594 


1551 


2405601 


3731087151 


39.3827 


11.5754 


1586 


2515396 3989418056 


39.8246 11.6619 


155-2 


2408704 


373830860* 


39.3954 


11.5779 


1587 


25185693996969003 


39.8372 11.6643 


1553 


2411809 


374553937" 


39.4081 


11.5804 


1588 


2521744 4004529472 


39.8497 11.6668 


1554 


2414916 


3752779464 


39.4208 


11.5829 


1589 


25249214012099469 


39.8623 11.6692 


1 555 


2418025 


376002887c 


39.4335 


11.5854 


1590 


2528100 4019679000 


39.8748 11.6717 


5; 


2421136 


3767287616 


39.4462 


11.5879 


1591 


2531281 14027268071 


39.8873 11.6741 


1 557 


2424249 


3774555693 39.4588 


11.5903 


1592 


2534464 '4034866688 


39.8999 11.6765 




2427364 


3781833112 39.4715 


11.5928 


1593 


2537049 ! 4042474857 


39. 9124 11. 6790 


1559 


2430481 


37891 1987C 


39.4842 


11.5953 


1594 


25408364050092584 


39.9249,11.6814 


1560 


2433600 


37964 1600( 


39 4968 


11.5978 


1595 


2544025 4057719875 


39.9375 11.7839 


1561 


2436721 


380372148 


39.5095 


11.6003 


1596 


2547216 4065356736 


39.9500 11.6863 


1564 


2439844 


381103632* 


39.5221 


11.6027 


1597 


2550409 4073003173 


39.9625 11.6888 


1563 


2442969 


381836054' 


39.5348 


11.6052 


1598 


2553604 4080659192 


39.9750,11.6912 


1564 


2446096 


382569414 


39.5474 


11.6077 


1599 
1600 


25568014088324799 
2560000 4096000000 


39.9875 11.6936 
40.0000 11.6961 


SQUARES AND CUBES OF DECIMALS. 


No. 


Square. 


Cube. 


No. 


Square. 


Cube. 


No 


Square. 


Cube. 


.1 


.01 


.001 


.01 


.0001 


.000 001 


.001 


.00 00 01 


.000 000 001 


.2 


.04 


.008 


.02 


.0004 


.000 008 






00 00 04 


.000 000 008 


.3 


.09 


.027 


.03 


.0009 


.000 027 






00 00 09 


.000 000 027 


.4 


.16 


.064 


.04 


.0016 


.000 064 


.004 




00 00 16 


.000 000 064 


.5 


.25 


.125 


.05 


.0025 


.000 125 


.00.: 




00 00 25 


.000 000 125 


.6 


.36 


.216 


.06 


.0036 


.000 216 






00 00 36 


.000 000 216 


7 


.49 


.343 


.07 


.0049 


.000 343 






00 00 49 


.000 000 343 


.8 


.64 


.512 


.08 


.0064 


.000 512 


.00* 




00 00 64 


.000 000 512 


.9 


.81 


.729 


.09 


.0081 


.000 729 






00 00 81 


.000 000 729 


1.0 


1.00 


1.000 


.10 


.0100 


.001 000 


.OK 




00 01 00 


.000 001 000 


1.2 


1.44 


1.728 


.12 


.0144 


.001 728 


.01; 




00 01 44 


.000 001 728 



Note that the square has twice as many decimal places, and the cube three 
times as many decimal places, as the root. 



102 



MATHEMATICAL TABLES. 



FIFTH ROOTS AND FIFTH POWERS. 

(Abridged from Trautwine.) 



u 




o ° 


Power. 


£« 




.10 


.000010 


.15 


.000075 


.20 


.000320 


.25 


.000977 


.30 


.002430 


.35 


.005252 


.40 


.010240 


.45 


.018453 


.50 


.031250 


.55 


.050328 


.60 


.077760 


.65 


.116029 


.70 


.168070 


.75 


.237305 


.80 


.327680 


.85 


.443705 


.90 


.590490 


.95 


.773781 


1.00 


1.00000 


1.05 


1.27628 


1.10 


1.61051 


1.15 


2.01135 


1.20 


2.48832 


1.25 


3.05178 


1.30 


3.71293 


1.35 


4.48403 


1.40 


5.37824 


1.45 


6.40973 


1.50 


7.59375 


1.55 


8.94661 


1.60 


10.4858 


1.65 


12.2298 


1.70 


14.1986 


1.75 


16.3141 


1.80 


18.8957 


1.85 


21.6700 


1.90 


24.7610 


1.95 


28.1951 


2.00 


32.0C00 


2.05 


36.2051 


2.10 


40.8410 


2.15 


45.9401 


2.20 


51.5363 


2.25 


57.6650 


2.30 


64.3634 


2.35 


71.6703 


2.40 


79.6262 


2.45 


88.2735 


2.50 


97.6562 


2.55 


107.820 


2.60 


118.814 


2.70 


143.489 


2.80 


172.104 


2.90 


205.111 


3.00 


243.000 


3.10 


286.292 


3.20 


335.541 


3.30 


391.354 


3.40 


454.354 


3.50 


525.219 


3.60 


604.662 



693.440 
792.352 
902.242 
1024.00 
1158.56 
1306.91 
1470.08 
1649.16 
1845.28 
2059.63 
2293.45 
2548.04 
2824.75 
3125.00 
3450.25 
3802.04 
4181.95 
4591.65 
5032.84 
5507.32 
6016.92 
6563.57 
7149.24 
7776.00 
8445.96 
9161.33 
9924.37 

10737 

11603 

12523 

13501 

14539 

15640 

16807 

18042 

19349 

20731 

22190 

23730 

25355 

27068 

28872 

30771 

32768 

34868 

37074 

39390 

41821 

44371 

47043 

49842 

52773 

55841 

59049 

62403 

65908 

69569 



81537 

85873 



17.0 

17.8 

18.0 

18.2 

18.4 

18. G 

18 

19.0 

19.2 

19.4 

19.6 

lli.S 

20.0 

20.2 

20.4 

20.6 

20. 

21.0 

21.2 

21.4 

21.6 



90392 
95099 
100000 
110408 
121665 
138823 
146933 
161051 
176234 
192541 
210131 
228776 
24SN82 
270271 
2981(33 
317580 
343597 
371293 
400746 
43.2040 
465259 
500490 
537824 
577353 
619174 



710082 
759375 
811368 
866171 



1048576 
1115771 
1186367 
1260493 
1338278 
1419857 
1505366 
1594947 
1688742 
1786899 
1889568 



2109061 
2226203 
2348493 
2476099 
2609193 
2747949 
2892547 
3043168 
3200000 
3363232 
3533059 
3709677 



4084101 
4282322 
4488166 
4701850 



21.8 

22.0 
22.2 
22.4 
22.6 
22.8 
28.0 
'8.2 
28.4 
23.6 
28 . 8 
24.0 
24.2 
24.4 
24.6' 
24. 8 ! 
25. 1 
25. 2' 
25.4 
25.6 
25.8. 
.0 
.2 
26.4 
.6 
20.8 
27.0 

27\4 
27.6 

27.8 
28.0 



4923597 
5153632 

58!i2180 
5639493 
5895793 
6161327 
6436343 
6721093 
7015834 
7 32(1825 
7630332 
7962624 
8299976 
8641-666 
'.I0IWI7S 
9381200 
9765625 
10162550 
10572278 
10995116 
11431377 
11881376 
12345437 
12823886 
13317055 
13825281 
14348907 



15443752 

16015681 

16604430 

17210368 

17833868 

18475309 

19135075 

19813557 

20511149 

21228253 

21965275 

22722628 

23500728 

24300000 

26393634 

28629151 

31013642 

33554432 

36259082 , ov 

39135393 J 87 

42191410 

45435424 



52521875 
56382167 
60466176 
64783487 
69343957 
74157715 
79235168 
84587005 
90224199 
9615801.2 



CIRCUMFERENCES AND AREAS OF CIRCLES. 103 
CIRCUMFERENCES AND AREAS OF CIRCL.ES. 



Diam. 


Circum. 


Area. 


1 


3.1416 


0.7854 


2 


6.2832 


3.1416 


3 


9.4248 


7.0686 


4 


12.5664 


12.5664 


5 


15.7080 


19.635 


6 


18.850 


28.274 


7 


21.991 


38.485 


8 


25.133 


50.266 


9 


28.274 


63.617 


10 


31.416 


78.540 


11 


34.558 


95.033 


12 


37.699 


113.10 


13 


40.841 


132.73 


14 


43.982 


153.94 


15 


47.124 


176.71 


16 


50.265 


201.06 


17 


53.407 


226.98 


18 


56.549 


254.47 


19 


59.690 


283.53 


20 


62.832 


314.16 


21 


65.973 


346.36 


22 


69.115 


380.13 


23 


72.257 


415.48 


24 


75.398 


452.39 


25 


78.540 


490.87 


26 


81.681 


530.93 


27 


84.823 


572.56 


28 


87.965 


615.75 


29 


91.106 


660.52 


30 


94.248 


706.86 


31 


97.389 


754.77 


32 


100.53 


804.25 


33 


103.67 


855.30 


34 


106.81 


907.92 


35 


109.96 


962.11 


36 


113.10 


1017.88 


37 


116.24 


1075.21 


38 


119.38 


1134.11 


39 


122.52 


1194.59 


40 


125.66 


1256.64 


41 


128.81 


1320.25 


42 


131.95 


1385.44 


43 


135.09 


1452.20 


44 


138.23 


1520.53 


45 


141.37 


1590.43 


46 


144.51 


1661.90 


47 


147.65 


1734.94 


48 


150.80 


1809.56 


49 


153 94 


1885.74 


50 


157.08 


1963.50 


51 


160.22 


2042.82 


52 


163.36 


2123.72 


53 


166.50 


2206.18 


54 


169.65 


2290.22 


55 


172.79 


2375.83 


56 


175.93 


2463.01 


57 


179.07 


2551.76 


58 


182.21 


2642.08 


59 


185.35 


2733.97 


60 


188.50 


2827.43 


61 


191.64 


2922.47 


62 


194.78 


3019.07 


63 


197.92 


3117.25 


64 


201.06 


3210.99 



Diam. 


Circum. 


Area. 


65 


204.20 


3318.31 


66 


207.34 


3421.19 


67 


210.49 


3525.65 


68 


213.63 


3631.68 


69 


216.77 


3739.28 


70 


219.91 


3848.45 


71 


223 05 


3959.19 


72 


226.19 


4071.50 


73 


229.34 


4185.39 


74 


232.48 


4300 84 


75 


235.62 


4417.86 


76 


238.76 


4536.46 


77 


241.90 


4656.63 


78 


245.04 


4778.36 


79 


248.19 


4901.67 


80 


251.33 


5026.55 


81 


254.47 


5153.00 


82 


257.61 


5281.02 


83 


260.75 


5410.61 


84 


263.89 


5541.77 


85 


267.04 


5674 50 


86 


270.18 


5808.80 


87' 


273.32 


5944.68 


88 


276.46 


6082.12 


89. 


279.60 


6221.14 


90 


282.74 


6361.73 


91 


285.88 


6503.88 


92 


289.03 


6647.61 


93 


292.17 


6792.91 


94 


295.31 


6939.78 


95 


298.45 


7088.22 


96 


301.59 


7238.23 


97 


304.13 


7389.81 


98 


307.88 


7542.96 


99 


311.02 


7697.69 


100 


314.16 


7853.98 


101 


317.30 


8011.85 


102 


320.44 


8171.28 


103 


323.58 


8332.29 


104 


326.73 


8.494.87 


105 


329 87 


8659.01 


106 


333.01 


8824.73 


107 


336.15 


8992.02 


108 


339.29 


9160.88 


109 


342.43 


9331.32 


110 


345.58 


9503.32 


111 


348.72 


9676.89 


112 


351.86 


9852.03 


113 


355.00 


10028.75 


114 


358.14 


10207.03 


115 


361.28 


10386 89 


116 


364.42 


10568.32 


117 


367.57 


10751.32 


118 


370.71 


10935.88 


119 


373.85 


11122.02 


120 


376.99 


11309.13 


121 


380.13 


11499.01 


122 


383.27 


11689.87 


123 


386.42 


11882.29 


124 


389.56 


12076.28 


125 


392.70 


12271.85 


126 


395.84 


12468.98 


127 


398.98 


12667.69 


128 


402.12 


12867.96 



Diam. 


Circum. 


Area. 


129 


405.27 


13069.81 


130 


408.41 


13273.23 


131 


411.55 


13478.22 


132 


414.69 


13684 78 


133 


417.83 


13892.91 


134 


420.97 


14102.61 


135 


424.12 


14313.88 


136 


427.26 


14526.72 


137 


430.40 


14741.14 


138 


433.54 


14957.12 


139 


436.68 


15174.68 


140 


439.82 


15393.80 


141 


442.96 


15614.50 


142 


446.11 


15836.77 


143 


449.25 


16060.61 


144 


452.39 


16286.02 


145 


455.53 


16513.00 


146 


458.67 


16741.55 


147 


461.81 


16971.67 


148 


464.96 


17203.36 


149 


468.10 


17436.62 


150 


471.24 


17671.46 


151 


474.38 


17907 86 


152 


477.52 


18145.84 


153 


480.66 


18385.39 


154 


483.81 


18626.50 


155 


486.95 


18869.19 


156 


490.09 


19113.45 


157 


493.23 


19359.28 


158 


496.37 


19606.68 


159 


499.51 


19855.65 


160 


502.65 


20106.19 


161 


505.80 


20358.31 


162 


508.94 


20611.99 


163 


512.08 


20867.24 


164 


515.22 


21124.07 


165 


518.36 


21382.46 


166 


521.50 


21642.43 


167 


524.65 


21903 97 


168 


527.79 


22167 08 


169 


530.93 


22431.76 


170 


534.07 


22698.01 


171 


537.21 


22965.83 


172 


540.35 


23235:22 


173 


543.50 


23506.18 


174 


546.64 


23778.71 


175 


549.78 


24052.82 


176 


552.92 


24328.49 


177 


556.06 


24605.74 


178 


559.20 


24884.56 


179 


562.35 


25164.94 


180 


565.49 


25446 90 


181 


568.63 


25730.43 


182 


571.77 


26015.53 


183 


574.91 


26302.20 


184 


578.05 


26590.44 


185 


581.19 


26880.25 


186 


584.34 


27171.63 


187 


587.48 


27464.59 


188 


590.62 


27759.11 


189 


593.76 


28055.21 


190 


596.90 


28352.87 


191 


600.04 


28652.11 


192 


603.19 


28952.92 



104 



MATHEMATICAL TABLES. 



Diam. 


Circum. 


Area. 


Diam. 


Circum. 


Area. 


Diam. 


Circum. 


Area. 


193 


606.33 


29255.30 


260 


816.81 


53092.92 


327 


1027.30 


83981.84 


194 


609.47 


29559.25 


261 


819.96 


53502.11 


328 


1030.44 


84496.28 


195 


612.61 


29864.77 


262 


823.10 


53912.87 


329 


1033.58 


' 85012.28 


196 


615.75 


30171.86 


263 


826.24 


54325.21 


330 


1036.73 


85529.86 


197 


618.89 


30480.52 


264 


829.38 


54739.11 


331 


1039.87 


86049.01 


198 


622.04 


30790.75 


265 


832.52 


55154.59 


332 


1043.01 


86569.73 


199 


625.18 


31102.55 


266 


835.66 


55571.63 


333 


1046.15 


87092.02 


200 


(528.32 


31415.93 


267 


838.81 


55990.25 


334 


1049.29 


87615.88 


201 


631.46 


31730.87 


268 


841.95 


56410.44 


335 


1052.43 


88141.31 


202 


634.60 


32047.39 


269 


845.09 


56832.20 


336 


1055.58 


88668.31 


203 


637.74 


32365.47 


270 


848.23 


57255.53 


337 


1058.72 


89196.88 


204 


'640.88 


32685.13 


271 


851.37 


57680.43 


338 


1061.86 


89727.03 


205 


644.03 


33006.36 


272 


854.51 


58106.90 


339 


1065.00 


90258.74 


206 


647.17 


33329.16 


273 


857.65 


58534.94 


340 


1068.14 


90792. 1)3 


207 


650.31 


33653.53 


274 


860.80 


58964.55 


341 


1071.28 


91326.88 


208 


653.45 


33979.47 


275 


863.94 


59395.74 


342 


1074.42 


91863.31 


209 


656.59 


34306.98 


276 


867.08 


59828.49 


343 


1077.57 


92401.31 


210 


659.73 


34636.06 


277 


870.22 


60262.82 


344 


1080.71 


92940.88 


211 


662.88 


34966.71 


278 


873.36 


60698.71 


345 


1083.85 


93482.02 


212 


666.02 


35298.94 


279 


876.50 


61136.18 


346 


1086.99 


94024.73 


213 


669.16 


35632.73 


280 


879.65 


61575.22 


347 


1090.13 


94569.01 


214 


672.30 


35968.09 


281 


882.79 


62015.82 


348 


1093.27 


95114.86 


215 


675.44 


36305.03 


282 


885.93 


62458.00 


349 


1096.42 


95662.28 


216 


678.58 


36643.54 


283 


889.07 


62901.75 


350 


1099.56 


96211.28 


217 


681.73 


36983.61 


284 


892.21 


63347.07 


351 


1102.70 


96761.84 


218 


684.87 


37325.26 


285 


895.35 


63793.97 


352 


1105.84 


97313.97 


219 


688.01 


37668.48 


286 


898.50 


64242.43 


353 


1108.98 


97867.68 


220 


091.15 


38013.27 


287 


901.64 


64692.46 


354 


1112.12 


98422.96 


221 


694.29 


38359.63 


288 


904.78 


65144.07 


355 


1115.27 


98979.80 


222 


697.43 


38707.56 


289 


907.92 


65597.24 


356 


1118.41 


- 99538.22 


223 


700.58 


39057.07 


290 


911.06 


66051.99 


357 


1121.55 


100098.21 


224 


703.72 


39408.14 


291 


914.20 


66508.30 


358 


1124.69 


100659.77 


225 


706.86 


39760.78 


292 


917.35 


669C6.19 


359 


1127.83 


101222.90 


226 


710.00 


40115.00 


293 


920.49 


67425.65 


360 


1130.97 


101787.60 


227 


713.14 


40470.78 


294 


923.63 


67886.68 


361 


1134.11 


102353.87 


228 


716.28 


40828.14 


295 


926.77 


68349.28 


362 


1137.26 


102921. 7'2 


229 


719.42 


41187.07 


296 


929.91 


68813.45 


363 


1140.40 


103491.13 


230 


722.57 


41547.56 


297 


933.05 


69279.19 


364 


1143.54 


104062.12 


231 


725.71 


41909.63 


298 


936.19 


69746.50 


365 


1146.68 


104634.67 


232 


728.85 


42273.27 


299 


939.34 


70215.38 


366 


1149.82 


105208.80 


233 


731.99 


42638.48 


300 


942.48 


70685.83 


367 


1152.96 


105784.49 


234 


735.13 


43005.26 


301 


945.62 


71157.86 


368 


1156.11 


106361.76 


235 


738.27 


43373.61 


303 


948.76 


71631.45 


369 


1159.25 


106940.60 


236 


741.42 


43743.54 


303 


951.90 


72106.62 


370 


1162.39 


107521.01 


237 


744.56 


44115.03 


304 


955.04 


72583.36 


371 


1165.53 


108102.99 


238 


747.70 


44488.09 


305 


958.19 


73061.66 


372 


1168.67 


108686.54 


239 


750.84 


44862.73 


306 


961.33 


73541.54 


373 


1171.81 


109271.66 


240 


753.98 


45238.93 


307 


964.47 


74022.99 


374 


1174.96 


109858.35 


241 


757.12 


45616.71 


308 


967.61 


74506.01 


375 


1178.10 


110446.62 


242 


760.27 


45996.06 


309 


970.75 


74990.60 


376 


1181.24 


111036.45 


243 


763.41 


46376.98 


310 


973.89 


75476.76 


377 


1184.38 


111627.86 


244 


766.55 


46759.47 


311 


977.04 


75964.50 


378 


1187.52 


112220.83 


245 


769.69 


47143.52 


312 


980.18 


76453.80 


379 


1190.66 


112815.38 


246 


772.83 


47529.16 


313 


983.32 


76944.67 


380 


1193.81 


113411.49 


247 


775.97 


47916.36 


314 


986.46 


77437.12 


381 


1196.95 


114009.18 


248 


779.11 


48305.13 


315 


989.60 


77931.13 


382 


1200.09 


114608.44 


249 


782.26 


48695.47 


316 


992.74 


78426.72 


383 


1203.23 


115209.27 


250 


785.40 


49087.39 


317 


995.88 


78923.88 


384 


1206.37 


115811.67 


251 


788.54 


49480.87 


318 


999.03 


79422.60 


385 


1209.51 


116415.64 


252 


791.68 


49875.92 


319 


1002.17 


79922.90 


386 


1212.65 


117021.18 


253 


794.82 


50272.55 


320 


1005.31 


80424.77 


387 


1215.80 


117628.30 


254 


797.96 


50670.75 


321 


1008.45 


80928.21 


388 


1218.94 


118236.98 


255 


801.11 


51070.52 


322 


1011.59 


81433 22 


389 


1222.08 


118847.24 


256 


804.25 


51471.85 


323 


1014.73 


81939.80 


390 


1225.22 


119459.06 


257 


807.39 


51874.76 


324 


1017.88 


82447.96 


391 


1228.36 


1 20072. 4ft 


258 


810.53 


52279.24 


325 


1021.02 


82957.68 


392 


1231.50 


12068742 


259 


813.67 


52685.29 


326 


1024.16 


83468.98 


393 


1234.65 


121303.96 



CIRCUMFERENCES AND AREAS OF CIRCLES. 



105 



1237 

1240.93 
1244.07 
1247.21 
1250.35 
1253.50 
1256.64 



1266 

1269.20 

1272.35 

1275.49 

1278.63 

1281.77 

1284.91 

1288.05 

1291.19 

1294.34 

1297.48 

1300.6.2 

1303.76 

1306.90 

1310.04 

1313.19 

1316.33 

1319.47 

1322.61 

1325.75 

1328.89 

1332.04 

1335.18 

1338.32 

1341.46 

1344.60 

1347.74 

1350.88 

1354.03 

1357.17 

1360.31 

1363.45 

1366.59 

1369.73 

1372.88 

1376.02 

1379.16 

1382.30 

1385.44 

1388.58 

1391.73 

1391.87 

1398.01 

1401.15 

1404.29 

1407.43 

1410.58 

1413.72 

1416 

1420.00 

1423.14 

1426.28 

1429.42 

1432.5' 

1435.7 

1438.85 

1441.99 

1445.13 



121922.0' 
122541.7; 
123163.00 
123785.82 
124410.21 
125036.17 
125663.71 
126292.81 
126923.48 
127555.73 
128189.55 
128824 "" 
129161.89 
130100.42 
130740.52 
131382.19 
132025.43 
132670.24 
133316.63 
133964.58 
134614.10 
135265.20 
135917.86 
136572.10 
137227.91 
137885.29 
138544.24 
139204.76 
139866.85 
140530.51 
141195.74 
141862.54 
142530.92 
143200.86 
143872.38 
144545.46 
145220.12 
145896.35 
146574 15 
147253.52 
147934.46 
148616.97 
149301.05 
149986.70 
150673.93 
151362.72 
152053.08 
152745.02 
153438 53 
154133.60 
154830.25 
155528.47 
156228.26 
156929.62 
157632.55 
158337.06 
159043.13 
159750.77 
160459.99 
161170.77 
161883.13 
162597.05 
163312.55 
164029.6) 
164748.26 
165468.47 
166190.25 



Diam. 


Circum. 


Area. 


461 


1448.27 


166913.60 


462 


1451.42 


167638 53 


463 


1454.56 


16S365.02 


464 


1457.70 


169093.08 


465 


1460.84 


169822.72 


466 


1463.98 


170553.92 


467 


1467.12 


171286.70 


468 


1470.27 


172021.05 


469 


1473 41 


172756.97 


470 


1476.55 


173494.45 


471 


1479.69 


174233.51 


472 


1482.83 


174974.14 


473 


1485.97 


175716.35 


474 


1489.11 


176460 12 


475 


1492.26 


177205.46 


476 


1195.40 


177952.37 


477 


1498.54 


178700.86 


478 


1501.68 


179450.91 


479 


1504.82 


180202.54 


480 


1507.96 


180955.74 


481 


1511.11 


181710.50 


482 


1514.25 


182466.84 


483 


1517.39 


183224.75 


484 


1520.53 


183984.23 


485 


1523.67 


184745.28 


486 


1526.81 


185507.90 


487 


1529.96 


186272.10 


488 


1533.10 


187037.86 


489 


1536.24 


187805.19 


490 


1539.38 


188574.10 


491 


1542.52 


189344.57 


492 


1545.66 


190116.62 


493 


1548.81 


190890.24 


494 


1551.95 


191665.43 


495 


1555 09 


192442.18 


496 


1558.23 


193220.51 


497 


1561.37 


194000.41 


498 


1564.51 


194781.89 


499 


1567.65 


195564.93 


500 


1570.80 


196349.54 


501 


1573.94 


197135.72 


502 


1577.08 


197923.48 


503 


1580.22 


198712.80 


504 


1583.36 


199503.70 


505 


1586 50 


200296.1? 


506 


1589.65 


201090.20 


507 


1592.79 


201885.81 


508 


1595.93 


202682.99 


509 


1599.07 


203481.74 


510 


1602.21 


204282.06 


511 


1605.35 


205083.95 


512 


1608.50 


205887.42 


513 


1611.64 


206692.45 


514 


1614.78 


207499.05 


515 


1617.92 


208307.23 


516 


1621.06 


209116.97 


517 


1624.20 


209928.29 


518 


1627.34 


210741.18 


519 


1630.49 


211555.63 


520 


1633.63 


212371.66 


521 


1636.77 


213189.26 


522 


1639.91 


214008.43 


523 


1643.05 


214829.17 


524 


1646.19 


215651.49 


525 


1649.34 


216475.37 


526 


1652.48 


217300.82 


527 


1655.62 


218127.85 



Diam. 


Circum. 


Area. 


528 


1658.76 


218956.44 


529 


1661.90 


219786.61 


530 


1665.04 


220618.34 


531 


1668.19 


221451.65 


532 


1671.33 


222286.53 


533 


1674.47 


223122.98 


534 


1677.61 


223961.00 


535 


16S0.75 


224800.59 


536 


1683.89 


225641.75 


537 


1687.04 


226484.48 


538 


1690.18 


227328.79 


539 


1693.32 


228174.66 


540 


1696.46 


229022.10 


541 


1699.60 


229871.12 


542 


1702.74 


230721.71 


543 


1705.88 


231573.86 


544 


1709.03 


232427.59 


545 


1712.17 


233282.89 


546 


1715.31 


234139.76 


• 547 


1718.45 


234998.20 


548 


1721.59 


235858.21 


549 


1724.73 


236719.79 


550 


1727.88 


237582.94 


551 


1731.02 


238447.6? 


552 


1734.16 


239313.96 


553 


1737.30 


240181.83 


554 


1740.44 


241051.26 


555 


1743.58 


241922.27 


556 


1746.73 


242794. 85 


557 


1749.87 


243668.99 


55S 


1753.01 


244544.71 


559 


1756.15 


245422 00 


560 


1759.29 


246300.86 


561 


1762.43 


247181.30 


562 


1765.58 


248063.30 


563 


176S.72 


248946.87 


564 


1771.86 


249832.01 


565 


1775.00 


250718,73 


566 


1778.14 


251607.01 


56? 


1781.28 


252496.87 


568 


1784.42 


253388.30 


569 


1787.57 


254281.29 


570 


1790.71 


255175.86 


571 


1793.85 


256072.00 


572 


1796.99 


256969.71 


573 


1800.13 


257868.99 


574 


1803.27 


258769.85 


575 


1806.42 


259672.27 


576 


1809.56 


260576.26 


577 


1812.70 


261481.83 


578 


1815 84 


262388.96 


579 


1818.98 


263297.67 


580 


1822.12 


264207.91 


581 


1825.27 


265119.79 


582 


1828.41 


266033.21 


583 


1831.55 


266948.20 


584 


1834.69 


267864.76 


585 


1837.83 


268782.89 


586 


1840.97 


269702.59 


587 


1844.11 


270623.86 


588 


1847.26 


271546.70 


589 


1850.40 


272471.12 


590 


1853.54 


273397.10 


591 


1856.68 


274324.06 


592 


1859.82 


275253.78 


593 


1862.96 


276184.48 


594 


1866.11 


277116.75 



106 



MATHEMATICAL TABLES. 



Diain. 


Circum. 


Area. 


Diam 


Circum. 


Area. 


Diam- 


Circum. 


Area. 


595 


1869.25 


278050.58 


603 


2082.88 


345236.69 


731 


2296.50 


419686.15 


596 


1872.39 


278985.99 


664 


2086.02 


346278.91 


732 


2299.65 


420835.19 


597 


1875.53 


279922.97 


665 


2089.16 


347322.70 


733 


2302.79 


421985.79 


598 


1878.67 


280861.52 


666 


2092.30 


348368.07 


734 


2305.93 


423137.97 


599 


1881.81 


281801 65 


667 


2095.44 


349415.00 


735 


2309.07 


424291.72 


600 


1884.96 


282743.34 


668 


2098.58 


350463.51 


736 


2312.21 


425447.04 


601 


1888.10 


283686.60 


609 


2101.73 


351513.59 


737 


2315.35 


426603.94 


602 


1891.24 


284631.44 


670 


2104.87 


352565.24 


738 


2318.50 


427762.40 


603 


1894.38 


285577.84 


671 


2108.01 


353618.45 


739 


2321.64 


428922.43 


604 


1897.52 


286525.82 


672 


2111.15 


354673 24 


740 


2324.78 


430084.03 


605 


1900.66 


287475.36 


673 


2114.29 


355729.60 


741 


2327.92 


431247.21 


606 


1903.81 


288426.48 


674 


2117.43 


356787.54 


742 


2331.06 


432411.95 


607 


1906.95 


289379.17 


675 


2120.58 


357847.04 


743 


2334.20 


433578.27 


608 


1910.09 


290333.43 


676 


2123.72 


358908.11 


744 


2337.34 


434746.16 


609 


1913.23 


291289.26 


677 


2126.86 


359970.75 


745 


2340.49 


435915.62 


610 


1916.37 


292246.66 


678 


2130.00 


361034.97 


746 


2343.03 


437086.64 


611 


1919 51 


293205.63 


679 


2133.14 


362100.75 


747 


2346.77 


438259.24 


612 


1922.65 


294166.17 


680 


2136.28 


363168.11 


748 


2349.91 


439433.41 


613 


1925.80 


295128.28 


681 


2139.42 


364237.04 


749 


2353.05 


440609 16 


614 


1928.94 


296091.97 


682 


2142.57 


365307.54 


750 


2356.19 


441786.47 


615 


1932.08 


297057.22 


683 


2145.71 


360379.60 


751 


2359.34 


442965.35 


616 


1935.22 


298024.05 


684 


2148.85 


367453.24 


752 


2362.48 


444145.80 


617 


1938.36 


298992.44 


685 


2151.99 


368528.45 


753 


2365.02 


445327.83 


618 


1941.50 


299962.41 


686 


2155.13 


369005.23 


754 


2368.76 


446511.42 


619 


1944.65 


300933.95 


687 


2158.27 


370683.59 


755 


2371.90 


447696.59 


620 


1947.79 


301907.05 


688 


2161.42 


371763.51 


756 


2375.04 


448883.32 


621 


1950.93 


302881.73 


689 


2164.56 


372845.00 


757 


2378.19 


450071.63 


622 


1954.07 


303857.98 


690 


2167.70 


373928.07 


758 


2381.33 


451261.51 


623 


1957.21 


304835.80 


691 


2170.84 


375012.70 


759 


2384.47 


452452.96 


624 


1960.35 


305815.20 


692 


2173.98 


376098.91 


760 


2387.61 


453645.98 


625 


1963.50 


306796.16 


693 


2177.12 


377186.68 


761 


2390.75 


454840.57 


626 


1966.64 


307778.69 


694 


2180.27 


378276.03 


762 


2393.89 


456036.73 


627 


1969.78 


308762.79 


695 


2183.41 


379366.95 


763 


2397.04 


457234.46 


628 


1972.92 


309748.47 


696 


2186.55 


380459.44 


764 


2400.18 


458433.77 


629 


1976.06 


310735.71 


697 


2189.69 


381553.50 


765 


2403.32 


459034.64 


630 


1979.20 


311724.53 


698 


2192.83 


382649.13 


766 


2406.46 


460837.08 


631 


1982.35 


312714.92 


699 


2195.97 


383746.33 


767 


2409.60 


462041.10 


632 


1985.49 


313706.88 


700 


2199.11 


384845.10 


768 


2412.74 


463246.69 


633 


1988.63 


314700.40 


701 


2202.26 


385945.44 


769 


2415.88 


464453.84 


634 


1991.77 


315695.50 


702 


2205.40 


387047.36 


770 


2419.03 


465662.57 


635 


1994.91 


316692.17 


703 


2208.54 


388150.84 


771 


2422.17 


466872.87 


636 


1998.05 


317690.42 


704 


2211.68 


389255.90 


772 


2425.31 


468084.74 


637 


2001.19 


318690.23 


705 


2214.82 


390362.52 


773 


2428.45 


469298.18 


638 


2004.34 


319691.61 


706 


2217.96 


391470.72 


774 


2431.59 


470513.19 


639 


2007.48 


320694.56 


707 


2221.11 


392580.49 


775 


2434.73 


471729.77 


640 


2010.62 


321699.09 


708 


2224 25 


393091.82 


776 


2437.88 


472947.92 


641 


2013.76 


322705.18 


709 


2227.39 


394804.73 


777 


2441.02 


474167.05 


642 


2016.90 


323712.85 


710 


2230.53 


395919.21 


778 


2444.16 


475388.94 


643 


2020.04 


324722.09 


711 


2233.07 


397035 26 


779 


2447.30 


476611.81 


644 


2023.19 


325732.89 


712 


2236.81 


398152.89 


780 


2450.44 


477836.24 


645 


20-20.33 


326745.27 


713 


2239.96 


399272.08 


781 


2453.58 


479062.25 


646 


2029.47 


327759.22 


714 


2243.10 


400392.84 


782 


2456.73 


480289.83 


647 


2032.61 


328774.74 


715 


2240.24 


401515.18 


783 


2459.87 


481518.97 


648 


2035.75 


329791.83 


716 


2249.38 


402639.08 


784 


2463.01 


482749.69 


649 


2038.89 


330810.49 


717 


2252.52 


403764.56 


785 


2466.15 


483981.98 


650 


2042.04 


331830.72 


718 


2255.66 


404891.60 


786 


2469.29 


485215.84 


651 


2045.18 


332852.53 


719 


2258.81 


406020.22 


787 


2472.43 


486451.28 


652 


2048.32 


333875.90 


720 


2261.95 


407150.41 


788 


2475.58 


487688.28 


653 


2051.46 


334900.85 


721 


2265.09 


408282.17 


789 


2478.72 


488926.85 


654 


2054.60 


335927.36 


722 


2268.23 


409415.50 


790 


2481.86 


490166.99 


655 


2057.74 


336955.45 


723 


2271.37 


410550.40 


791 


2485.00 


491408.71 


656 


2060.88 


337985.10 


724 


2274.51 


411686.87 


792 


2488.14 


492651.99 


657 


2064.03 


339016.33 


725 


2277.05 


412824.91 


793 


2491.28 


493896.85 


658 


2007.17 


340049.13 


726 


2280.80 


413964.52 


794 


2494.42 


495143.28 


659 


2070.31 


341083.50 


727 


22S3.94 


415105.71 


795 


2497.57 


496391.27 


660 


2073.45 


342119.44 


728 


2287.08 


416248.46 


796 


2500.71 


497640.84 


661 


2076.59 


343156.95 


729 


2290.22 


417392.79 


797 


2503.85 


498891.98 


602 


2079.73 


344196.03 


730 


2293.36 


418538.68 


798 


2506.99 


500144.69 



CIRCUMFERENCES AND AREAS OF CIRCLES. 107 



Diam. 


Circum. 


Area. 


Diam. 


Circum. 


Area. 


Diam. 


Circum. 


Area. 


799 


2510.13 


501398.97 


867 


2723.76 


590375.16 


935 


2937.39 


686614.71 


800 


2513.27 


502654.82 


868 


2726.90 


591737.83 


936 


2940.53 


688084.19 


801 


2516.42 


503912.25 


869 


2730.04 


593102 06 


937 


2! 143.67 


689555.24 


802 


2519.56 


505171.24 


S70 


2733.19 


594467.87 


938 


2946 81 


691027.86 


803 


2522.70 


506431.80 


871 


2736.33 


595835.25 


939 


2949.96 


692502.05 


804 


2525.84 


507693.94 


872 


2739.47 


597204.20 


940 


2953.10 


693977.82 


805 


2528.98 


508957.64 


873 


2742.61 


598574.72 


941 


2956.24 


695455.15 


806 


2532.12 


510222.92 


874 


2745.75 


599946.81 


942 


2959.38 


696934.06 


807 


2535.27 


511489.77 


875 


2748.89 


601320.47 


943 


2962.52 


698414.53 


808 


2538.41 


512758.19 


876 


2752.04 


602695.70 


944 


2965.66 


699896.58 


S09 


2541.55 


514028.18 


877 


2755.18 


604072.50 


945 


2968.81 


701380.19 


810 


2544.69 


515299.74 


878 


2758.32 


605450.88 


946 


2971.95 


702865.38 


811 


2547.83 


516572.87 


879 


2761.46 


606830.82 


947 


2975.09 


704352.14 


812 


2550.97 


517847.57 


880 


2764.60 


608212.34 


948 


2>)78.23 


705840.47 


813 


2554.11 


519123.84 


881 


2767.74 


609595.42 


949 


2981.37 


707330.37 


814 


2557.26 


520401.68 


882 


2770.88 


610980.08 


950 


2984.51 


708821.84 


815 


2560.40 


521681.10 


883 


2774.03 


612366.31 


951 


2987.65 


710314.88 


816 


2563.54 


522962.08 


884 


2777.17 


613754.11 


952 


2990.80 


711809.50 


817 


2566.68 


524244.63 


885 


2780.31 


615143.48 


953 


2993.94 


713305.68 


818 


2569.82 


525528.76 


886 


2783.45 


616534.42 


954 


2997.08 


714803.43 


819 


2572.96 


526814.46 


887 


2786.59 


617926.93 


955 


3000.22 


716302.76 


820 


2576.11 


528101.73 


888 


2789.73 


619321.01 


956 


3003.36 


717803.66 


821 


2579.25 


529390.56 


889 


2792.88 


620716.66 


957 


3006.50 


719306.12 


822 


2582.39 


530680.97 


890 


2 71)0.02 


622113.89 


958 


3009.65 


720810.16 


823 


2585.53 


531972.95 


891 


2799.16 


623512.68 


959 


3012.79 


722315.77 


824 


2588.67 


533266.50 


892 


2802.30 


624913.04 


960 


3015.93 


723822.95 


825 


2591.81 


534561.62 


893 


2805.44 


626314.98 


961 


3019.07 


725331.70 


826 


2594.96 


535858.32 


894 


2808.58 


627718.49 


962 


3022.21 


726842.02 


827 


2598.10 


537156.58 


895 


2811.73 


629123.56 


963 


3025.35 


728353.91 


828 


2601.24 


538456 41 


896 


2814.87 


630530.21 


964 


3028.50 


729867.37 


829 


2604.38 


539757.82 


897 


2818.01 


631938.43 


965 


3031.64 


731382.40 


830 


2607.52 


541060.79 


898 


2821.15 


633348.22 


966 


3034.78 


732899.01 


831 


2610.66 


542365.34 


899 


2824.29 


634759.58 


967 


3037.92 


734417.18 


832 


2613.81 


543671.46 


900 


2827.43 


636172.51 


968 


3041.06 


735936.93 


833 


2616.95 


544979.15 


901 


2830.58 


637587.01 


969 


3044.20 


737458.24 


834 


2620.09 


546288.40 


902 


2833.72 


639003.09 


970 


3047.34 


738981.13 


835 


2023.23 


547599.23 


903 


2836.86 


640420.73 


971 


3050.49 


740505.59 


836 


2626.37 


548911.63 


904 


2840.00 


641839.95 


972 


3053.63 


742031.62 


S37 


2629.51 


550225.61 


905 


2843.14 


643260.73 


973 


3056.77 


743559.22 


838 


2032,65 


551541.15 


906 


2846.28 


644683.09 


974 


3059.91 


745088.39 


839 


2635.80 


552858.26 


907 


2849.42 


646107.01 


975 


3063.05 


746619.13 


840 


2638.94 


554176.94 


908 


2852.57 


647532.51 


976 


3066.19 


748151.44 


: 841 


2642.08 


555497.20 


909 


2855.71 


648959.58 


977 


3069.34 


749685.32 


842 


2645.22 


556819.02 


910 


2858.85 


6503S8.22 


978 


3072.48 


751220.78 


843 


2648 36 


558142.42 


911 


2861.99 


651818.43 


979 


3075.62 


752757.80 


844 


2651.50 


559467.39 


912 


2865.13 


653250.21 


980 


3078.76 


754296.40 


845 


2654.65 


560793.92 


913 


2868.27 


654683.56 


981 


3081.90 


755836.56 


846 


2657.79 


562122.03 


914 


2871.42 


656118.48 


982 


3085.04 


757378.30 


847 


2660.93 


563451.71 


915 


2874.56 


657554.98 


983 


3088.19 


758921.61 


848 


2664.07 


564782.96 


916 


2877.70 


658993.04 


984 


3091.33 


760466.48 


849 


2667.21 


566115.78 


917 


2880.84 


660432.68 


985 


3094.47 


762012.93 


850 


2670.35 


567450.17 


918 


2883.98 


661873.88 


986 


3097.61 


763560.95 


851 


2673.50 


568786.14 


919 


2887.12 


663316 66 


987 


3100.75 


765110.54 


852 


2676.64 


570123.67 


920 


2890.27 


664761.01 


988 


3103.89 


766661.70 


853 


2679.78 


571462.77 


921 


2893.41 


666206.92 


989 


3107.04 


768214.44 


854 


2682.92 


572803.45 


922 


2896.55 


667654.41 


990 


3110.18 


769768.74 


855 


2686.06 


574145.69 


923 


2899.69 


669103.47 


991 


3113.32 


771324.61 


856 


2689.20 


575489.51 


924 


2902.83 


670554.10 


992 


3116.46 


772882.06 


857 


2692.34 


576834.90 


925 


2905.97 


672006.30 


993 


3119.60 


774441.07 


858 


2695.49 


578181.85 


926 


2909.11 


673460.08 


994 


3122.74 


776001.66 


859 


2698.63 


579530.38 


927 


2912.26 


674915.42 


995 


3125.88 


777563.82 


860 


2701.77 


580880.48 


928 


2915.40 


676372.33 


996 


3129.03 


779127.54 


861 


2704.91 


582232.15 


929 


2918.54 


677830.82 


997 


3132.17 


780692.84 


862 


2708.05 


583585.39 


930 


2921.68 


679290.87 


998 


3135.31 


782259.71 


863 


2711.19 


584940.20 


931 


2924.82 


680752.50 


999 


3138.45 


783828.15 


864 


2714.34 


586296.59 


932 


2927.96 


682215.69 


1000 


3141.59 


785398.16 


865 


2717.48 


587654.54 


933 


2931.11 


683680.46 








866 


2720 62 


589014.07 


934 


2934.25 


685146.80 

























108 




MATHEMATICAL 


TABLES. 






CIRCUMFERENCES 


AND AREAS OF 


CIRCLES 






Advancing l>y Eighths. 






Diam. 


Circum. 


Area. 


Diam. 


Circum. 


Area. 


Diain. 


Circum. 


Area, j 


1/64 


.04909 


.00019 


2 Va 


7.4613 


4.4301 


6 Va 


19.242 


29.465 1 


1/32 


.09818 


.00077 


7/16 


7.6576 


4.6664 


Va 


19.635 


30.680 


3/64 


.14726 


.00173 


H 


7.8540 


4.9087 


Va 


20.028 


31.919 


1/16 


.19635 


.00307 


9/16 


8.0503 


5.1572 


Vz 


20.420 


33.183 


3/32 


.29452 


.00690 


Va 


8.2467 


5.4119 


% 


20.813 


34.472 


H 


.39270 


.01227 


11/16 


8.4430 


5.6727 


H 


21 206 


35.785 


5/32 


.49087 


.01917 


Va 


8.6394 


5.9396 


Va 


21.598 


37.122 


3/16 


.58905 


.02761 


13/16 


8.8357 


6.2126 


7. 


21.991 


38.485 


7/32 


.68722 


.03758 


% 


9.0321 


6.4918 


H 


22.384 


39.871 








15/16 


9.2284 


6.7771 


H 


22.776 


41.282 


H 


.78540 


.01909 








Va 


23.169 


42.718 


9/32 


.88357 


.06213 


3. 


9.4248 


7.0686 


H 


23.562 


44.179 


5/16 


.98175 


.07670 


1/16 


9.6211 


7.3662 


Va 


23.955 


45 664 


11/32 


1.0799 


.09281 


Va 


9.8175 


7.6699 


H 


24.347 


47.173 


Va 


1.1781 


.11045 


3/16 


10.014 


7.9798 


Va 


24.740 


48.707 


13/32 


1.2763 


.12962 


H 


10.210 


8.2958 


8. 


25.133 


50.265 


7/16 


1.3744 


.15033 


5/16 


10.407 


8.6179 


H 


25.525 


51.849 


15/32 


1.4726 


. 17257 


% 


10.603 


8.9462 


Va 


25.918 


53.456 








7/16 


10.799 


9.2806 


% 


26.311 


55.088 


M 


1 5708 


.19635 


y* 


10.996 


9.6211 


% 


26.704 


56.745 


17/32 


1.6690 


.22160 


9/16 


11.192 


9.9678 


Va 


27.096 


58.426 


9/16 


1.7671 


.24850 


Va 


11.388 


10.321 


¥a 


27.489 


60.132 


19/32 
Va 


1.8653 


.27688 


11/16 


11.585 


10.680 


Va 


27.882 


61.862 


1.9635 


.30680 


u 


11.781 


11.045 


9. 


28.274 


63.617 


21/32 


2.0617 


.33824 


13/16 


11.977 


11.416 


Va 


28.667 


65.397 


11/16 


2.1598 


.37122 


Va 


12.174 


11.793 


X A 


29.060 


67.201 


23/32 


2.2580 


.40574 


15/16 


12.370 


12.177 


Va 


29.452 


69.029 








4. 


12.566 


12.566 


y* 


29.845 


70.SS2 


Va 

25/32 


2.3562 


.44179 


1/16 


12.763 


12.962 


Va 


30.238 


72.760 


2.4544 


.47937 


H 


12.959 


13.364 


8 


30.631 


74.662 


13/16 


2.5525 


.51849 


3/16 


13.155 


13.772 


31.023 


76.589 


27/32 


2.6507 


.55914 


H 


13.352 


14.186 


10. 


31.416 


78.540 


% 


2.7489 


.60132 


5/16 


13.548 


14.607 


H 


31.809 


80.516 


29/32 


2.8471 


.64504 


Va 


13.744 


15.033 


% 


32.201 


82.516 


15/16 


2.9452 


.69029 


7/16 


13.941 


15.466 




32.594 


84.541 


31/82 


3.0434 


.73708 


% 


14.137 


15.904 


Va 


32.987 


86.590 








9/16 


14.334 


16.349 


33.379 


88.664 


1. 


3.1416 


.7854 


Va 


14.530 


16.800 


Va 


33.772 


90.763 


1/16 


3.3379 


.8866 


11/16 


14.726 


17.257 


Va 


34.165 


92.886 


Va 


3.5343 


.9940 


Va 


14.923 


17.728 


11. 


34.558 


95.033 


3/16 


3.7306 


1.1075 


13/16 


15.119 


18.190 


Va 


34.950 


97.205 


k 


3.9270 


1.2272 


Va 


15.315 


18.665 


Va 

i 


35.343 


99.402 


5/16 


4.1233 


1.3530 


15/16 


15 512 


19.147 


35.736 


101.62 


% 


4.3197 


1.4849 


5. 


15.708 


19.635 


36.128 


103.87 


7/16 


4.5160 


1.6230 


1/16 


15.904 


20.129 


Va 


36.521 


106.14 


^ 


4.7124 


1.7671 


H 


16.101 


20.629 


Va 


36.914 


108.43 


9/16 


4.9087 


1.9175 


3/16 


16.297 


21.135 


Vs 


37.306 


110.75 


Va 


5.1051 


2.0739 


H 


16.493 


21.648 


12. 


37.699 


113.10 


11/16 


5.3014 


2.2365 


5/16 


16.690 


22.166 


Va 


38.092 


115.47 


% 


5.4978 


2.4053 


Va 


16.886 


22.691 


Va 


38.485 


117.86 


13/16 


5.6941 


2.5S02 


7/16 


17.082 


23.221 


Va 


38.877 


120.28 


Va 


5.8905 


2.7612 


M 


17.279 


23.758 


Vz 


39.270 


122.72 


15/16 


6.0868 


2.9483 


9/16 


17.475 


24.301 


Va 


39.663 


125.19 








Va 


17.671 


24.850 


Va 


40.055 


127.68 


o 


6.2832 


3.1416 


11/16 


17.868 ' 


25.406 


Va 


40.448 


130.19 


1/16 


6.4795 


3.3410 


n 


18.064 


25.967 


13. 


40.841 


132.73 


% 


6.6759 


3.5466 


13-16 


18.261 


26.535 


Va 


41.233 


135.30 


3/16 


6.8722 


3.7583 


% 


18.457 


27.109 


Va 


41.626 


137.89 


Va 


7.0686 


3.9761 


15-16 


18.653 


27.688 


Va 


42.019 


140.50 


5/16 


7.2649 


4.2000 


6. 


18.850 


28.274 


K 


42.412 


143.14 





















CIRCUMFERENCES AND AREAS OF CIRCLES. 



109 



Diam. 


Circum . 


Area. 


Diam. 


Circum. 


Area. 


Diam. 


Circum. 


Area. 


13% 


42.804 


145.80 


21% 


68.722 


375.83 


30% 


94.640 


712.76 


Va 


43.197 


148.49 


32. 


69.115 


380.13 


Va 


95.033 


728.69 


% 


43.590 


151.20 


% 


69.508 


384.46 


Vs 


95.426 


724.64 


14 


43.982 


153.94 


Va 


69.900 


388.82 


V* 


95.819 


730.62 


Vs 


44.375 


156.70 


Vs 


70.293 


393.20 


Vs 


96.211 


736.62 


Va 


44.768 


159.48 


Vq 


70.686 


397.61 


Va 


96.604 


742.64 


Vs 


45.160 


162.30 


Vs 


71.079 


402.04 


Vs 


96.997 


748.69 


% 


45 . 553 


165.13 


Va 


71.471 


406.49 


31. 


97.389 


754.77 


45.946 


167.99 


Vs 


71.864 


410.97 


Vs 


97.782 


760.87 


% 


46.338 


170.87 


23. 


72.257 


415.48 


Va 


98.175 


766.99 


Vs 


46.731 


173.78 


Vs 


72.649 


420.00 




98.567 


773.14 


15. 


47.124 


176.71 


Va 


73.042 


424.56 


V% 

Vs 


98.960 


779.31 


% 


47.517 


179.67 


Vs 


73.435 


429.13 


99.353 


785.51 


Va 


47.909 


182.65 


Vi 


73.827 


433.74 


1 


99.746 


791.73 


% 


48.302 


185.66 


Vs 


74.220 


438.36 


100.138 


797.98 


V* 


48.695 


188.69 


Va 


74.613 


443.01 


32. 


100.531 


804.25 


Vs 


49.087 


191.75 


Vs 


75.006 


447.69 


~ Vs 


100.924 


810.54 


Va 


49.480 


194.83 


24 


75.398 


452.39 


Va 


101.316 


816.86 


Vs 


49.873 


197.93 


Vs 


75.791 


457.11 


Vs 


101.709 


823.21 


16 


50.265 


201.06 


Va 


76.184 


461.86 


Vz 


102.102 


829.58 


% 


50.658 


204.22 


Vs 


76.576 


466.64 


Vs 


102.494 


835.97 


Va 


51.051 


207.39 


Vz 


76.969 


471.44 


¥ 


102.887 


842.39 


% 


51.414 


210.60 


Vs 


77.362 


476.26 




103.280 


848.83 


H 


51.836 


213.82 


Va 


77.754 


481.11 


33 8 


103.673 


855.30 


Vs 


52.229 


217.08 


Vs 


78.147 


485.98 


Vs 


104.065 


861.79 


K 


52.622 


220.35 


25. 


78.540 


490.87 


Va 


104.458 


868.31 


Vs 


53.014 


223.65 


Vs 


78.933 


495.79 


Vs 


104.851 


874.85 


17 


53.407 


226.98 


Va 


79.325 


500.74 


Vk 


105.243 


881.41 


% 


53.800 


230.33 


Vs 


79.718 


505.71 


Vs 


105.636 


888.00 


Va 


54.192 


233.71 


Mi 


80.111 


510.71 


Va 


106.029 


894.62 




54.585 


237.10 


Vs 


80.503 


515.72 


Vs 


106.421 


901.26 


vl 


04.978 


240.53 


Va 


80.896 


520.77 


34 


106.814 


907.92 




55.371 


243.98 


Vs 


81.289 


525.84 


Vs 


107.207 


914.61 


Va 


55 . 763 


247.45 


26 


81.681 


530.93 


1 


107.600 


921.32 


Vs 


56.156 


250.95 


Vs 


82.074 


536.05 


107.992 


928.06 


18 


56.549 


254.47 


Va 


82.467 


541.19 


Vz 


108.385 


934.82 


Vs 


56.941 


258.02 


Vs 


82 860 


546.35 




108.778 


941.61 


Va 


57.334 


261.59 


ri 


83.252 


551 . 55 


Va 


109.170 


948.42 


Vs 


57.727 


265.18 


83.645 


556.76 


Vs 


109.563 


955.25 


Vz 


58.119 


268.80 


Va 


84.038 


562.00 


35. 


109.956 


962.11 


Vs 


58.512 


272.45 


Vs 


84.430 


567.27 


Vs 


110.348 


969.00 


Va 


58.905 


276.12 


27 


84.823 


572.56 


Va 


110.741 


975.91 


Vs 


59.298 


279.81 


Vs 


85.216 


577.87 


Vs 


111.134 


982.84 


19. 


59.690 


283.53 


Va 


85.608 


583.21 


H 


111.527 


989.80 


H 


60.083 


287.27 


Vs 


86.001 


588.57 


Vs 


111.919 


996.78 


Va 


60.476 


291.04 


V2 


86.394 


593.96 


Va 


112.312 


1003.8 


Vs 


60.868 


294.83 


Vs 


86.786 


599.37 


Vs 


112.705 


1010.8 


% 


61.261 


298.65 


Va 


87.179 


604.81 


36. 


113.097 


1017.9 


Vs 


61.654 


302.49 


Vs 


87.572 


610.27 


Vs 


113.490 


1025.0 


Va 


62.046 


306.35 


28 


87.965 


615.75 


Va 


113.883 


1032.1 


Vs 


62.439 


310.24 


Vs 


88.357 


621.26 


Vs 


114.275 


1039.2 


20. 


62.832 


314.16 


Va 


88.750 


626 80 


V* 


114.668 


1046.3 


% 


63.225 


318.10 


Vs 


89.143 


632.36 


Vs 


115.061 


1053.5 


% 


63.617 


322.06 


H 


89.535 


637.94 


Va 


115.454 


1060.7 


64.010 


326.05 


89.928 


643.55 


Vs 


115.846 


1068.0 


Vz 


64.403 


330.06 


Va 


90.321 


649.18 


37 


116.239 


1075.2 


% 


64.795 


334.10 


Vs 


90.713 


654.84 


Vs 


116.632 


1082.5 


Va 


65.188 


338.16 


29 


91.106 


660.52 


Va 


117.024 


1089.8 


Vs 


65.581 


342.25 


Vs 


91 .499 


666.23 


Vs 


117.417 


1097.1 


21. 


65.973 


346.36 


Vs 


91.892 


671.96 


Vz 


117.810 


1104.5 


% 


66.366 


350.50 


92.284 


677.71 




118.202 


1111.8 


Va 


66.759 


354.66 


Vs 


92.677 


683.49 


Va 


118.596 


1119.2 


Vs 


67.152 


358.84 


93.070 


689.30 


Vs 


118.988 


1126.7 


H 


67.544 


363.05 


Va 


93.462 


695.13 


38. 


119.381 


1134.1 


Vs 


67.937 


367.28 


Vs 


93.855 


700.98 


Vs 


119.773 


1141.6 


■ Va 


68.330 


371.54 


30. 


94.248 


706 86 


Va 


120.166 


1149.1 



110 



MATHEMATICAL TABLES. 



Diam. 


Circum. 


Area. 


Diam. 


Circum. 


Area. 


Diam. 


Circum. 


Area. 


s$% 


120.559 


1156.6 


46% 


146.477 


1707.4 


54% 


172.395 


2365.0 


y a 


120.951 


1164.2 


Va 


146.869 


1716.5 


55. 


172.788 


2375.8 


% 


121.344 


1171.7 


Va 


147.262 


1725.7 


Va 


173.180 


2386.6 


u 


121.737 


1179.3 


47. 


147.655 


1734.9 


Va 


173.573 


2397.5 


Va 


122.129 


1186.9 


Va 


148.048 


1744.2 


Va 


173.966 


2408.3 


39. 


122.522 


1194.6 


Va 


148.440 


1753.5 


3 


174.358 


2419.2 


v& 


122.915 


1202.3 


148.833 


1762.7 


174.751 


2430.1 


v± 


128.308 


1210.0 


Mi 


149.226 


1772.1 


Va 


175.144 


2441.1 


v& 


123.700 


1217.7 


Va 


149.618 


1781.4 


Va 


175.536 


2452.0 


g 


124.093 


1225.4 


Va 


150.011 


1790.8 


56. 


175.929 


2463.0 


124.486 


1233.2 


Va 


150.404 


1800.1 


Va 


176.322 


2474.0 


% 


124.878 


1241.0 


48. 


150.796 


1809.6 


Va 


176.715 


2185.0 


% 


125.271 


1248.8 


Va 


151.189 


1819.0 


Va 


177.107 


2196.1 


40. 


125.664 


1256.6 


' Va 


151.582 


1828.5 


It 


177.500 


2507.2 


Va 


126.056 


1264.5 


Va 


151.975 


1837.9 


177.893 


2518.3 


Va 


126.449 


1272.4 


II 


152.367 


1847.5 


Va 


178.285 


2529.4 


Va 


126.842 


12S0.3 


152.760 


1857.0 


Va 


178.678 


2540.6 


t 


127.235 


1288.2 


Va 


153.153 


1866.5 


57. 


179.071 


2551.8 


127.627 


1296.2 


Va 


153.545 


1876.1 


Va' 


179.463 


2563.0 


H 


128.020 


1304.2 


49. 


153.938 


1885.7 


Va 


179.856 


2574.2 


Va 


128.413 


1312.2 


Va 


154.331 


1895.4 


Va 


180.249 


2585.4 


41. 


128.805 


1320.3 


Va 


154.723 


1905.0 


Vz 


180.642 


2596.7 


Va 


129.198 


1328.3 


Va 


155.116 


1914.7 


Va 


181.034 


2608.0 


Va 


129.591 


1336.4 


V* 


155.509 


1924.4 


Va 


181.427 


2619.4 


a 


129.983 


1344.5 


Va 


155.902 


1934.2 


Va 


181.820 


2630.7 


130.376 


1352.7 


Va 


156.294 


1943.9 


58. 


182.212 


2642.1 


% 


130.769 


1360.8 


Va 


156 687 


1953.7 


Va 


182.605 


2653.5 


% 


131.161 


1369.0 


50. 


157.080 


1963.5 


Va 


182.998 


2664.9 


% 


131.554 


1377.2 


Va 


157.472 


1973.3 


Va 


183.390 


2676.4 


42. 


131.947 


1385.4 


Va 


157.865 


1983.2 


V2 


183.783 


2687.8 


Va 


132.340 


1393.7 


Va 


158.258 


1993.1 


Va 


184.176 


2699.3 


a 


132.732 


1402.0 


Mi 


158.650 


2003.0 


Va 


184.569 


2710.9 


133.125 


1410.3 


Va 


159.043 


2012.9 


Va 


184.961 


2722.4 


l A 


133.518 


1418.6 


Va 


159.436 


2022.8 


59. 


185.354 


2734.0 


Va 


133.910 


1427.0 


Va 


159.829 


2032.8 


Va 


185.747 


2745.6 


Va 


134.303 


1435.4 


51 


160.221 


2042.8 


Va 


186.139 


2757.2 


Va 


134.696 


1443.8 


Va 


160.614 


2052.8 


Va 


186.532 


2768.8 


43. 


135.088 


1452.2 


Va 


161.007 


2062.9 


Va 


1S6.925 


2780.5 


Va 


135.481 


1460.7 




161.399 


2073.0 


187.317 


2792.2 


Va 


135.874 


1469.1 


4 
Va 


161.792 


2083.1 


Va 


187.710 


2803.9 


Va 


136.267 


1477.6 


162.185 


2093.2 


Va 


188.103 


2815.7 


l A 


136.659 


1486.2 


Va 


162.577 


2103.3 


60. 


188.496 


2827.4 


Va 


137.052 


1494.7 


Va 


162.970 


2113.5 


Va 


188.888 


2839.2 


Va 


137.445 


1503.3 


53. 


163.363 


2123.7 


Va 


189.281 


2851.0 


Va 


137.837 


1511.9 


Va 


163.756 


2133.9 


Va 


189.674 


2862.9 


44 


138.230 


1520.5 


Va 


164.148 


2144.2 


Mi 


190.066 


2874.8 


X A 


138.023 


1529.2 


164.541 


2154.5 


Va 


190.459 


2886.6 


Va 


139.015 


1537.9 


II 


164.934 


2164.8 


Va 


190.852 


2898.6 


Vs. 


139.408 


1546.6 


165.326 


2175.1 


Va 


191.244 


2910.5 


Vz 


139.801 


1555.3 


% 


165.719 


2185.4 


61. 


191.637 


2922.5 


Va 


140.194 


1564.0 


Va 


166.112 


2195.8 


Va 


192.030 


2934.5 


Va 


140.586 


1572.8 


53. 


166.504 


2206.2 


Va 


192.423 


2946.5 


Va 


140.979 


1581.6 


Va 


166.897 


2216.6 


Va 


192.815 


2958.5 


45 


141.372 


1590.4 


Va 


167.290 


2227.0 


Vz 


193.208 


2970.6 


Va 


141.764 


1599.3 


Va 


167.683 


2237.5 


Va 


193.601 


2982.7 


Va 


142.157 


1608.2 


H 


168.075 


2248.0 


Va 


193.993 


2994.8 


Va 


142.550 


1617.0 


Va 


168.468 


2258.5 


% 


194.386 


3006 9 


Mi 


142.942 


1626.0 


Va 


168.861 


2269.1 


62. 


194.779 


3019.1 


Va 


143.335 


1634.9 


% 


169.253 


2279.6 


Va 


195.171 


3031.3 


Va 


143.728 


1643.9 


54. 


169.646 


2290.2 


Va 


195.564 


3043.5 


Va 


144.121 


1652.9 


Va 


170.039 


2300.8 


Va 


195.957 


3055.7 


46 


144.513 


1661.9 


Va 


170.431 


2311.5 


V% 


196.350 


3068.0 


Va 


144.906 


1670.9 




170.824 


2322.1 


Va 


196.742 


3080.3 


Va 


145.299 


1680.0 


m\ 


171.217 


2332.8 


Va 


197.135 


3092.6 


Va 


145.691 


1689.1 


Va 


171.609 


2343.5 


Va 


197.528 


3104.9 


y* 


146.084 


1698.2 


Va 


-172.002 


2354.3 


63 


197.920 


3117.2- 



CIRCUMFERENCES AND AREAS OF CIRCLES. 



ill 



Diam. 


Circum. 


Area. 


Diam. 


Circum. 


Area. 


Diam. 


Circum. 


Area. 


63^ 


198.313 


3129.6 


71% 


224.231 


4001.1 


79% 


250.149 


4979.5 


Va 


198.706 


3142.0 


Vz 


224.624 


4015.2 


Va 


250 542 


4995.2 


% 


199.098 


3154.5 


Vs 


225.017 


4029.2 


Vs 


250.935 


5010.9 


- Vz 


199.491 


3166.9 


u 


225.409 


4043.3 


80 


251.3-27 


5026.5 


% 


199.884 


3179.4 


Vs 


225.802 


4057.4 


Vs 


251.720 


5042.3 


u 


200.277 


3191.9 


72. 


226.195 


4071.5 


Va 


252.113 


5058.0 


Vs 


200.669 


3204.4 


Vs 


226.587 


4085.7 


Vs 


252.506 


5073.8 


64. 


201.062 


3217.0 


Va 


226.980 


4099.8 




252.898 


5089.6 


Vs 


201.455 


3229.6 


Vs 


227.373 


4114.0 


% 


253.291 


5105.4 


Va 


201.847 


3242.2 




227.765 


4128.2 


Va 


253.684 


5121.2 


Vs 


202.240 


3254.8 


% 


228.158 


4142.5 


Vs 


254.076 


5137.1 


Vz 


202.633 


3267.5 


8 


228.551 


4156.8 


81. 


254.469 


5153.0 


Vs 


203.025 


3280.1 


228.944 


4171.1 


Vs 


254.862 


5168.9 


Va 


203.418 


3292.8 


73 


229.336 


4185.4 


Va 


255.254 


5184.9 


Vs 


203.811 


3305.6 


Vs 


229.729 


4199.7 


Vs 


255.647 


5200.8 


65. 


204.204 


3318.3 


Va 


230.122 


4214.1 


g 


256.040 


5216.8 


Vs 


204.596 


3331.1 


Vs 


230.514 


4228.5 


256.433 


5232.8 


Va 


204.989 


3343.9 


Vz 


230.907 


4242.9 


S 


256.825 


5248.9 


Vs 


205.382 


3356.7 


Vs 


231.300 


4257.4 


257.218 


5264.9 


Vz 


205.774 


3369.6 


g 


231.692 


4271.8 


82. 


257.611 


5281.0 


% 


206.167 


3382.4 


232.085 


4286.3 


% 


258.003 


5297.1 




206.560 


3395.3 


74. 


232.478 


4300.8 


Va 


258.396 


5313.3 


% 


206.952 


3408.2 


Vs 


232.871 


4315.4 


Vs 


258.789 


5329.4 


66. 


207.345 


3421.2 


Va 


233.263 


4329.9 


Vz 


259.181 


5345.6 


Vs 


207.738 


3434.2 


Vs 


233 656 


4344.5 


Vs 


259.574 


5361.8 


Va 


208.131 


3447.2 


Vz 


234.019 


4359.2 


M 


259.967 


5378.1 


Vs 


208.523 


3460.2 


Vs 


234.441 


4373.8 


% 


260.359 


5394.3 


Vs 


208.916 


3473.2 


1 


234.834 


4388.5 


83. 


260.752 


5410.6 


209.309 


3486.3 


235.227 


4403.1 


Vs 


261.145 


5426.9 


Va 


209.701 


3499.4 


75. 


235.619 


4417.9 


Va 


261.538 


5443.3 


Vs 


210.094 


3512.5 


Vs 


236.012 


4432.6 


Vs 


261.930 


5459.6 


67. 


210.487 


3525 7 


Va 


236 . 405 


4447.4 


Vz 


262.323 


5476.0 


% 


210.879 


3538.8 


Vs 


236.798 


4462.2 


Vs 


262.716 


5492.4 


Va 


211.272 


3552.0 


Vz 


237.190 


4477.0 


M 


263.108 


5508.8 


Vs 


211.665 


3565.2 


Vs 


237.583 


4491.8 


Vs 


263.501 


5525.3 


H 


212.058 


3578.5 


M 


237.976 


4506.7 


84. 


263.894 


5541.8 


Vs 


212.450 


3591.7 


Vs 


238.368 


4521.5 


Vs 


264.286 


5558.3 


Va 


212.843 


3ti05.0 


76. 


238.761 


4536.5 


Va 


264.679 


5574.8 


Vs 


213.236 


3618.3 


Vs 


239.154 


4551.4 


Vs 


265.072 


5591.4 


68. 


213.628 


3631.7 


Va 


239.546 


4566.4 


Vz 


265.465 


5607.9 


Vs 


214.021 


3645.0 


Vs 


239.939 


4581.3 


Vs 


265.857 


5624.5 


Va 


214.414 


3658.4 


Vz 


240.332 


4596.3 


Va 


266.250 


5641.2 


Vs 


214.806 


3671.8 


Vs 


240.725 


4611.4 


Vs 


266.643 


5657.8 


Vz 


215.199 


3685.3 


Va 


241.117 


4626.4 


85. 


267.035 


5674.5 


% 


215.592 


3698.7 


Vs 


241.510 


4641.5 


Vs 


267.428 


5691.2 


Va 


215.984 


3712.2 


77. 


241.903 


4656.6 


I 


267.821 


5707.9 


% 


216.377 


3725.7 


Vs 


242.295 


4671.8 


268.213 


5724.7 


69. 


216.770 


3739.3 


Va 


242.688 


4686.9 


Vz 


268.606 


5741.5 


Vs 


217.163 


3752.8 


Vs 


243.081 


4702.1 


Vs 


268.999 


5758.3 


Va 


217.555 


3766.4 


Vz 


243.473 


4717.3 


S 


269.392 


5775.1 


M 


217.948 


3780.0 


Vs 


243.866 


4732.5 


269.784 


5791.9 


Vz 


218.341 


3793.7 


Va 


244.259 


4747.8 


86. 


270.177 


5808.8 


Vs 


218.738 


3807.3 


Vs 


244 . 652 


4763.1 


Vs 


270.570 


5825.7 


Va 


219.126 


3821.0 


78. 


245.044 


4778.4 


8 


270.962 


5842.6 


Vs 


219.519 


3834.7 


Vs 


245 437 


4793.7 


271.355 


5859.6 


70. 


219.911 


3S48.5 


% 


245.830 


4809.0 


II 


271.748 


5876.5 


Vs 


220.304 


3862.2 


246.222 


4824.4 


272.140 


5893.5 


Va 


220.697 


3876.0 


Vz 


246.615 


4839 8 


M 


272.533 


5910.6 


Vs 


221.090 


3889.8 


Vs 


247.008 


4855.2 


Vs 


272.926 


5927.6 


Vz 


221.482 


3903.6 


Va 


247.400 


4870.7 


87. 


273.319 


5944.7 


Vs 


221.875 


3917.5 


Vs 


247.793 


4886. 2 


Vs 


273.711 


5961.8 


Va 


222.268 


3931.4 


79. 


248.186 


4901.7 


Va 


274.104 


5978.9 


Vs 


222.660 


3945.3 


Vs 


248.579 


4917.2 


Vs 


274.497 


5996.0 


71. 


223.053 


3959.2 


Va 


248.971 


4932.7 


Vz 


274.889 


6013.2 


Vs 


223.446 


3973.1 


Vs 


249.364 


4948.3 


Vs 


275.282 


6030.4 


Va 


223.838 


3987.1 


Vz 


249.757 


4963.9 


Va 


275.675 


6047.6 



112 



MATHEMATICAL TABLES. 



Diam. 


Circum. 


Area. 


Diam. 


Circum. 


Area. 


Diam. 


Circum. 


Area. 


87% 


276,067 


6064.9 


92. 


289.027 


6647.6 


96^ 


301.986 


7257.1 


88. 


276.460 


6082.1 


Va 


289.419 


6665.7 


Va 


302.378 


7276.0 


H 


276.853 


6099.4 


Va 


289.812 


6683.8 


302.771 


7294.9 


U 


277.246 


6116.7 


Va 


290.205 


6701.9 


y% 


303.164 


7313.8 




277.638 


6134.1 


Va 


290.597 


6720.1 


Va 


303.556 


7332.8 


M& 


278.031 


6151.4 


290.990 


6738.2 


Va 


303.949 


7351.8 


% 


278.424 


6168.8 


% 


291.383 


6756.4 


Va 


304.342 


7370.8 


% 


278.816 


6186.2 


% 


291.775 


6774.7 


97. 


304.734 


7389.8 


% 


279.209 


6203.7 


93. 


292.168 


6792.9 


Va 


305.127 


7408.9 


89. 


279.602 


6221.1 


Va 


292.561 


6811.2 


Va 


305.520 


7428.0 


Va 


279.994 


6238.6 


H 


292.954 


6829.5 


% 


305.913 


7447.1 


8 


280.387 


6256.1 


Va 


293.346 


6847.8 


V% 


306.305 


7466.2 


280.780 


6273.7 


y* 


293.739 


6866.1 


Va 


306.698 


7485.3 


M 


281.173 


6291.2 


% 


294.132 


6884.5 


Va 


307.091 


7504.5 


281.565 


6-08.8 


n 


294.524 


6902.9 


Va 


307.483 


7523.7 


S 


281.958 


6326.4 


294.917 


6921.3 


98 


307.876 


7543.0 


282.351 


6344.1 


94 


295.310 


6939.8 


Va 


308.269 


7562.2 


90. 


282.743 


6361.7 


y& 


295.702 


0958.2 


Va 


308.661 


7581.5 


Va 


283.136 


6379.4 


Va 


296.095 


6976.7 


% 


309.054 


7600.8 


a 


283.529 


6397.1 


Va 


296.488 


6995.3 


8 


309.447 


7620.1 


283.921 


6414.9 


M 


296.881 


7013.8 


309.840 


7639.5 


Ps 


284.314 


6432.6 


Va 


297.273 


703^.4 


Va 


310.232 


7658.9 


284.707 


6450.4 


s 


297.666 


7051.0 


Va 


310.625 


7678.3 


% 


285.100 


6468.2 


298.059 


7069.6 


99. 


311.018 


7697.7 


285.492 


6486.0 


95. 


298.451 


7088.2 


Va 


311.410 


7717.1 


91. 


285.885 


6503.9 


n 


298.844 


7106.9 


Va 


311.803 


7736.6 


Va 


286.278 


6521.8 


299.237 


7125.6 


Va 


312.196 


7756.1 


M 


286.670 


6539.7 


% 


299.629 


7144.3 


8 


312.588 


7775.6 


% 


287.063 


6557.6 


Vt 


300.022 


7163.0 


312.981 


7795.2 


i^ 


287.456 


6575.5 


Va 


300.415 


7181.8 


S 


313.374 


7814.8 


% 


287.848 


6593.5 


Va 


300.807 


7200.6 


313.767 


7834.4 


1 


288.241 


6611.5 


Va 


301.200 


7219.4 


100. 


314.159 


7854.0 


288.634 


6629.6 


96. 


301.593 


7238.2 









DECIMALS OF A FOOT EQUIVALENT TO INCHES 

AND FRACTIONS OF AN INCH. 



Inches. 





Va 


Va 


Va 


X 


Va 


Va 


Va 








.01042 


.02083 


.03125 


.04166 


05208 


.06250 


07292 


1 


.0833 


.0937 


.1042 


.1146 


.1250 


1354 


.1459 


1563 


2 


.1667 


.1771 


.1875 


.1979 


.2083 


2188 


.2292 


2396 


3 


.2500 


.2604 


.2708 


.2813 


.2917 


3021 


.3125 


3229 


4 


.3333 


.3437 


.3542 


.3646 


.3750 


3854 


.3958 


4063 


5 


.4167 


.4271 


.4375 


.4479 


.4583 


4688 


.4792 


4896 


6 


.5000 


.5104 


.5208 


.5313 


.5417 


5521 


.5625 


5729 


7 


.5833 


.5937 


.6042 


.6146 


.6250 


6354 


.6459 


6563 


8 


.6667 


.6771 


.6875 


.6979 


.7083 


7188 


.7292 


7396 


9 


.7500 


.7604 


.7708 


.7813 


.7917 


8021 


.8125 


8229 


10 


.8333 


.8437 


.8542 


.8646 


.8750 


8854 


.8958 


9063 


11 


.9167 


.9271 


.9375 


.9479 


.9583 


9688 


.9792 


9896 



CIRCUMFERENCES OE CIRCLES. 



113 



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114 



MATHEMATICAL TABLES. 



LENGTHS OF CIRCULAR ARCS. 
(Degrees being given. Radius of Circle =1.) 

Formula.— Length of arc — — X radius X number of degrees. 

Rule. — Multiply the factor in table for any given number of degrees by 
the radius. 

Example. — Given a curve of a radius of 55 feet and an angle of 78° 20'. 
What is the length of same in feet ? 

Factor from table for 78° 1.3613568 

Factor from table for 20' .0058178 

Factor 1.3671746 

1.3671746 X 55 = 75.19 feet. 







. Degrees. 






Minutes. 


1 


.0174533 


61 


1.0646508 


121 


2.1118484 


1 


.0002909 


2 


.0349066 


62 


1.0821041 


122 


2.1293017 


2 


.0005818 


3 


.0523599 


63 


1.0995574 


123 


2.1467550 


3 


.0008727 


4 


.0698132 


64 


1.1170107 


124 


2.1642083 


4 


.0011636 


5 


.0872665 


65 


1.1344640 


125 


2.1816616 


5 


.0014544 


6 


.1047198 


66 


1.1519173 


126 


2.1991149 


6 


.0017453 


7 


.1221730 


67 


1.1693706 


127 


2.2165682 


7 


.0020362 


8 


.1396263 


68 


1.1868239 


128 


2.2340214 


8 


.0023271 


9 


.1570796 


69 


1.2042772 


129 


2.2514747 


9 


.0026180 


10 


.1745329 


70 


1.2217305 


130 


2.2689280 


10 


.0029089 


11 


.1919862 


71 


1.2391838 


131 


2.2863813 


11 


.0031998 


12 


.2094395 


72 


1.2566371 


132 


2.3038346 


12 


.0034907 


13 


.2268928 


73 


1.2740904 


133 


2.3212879 


13 


.0037815 


14 


.2443461 


74 


1.2915436 


134 


2.3387412 


14 


.0040724 


15 


.2617994 


75 


1.3089969 


135 


2.3561945 


15 


.0043633 


16 


.2792527 


76 


1.3264502 


136 


2.3736478 


16 


.0046542 


17 


.2967060 


77 


1.3439035 


137 


2.3911011 


17 


.0049451 


18 


.3141593 


78 


1.3613568 


138 


2.4085544 


18 


.0052360 


19 


.3316126 


79 


1.3788101 


139 


2.4260077 


19 


.0055269 


20 


.3490659 


80 


1.3962634 


140 


2.4434610 


20 


.0058178 


21 


.3665191 


81 


1.4137167 


141 


2.4609142 


21 


.0061087 


22 


.3839724 


82 


1.4311700 


142 


2.4783675 


22 


.0063995 


23 


.4014257 


83 


1.4486233 


143 


2.4958208 


23 


.0066904 


24 


.4188790 


84 


1.4660766 


144 


2.5132741 


24 


.0069813 


25 


.4363323 


85 


1.4835299 


145 


2.5307274 


25 


.0072722 


26 


.4537856 


86 


1.5009832 


146 


2.5481807 


26 


.0075631 


27 


.4712389 


87 


1.5184364 


147 


2.5656340 


.. "27 


.0078540 


28 


.4886922 


88 


1.5358897 


148 


2.5830873 


28 


.0081449 


29 


.5061455 


89 


1.5533430 " 


149 


2.6005406 


29 


.0084358 


30 


.5235988 


90 


1.5707963 


150 


2.6179939 


30 


.0087266 


31 


.5410521 - 


' 91 


1.5882496 


151 


2.6354472 


31 


.0090175 


32 


.5585054 


92 


1.6057029 


152 


2.6529005 


32 


.0093084 


33 


.5759587 


93 


1.6231562 


153 


2.6703538 


33 


.0095993 


34 


.5934119 


94 


1.6406095 


154 


2.6878070 


34 


.0098902 


35 


.6108052 


95 


1.6580628 


155 


2.7052603 


35 


.0101811 


36 


.6283185 


96 


1.6755161 


156 


2.7227136 


36 


.0104720 


37 


.6457718 


97 


1.6929694 


157 


2.7401669 


37 


.0107629 


38 


.6632251 


98 


1.7104227 


158 


2.7576202 


38 


.0110538 


39 


.6806784 


99 


1.7278760 


159 


2.7750735 


39 


.0113446 


40 


.6981317 


100 


1.7453293 


160 


2.7925268 


40 


.0116355 


41 


.7155850 


101 


1.7627825 


161 


2.8099801 


41 


.0119264 


42 


.7330383 


102 


1.7802358 


162 


2.8274334 


42 


.0122173 


43 


.7504916 


103 


1.7976891 


163 


2.8448867 


43 


.0125082 


44 


.7679449 


104 


1.8151424 


164 


2.8623400 


44 


.0127991 


45 


.7853982 


105 


1.8325957 


165 


2.8797933 


45 


.0130900 


46 


.8028515 


106 


1.8500490 


166 


2.8972466 


46 


.0133809 


47 


.8203047 


107 


1.8675023 


167 


2.9146999 


47 


.0136717 


48 


.8377580 


108 


1.8S49556 


168 


2.9321531 


48 


.0139626 


49 


.8552113 


109 


1.9024089 


169 


2.9496064 


49 


.0142535 


50 


.8726646 


110 


1.9198622 


170 


29670597 


50 


.0145444 


51 


.8901179 


111 


1.9373155 


171 


2.9845130 


51 


.0148353 


52 


.9075712 


112 


1.9547688 


172 


3.0019663 


52 


.0151262 


53 


.9250245 


113 


1.9722221 


173 


3.0194196 


53 


.0154171 


54 


.9424778 


114 


1.9896753 


174 


3.0308729 


54 


.0157080 


55 


.9599311 


115 


2.0071286 


175 


3.0543262 


55 


.0159989 


56 


.9773844 


116 


2.0245819 


176 


3.0717795 


56 


.0162897 


57 


.9948377 


117 


2 0420352 


177 


3.0892328 


57 


.0165806 


58 


1.0122910 


118 


2.0594885 


178 


3.1066861 


58 


.0168715 


59 


1.0297443 


119 


2.0769418 


179 


3.1241394 


59 


.0171624 


60 


1.0471976 


120 


2.0943951 


180 


3.1415927 


60 


.0174533 



LENGTHS OF CIRCULAR ARCS. 



115 



LENGTHS OF CIRCXJL.AR ARCS. 

(Diameter = 1. Given the Chord and Height of the Arc.) 

Rule for Use op the Table.— Divide the height by the chord. Find in the 
column of heights the number equal to this quotient. Take out the corre- 
sponding number from the column of lengths. Multiply this last number 
by the length of the given chord; the product will be length of the arc. 

' If the arc is greater than a semicircle, first find the diameter from the 
formula, Diam. = (square of half chord -4- rise) + rise; the formula is true 
whether the arc exceeds a semicircle or not. Then find the circumference. 
From the diameter subtract the given height of arc, the remainder will be 
height of the smaller arc of the circle; find its length according to the rule, 
and subtract it from the circumference. 



Hgts. 


Lgths. ! 


Hgts. 


Lgths. 

1.05896 


Hgts. 


Lgths. 


Hgts. 


Lgths. 


Hgts. 


Lgths. 


.001 


1. 00002 1 


.15 


.238 


1.14480 


.326 


1.26288 


.414 


1.40788 


.005 


1.00007' 


.152 


1.06051 


.24 


1.14714 


.328 


1 . 26588 


.416 


1.41145 


.01 


1.00027 


.154 


1.06209 


.242 


1.14951 


.33 


1.26892 


.418 


1.41503 


.015 


1.00061 


.156 


1.06368 


.244 


1.15189 


.332 


1.27196 


.42 


1.41861 


.02 


1.00107 


.158 


1.06530 


.246 


1.15428 


.334 


1.27502 


.422 


1.42221 


.0-25 


1.00167 


.16 


1.06693 


.248 


1.15670 


.336 


1.27810 


.424 


1.42583 


.03 


1.00240 


.162 


1.06858 


.25 


1.15912 


.338 


1.28118 


.426 


1.42945 


.035 


1.00327 


.164 


1.07025 


.252 


1.16156 


.34 


1.28428 


.428 


1.43309 


.04 


1.00426 


.166 


1.07194 


.254 


1.16402 


.342 


1.28739 


.43 


1.43673 


.045 


1.00539 


.168 


1.07365 


.256 


1.16650 


.344 


1.29052 


.432 


1.44039 


.05 


1.00665 


.17 


1.07537 


.258 


1.16899 


.346 


1.29366 


.434 


1.44405 


.055 


1.00805 


.172 


1.07711 


.26 


1.17150 


.348 


1.29681 


.436 


1.44773 


.06 


1.009571 


.174 


1.07888 


.262 


1.17403 


.35 


1.29997 


.438 


1.45142 


.065 


1.01123 


.176 


1.08066 


.264 


1.17657 


.352 


1.30315 


.44 


1.45512 


.07 


1.01302 


.178 


1.08246 


.266 


1.17912 


.354 


1.30634 


.442 


1.45883 


.075 


1.01493 


.18 


1.08428 


.268 


1.18169 


.356 


1.30954 


.444 


1.46255 


.08 


1.01698 


.182 


1.08611 


.27 


1.18429 


.358 


1.31276 


.446 


1.46628 


.085 


1.01916 


.184 


1.08797 


.272 


1.18689 


.36 


1.31599 


.448 


1.47002 


.09 


1.02146 


.186 


1.08984 


.274 


1.18951 


.362 


1.31923 


.45 


1.47377 


.095 


1.02389 


.188 


1.09174 


.276 


1.19214 


.364 


1.32249 


.452 


1.47753 


.10 


1.02646 


.19 


1.09365 


.278 


1.19479 


.366 


1.32577 


.454 


1.4S131 


.102 


1.02752 


.192 


1.09557 


.28 


1.19746 


.368 


1.32905 


.456 


1.48509 


.104 


1.02860 


.194 


1.09752 


.282 


1.20014 


.37 


1.33234 


.458 


1.48889 


.106 


1.02970 


.196 


1.09949 


.284 


1.20284 


.372 


1.33564 


.46 


1.49269 


.108 


1.03082 


.198 


1.10147 


.286 


1.20555 


.374 


1.33896 


.462 


1.49651 


.11 


1.03196 


.20 


1.10347 


.288 


1.20827 


.376 


1.34229 


.464 


1.50033 


.112 


1.03312 


.202 


1.10548 


.29 


L 21 102 


.378 


1.34563 


.466 


1.50416 


.114 


1.03430 


.204 


1.10752 


.292 


1.21377 


.38 


1.34899 


.468 


1.50800 


.116 


1.03551 


.206 


1.10958 


.294 


1.21654 


.382 


1.35237 


.47 


1.51185 


.118 


1.03672 


.208 


1.11165 


.296 


1.21933 


.384 


1.35575 


.472 


1.51571 


.12 


1.03797 


.21 


1.11374 


.298 


1.22213 


.386 


1.35914 


.474 


1.51958 


.122 


1.03923 


.212 


1.11584 


.30 


1.22495 


.388 


1.36254 


.476 


1.52346 


.124 


1.04051 


.214 


1.11796 


.302 


1.22778 


.39 


1.36596 


.478 


1.52736 


.126 


1.04181 


.216 


1.12011 


.304 


1.23063 


.392 


1.36939 


.48 


1.53126 


.128 


1.04313 


.218 


1.12225 


.306 


1.23349 


.394 


1.37283 


.482 


1.53518 


.13 


1.04447 


.22 


1.12444 


.308 


1.23636 


.396 


1.37528 


.484 


1.53910 


.132 


1.04584 


.222 


1.12664 


.31 


1.23926 


.398 


1.37974 


.486 


1.54302 


.134 


1.04722 


.224 


1.12885 


.312 


1.24216 


.40 


1.38322 


.488 


1.54696 


.136 


1.04862 


.226 


1.13108 


.314 


1.24507 


.402 


1.38671 


.49 


1.55091 


.138 


1.05003 


.228 


1.13331 


.316 


1.24801 


.404 


1.39021 


.492 


1.55487 


.14 


1.05147 


.23 


1.13557 


.318 


1.25095 


.406 


1.39372 


.494 


1.55854 


.142 


1.05293 


.232 


1.13785 


.32 


1.25391 


.408 


1.39724 


.496 


1.56282 


.144 


1.05441 


.234 


1.14015 


.322 


1.25689 


.41 


1.40077 


.498 


1.56681 


.146 


1.05591 


.236 


1.14247 


.324 


1.25988 


.412 


1.40432 


.5 


1.57080 


.148 


1 1.05743 



















116 



MATHEMATICAL TABLES. 



AREAS OF THE SEGMENTS OF A CIRCL.E. 

(Diameter = 1 ; Rise or Versed Sine in parts of Diameter 
being given.) 

Rule for Use of the Table,— Divide the rise or height of the segment by 
the diameter to obtain the versed sine. Multiply the area in the table cor- 
responding to this versed sine by the square of the diameter. 

If the segment exceeds a semicircle its area is area of circle— area of seg- 
ment whose rise is (diam. of circle— rise of giveu segment). 

Given chord and rise, to find diameter. Diam. = (square of half chord -*- 
rise) 4- rise. The half chord is a mean proportional between the two parts 
into which the chord divides the diameter which is perpendicular to it. 



Versed 




Verged 




Versed 




Versed 




Versed 




Sine. 


Area. 


Sine. 


Area. 




Area. 


Sine. 


Area. 


Sine. 


Area. 


.001 


.00004 


.054 


.01646 


.107 


.04514 


.16 


.08111 


.213 


.12235 


.002 


.00012 


.055 


.01691 


.108 


.04576 


.161 


.08185 


.214 


.12317 


.003 


.00022 


.056 


.01737 


.109 


.04638 


.162 


.08258 


.215 


.12399 


.004 


.00034 


.057 


.01783 


.11 


.04701 


.163 


.08332 


.216 


.12481 


.005 


.00047 


.058 


.01830 


.111 


.04763 


.164 


.08406 


.217 


.12563 


.006 


.00062 


.059 


.01877 


.112 


.04826 


.165 


.08480 


.218 


.12646 


.007 


.00078 


.06 


.01924 


.113 


.04889 


.166 


.08554 


.219 


.12729 


.008 


.00095 


.061 


.01972 


.114 


.04953 


.167 


.08629 


.22 


.12811 


.009 


.00113 


.062 


.02020 


.115 


.05016 


.168 


.08704 


.221 


.12894 


.01 


.00133 


.063 


.02068 


.116 


.05080 


.169 


.08779 


.222 


.12977 


.011 


.00153 


.064 


.02117 


.117 


.05145 


.17 


.08854 


.223 


.13060 


.012 


.00175 


.065 


.02166 


.118 


.05209 


.171 


.08929 


.224 


.13144 


.013 


.00197 


.066 


.02215 


.119 


.05274 


.172 


.09004 


.225 


.13227 


.014 


.0022 


.067 


.02265 


.12 


.05338 


.173 


.09080 


.226 


.13311 


.015 


.00244 


.068 


.02315 


.121 


.05404 


.174 


.09155 


.227 


.13395 


.016 


.00268 


.069 


.02366 


.122 


.05469 


.175 


.09231 


.228 


.13478 


.017 


.00294 


.07 


.02417 


.123 


.05535 


.176 


.09307 


.229 


.13562 


.018 


.0032 


.071 


.02468 


.124 


.05600 


.177 


.09384 


.23 


.13646 


.019 


.00347 


.072 


.02520 


.125 


.05666 


.178 


.09460 


.231 


.13731 


.02 


.00375 


.073 


.02571 


.126 


.05733 


.179 


.09537 


.232 


.13815 


.021 


.00403 


.074 


.02624 


.127 


.05799 


.18 


.09613 


.233 


.13900 


.022 


.00432 


.075 


.02676 


.128 


.05866 


.181 


.09690 


.234 


.13984 


.023 


.00462 


.076 


.02729 


.129 


.05933 


.182 


.09767 


.235 


. 14069 


.024 


.00492 


.077 


.02782 


.13 


.06000 


.183 


.09845 


.236 


.14154 


.025 


.00523 


.078 


.02836 


.131 


.06067 


.184 


.09922 


.237 


. 14239 


.026 


.00555 


.079 


.02889 


.132 


.06135 


.185 


. 10000 


.238 


.14324 


.027 


.00587 


.08 


.02943 


.133 


.06203 


.186 


.10077 


.239 


.14409 


.028 


.00619 


.081 


.02998 


.134 


.06271 


.187 


.10155 


.24 


.14494 


.029 


.00653 


.082 


.03053 


.135 


.06339 


.188 


.10233 


.241 


. 14580 


.03 


.00687 


.083 


.03108 


.136 


.06407 


.189 


.10312 


.242 


.14666 


.031 


.00721 


.084 


.03163 


.137 


.06476 


.19 


. 10390 


.243 


.14751 


.032 


.00756 


.085 


.03219 


.138 


.06545 


.191 


.10169 


.244 


.14837 


.033 


.00791 


.086 


.03275 


.139 


.06614 


.192 


.10547 


.245 


.14923 


.034 


.00827 


.087 


.03331 


.14 


.06683 


.193 


.10626 


.246 


.15009 


.035 


.00864 


.08S 


.03387 


.141 


.06753 


.194 


.10705 


.247 


.15095 


.036 


.00901 


.089 


.03444 


.142 


.06822 


.195 


.10784 


.248 


.15182 


.037 


.00938 


.09 


.03501 


.143 


.06892 


.196 


. 10864 


.249 


.15268 


.038 


.00976 


.091 


.03559 


.144 


.06963 


.197 


.10943 


.25 


.15355 


.039 


.01015 


.092 


.03616 


.145 


.07033 


.198 


.11023 


.251 


.15441 


.04 


.01054 


.093 


.03674 


.146 


.07103 


.199 


.11102 


.252 


.15528 


.041 


.01093 


.094 


.03732 


.147 


.07174 


.2 


.11182 


.253 


.15615 


.042 


.01133 


.095 


.03791 


.148 


.07245 


.201 


.11262 


.254 


.15702 


.043 


.01173 


.096 


.03850 


.149 


.07316 


.202 


.11343 


.255 


.15789 


.044 


.01214 


.097 


.03909 


.15 


.07387 


.203 


.11423 


.256 


. 15876 


.045 


.01255 


.098 


.03968 


.151 


.07459 


.204 


.11504 


.257 


.15964 


.046 


.01297 


.099 


.01028 


.152 


.07531 


.205 


.11584 


.258 


.16051 


.047 


.01339 


.1 


.04087 


.153 


.07603 


.206 


.11665 


.259 


.16139 


.048 


.01382 


.101 


.04148 


.154 


.07675 


.207 


.11746 


.26 


.16226 


.049 


.01425 


.102 


.04208 


.155 


.07747 


.208 


.11827 


.261 


.16314 


.05 


.01468 


.103 


.04269 


.156 


.07819 


.209 


.11908 


.262 


.16402 


.051 


.01512 


.104 


.04330 


.157 


.07892 


.21 


.11990 


.263 


.16490 


.052 


.01556 


.105 


.04391 


.158 


.07965 


.211 


.12071 


.264 


.16578 


.053 


.01601 


.106 


.04452 


.159 


.08038 


.212 


.12153 


.265 


.16666 



AREAS OF THE SEGMENTS OF A CIRCLE. 



117 



Versed 


*,,. 


Versed 
Sine. 


A ... 


Versed 
Sine. 


Area. 


Versed 
Sine. 


A ,„. 


Sine. 


,„,. 


.266 


16755 


.313 


.21015 


.36 


.25455 


407 


.30024 


.454 


.34676 


.267 


16843 


.314 


.21108 


.361 


.25551 


.408 


.30122 


.455 


.34776 


.268 


16932 


.315 


.21201 


.362 


.25647 


.409 


.30220 


.456 


.34876 


.269 


17020 


.316 


.21294 


.363 


.25743 


.41 


.30319 


.457 


.34975 


.27 


17109 


.317 


.21387 


.364 


.25839 


.411 


.30417 


.458 


.35075 


.271 


17198 


.318 


.21480 


.365 


.25936 


412 


.30516 


.459 


.35175 


.272 


17287 


.319 


.21573 


.366 


.26032 


.413 


.30614 


.46 


.35274 


.273 


17376 


.32 


.21667 


.367 


.26128 


.414 


.30712 


.461 


.35374 


.274 


17465 


.321 


.21760 


.368 


.26225 


.415 


.30811 


.462 


.35474 


.275 


17554 


.322 


.21853 


.369 


.26321 


.416 


.30910 


.463 


.35573 


.276 


17644 


.323 


.21947 


.37 


.26418 


.417 


.31008 


.464 


.35673 


277 


17733 


.324 


.22040 


.371 


.26514 


.418 


.31107 


.465 


.35773 


.278 


17823 


.325 


.22134 


.372 


.26611 


.419 


.31205 


.466 


.35873 


.27£ 


17912 


.326 


.22228 


. 73 


.26708 


.42 


.31304 


.467 


.35972 


.28 


18002 


.327 


.22322 


.374 


.26805 


.421 


.31403 


.468 


.36072 


.281 


18092 


.328 


.22415 


375 


.26901 


.422 


.31502 


.469 


.36172 


.282 


18182 


.329 


.22509 


i376 


.26998 


.423 


.31600 


.47 


.36272 


.283 


18272 


.S3 


.22603 


.377 


.27095 


.424 


.31699 


.471 


.36372 


.284 


18362 


.331 


.22697 


.378 


.27192 


.425 


.31798 


.472 


.36471 


.285 


18152 


.332 


.22792 


.379 


.27289 


.426 


.31&97 


.473 


.36571 


.286 


18542 


.333 


.22886 


.38 


.27386 


.427 


.31996 


.474 


.36671 


.287 


18633 


.334 


.22980 


.381 


.27483 


.428 


.32095 


.475 


.36771 


.288 


18723 


.335 


.23074 


.382 


.27580 


.429 


.32194 


.476 


.36871 


.289 


18814 


.336 


.23169 


.383 


.27678 


.43 


.32293 


.477 


.36971 


.29 


18905 


.337 


.23263 


.384 


.27775 


.431 


.32392 


.478 


.37071 


.291 


18996 


.338 


.23358 


.385 


.27872 


.432 


.32491 


.479 


.37171 


.292 


19086 


.339 


.23453 


.386 


.27969 


.433 


.32590 


.48 


.37270 


.293 


19177 


.34 


.23547 


.387 


.28067 


.434 


.32689 


.481 


.37370 


.294 


19268 


.341 


.23642 


.388 


.28164 


.435 


.32788 


.482 


.37470 


.295 


19360 


.342 


.23737 


.389 


.28262 


.436 


.32887 


.483 


.37570 


.296 


19451 


: .343 


.23832 


.39 


.28359 


.437 


.32987 


.484 


.37670 


.297 


19542 


.344 


.23927 


.391 


.28457 


.438 


.33086 


.485 


.37770 


.298 


19634 


.345 


.24022 


.392 


.28554 


.439 


.33185 


.486 


.37870 


.299 


19725 


.346 


.24117 


.393 


.28652 


.44 


.33284 


.487 


.37970 


.3 


19817 


.347 


2421 2 


.394 


.28750 


.441 


.33384 


.488 


.38070 


.301 


19908 


.348 


>24307 


.395 


.28848 


.442 


.33483 


.489 


.38170 


.302 


20000 


.349 


.24403 


.396 


.28945 


.443 


.33582 


.49 


.38270 


.303 


20092 


.35 


.24498 


.397 


.29043 


.444 


.33682 


.491 


.38370 


.304 


20184 


1 .351 


.24593 


.398 


.29141 


.445 


.33781 


.492 


.38470 


.305 


20276 


.352 


.24689 


.399 


.29239 


.446 


.33880 


.493 


.38570 


.306 


20368 


.353 


.24784 


.4 


.29337 


.447 


.33980 


.494 


.38670 


.307 


20460 


.354 


.24880 


.401 


.29435 


.448 


.34079 


.495 


.38770 


.308 


20553 


.355 


.24976 


.402 


.29533 


.449 


.34179 


.496 


.38870 


.309 


20645 


.356 


.25071 


.403 


.29631 


.45 


.34278 


.497 


.38970 


.31 


20738 


.357 


.25167 


.404 


.29729 


.451 


.34378 


.498 


.39070 


.311 


20830 


.358 


.25263 


.405 


.29827 


.452 


.34477 


.499 


.39170 


.312 


20923 


.359 


.25359 


.406 


.29926 


.453 


.34577 


.5 


.39270 



For rules for finding the area of a segment see Mensuration, page 59. 



118 



MATHEMATICAL TABLES. 



SPHERES. 

(Some errors of 1 in the last figure only. From Trautwine.) 



Diam. 


Sur- 
face. 


Solid- 
ity. 


Diam. 


Sur- 
face. 


Solid- 
ity. 


Diam. 


Sur- 
face. 


Solid- 
ity. 


1-32 


.00307 


.00002 


3 Ya 

5-16 


33.183 


17.974 


9 Va 


306.36 


504.21 


1-16 


.01227 


.00013 


34.472 


19.031 


10. 


314.16 


523.60 


3-32 


.02761 


.00043 


% 


35.784 


20.129 


Ya 


322.06 


543 48 


Ya 


.04909 


.00102 


7-16 


37.122 


21.268 


I 


330.06 


563.86 


5-32 


.07670 


.00200 


»J 


38.484 


22.449 


338.16 


584.74 


3-16 


.11045 


.00345 


39.872 


23.674 




346.36 


606.13 


7-32 


.15033 


.00548 


% 


41.283 


24.942 


% 


354.66 


628.04 


H 


.19635 


.00818 


11-16 


42.719 


26.254 


Ya 


363.05 


650.46 


9-32 


.24851 


.01165 


Ya 


44.179 


27.611 


Va 


371.54 


673.42 


5-16 


.30680 


.01598 


13-16 


45.664 


29.016 


11. 


380.13 


696.91 


11-32 


.37123 


.02127 


Va 


47.173 


30.466 


Ya 


388.83 


720.95 


Ya 


.44179 


.02761 


15-16 


48.708 


31.965 


Ya 


397.61 


745.51 


13-32 


.51848 


.03511 


4. 


50.265 


33.510 


Va 


406.49 


770.64 


7-16 


.60132 


.04385 


Ya 


53.456 


36.751 


II 


415.48 


796.33 


15-32 


.69028 


.05393 


Ya 


56.745 


40.195 


424.50 


822.58 


y* 


.78540 


.06545 


Ya 


60.133 


43.847 


Va 


433.73 


849.40 


9-16 


.99403 


.09319 


Yz 


63.617 


47.713 


Va 


443.01 


876.79 


% 


1.2272 


.12783 


% 


67.201 


51.801 


12. 


452.39 


904.78 


11-16 


1.4849 


.17014 


Ya 


70.883 


56.116 


Ya 


471.44 


962.52 


Ya 


1.7671 


.22089 


Va 


74.663 


60.663 




490.87 


1022.7 


13-16 


2.0739 


.28084 


5. 


78.540 


65.450 


Ya 


510.71 


1085.3 


Va 
15-16 


2.4053 


.35077 


H 


82.516 


70.482 


13. 


530.93 


1150.3 


2.7611 


.43143 


H 


86.591 


75.767 


Ya 


551.55 


1218.0 


1. 


3.1416 


.52360 


Ya 


90.763 


81.308 


Y* 


572.55 


1288.3 


1-16 


3.5466 


.62804 


H 


95.033 


87.113 


Ya 


593.95 


1361.2 


Ya 


3.9761 


.74551 


% 


99.401 


93.189 


14. 


615.75 


1436.8 


3-16 


4.4301 


.87681 


H 


103.87 


99.541 


Ya 


637.95 


1515.1 


Ya 


4.9088 


1.0227 


% 


108.44 


106.18 




660.52 


1596.3 


5-16 


5.4119 


1.1839 


6. 


113.10 


113.10 


% 


683.49 


1680.3 


% 


5.9396 


1.3611 


Ya 


117.8? 


120.31 


15. 


706 85 


1767.2 


7-16 


6.4919 


1.5553 


Ya 


122.72 


127.83 


Ya 


730.63 


1857.0 


Y2 

9-16 


7.0686 


1.7671 


% 


127.68 


135.66 


Vk 


754.77 


1949.8 


7.6699 


1.9974 


Y2 


132.73 


143.79 


Va 


779.32 


2045.7 


% 


8.2957 


2.2468 


% 


137.89 


152.25 


16. 


804.25 


2144.7 


11-16 


8.9461 


2.5161 


Ya 


143.14 


161.03 


Ya 


829.57 


2246.8 


Ya 
13-16 


9.6211 


2.8062 


Va 


148.49 


170.14 


Ya 


855.29 


2352.1 


10.321 


3.1177 


7. 


153.94 


179.59 


Va 


881.42 


2460.6 


Va 


11.044 


3.4514 


Ys 


159.49 


189.39 


17. 


907.93 


2572.4 


15-16 


11.793 


3.8083 


Ya 


165.13 


199.53 


Ya 


934.83 


2687.6 


2. 


12.566 


4.1888 


Ya 


170.87 


210.03 


Ya 


962.12 


2806.2 


1-16 


13.364 


4.5939 


Yi 


176.71 


220.89 


Va 


989.80 


2928.2 


Ya 


14.186 


5.0243 


% 


182.66 


232.13 


18. 


1017.9 


3053.6 


3-16 


15.033 


5.4809 


Ya 


188.69 


243.73 


Ya 


1046.4 


3182.6 


Va 


15.904 


5.9641 


Va 


194.83 


255.72 


8 


1075.2 


3315.3 


5-16 


16.800 


6.4751 


8. 


201.06 


268.08 


Ya 


1104.5 


3451.5 


% 


17.721 


7.0144 


Ys 


207.39 


280.85 


19. 


1134.1 


3591.4 


7-16 


18.666 


7.5829 


Ya 


213.82 


294.01 


Ya 


1164.2 


3735.0 


Vi 


19.635 


8.1813 


% 


220.36 


307.58 


A 


1194.6 


3882.5 


9-16 


20.629 


8.8103 


K 


226.98 


321.56 


Ya 


1225.4 


4033.7 


% 


21.648 


9.4708 


233.71 


335.95 


20. 


1256.7 


4188.8 


11-16 


22.691 


10.164 


Ya 


240.53 


350.77 


Va 


1288.3 


4347.8 


u 


23.758 


10.889 


Va 


247.45 


360.02 


Y* 


1320.3 


4510.9 


13-16" 


24.850 


11.649 


9. 


254.47 


381.70 


Ya 


1352.7 


4677.9 


Va 


25.967 


12.443 


Ys 


261.59 


397.83 


21. 


1385.5 


4849.1 


15-16 


27.109 


13.272 


8 


268.81 


414.41 


Ya 


1418.6 


5024.3 


3. 


28.274 


14.137 


270.12 


431.44 


A 


1452.2 


5203.7 


1-16 


29.465 


15.039 


Ya 


283.53 


448.92 


Ya 


1486.2 


5387.4 


% 


30.680 


15.979 


% 


291.04 


466.87 


22. 


1520.5 


5575.3 


3-16 


31.919 


16.957 


u 


298.65 


485.31 


Ya 


1555.3 


5767.6 



SPHERES. 
SPHERES— (Contained.) 



119 



Diam. 


Sur- 
face. 


Solid- 
ity. 


Diam. 


Sur- 
face. 


Solid- 
ity. 


Diam. 


Sur- 
face. 


Solid- 
ity. 


*$ 


1590.4 


5964.1 


40 


Yz 


5153.1 


34783 


70 


Yz 


15615 


183471 


1626.0 


6165.2 


41. 




5281 . 1 


36087 


71 




15837 


187402 


23. 


1661.9 


6370.6 




Mi 


5410.7 


37423 




Yz 


16061 


191389 


H 


1698.2 


6580.6 


42. 




5541.9 


38792 


72 




16286 


195,133 


y% 


1735.0 


6795.2 




Yz 


5674.5 


40194 




Yz 


16513 


199532 


% 


1772.1 


7014.3 


43. 




5808.8 


41630 


73 




16742 


203689 


24. 


1809.6 


7238.2 




}4 


5944.7 


43099 




Yz 


16972 


207903 


Ya 


1847.5 


7466.7 


44. 




6082.1 


44602 


74 




17204 


212175 


1 


1885.8 


7700.1 




Yz 


6221.2 


46141 




Yz 


17437 


216505 


1924.4 


7938.3 


45. 




6361.7 


47713 


75 




17672 


220894 


25. 


1963.5 


8181.3 




Yz 


6503.9 


49321 




Yz 


17908 


225341 


i 


2002.9 


8429.2 


46. 




6647.6 


50965 


76 




18146 


229848 


2042.8 


8682.0 




Yz 


6792.9 


52645 




Yz 


18386 


234414 


M 


2083.0 


8939.9 


47. 




6939.9 


54362 


77 




18626 


239041 


26. 


2123.7 


9202.8 




% 


7088.3 


56115 




Yz 


18869 


243728 


J4 


2164.7 


9470.8 


48. 




7238.3 


57906 


7S 




19114 


248475 


^ 


2206.2 


9744.0 




% 


7389.9 


59734 




Yz 


19360 


253284 


94 


2248.0 


10022 


49. 




7543.1 


61601 


79 




19607 


258155 


27. 


2290.2 


10306 




H 


7697.7 


63506 




Yz 


19856 


263088 


Ya 


2332.8 


10595 


50. 




7854.0 


65450 


80 




20106 


268083 


y* 


2375.8 


10889 




Yz 


8011.8 


67433 




Yz 


20358 


273141 


H 


2419.2 


11189 


51. 




8171.2 


69456 


81 




20612 


278263 


28. 


2463.0 


11494 




y% 


8332.3 


71519 




Yz 


20867 


283447 


S 


2507.2 


11805 


52. 




8494.8 


73622 


82 




21124 


288696 


2551.8 


12121 




Yz 


8658.9 


75767 




Yz 


21382 


294010 


% 


2596.7 


12443 


53. 




8824.8 


77952 


83 




21642 


299388 


29. 


2642.1 


12770 




Yz 


8992.0 


80178 




Yi 


21904 


304831 


M 


2687.8 


13103 


54. 




9160.8 


82448 


84 




22167 


310340 


*6 


2734.0 


13442 




Ya 


9331.2 


84760 




Yz 


22432 


315915 


H 


2780.5 


13787 


55. 




9503.2 


87114 


85 




22698 


321556 


30. 


2827.4 


14137 




Yi 


9676.8 


89511 




Yz 


22966 


327264 


J4 


2874.8 


14494 


56. 




9852.0 


91953 


80 




23235 


333039 




2922.5 


14856 




H 


10029 


94438 




Y-2 


23506 


338882 


M 


2970.6 


15224 


57. 




10207 


96967 


87 




23779 


344792 


31. 


3019.1 


15599 




Yz 


10387 


99541 




Yz 


24053 


350771 


M 


3068.0 


15979 


58 




10568 


102161 


88 




24328 


356819 


% 


3117.3 


16366 




k 


10751 


104826 




Yz 


24606 


362935 


Y\ 


3166.9 


16758 


59 




10936 


107536 


89 




24885 


369122 


32. 


3217.0 


17157 




% 


11122 


110294 




Yz 


25165 


375378 


a 


3267.4 


17563 


60 




11310 


113098 


90 




25447 


381704 


3318.3 


17974 


"" 


y* 


11499 


115949 




Yz 


25730 


388102 


% 


3369.6 


18392 


61 




11690 


118847 


91 




26016 


394570 


33. 


3421.2 


18817 




X 


11882 


121794 




Yz 


26302 


401109 


H 


3473.3 


19248 


62 




12076 


124789 


92 




26590 


407721 


14 


3525.7 


19685 




Yz 


12272 


127832 




Yz 


26880 


414405 


U 


3578.5 


20129 


63. 




12469 


130925 


93 




27172 


421161 


34. 


3631.7 


20580 




Yz 


12668 


134067 




Yz 


27464 


427991 


» 


3685.3 


21037 


64. 




12868 


137259 


94 




27759 


434894 


» 


3739.3 


21501 




Yz 


13070 


140501 




Yz 


28055 


441871 


35. 


3848.5 


22449 


65. 




13273 


143794 


95 




28353 


448920 


^ 


3959.2 


23425 




y> 


13478 


147138 




Yz 


28652 


456047 


36. 


4071.5 


24429 


66 




13685 


150533 


96 




28953 


463248 


^ 


4185.5 


25461 




Yz 


13893 


153980 




Yz 


29255 


470524 


37. 


4300.9 


26522 


67. 




14103 


157480 


97 




29559 


477874 


H 


4417.9 


27612 




Yz 


14314 


161032 




Yz 


29865 


485302 


38. 


4536.5 


28731 


6S 




14527 


164637 


98 




30172 


492808 


^ 


4656.7 


29880 




Yz 


14741 


168295 




Yz 


30481 


500388 


39. 


4778.4 


31059 


69. 




14957 


172007 


99 




30791 


508047 


3£ 


4901.7 


32270 




Yz 


15175 


175774 




Yz 


31103 


515785 


40. 


5026.5 


33510 


70. 




15394 


179595 


100 




31416 


523598 



120 



MATHEMATICAL TABLES. 



CONTENTS IN CUBIC FEET AND U. S. GALLONS OF 
PIPES AND CYLINDERS OF VARIOUS DIAMETERS 
AND ONE FOOT IN LENGTH. 





1 gallon = 231 cubic inches. 1 cubic foot 


- 7.4805 gallons. 






For 1 Foot in 




For 1 Foot in 




For 1 Foot in 


a 


Length. 


a 


Length. 


a 


Length. 


3 & 

§1 

5 


Cubic Ft. 
also Area 
in Sq. Ft. 


U.S. 

Gals., 

231 
Cu. In. 


& a> 

11 

C3hH 

s 


Cubic Ft. 
also Area 
in Sq. Ft. 


U.S. 
Gals., 

231 
Cu. In. 


5 


Cubic Ft. 
also Area 
in Sq. Ft. 


U.S. 
Gals., 

231 
Cu. In. 


H 


.0003 


.0025 


m 


.2485 


1.859 


19 


1.969 


14.73 


5-16 


.0005 


.004 




.2673 


1.999 


19^ 


2.074 


15.51 


% 


.0008 


.0057 


k 


.2867 


2.145 


20 


2.182 


16.32 


7-16 


.001 


.0078 


.3068 


2.295 


20^ 


2.292 


17.15 


H 


,0014 


.0102 


Wa 


.3276 


2.45 


21 


2.405 


17.99 


9-16 


.0017 


.0129 


8 


.3491 


2.611 


21^2 


2.521 


18.86 


% 


.0021 


.0159 


m 


.3712 


2.777 


22 


2.640 


19.75 


11-16 


.0026 


.0193 


sy 2 


.3941 


2.948 


22^ 


2.761 


20.66 


% 


.0031 


.0230 


8% 


.4176 


3.125 


23 


2.885 


21.58 


13-16 


.0036 


.0269 


9 


.4418 


3.305 


23^ 


3.012 


22.53 


Vs 


.0042 


.0312 


m 


.4667 


3.491 


24 


3.142 


23.50 


15-16 


.0048 


.0359 


9^ 


.4922 


3.682 


25 


3.409 


25.50 


1 


.0055 


.0408 


9% 


.5185 


3.879 


26 


3.687 


27.58 


M 


.0085 


.0638 


10 


.5454 


4.08 


27 


3.976 


29.74 


1H 


.0123 


.0918 


10M 


.5730 


4.286 


28 


4.276 


31.99 


m 


.0167 


.1249 


10^ 


.6013 


4.498 


29 


4.587 


34.31 


2 


.0218 


.1632 


10% 


.6303 


4.715 


30 


4.909 


36.72 




.0276 


.2066 


11 


.66 


4.937 


31 


5.241 


39.21 


.0341 


.2550 




.6903 


5.164 


32 


5.585 


41.7S 


*4 


.0412 


.3085 


.7213 


5.396 


33 


5.940 


44.43 


3 


.0491 


.3672 


u% 


.7530 


5.633 


34 


6.305 


47.16 


3M 


.0576 


.4309 


12 


.7854 


5.875 


35 


6.681 


49.98 


.0668 


.4998 


12J6 


.8522 


6.375 


36 


7.069 


52.88 


m 


.0767 


.5738 


13 


.9218 


6.895 


37 


7.467 


55.86 


4 


.0873 


.6528 


13J^ 


.994 


7.436 


38 


7.876 


58.92 


QA 


.0985 


.7369 


14 


1.069 


7.997 


39 


8.296 


62.06 


V/z 


.1134 


.8263 


14J4 


1 147 


8.578 


40 


8.727 


65.28 


4% 


.1231 


.9206 


15 


1.227 


9.180 


41 


9.168 


68.58 


5 


.1364 


1.020 


15J4 


1.310 


9.801 


42 


9.621 


71.97 


m 


.1503 


1.125 


16 


1.396 


10.44 


43 


10.085 


75.44 


5V 2 


.1650 


1.234 


16^ 


1.485 


11.11 


44 


10.559 


78.99 


Wa. 


.1803 


1.349 


17 


1.576 


11.79 


45 


11.045 


82.62 


6 


.1963 


1.469 


17^ 


1.670 


12.49 


46 


11.541 


86.33 


6M 


.2131 


1.594 


18 


1.768 


13.22 


47 


12.048 


90.13 


6^ 


.2304 


1.724 


18V£ 


1.867 


13.96 


48 


12.566 


94.00 



To find the capacity of pipes greater than the largest given in the table, 
look in the table for a pipe of one half the given size, and multiply its capac- 
ity by 4; or one of one third its size, and multiply its capacity by 9, etc. 

To find the weight of water in any of the given sizes multiply the capacity 
in cubic feet by 62)4 or the gallons by 8*4 or, if a closer approximation is 
required, by the weight of a cubic foot of water at the actual temperature in 
the pipe. 

Given the dimensions of a cylinder in inches, to find its capacity in U. S. 
gallons: Square the diameter, multiply by the length and by .0034. If d '= 

d 2 X 7854 X I 
diameter, I — length, gallons = ^ = .0034d 2 l. 



CAPACITY OF CYLINDKICAL VESSELS. 



121 



CYLINDRICAL. VESSELS, TANKS, CISTERNS, ETC. 

Diameter in Feet and Inches, Area in Square Feet, and 
U. S. Gallons Capacity for One Foot in Depth. 

1 gallon = 231 cubic inches = ° U „ 1q 05 °° = 0.13368 cubic feet. 



Diani. 


Area. 


Gals. 


Diam. 


Area. 


Gals. 


Diam. 


Area. 


Gals. 


Ft. In. 


Sq. ft. 


lfoot 
depth. 


Ft. In. 


Sq. ft. 


lfoot 
depth. 


Ft. In. 


Sq. ft. 


1 foot 
depth. 


1 


.785 


5.87 


5 8 


25.22 


188.66 


19 




283.53 


2120.9 


1 1 


.922 


6.89 


5 9 


25.97 


194.25 


19 


3 


291.04 


2177.1 


1 2 


1.069 


8.00 


5 10 


26.73 


199.92 


19 


6 


298.65 


2234.0 


1 3 


1.227 


9.18 


5 11 


27.49 


205.67 


19 


9 


306.35 


2291.7 


1 4 


1.396 


10.44 


6 


28.27 


211.51 


20 




314.16 


2350.1 


1 5 


1.576 


11.79 


6 3 


30.68 


229.50 


20 


3 


322.06 


2409.2 


1 6 


1.767 


13.22 


6 6 


33.18 


248.23 


20 


6 


330.06 


2469.1 


1 7 


1.969 


14.73 


6 9 


35.78 


267.69 


20 


9 


338.16 


2529.6 


1 8 


2.182 


16.32 


7 


38.48 


287.88 


21 




346.36 


2591.0 


1 9 


2.405 


17.99 


7 3 


41.28 


308.81 


21 


3 


354.66 


2653.0 


1 10 


2.640 


19.75 


7 6 


44.18 


330.43 


21 


6 


363.05 


2715.8 


1 11 


2.885 


21.58 


7 9 


47.17 


352.88 


21 


9 


371.54 


2779.3 


2 


3.142 


23.50 


8 


50.27 


376.01 


22 




380.13 


2843.6 


2 1 


3.409 


25.50 


8 3 


53.46 


399.88 


22 


3 


388.82 


2908.6 


2 2 


3.687 


27.58 


8 6 


56.75 


424.48 


22 


6 


397.61 


2974.3 


2- 3 


3.976 


29.74 


8 9 


60.13 


449.82 


22 


9 


406.49 


3040.8 


2 4 


4.276 


31.99 


9 


63.62 


475.89 


23 




415.48 


3108.0 


2 5 


4.587 


34.31 


9 3 


67.20 


502.70 


23 


3 


424.56 


3175.9 


2 6 


4.909 


36.72 


9 6 


70.88 


530 24 


23 


6 


433.74 


3244.6 


2 7 


5.241 


39.21 


9 9 


74.66 


558.51 


23 


9 


443.01 


3314.0 


2 8 


5.585 


41.78 


10 


78.54 


587.52 


24 




452.39 


3384.1 


2 9 


5.940 


44.43 


10 3 


82.52 


617.26 


24 


3 


461.86 


3455.0 


2 10 


6.305 


47.16 


10 6 


86.59 


647.74 


24 


6 


471.44 


3526.6 


2 11 


6.681 


49.98 


10 9 


90.76 


678.95 


24 


9 


481.11 


3598.9 


3 


7.069 


52.88 


11 


95.03 


710.90 


25 




490.87 


3672.0 


3 1 


7.467 


55.86 


11 3 


99.40 


743.58 


25 


3 


500.74 


3745.8 


3 2 


7.876 


58.92 


11 6 


103.87 


776.99 


25 


6 


510.71 


3820.3 


3 3 


8.296 


62.06 


11 9 


108.43 


811.14 


25 


9 


520.77 


3895.6 


3 4 


8.727 


65.28 


12 


113.10 


846.03 


26 




530.93 


3971.6 


3 5 


9.168 


68.58 


12 3 


117.86 


881.65 


26 


3 


541.19 


4048.4 


3 6 


9.621 


71.97 


12 6 


122.72 


918.00 


26 


6 


551.55 


4125.9 


3 7 


10.085 


75.44 


12 9 


127.68 


955.09 


26 


9 


562.00 


4204.1 


3 8 


10.559 


78.99 


13 


132.73 


992.91 


27 




572.56 


4283.0 


3 9 


11.045 


82.62 


13 3 


137.89 


1031.5 


27 


3 


583.21 


4362.7 


3 10 


11.541 


86.33 


13 6 


143.14 


1070.8 


27 


6 


593.96 


4443.1 


3 11 


12.048 


90.13 


13 9 


148.49 


1110.8 


27 


9 


604.81 


4524.3 


4 


12.566 


94.00 


14 


153.94 


1151.5 


28 




615.75 


4606.2 


4 1 


13.095 


97.96 


14 3 


159.48 


1193.0 


28 


3 


626.80 


4688.8 


4 2 


13.635 


102.00 


14 6 


165.13 


1235.3 


28 


6 


637.94 


4712.1 


4 3 


14.186 


106.12 


14 9 


170.87 


1278.2 


28 


9 


649.18 


4856.2 


4 4 


14.748 


110.32 


15 


176.71 


1321.9 


29 




660.52 


4941.0 


4 5 


15.321 


114.61 


15 3 


182 65 


1366.4 


29 


3 


671.96 


5026.6 


4 6 


15.90 


118.97 


15 6 


188.69 


1411.5 


29 


6 


683.49 


5112.9 


4 7 


16.50 


123.42 


15 9 


194.83 


1457.4 


29 


9 


695.13 


5199.9 


4 8 


17.10 


127.95 


16 


201.06 


1504.1 


30 




706.86 


5287.7 


4 9 


17.72 


132.56 


16 3 


207.39 


1551.4 


30 


3 


718.69 


5376.2 


4 10 


18.35 


137.25 


16 6 


213 82 


1599.5 


30 


6 


730.62 


5465.4 


4 11 


18.99 


142.02 


16 9 


220.35 


1648.4 


30 


9 


742.64 


5555.4 


5 


39.63 


146.88 


17 


226.98 


1697.9 


31 




754.77 


5646.1 


5 1 


20.29 


151.82 


17 3 


23'171 


1748.2 


31 


3 


766.99 


5737.5 


5 2 


20.97 


156.83 


17 6 


240.53 


1799.3 


31 


6 


779.31 


5829.7 


5 3 


21.65 


161.93 


17 9 


247.45 


1851.1 


31 


9 


791.73 


5922.6 


5 4 


22.34 


167.12 


18 


254.47 


1903.6 


32 




804.25 


6016.2 


5 5 


23.04 


172.38 


18 3 


261.59 


1956.8 


32 


3 


816.86 


6110.6 


5 6 


23.76 


177.72 


18 6 


268.80 


2010.8 


32 


6 


829.58 


6205.7' 


5 7 


24.48 


183.15 


18 9 


276.12 


2065.5 


32 


9 


842.39 


6301.5 



122 MATHEMATICAL TABLES. 

GALLONS AND CUBIC FEET. 
United States Gallons in a given Number of Cubic Feet. 

1 cubic foot = 7.480519 U. S. gallons; 1 gallon = 231 cu. in. = .13368056 cu. ft. 



Cubic Ft. 


Gallons. 


Cubic Ft. 


Gallons. 


Cubic Ft. 


Gallons. 


0.1 
02 
0.3 
0.4 
0.5 


0.75 
1.50 
2.24 
2.99 
3.74 


50 
60 
70 
80 
90 


374.0 
448.8 
523.6 
598.4 
673.2 


8,000 
9,000 
10,000 
20,000 
30,000 


59,844.2 
67,324.7 
74,805.2 
149,610.4 
224,415.6 


0.6 
0.7 
0.8 
0.9 

1 


4.49 
5.24 

5.98 
6.73 

7.48 


100 
200 
300 
400 
500 


748.0 
1,496.1 
2,244.2 
2,992.2 
3,740.3 


40,000 
50,000 
60,000 
70,000 
80,000 


299,220.8 
374,025.9 
448,831.1 
523,636.3 
598,441.5 


2 
3 
4 
5 
6 


14.96 
22.44 
29.92 
37.40 

44.88 


600 
700 
800 
900 
1,000 


4,488.3 
5,236.4 
5,984.4 
6,732.5 
7,480.5 


90,000 
100,000 
200,000 
300,000 
400,000 


673,246.7 

748,051.9 

1,496,103.8 

2,244,155.7 

2,992,207.6 


7 
8 
9 
10 
20 


52.36 
59.84 
67.32 
74.80 
149.6 


2,000 
3,000 
4,000 
5,000 
6,000 


14,961.0 
22,441.6 
29,922.1 
37,402.6 
44,883.1 


500,000 
600,000 
700,000 
800,000 
900,000 


3,740,259.5 
4.488,311.4 
5,236,363.3 
5,984,415.2 
6,732,467.1 


30 
40 


224.4 
299.2 


7,000 


52,363.6 


1,000,000 


7,480,519.0 



Cubic Feet in a given Number of Gallons. 



Gallons. 


Cubic Ft. 


Gallons. 


Cubic Ft. 


Gallons. 


Cubic Ft. 


1 
2 
3 
4 
5 

6 

8 
9 
10 


.134 
.267 
.401 
.535 

.668 

.802 
.936 
1.069 
1.203 
1.337 


1,000 
2,000 
3,000 
4,000 
5,000 

6,000 

7,000 
8,000 
9,000 
10,000 


133.681 
267.361 
401.042 
534.722 
668.403 

802.083 

935.764 

1,060.444 

1,203.125 

1,336.806 


1,000,000 
2,000,000 
3,000,000 
4,000,000 
5,000,000 

6,000,000 
7,000,000 
8,000,000 
9,000,000 
10,000,000 


133,680.6 
267,361.1 - 
401,041.7 
534,722.2 
668,402.8 

802,083.3 

935,763.9 

1,069,444.4 

1,203,125.0 

1,336,805.6 



DUMBER OF SQUARE FEET IN" PLATES. 



123 



NUMBER OF SQUARE FEET IN PLATES 3 TO 32 
FEET LONG, AND 1 INCH WIDE. 

For other widths, multiply by the width in inches. 1 sq. in. = .0069f sq. ft. 



Ft. and 

in. 

Long. 


Ins. 
Long. 


Square 
Feet. 


Ft. and 
Ins. 
Long. 


Ins. 
Long. 


Square 
Feet. 


Ft. and 
Ins. 
Long. 


Ins. 
Long. 


Square 
Feet. 


3. 


36 


.25 


7.10 


94 


.6528 


13. 8 


152 


1.056 


1 


37 


.2569 


11 


95 


.6597 


9 


153 


1.063 


2 


38 


.2639 


8. 


96 


.6667 


10 


154 


1.069 


3 


39 


.2708 


1 


97 


.6736 


11 


155 


1.076 


4 


40 


.2778 


2 


98 


.6806 


13. 


156 


1.083 


5 


41 


!2847 


3 


99 


.6875 


1 


157 


1.09 


6 


42 


.2917 


4 


100 


.6944 


2 


158 


1.097 


7 


43 


.2986 


5 


101 


.7014 


3 


159 


1.104 


8 


44 


.3056 


• 6 


102 


.7083 


4 


160 


1.114 


9 


45 


.3125 


7 


103 


.7153 


5 


161 


1.118 


10 


46 


.3194 


8 


104 


.7222 


6 


162 


1.125 


11 


47 


.3264 


9 


105 


.7292 


7 


163 


1.132 


4. 


48 


.3333 


10 


106 


.7361 


8 


164 


1.139 


1 


49 


.3403 


11 


107 


.7431 


9 


165 


1.146 


2 


50 


.3472 


9. 


108 


.75 


10 


166 


1.153 


3 


51 


.3542 


1 


109 


.7569 


11 


167 


1.159 


4 


52 


.3611 


2 


no 


.7639 


14.0 


168 


1.167 


5 


53 


.3681 


3 


in 


.7708 


1 


169 


1.174 


6 


54 


.375 


4 


112 


.7778 


2 


170 


1.181 


7 


55 


.3819 


5 


113 


.7847 


3 


171 


1.188 


8 


56 


.3889 


6 


114 


.7917 


4 


172 


1.194 


9 


57 


.3958 


7 


115 


.7986 


5 


173 


1.201 


10 


58 


.4028 


8 


116 


.8056 


6 


174 


1.208 


11 


59 


.4097 


9 


117 


.8125 


7 


175 


1.215 


5. 


60 


.4167 


10 


118 


.8194 


8 


176 


1.222 


1 


61 


.4236 


11 


119 


.8264 


9 


177 


1.229 


2 


62 


.4306 


10.0 


120 


.8333 


10 


178 


1.236 


3 


63 


.4375 


1 


121 


.8403 


11 


179 


1.243 


4 


64 


.4444 


2 


122 


.8472 


15.0 


180 


1.25 


5 


65 


.4514 


3 


123 


.8542 


1 


181 


1.257 


6 


66 


.4583 


4 


124 


.8611 


2 


182 


1.264 


7 


67 


.4653 


5 


125 


.8681 


3 


183 


1.271 


8 


68 


.4722 


6 


126 


.875 


4 


184 


1.278 


9 


69 


.4792 


7 


127 


.8819 


5 


185 


1.285 


10 


70 


.4861 


8 


128 


.8889 


6 


186 


1.292 


11 


71 


.4931 


9 


129 


.8958 


7 


187 


1.299 


6. 


72 


.5 


10 


130 


.9028 


8 


188 


1.306 


1 


73 


.5069 


11 


131 


.9097 


9 


189 


1.313 


2 


74 


.5139 


11.0 


132 


9167 


10 


190 


1.319 


3 


75 


.5208 


1 


133 


.9236 


11 


191 


1.326 


4 


76 


.5278 


2 


134 


.9306 


16.0 


192 


1.333 


5 


77 


.5347 


3 


135 


.9375 


1 


193 


1.34 


6 


78 


.5417 


4 


136 


.9444 


2 


194 


1.347 


7 


79 


.5486 


5 


137 


.9514 


3 


195 


1.354 


8 


80 


.5556 


6 


138 


.9583 


4 


196 


1 361 


9 


81 


.5625 


7 


139 


.9653 


5 


197 


1.368 


10 


82 


.5694 


8 


140 


.9722 


6 


198 


1.375 


11 


83 


.5764 


9 


141 


.9792 


7 


199 


1.382 


1. 


84 


.5834 


10 


142 


.9861 


8 


200 


1.389 


1 


85 


.5903 


11 


143 


.9931 


9 


201 


1.396 


2 


86 


.5972 


12.0 


144 


1.000 


10 


202 


1.403 


3 


87 


.6042 


1 


145 


1.007 


11 


203 


1.41 


4 


88 


.6111 


2 


146 


1.014 


17.0 


204 


1.417 


5 


89 


.6181 


3 


147 


1.021 


1 


205 


1.424 


6 


90 


.625 


4 


148 


1.028 


2 


206 


1.431 


7 


91 


.6319 


5 


149 


1.035 


3 


207 


1.438 


8 


92 


.6389 


6 


150 


1.042 


4 


208 


1.444 


9 


93 


.6458 


7 


151 


1.049 


5 


209 


1.451 



124 



MATHEMATICAL TABLES. 





SQUARE FEET 


IN PliATES- 


-(Continued.) 




Ft. and 
Ins. 
Long. 


Ins. 


Square 


Ft. and 


Ins. 


Square 
Feet. 


Ft. and 


Ins. 


Square 
Feet. 


Long. 


Feet. 


Long. 


Long. 


Long. 


Long. 


17.6 


210 


1.458 


22.5 


269 


1.868 


27.4 


328 


2.278 


7 


211 


1.465 


6 


270 


1.875 


5 


329 


2.285 


8 


212 


1.472 


7 


271 


1.882 


6 


330 


2.292 


9 


213 


1.479 


8 


272 


1.889 


7 


331 


2.299 


10 


214 


1.486 


9 


273 


1.896 


8 


332 


2.306 


11 


215 


1.493 


10 


274 


1.903 


9 


333 


2.313 


18.0 


216 


1.5 


11 


275 


1.91 


10 


334 


2.319 


1 


217 


1.507 


33. 


276 


1.917 


11 


335 


2.326 


2 


218 


1.514 


1 


277 


1.924 


28.0 


336 


2.333 


3 


219 


1.521 


2 


278 


1.931 


1 


337 


2.34 


4 


220 


1.528 


3 


279 


1.938 


2 


338 


2.347 


5 


221 


1.535 


4 


280 


1.944 


3 


339 


2.354 


6 


222 


1.542 


5 


281 


1.951 


4 


340 


2.361 


7 


223 


1.549 


6 


282 


1.958 


5 


341 


2.368 


8 


224 


1.556 


7 


283 


1.965 


6 


342 


2.375 


9 


225 


1.563 


8 


284 


1.972 


7 


343 


2.382 





226 


1.569 


9 


285 


1.979 


8 


344 


2.389 


11 


227 


1.576 


10 


286 


1.986 


9 


345 


2.396 


19.0 


228 


1.583 


11 


287 


1.993 


10 


346 


2.403 


1 


229 


1.59 


24.0 


288 


2 


11 


347 


2.41 


2 


230 


1.597 


1 


289 


2.007 


29.0 


348 


2.417 


3 


231 


1.604 


2 


290 


2.014 


1 


349 


2.424 


4 


232 


1.611 


3 


291 


2.021 


2 


350 


2.431 


5 


233 


1.618 


4 


292 


2.028 


3 


351 


2.438 


6 


234 


1.625 


5 


293 


2.035 


4 


352 


2.444 


7 


235 


1.632 


6 


294 


2.042 


5 


353 


2.451 


8 


236 


1.639 


7 


295 


2.049 


6 


354 


2.458 


9 


237 


1.645 


8 


296 


2.056 


7 


355 


2.465 


10 


238 


1.653 


9 


297 


2.063 


8 


356 


2.472 


11 


239 


1.659 


10 


298 


2.069 


9 


357 


2.479 


20.0 


240 


1.667 


11 


299 


2.076 


10 


358 


2.486 


1 


241 


1.674 


25.0 


300 


2.083 


11 


359 


2.493 


2 


242 


1.681 


1 


301 


2.09 


30.0 


360 


2.5 


3 


243 


1.688 


2 


302 


2.097 


1 


361 


2.507 


4 


244 


1.694 


3 


303 


2.104 


2 


362 


2.514 


5 


245 


1.701 


4 


304 


2.111 


3 


363 


2.521 


6 


246 


1.708 


5 


305 


2.118 


4 


364 


2.528 


7 


247 


1.715 


6 


306 


2.125 


5 


365 


2.535 


8 


.248 


1.722 


7 


307 


2.132 


6 


366 


2.542 


9 


249 


1.729 


8 


308 


2.139 


7 


367 


2.549 


10 


250 


1.736 


9 


309 


2.146 


8 


368 


2.556 


11 


251 


1.743 


10 


310 


2.153 


9 


369 


2.563 


21.0 


252 


1.75 


11 


311 


2.16 


10 


370 


2.569 


1 


253 


1.757 


26.0 


312 


2.167 


11 


371 


2.576 


2 


254 


1.764 


1 


313 


2.174 


31.0 


372 


2.583 


3 


255 


1.771 


2 


314 


2.181 


1 


373 


2.59 


4 


256 


1.778 


3 


315 


2.188 


2 


374 


2.597 


5 


257 


1.785 


4 


316 


2.194 


3 


375 


2.604 


6 


258 


1.792 


5 


317 


2.201 


4 


376 


2.611 


7 


259 


1.799 


6 


318 


2.208 


5 


377 


2.618 


8 


260 


1.806 


7 


319 


2.215 


6 


378 


2.625 


9 


261 


1.813 


8 


320 


2.222 


7 


379 


2.632 


10 


262 


1.819 


9 


321 


2.229 


8 


380 


2.639 


11 


263 


1.826 


10 


322 


2.236 


9 


381 


2.646 


22.0 


264 


1.833 


11 


323 


2.243 


10 


382 


2 653 


1 


265 


1.84 


27.0 


324 


2.25 


11 


383 


2.66 


2 


266 


1.847 


1 


325 


2.257 


32.0 


384 


2.667 


3 


267 


1.854 . 


2 


326 


2.264 


1 


385 


2.674 


4 


268 


1.861 


1 3 


337 


2.271 


2 


386 


2.681 



CAPACITY OF KECTAtfGtJLAft TAtfES. 



125 



CAPACITIES OF RECTANGULAR TANKS IN U. S. 
GALLONS, FOR EACH FOOT IN DEPTH. 

1 cubic foot = 7.4805 U. S. gallons. 



Width 


Length of Tank. 


of 
Tank. 


feet. 
2 


ft. in. 
2 6 


feet. 
3 

44.88 
56.10 
67.32 


ft. in. 
3 6 


feet. 
4 


ft. in. 
4 6 


feet. 
5 


ft. in. 
5 6 

82.29 
102.86 
123.43 
144.00 
164.57 

185.14 
205.71 
226.28 


feet. 
6 


ft. in. 
6 6 


feet. 

1 


ft, in. 

2 

2 G 
3 


29.92 


37.40 

46.75 


52.36 

65.45 
78.54 
91.64 


59.84 

74.80 
89.77 
104.73 
119.69 


67.32 

84.16 
100.99 
117.82 
134.65 

151.48 


74.81 
93.51 

112.21 
130.91 
149.61 

168.31 
187.01 


89.77 
112.21 

157.09 
179.53 

201.97 
224.41 
246.86 
269.30 


97.25 

121.56 
145.87 
170.18 
194.49 

218.80 
243.11 
267.43 
291.74 
316.05 


104.73 
130.91 
157.09 


3 6 






183.27 


4 
4 6 








209.45 
235.63 


5 












261.82 


5 6 














288.00 


6 
















314 18 


6 6 


















340.36 
366.54 



Width 








Length of Tank. 








of 
Tank. 


ft. in. 
7 6 


feet. 
8 


ft. in. 
8 6 


feet. 
9 


ft. in. 
9 6 


feet. 
10 


ft. in. 
10 6 


feet. 
11 


ft. in. 
11 6 


feet. 
12 


ft. in. 
2 

2 6 
3 

3 6 
4 

4 6 
5 

5 6 
6 

6 6 

7 6 
8 

8 6 


112.21 
140.26 
168.31 
196.36 
224.41 

252.47 

280.52 
308.57 
336.62 
364.67 

392.72 

420.78 


119.69 
149.61 
179.53 
209.45 
239.37 

269.30 
299.22 
329.14 
359.06 
388.98 

418.91 

448.83 
478.75 


127.17 

158.96 
190.75 

254.34 

286.13 
317.92 
349.71 
381.50 
413.30 

445 09 

476.88 
508.67 
540.46 


134.65 

168.31 
202.97 
235.63 
269.30 

302.96 

336.62 
370.28 
403.94 
437.60 

471 27 
504.93 
538.59 
572.25 
605.92 


142 13 

177.66 
233.19 

248.73 
284.26 

319.79 

390.85 
426.39 
461.92 

497.45 

568.51 
604.05 
639.58 

675.11 


149.61 

187.01 
224.41 
261.82 
299.22 

336.62 
374.03 
411.43 
448.83 
486.23 

523.64 
561 04 
598.44 
685.84 
673.25 

710.65 
748.05 


157.09 
196.36 
235.63 
274.90 
314.18 

353.45 

392.72 
432.00 
471.27 
510.54 

549.81 
589.08 
628.36 
667 63 
706.90 

746.17 

785.45 
824.73 


164.57 

205.71 
246.86 
288.00 
329.14 

370.28 
411.43 
452.57 
493.71 
534.85 

575.99 
617.14 

- 
699.42 
740.56 

781.71 

822.86 
864.00 
905.14 


172.05 

215.06 
258.07 
301.09 
344.10 

387.11 
430.13 
473.14 
516.15 
559.16 

602.18 
645.19 
688.20 
731.21 
774.23 

817.24 
860.26 
903.26 
946.27 
989.29 


179 53 
224.41 

269.30 
314.18 
359.06 

403.94 
448.83 
493.71 
538.59 
583.47 

628 36 
673.24 
718.12 
763.00 


9 






807 89 


9 6 








852 77 


10 










897.66 


10 6 












942 56 


11 














987 43 


11 6 
















1032.3 


12 


















1077 2 

























12G 



MATHEMATICAL TABLES. 



NUMBER OF BARRELS (31 1-2 GAL.L.ONS) IN 

CISTERNS AND TANKS. 



= 4.21094 cubic feet. Reciprocal = 



Depth 


Diameter in Feet. 


Feet. 


5 


6 


7 


8 


9 


10 


11 


12 


13 


14 


1 


4.663 


6.714 


9.139 


11.937 


15.10S 


18.652 


22.569 


26.859 


31.522 


36.557 


5 


23.3 


33.6 


45.7 


59.7 


75.5 


93.3 


112.8 


134.3 


157.6 


182.8 


6 


28.0 


40.3 


54.8 


71.6 


90.6 


111.9 


135.4 


161.2 


189.1 


219 3 


7 


32.6 


47.0 


64.0 


83.6 


105.8 


130.6 


158.0 


188.0 


220.7 


255.9 


8 


37.3 


53.7 


73.1 


95.5 


120.9 


149.2 


180.6 


214.9 


252.2 


292.5 


9 


42.0 


60.4 


82.3 


107.4 


136.0 


167.9 


203.1 


241.7 


283.7 


329.0 


10 


46.6 


67.1 


91.4 


119.4 


151.1 


186.5 


225.7 


268.6 


315.2 


365.6 


11 


51.3 


73.9 


100.5 


131.3 


166.2 


205.2 


248 3 


295.4 


346.7 


402.1 


12 


56.0 


80.6 


109.7 


143.2 


181.3 


223.8 


270.8 


322.3 


378.3 


438.7 


13 


60.6 


87.3 


118.8 


155.2 


196.4 


242.5 


293.4 


349.2 


409.8 


475.2 


14 


65.3 


94.0 


127.9 


167.1 


211.5 


261.1 


316.0 


376.0 


441.3 


511.8 


15 


69.9 


100.7 


137.1 


179.1 


226.6 


289.8 


338.5 


402.9 


472.8 


548.4 


16 


74.6 


107.4 


146.2 


191.0 


241.7 


298.4 


361.1 


429.7 


504.4 


5S4.9 


17 


79.3 


114.1 


155.4 


202.9 


256.8 


317.1 


383.7 


456.6 


535.9 


621 .5 


18 


83.9 


120.9 


164.5 


214.9 


271.9 


335.7 


406.2 


483.5 


567.4 


658.0 


19 


88.6 


127.6 


173.6 


226.8 


287.1 


354.4 


428.8 


510.3 


598.9 


694.6 


20 


93 3 


134.3 


182.8 


238.7 


302.2 


373.0 


451.4 


537.2 


630.4 


731.1 



Depth 


Diameter in Feet. 


Feet. 


15 


16 


17 


18 


19 


20 


21 


22 


1 


41.966 


47.748 


53.903 


60.431 


67.332 


74.606 


82.253 


90.273 


5 


209.8 


238.7 


269.5 


302.2 


336.7 


373.0 


411.3 


451.4 


6 


251.8 


286.5 


323.4 


362 6 


404.0 


447.6 


493.5 


541.6 


7 


293.8 


334.2 


377.3 


423 


471.3 


522.2 


575.8 


631.9 


8 


335.7 


382.0 


431.2 


483.4 


538.7 


596.8 


658.0 


722.2 


9 


377.7 


429.7 


485 1 


543.9 


606.0 


671.5 


740.3 


812.5 


10 


419.7 


477.5 


539.0 


604.3 


673.3 


746.1 


822.5 


902.7 


11 


461.6 


525.2 


592.9 


664.7 


740.7 


820.7 


904.8 


993.0 


12 


503.6 


573.0 


646.8 


725.2 


808.0 


895.3 


987.0 


1083.3 


13 


545.6 


620.7 


700.7 


785.6 


875.3 


969.9 


1069.3 


1173.5 


14 


587.5 


668.5 


754.6 


846.0 


942.6 


1044.5 


1151.5 


1263.8 


15 


629.5 


716.2 


808.5 


906.5 


1010.0 


1119.1 


1233.8 


1354.1 


16 


671.5 


764.0 


862.4 


966.9 


1077.3 


1193.7 


1316.0 


1444.4 


17 


713.4 


811.7 


916.4 


1027.3 


1144.6 


1268.3 


1398.3 


1534.5 


18 


755.4 


859.5 


970.3 


1087.8 


1212.0 


1342.9 


1480.6 


1624.9 


19 


797.4 


907.2 


1024.2 


1148.2 


1279.3 


1417.5 


1562.8 


1715.2 


20 


839.3 


955.0 


1078.1 


1208.6 


1346.6 


1492.1 


1645.1 


1805.5 



LOGARITHMS. 



127 



NUMBER OF BARRELS (31 1-2 GALLONS) IN 
CISTERNS AND TANKS.— Contained. 



Depth 


Diameter in Feet. 


in 
Feet. 


23 


24 


25 


26 


27 


28 


29 


30 


1 


98.666 


107.482 


116.571 


120.083 


135.968 


146.226 


157.858 


167.86? 


5 


493.3 


537.2 


582.9 


630.4 


679.8 


731.1 


784.3 


839.3 


6 


592.0 


644.6 


699.4 


756.5 


815.8 


877.4 


941.1 


1007.2 


7 


690.7 


752.0 


816.0 


882.6 


951.8 


1023.6 


1098.0 


1175.0 


8 


789.3 


859.5 


932.6 


1008.7 


1087.7 


1169.8 


1254.9 


1342.9 


9 


888.0 


966.9 


1049.1 


1134.7 


1223.7 


1316.0 


1411.7 


1510.8 


10 


986.7 


1074.3 


1165.7 


1260.8 


1359.7 


1462.2 


1568.6 


1678.6 


11 


1085.3 


1181.8 


1282.3 


1386.9 


1495.6 


1608.5 


1725.4 


1846.5 


12 


1184.0 


1289.2 


1398.8 


1513.0 


1631.6 


1754.7 


1882.3 


2014.4 


13 


1282.7 


1396.6 


1515.4 


1639.1 


1767.6 


1900.9 


2039.2 


2182.2 


14 


1381.3 


1504 


1632.0 


1765.2 


1903.6 


2047.2 


2196.0 


2350.1 


15 


1480.0 


1611.5 


1748.6 


1891.2 


2039.5 


2193.4 


2352.9 


2517.9 


16 


1578.7 


1718.9 


1865.1 


2017.3 


2175.5 


2339.6 


2509.7 


2685.8 


17 


1677.3 


1826.3 


1981.7 


2143.4 


2311.5 


2485.8 


2666.6 


2853.7 


18 


1776.0 


1933.8 


2098.3 


2269.5 


2447.4 


2632.0 


2823.4 


3021.5 


19 


1874.7 


2041.2 


2214.8 


2395.6 


2583.4 


2778.3 


2980.3 


3189.4 


20 


1973.3 


2148.6- 


2321.4 


2521.7 


2719.4 


2924.5 


3137.2 


3357.3 



LOGARITHMS. 



Logarithms (abbreviation log).— The log of a number is the exponent 
of the power to which it is necessary to raise a fixed number to produce the 
given number. The fixed number is called the base. Thus if the base is 10, 
the log of 1000 is 3, for 10 3 = 1000. There are two systems of logs in general 
use, the common, in which the base is 10, and the Naperian, or hyperbolic, 
in which the base is 2.718281828 .... The Naperian base is commonly de- 
noted by e, as in the equation e y = x, in which y is the Nap. log of x. 

In any system of logs, the log of 1 is 0; the log of the base, taken in that 
system, is 1. In any system the base of which is greater than 1, the logs of 
all numbers greater than 1 are positive and the logs of all numbers less than 
1 are negative. 

The modulus of any system is equal to the reciprocal of the Naperian log 
of the base of that system. The modulus of the Naperian system is 1, that 
of the common system is .4342945. 

The log of a number in any system equals the modulus of that system X 
the Naperian log of the number. 

The hyperbolic or Naperian log of any number equals the common log 
X 2.30 5851. 

Every log consists of two parts, an entire part called the characteristic, or 
index, and the decimal part, or mantissa. The mantissa only is given in the 
usual tables of common logs, with the decimal point omitted. The charac- 
teristic is found by a simple rule, viz., it is one less than the number of 
figures to the left of the decimal point in the number whose log is to be 
found. Thus the characteristic of numbers from 1 to 9.99 + is 0, from 10 to 
99.99 + is 1, from 100 to 999 -f- is 3, from .1 to .99 + is - 1, from .01 to .099 4- 
is - 2, etc. Thus 

log of 2000 is 3.30103; log of .2 is - 1.30103; 

" " 200 " 2.30103; " " .02 " - 2.30103; 

" " 20 "1.30103; " " .002 "- 3.30103; 

** " 2 " 0.30103; *' " ,0002 " - 4.30103,- 



128 MATHEMATICAL TABLES. 

The minus sign is frequently written above the characteristic thus: 
log .002 = 3.30103. The characteristic only is negative, the decimal part, or 
mantissa, being always positive. 

When a log consists of a negative index and a positive mantissa, it is usual 
to write the negative sign over the index, or else to add 10 to the index, and 
to indicate the subtraction of 10 from the resulting logarithm. 

Thus log .2 = T.30103, and this may be written 9.30103 - 10. 

In tables of logarithmic sines, etc., the — 10 is generally omitted, as being 
understood. 

Rules for use of the table of Logarithms. -To find the 
log of any whole number.— For 1 to 100 inclusive the log is given 
complete in the small table on page 129. 

For 100 to 999 inclusive the decimal part of the log is given opposite the 
given number in the column headed in the table (including the two figures 
to the left, making six figures). Prefix the characteristic, or index, 2. 

For 1000 to 9999 inclusive : The last four figures of the log are found 
opposite the first three figures of the given number and in the vertical 
column headed with the fourth figure of the given number ; prefix the two 
figures under column 0, and the index, which is 3. 

For numbers over 10,000 having five or more digits : Find the decimal part 
of the log for the first four digits as above, multiply the difference figure 
in the last column by the remaining digit or digits, and divide by 10 if there 
be only one digit more, by 100 if there be two more, and so on ; add the 
quotient to the log of the first four digits and prefix the index, which is 4 
if there are five digits, 5 if there are six digits, and so on. The table of pro- 
portional parts may be used, as shown below. 

To find the log of a decimal fraction or of a whole 
number and a decimal.— First find the log of the quantity as if there 
were no decimal point, then prefix the index according to rule ; the index is 
one less than the number of figures to the left of the decimal point. 

Required log of 3.141593. 

log of 3.141 =0.497068. Diff . = 138 

From proportional parts 5 = 690 

" " 09 = 1242 

003 = 041 



log 3.141593 0.4971498 

To find the number corresponding to a given log.— Find 

in the table the log nearest to the decimal part of the given log and take the 
first four digits of the required number from the column N and the top or 
foot of the column containing the log which is the next less than the given 
log:. To find the 5th and 6th digits subtract the log in the table from the 
given log, multiply the difference by 100, and divide by the figure in the 
Diff. column opposite the log ; annex the quotient to the four digits already 
found, and place the decimal point according to the rule ; the number of 
figures to the left of the decimal point is one greater than the index. 

Find number corresponding to the log 0.497150 

Next lowest log in table corresponds to 3141 497068 

Diff. = 82 

Tabular diff. = 138; 82 ~ 138 = .59 + 

The index being 0, the nvimber is therefore 3.14159 -4-. 

To multiply two numbers by the use of logarithms.— 

Add together the logs of the two numbers, and find the number whose log 
is the sum. 

To divide two numbers.— Subtract the log of the less from the 
log of the greater, and find the number whose log is the difference. 

To raise a number to any given power.— Multiply the log of 
the number by the exponent of the power, and find the number whose log is 
the product. 

To find any root of a given number.— Divide the log of the 
number by the index of the root. The quotient is the log of the root. 

To find the reciprocal of a number.— Subtract the decimal 
part of the log of the number from 0, add 1 to the index and change the sign 
of the index, The result is the log of the reciprocal. 



LOGARITHMS. 



129 



Required the reciprocal of 3.141593. 

Log of 3.141593, as found above 0.4971498 

Subtract decimal part from gives _0.5028502 

Add 1 to the index, and changing sign of the index gives. . l .5028502 
which is the log of 0.31831 . 

To find the fourth term of a proportion by logarithms. 
—Add the logarithms of the second and third terms, aud from their sum 
subtract the logarithm of t'ne first term. 

When one logarithm is to be subtracted from another, it may be more 
convenient to convert the subtraction into an addition, which may be done 
by first subtracting th^ given logarithm from 10, adding the difference tc the 
other logarithm, and afterwards rejecting the 10. 

The difference between a given logarithm and 10 is called its arithmetical 
complement, or cologarithm. 

To subtract one logarithm from another is the same as to add its comple- 
ment and then reject 10 from the result. For a — b = 10 — b -j- a — 10. 

To work a proportion, then, by logarithms, add the complement of the 
logarithm of the first term to the logarithms of the second and third terms. 
The characteristic must afterwards be diminished by 10. 

Example in logarithms with a negative index. —Solve by 

which means divide 526 by 1011 and raise the quotient 



logarithms (^ m j 
to the 2.45 power. 



log 526 = 
log 1011 = 
log of quotient = 
Multiply by 



2.7 

3.004751 



716235 
2.45 



- 2.581175 
- 2.8 64940 

- 1.43 2470 

- 1.30 477575 = .20173, Ans. 

In multiplying - 1.7 by 5, we say: 5 x 7 - 35, 3 to carry; 5 x — 1 = — 5 less 
+ 3 carried — — 2. In adding -2-j-8-f-3 + l carried from previous column, 
we say: 1 -j- 3 + 8 = 12, minus 2 = 10, set down and carry 1 ; 1 -fc- 4 — 2 = 3. 

Logarithms of Numbers from 1 to 100. 



Log. 



0.000000 
0.301030 
0.477121 
0.602060 
0.698970 

0.778151 

0. 

0. 

0.954243 

1.000000 

1.041393 
1.079181 
1.113943 
1.146128 
1.176091 

1.204120 
1.230449 
1.255273 

1.278754 
1. 



N. 


Log. 


N. 


Log. 


N. 




Log. 




21 


1.322219 


41 


1.612784 


61 


1.785330 


22 


1.342423 


42 


1.623249 


62 


1 


792392 


82 


23 


1.361728 


43 


1.633468 


63 


1 


799341 


83 


24 


1.380211 


44 


1.643453 


64 


1 


806180 


84 


25 


1.397940 


45 


1.653213 


65 


1 


812913 


85 


26 


1.414973 


46 


1.662758 


66 


1 


819544 


86 


| 27 


1.431364 


47 


1.672098 


67 


1 


826075 


87 


28 


1.447158 


48 


1.681241 


68 


1 


832509 


88 


29 


1.462398 


49 


1.690196 


69 


1 


838849 


89 


30 


1.477121 


50 


1.698970 


70 


1 


845098 


90 


31 


1.491362 


51 


1.707570 


71 


1 


851258 


91 


32 


1.505150 


52 


1.716003 


72 


1 


857332 


92 


33 


1.518514 


53 


1.724276 


73 


1 


863323 


93 


34 


1.531,479 


54 


1.732394 


74 


1 


869232 


94 


35 


1.544068 


55 


1.740363 


75 


1 


875061 


95 


36 


1.556303 


56 


1.748188 


76 


1 


880814 


96 


37 


1.568202 


57 


1.755875 


77 


1 


886491 


97 


38 


1.579784 


58 


1.763428 


78 


1 


892095 


98 


39 


1.591065 j 


59 


1.770852 


79 


1 


897627 


99 


40 


1.602060 1 


60 


1.778151 


80 


1 


903090 


100 



1.908485 
1.913814 
1.919078 
1.924279 
1.929419 

1.934498 
1.939519 
1.944483 
1.949390 
1.954243 

1.959041 
1.9i - 



1.973128 

177724 

1.982271 
56772 
1.9! 
l.< 

2.000000 



130 



LOGARITHMS OF NUMBERS. 



No. 


100 L. 000.] 














[No. 109 L. 040. 


N. 





1 


2 


8 


4 


5 


6 7 

2598 3029 
6894 7321 

1147 1 1570 
5360 5779 
9532 . 9947 


8 


9 


Diff. 


100 
1 
2 


000000 
4321 
8600 


0434 
4751 
9026 


0868 
5181 


1301 

5609 


1734 
6038 


2166 
6466 


3461 

7748 


3891 

8174 


432 

428 




0300 
4521 
8700 


0724 
4940 
9116 


1993 
6197 


2415 
6616 


424 
420 


3 
4 


012837 
7033 


3259 

7451 


3680 
7868 


4100 

8284 


0361 
4486 
8571 

2619 
6629 


0775 
4896 
8978 


416 
412 
408 


5 
6 

7 


021189 
5306 
9384 


1603 
5715 
9789 


2016 
6125 


2428 
6533 


2841 
6942 


3252 
7350 


3664 

7757 

1812 
5830 
9811 


4075 
8164 


0195 

4227 
8223 


0600 
4628 
8620 


1004 
5029 
9017 


1408 
5430 
9414 


2216 
6230 


3021 
7028 

0998 


404 
400 


8 
9 


033424 

7426 
04 


3826 
7825 


0207 


0602 


397 



Proportional Parts. 



Diff. 

434 



422 
421 
420 
419 
418 
417 
416 
415 

414 
413 
412 
411 
410 
409 
408 
407 
406 
405 

404 
403 
402 
401 
400 



1 



43.4 
43.3 
43.2 
43.1 
43.0 
42.9 
42.8 
42.7 
42.6 
42.5 

42.4 
42.3 
42.2 
42.1 
42.0 
41.9 
41.8 
41.7 
41.6 
41.5 

41.4 
41.3 
41.2 
41.1 
41.0 
40.9 
40.8 
40.7 
40.6 
40.5 
40.4 
40.3 
40.2 
40.1 
40.0 



39.5 



86.4 
86.2 
86.0 
85.8 



85.2 
85.0 

84.8 
84.6 
84.4 
84.2 
84.0 



83.2 
83.0 



82.2 
82.0 
81.8 
81.6 
81.4 
81.2 
81.0 



79.2 
79.0 



129.6 
129.3 
129.0 
128.7 
128.4 
128.1 
127.8 
127.5 

127.2 
126.9 
126.6 
126.3 
126.0 
125.7 
125.4 
125.1 
124.8 
124.5 

124.2 
123.9 
123.6 
123.3 
123.0 
122.7 
122.4 
122.1 
121.8 
121.5 

121.2 
120.9 
120.6 
120.3 
120.0 
119.7 
119.4 
119.1 
118.8 
118.5 



173.6 

173.2 
172.8 
172.4 
172.0 
171.6 
171.2 
170.8 
170.4 
170.0 



168.8 
168.4 
168.0 
167.6 
167.2 
166.8 
166.4 
166.0 

165.6 
165.2 
164.8 
164.4 
164.0 
163.6 
163.2 
162.8 
162.4 
162.0 

161.6 
161.2 
160.8 
160.4 
160.0 
159.6 
159.2 
158.8 
158.4 
158.0 



5 


6 


7 


8 


217.0 


260.4 


303.8 


347.2 


216.5 


259.8 


303.1 


346.4 


216.0 


259.2 


302.4 


345.6 


215.5 


258.6 


301.7 


344.8 


215.0 


258.0 


301.0 


344.0 


214.5 


257.4 


300.3 


343.2 


214.0 


256.8 


299.6 


342.4 


213.5 


256.2 


298.9 


341.6 


213.0 


255.6 


298.2 


340.8 


212.5 


255.0 


297.5 


340.0 


212.0 


254.4 


296.8 


339.2 


211.5 


253.8 


296.1 


338.4 


211.0 


253.2 


295.4 


337.6 


210.5 


252.6 


294.7 


336.8 


210.0 


252.0 


294.0 


336.0 


209.5 


251.4 


293.3 


335.2 


209.0 


250.8 


292.6 


334.4 


208.5 


250.2 


291.9 


333.6 


208.0 


249.6 


291.2 


332.8 


207.5 


249.0 


290.5 


332.0 


207.0 


248.4 


289.8 


331.2 


206.5 


247.8 


289.1 


330.4 


206.0 


247.2 


288.4 


329.6 


205.5 


246.6 


287.7 


328.8 


205.0 


246.0 


287.0 


328.0 


204.5 


245.4 


286.3 


327.2 


204.0 


244.8 


285.6 


326.4 


203.5 


244.2 


284.9 


325.6 


203.0 


243 6 


284.2 


324.8 


202.5 


243.0 


283.5 


324.0 


202.0 


242.4 


282.8 


323.2 


201.5 


241.8 


282.1 


322.4 


201.0 


241 2 


281.4 


321.6 


200.5 


240.6 


280.7 


320.8 


200.0 


240.0 


280.0 


320.0 


199.5 


239.4 


279.3 


319.2 


199.0 


238.8 


278.6 


318.4 


198.5 


238.2 


277.9 


317.6 


198.0 


237.0 


277.2 


316.8 


197.5 


237.0 


276.5 


316 ; 



LOGARITHMS OP tttJMBERS. 



131 



No. 


110 L. 041.] 














[No 


119 L. 078. 


N. 





1 


2 


S 


4 


• 

3362 

7275 

1153 
4996 
8805 


6 

3755 
7664 

1538 
5378 
9185 


7 


8 


9 


Diff. 


110 
1 
2 


041393 
5323 
9218 


1787 
5714 
9606 


2182 
6105 
9993 


2576 
6495 


2969 

6885 


4148 
8053 


4540 
8442 


4932 

8830 


393 
390 




0380 
4230 
8046 


0766 
4613 
8426 


1924 
5760 
9563 


2309 
6142 
9942 


2694 
6524 


386 
383 


3 
4 


053078 
6905 


3463 

7286 


3846 
7666 




0320 
4083 
7815 


379 
376 
373 


5 
6 

7 


060698 
4458 
8186 


1075 
4832 
8557 


1452 
5206 
8928 


1829 
5580 
9298 


2206 
5953 
9668 


2582 
6326 


2958 
6699 


3333 
7071 


3709 
7443 




0038 
3718 
7368 


0407 
4085 
7731 


0776 
4451 
8094 


1145 
4816 
8457 


1514 

5182 
8819 


370 
366 
363 


8 
9 


071882 
5547 


2250 
5912 


2617 
6276 


2985 
6640 


3352 
7004 



Proportional Parts. 



Diff. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


395 


39.5 


79.0 


118.5 


158.0 


197.5 


237.0 


276.5 


316.0 


355.5 


394 


39.4 


78.8 


118.2 


157.6 


197.0 


236.4 


275.8 


315.2 j 354.6 


393 


39.3 


78.6 


117.9 


157.2 


196.5 


235.8 


275.1 


314.4 


353.7 


392 


39.2 


78.4 


117.6 


156.8 


196.0 


235.2 


274.4 


313.6 


352.8 


391 


39.1 


78.2 


117.3 


156.4 


195.5 


234.6 


273.7 


312.8 


351.9 


390 


39 


78.0 


117.0 


156.0 


195.0 


234.0 


273.0 


312.0 


351.0 


389 


38.9 


77.8 


116.7 


155.6 


194.5 


233.4 


272.3 


311.2 


350.1 


388 


38.8 


77.6 


116.4 


155.2 


194.0 


232.8 


271.6 


310.4 


349.2 


387 


38.7 


77.4 


116.1 


154.8 


193.5 


232.2 


270.9 


309.6 


348.3 


386 


38.6 


77.2 


115.8 


154.4 


193.0 


231.6 


270.2 


308.8 


347.4 


385 


38.5 


77.0 


115.5 


154.0 


192.5 


231.0 


269.5 


308.0 


346.5 


384 


38.4 


76.8 


115.2 


153.6 


192.0 


230.4 


268.8 


307.2 


345.6 


383 


38.3 


76.6 


114.9 


153.2 


191.5 


229.8 


268.1 


306.4 


344.7 


382 


38.2 


76.4. 


114.6 


152.8 


191.0 


229.2 


267.4 


305.6 


343.8 


381 


38.1 


76.2 


114.3 


152.4 


190.5 


228.6 


266.7 


304.8 


342.9 


380 


38.0 


76.0 


114.0 


152.0 


190.0 


228.0 


266.0 


304.0 


342.0 


379 


37.9 


75.8 


113.7 


151.6 


189.5 


227.4 


265.3 


303.2 


341.1 


378 


37.8 


75.6 


113.4 


151.2 


189.0 


226.8 


264.6 


302.4 


340.2 


377 


37.7 


75.4 


113.1 


150.8 


188.5 


226.2 


263.9 


301.6 


339.3 


376 


37.6 


75.2 


112.8 


150.4 


188.0 


225.6 


263.2 


300.8 


338.4 


375 


37.5 


75.0 


112.5 


150.0 


187.5 


225.0 


262.5 


300.0 


337.5 


374 


37.4 


74.8 


112.2 


149.6 


187.0 


224.4 


261.8 


299.2 


336.6 


373 


37.3 


74.6 


111.9 


149.2 


186.5 


223.8 


261.1 


298.4 


335.7 


372 


37.2 


74.4 


111.6 


148.8 


186.0 


223.2 


260.4 


297.6 


334.8 


371 


37.1 


74.2 


111.3 


148.4 


185.5 


222.6 


259.7 


296.8 


333.9 


370 


37.0 


74.0 


111.0 


148.0 


185.0 


222.0 


259.0 


296.0 


333.0 


369 


36.9 


73.8 


110.7 


147.6 


184.5 


221.4 


258.3 


295.2 


332.1 


368 


36.8 


73.6 


110.4 


147.2 


184.0 


220.8 


257.6 


294.4 


331.2 


367 


36.7 


73.4 


110.1 


146.8 


183.5 


220.2 


256.9 


293.6 


330.3 


366 


36.6 


73.2 


109.8 


146.4 


183.0 


219.6 


256.2 


292.8 


329.4 


365 


36.5 


73.0 


109.5 


146.0 


182.5 


219.0 


255.7 


292.0 


328.5 


364 


36.4 


72.8 


109.2 


145.6 


182.0 


218.4 


254.8 


291.2 


327.6 


363 


36.3 


72.6 


108.9 


145.2 


181.5 


217.8 


254.1 


290.4 


326.7 


362 


36.2 


72.4 


108.6 


144.8 


181.0 


217.2 


253.4 


289.6 


325.8 


361 


36.1 


72.2 


108.3 


144.4 


180.5 


216.6 


252.7 


288.8 


324.9 


360 


36.0 


72.0 


108.0 


144.0 


180.0 


216.0 


252.0 


288.0 


324.0 


359 


35.9 


71.8 


107.7 


143.6 


179.5 


215.4 


251.3 


287.2 


323.1 


a58 


35.8 


71.6 


107.4 


143.2 


179.0 


214.8 


250.6 


286.4 


322.2 


357 


35.7 


71.4 


107.1 


142.8 


178.5 


214.2 


249.9 


285.6 


321.3 


356 


35 .'6 


71.2 


106.8 


142.4 


178.0 


213.6 


249.2 


284.8 


320.4 



132 



LOGARITHMS OF KUMBERS. 



No. 


120 L. 079.] 














[N 


0. 134 L. 130. 


N. 


1 1 


2 


3 


i 4 II 6 
i 


6 


7 


8 


9 


Diff. 




079181 1 9543 


9904 












120 


0266 
3861 

7426 


0626 
4219 

7781 


1 0987 
4576 
8136 

1667 
5169 
8644 


1347 
4934 
8490 


1707 
5291 

8845 


2067 
5647 
9198 


2426 
6004 
9552 


360 


1 

2 
3 

4 
5 


082785 
6360 
9905 

093422 
6910 


3144 
6716 


3503 
7071 


357 
355 


0258 
3772 
7257 


0611 
4122 
7604 


0963 
4471 
7951 


1315 
4820 
8298 


2018 
5518 
8990 


2370 
5866 
9335 


2721 
6215 
9681 


3071 
6562 

0026 
3462 
6871 


352 
349 

346 
343 
341 


6 

8 

9 

130 
1 


100371 
3804 
7210 


0715 
4146 
7549 


1059 

4487 
7888 


1403 

4828 
8227 


1747 
5169 
8565 


2091 
5510 
8903 


2434 
5851 
9241 


2777 
6191 
9579 


3119 
6531 
9916 


0253 
3609 

6940 


338 
335 

333 


110590 

3943 
7271 


0926 

4277 
7603 


1263 

4611 
7934 


1599 
4944 

8265 


1934 

5278 
8595 


2270 

5611 

8926 


2605 

5943 
9256 


2940 
6276 
9586 


3275 

6608 
9915 


0245 
3525 
6781 


330 
328 
325 


2 
3 

4 


120574 
3852 
7105 

13 


0903 
4178 
7429 


1231 
4504 
7753 


1560 
4830 
8076 


1888 
5156 
8399 


2216 
5481 
8722 


2544 

5806 
9045 


2871 
6131 
9368 


3198 
6456 
9690 


0012 


323 



Proportional Parts. 



Diff. 


1 


2 


3 


4 


5 


6 


7 


355 


35.5 


71.0 


106.5 


142.0 


177.5 


213.0 


248.5 


354 


35.4 


70.8 


106.2 


141.6 


177.0 


212.4 


247.8 


353 


35.3 


70.6 


105.9 


141.2 


176.5 


211.8 


247.1 


.352 


35.2 


70.4 


105.6 


140.8 


176.0 


211.2 


246.4 


351 


35.1 


70.2 


105.3 


140.4 


175.5 


210.6 


245.7 


350 


35.0 


70.0 


105.0 


140.0 


175.0 


210.0 


245.0 


349 


34.9 


69.8 


104.7 


139.6 


174.5 


209.4 


244.3 


848 


34.8 


69.6 


104.4 


139.2 


174.0 


208.8 


243.6 


847 


34.7 


69.4 


104.1 


138.8 


173.5 


208.2 


242.9 


346 


34.6 


69.2 


103.8 


138.4 


173.0 


207.6 


242.2 


345 


34.5 


69.0 


103.5 


138.0 


172.5 


207.0 


241.5 


344 


34.4 


68.8 


103.2 


137.6 


172.0 


206.4 


240.8 


343 


34.3 


68.6 


102.9 


137.2 


171.5 


205.8 


240.1 


342 


34.2 


68.4 


102.6 


136.8 


171.0 


205.2 


239.4 


341 


34.1 


68.2 


102.3 


136.4 


170.5 


204.6 


238.7 


340 


34.0 


68.0 


102.0 


136.0 


170.0 


204.0 


238.0 


339 


33.9 


67.8 


101.7 


135.6 


169.5 


203.4 


237.3 


338 


33.8 


67.6 


101.4 


135.2 


169.0 


202.8 


236.6 


337 


33.7 


67.4 


101.1 


134.8 


168.5 


202.2 


235.9 


336 


33.6 


67.2 


100.8 


134.4 


168.0 


201.6 


235.2 


335 


33.5 


67.0 


100.5 


134.0 


167.5 


201.0 


234.5 


334 


33.4 


66.8 


100.2 


133.6 


167.0 


200.4 


233.8 


333 


33.3 


66.6 


99.9 


133.2 


166.5 


199.8 


233.1 


332 


33.2 


66.4 


99.6 


132.8 


166.0 


199.2 


232.4 


331 


33.1 


66.2 


99.3 


132.4 


165.5 


198.6 


231.7 


330 


33.0 


66.0 


99.0 


132.0 


165.0 


198.0 


231.0 


829 


32.9 


65.8 


98.7 


131.6 


164.5 


197.4 


230.3 


828 


32 8 


65.6 


98.4 


131.2 


164.0 


196.8 


229.6 


327 


32.7 


65.4 


98.1 


130.8 


163.5 


196.2 


228.9 


326 


32.6 


65.2 


97.8 


130.4 


163.0 


195.6 


228.2 


325 


32.5 


65.0 


97.5 


130.0 


162.5 


195.0 


227.5 


824 


32.4 


64.8 


97.2 


129.6 


162.0 


194.4 


226.8 


323 


32.3 


64.6 


96.9 


129.2 


161.5 


193.8 


226.1 


322 


32.2 


64.4 


96.6 


128.8 


161.0 


193.2 


225.4 



284.0 
283.2 
282.4 
281.6 



278.4 
277.6 
276.8 



276.0 


310.5 


275.2 


309.6 


274.4 


308.7 


273.6 


307.8 


272.8 


306.9 


272.0 


306.0 


271.2 


305.1 


270.4 


304.2 


269.6 


303.3 


268.8 


302.4 


268.0 


301.5 


267.2 


300.6 


266.4 


299.7 


265.6 


298.8 


264.8 


297.9 


264.0 


297.0 


263.2 


296.1 


262.4 


295.2 


261.6 


294.3 


260.8 


293.4 


260.0 


292.5 


259.2 


291.6 


25S.4 


290.7 


257.6 


289.8 



LOGARITHMS OP NUMBERS. 



133 



No. 135 L. 130.] 




[No. 149 L. 175. 


N. 


1 


2 


3 


4 


5 ! 

1 


6 


7 


8 


9 


Diff. 


135 i 130334 


0655 0977 


1298 


1619 


1939 ! 


2260 


2580 2900 


3219 


321 


6 3539 


3858 ! 4177 


4496 


4814 


5133 


5451 


5769 6086 


6403 


318 


7 ! 6721 

8 | 9879 


7037 | 7354 


7671 


7987 


! 8303 ! 


8618 


8934 9249 


9564 


316 


0194 


0508 ftftaa 


1136 


1450 


1763 


2076 2389 


2702 


314 


9 143015 


3327 


3639 


3951 


4263 


4574 


4885 


5196 i 5507 


5818 


311 


140 6128 


6438 


6748 


7058 


7367 


' 7676 : 


7985 


8294 8603 


8911 


309 


1 | 9219 


9527 


9835 
















0142 

3205 


0449 


' 0756 | 
3815 


1063 
4120 


1370 
4424 


1676 


1982 


307 


2 152288 


2594 


2900 


3510 


4728 


5032 


305 


3 5336 


5640 


5943 1 6246 


6549 


6852 


7154 


7457 


7759 


8061 


303 


4 8362 


8664 


8965 


9266 


9567 


9868 












0168 
3161 


0469 
3460 


0769 

3758 


1068 


301 


5 \ 161368 


1667 


1967 


2266 


2564 


2863 


4055 


299 


I 6 1 4353 


4650 


4947 


5244 


5541 


5838 


6134 


6430 


6726 


7022 


297 


I 7 


731? 


7613 


7908 


8203 


8497 


8792 
1726 1 


9086 
2019 


9380 


9674 
2603 


9968 
2895 


295 

293 


8 


170262 


0555 


0848 


1141 


1434 


2311 


9 


3186 


3478 


3769 


4060 


4351 


4641 


4932 


5222 5512 


5802 


291 






Proportio 


nal Pa 


aTS. 




Diff. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


321 


32.1 


64.2 


96.3 


128.4 


160.5 


192.6 


224.7 


256.8 


288.9 


320 


32.0 


64.0 


96.0 


128.0 


160.0 


192.0 


224.0 


256.0 


288.0 


319 


31.9 


63.8 


95.7 


127.6 


159.5 


191.4 


223.3 


255.2 


28Z.1 

286.2 


318 


31.8 


63.6 


95.4 


127.2 


159.0 


190.8 


222.6 


254.4 


317 


31.7 


63.4 


95.1 


126.8 


158.5 


190.2 


221.9 


253.6 


285.3 


316 


31.6 


63.2 


94.8 


126.4 


158.0 


189.6 


221.2 


252.8 


284.4 


315 


31.5 


63.0 


94.5 


126.0 


157.5 


189.0 


220.5 


252.0 


283.5 


314 


31.4 


62.8 


94.2 


125.6 


157.0 


188.4 


219.8 


251.2 


282.6 


313 


31.3 


62.6 


93.9 


125.2 


156.5 


187.8 


219.1 


250.4 


281.7 


312 


31.2 


62.4 


93.6 


124.8 


156.0 


187.2 


218.4 


249.6 


280.8 


311 


31.X 


62.2 


93.3 


124.4 


155.5 


186.6 


217.7 


248.8 


279.9 


310 


31.0 


62.0 


93.0 


124.0 


155.0 


186.0 


217.0 


248.0 


279.0 


309 


30.9 


61.8 


92.7 


123.6 


154.5 


185.4 


216.3 


247.2 


278.1 


308 


30.8 


61.6 


92.4 


123.2 


154.0 


184.8 


215.6 


246.4 


277.2 


307 


30.7 


61.4 


92.1 


122.8 


153.5 


184.2 


214.9 


245.6 


276.3 


306 


30.6 


61.2 


91.8 


122.4 


153.0 


183.6 


214.2 


244.8 


275.4 


1 305 


30.5 


61.0 


91.5 


122.0 


152.5 


183.0 


213.5 


244.0 


274,5 


304 


30.4 


60.8 


91.2 


121.6 


152.0 


182.4 


212.8 


243.2 


273.6 


303 


30.3 


60.6 


90.9 


121.2 


151.5 


181.8 


212.1 


242.4 


272.7 


302 


30.2 


60.4 


90.6 


120.8 


151.0 


181.2 


211.4 


241.6 


271.8 


301 


30.1 


60.2 


90.3 


120.4 


150.5 


180.6 


210.7 


240.8 


270.9 


300 


30.0 


60.0 


90.0 


120.0 


150.0 


180.0 


210.0 


240.0 


270.0 


299 


29.9 


59.8 


89.7 


119.6 


149.5 


179.4 


209.3 


239.2 


269.1 


298 


29.8 


59.6 


89.4 


119.2 


149.0 


178.8 


208.6 


238.4 


268.2 


297 


29.7 


59.4 


89.1 


118.8 


148.5 


178.2 


207.9 


237.6 


267.3 


296 


29.6 


59.2 


88.8 


118.4 


148.0 


177.6 


207.2 


236.8 


266.4 


295 


29.5 


59.0 


88.5 


118.0 


147.5 


177.0 


206.5 


236.0 


265.5 


294 


29.4 


58.8 


88.2 


117.6 


147.0 


176.4 


205.8 235.2 


264.6 


293 


29.3 


58.6 


87.9 


117.2 


146.5 


175.8 


205.1 ! 234.4 


263.7 


292 


29.2 


58.4 


87.6 


116.8 


146.0 


175.2 


204.4 


233.6 


262.8 


1 291 


29.1 


58.2 


87.3 


116.4 


145.5 


174.6 


203.7 


232.8 


261.9 


290 


29.0 


58.0 


87.0 


116.0 


145.0 


174.0 


203.0 


232.0 


261.0 


2S9 


28.9 


57.8 


86.7 


115.6 


144.5 


173.4 


202.3 


231.2 


260.1 


. 288 


28.8 


57.6 


86.4 


115.2 


144.0 


172.8 


201.6 


230.4 


259.2 


! 287 


28.7 


57.4 


86.1 


114.8 


143.5 


172.2 


200.9 


229.6 


258.3 


i 286 


28.6 


57.2 


85.8 


114.4 


143.0 


171.6 


200.2 


228.8 


257.4 



134 



LOGARITHMS OF KUMBKRS. 



No. 150 L. 176.1 


[N 


o. 169 L. 230. 


N. 





12 3 4 


5 


6 


7 


8 


9 


Diff. 


150 


176091 


6381 l 6070 : 6959 7248 


7536 


7825 


8113 


8401 


8689 


289 


1 


8977 


9264 i 9552 i 9839 

! A1 s>« 














0413 
3270 


0699 
3555 


0986 
3839 


1272 
4123 


1558 
4407 


287 
285 


2 


181844 


2129 


2415 ! 2700 ; 2985 


3 


4691 


4975 


5259 J 5542 j 5825 


6108 


6391 


6674 


6956 


7239 


283 


4 

5 


7521 


7803 


HlMiA fiXKfi Mftl7 


8928 


9209 


9490 


9771 








0051 


281 
279 


190332 


0612 


0892 ! 1171 ! 1451 


j 1730 


2010 


2289 


2567 


6 


3125 


3403 


3681 ! 3959 4237 


4514 


4792 


5069 


5346 i 5623 


878 


7 


5900 


6176 


6453 


6729 i 7005 


7281 


7556 


7832 


8107 i 8382 


276 


8 


8657 


8932 


9206 


9481 ] 9755 


1 










\ 0029 
| 2761 


0303 
3033 


0577 
3305 


0850 1124 
3577 3848 


274. 
272 


9 


201397 


1670 


1943 


2216 ■ 2488 


160 


4120 


4391 


4663 


4934 i 5204 


5475 


5746 


6016 


6286 | 6556 


■271 


1 


6826 


7096 


7365 


7634 j 7904 


8173 


8441 


8710 


8979 ! 9247 


269 


2 


9515 


9783 ' 
















0051 
2720 


0319 j 0586 
2986 , 3252 


! 0853 
3518 


1121 
37'83 


1388 
4049 


1654 ! 1921 
4314 ' 4579 


267 
266 


3 


212188 


2454 


4 


4844 


5109 


5373 


5638 5902 


6166 


6430 


6694 


6957 j 7221 


264 


5 
6 


7484 


7747 


8010 


8273 8536 


8798 
1414 


9060 


9323 


9585 j 9846 
2196 ! 2456 


262 


220108 


0370 


0631 


0892 


1153 


1675 


1936 


261 


7 


2716 


2976 


3236 


3496 


3755 


4015 


4274 


4533 


4792 ■ 5051 


259 


8 


5309 


5568 


5826 


6084 


6342 


6600 


6858 


7115 


7372 1 7630 


258 


9 


7887 


8144 


8400 


8657 


8913 


9170 


9426 


9682 


9938 ! 






23 














i 0193 


256 






Proportional Parts. 




Dift. 


1 


2 


3 


4 


5 


6 


7 8 


9 


285 


28.5 


57.0 


85.5 


114.0 


142.5 


171.0 


199.5 


228.0 


256.5 


284 


28.4 


56.8 


85.2 


113.6 


142.0 


170.4 


198.8 


227.2 


255.6 


283 


28.3 


56.6 


84.9 


113.2 


141.5 


169.8 


198.1 


226.4 


254.7 


282 


28.2 


56.4 


84.6 


112.8 


141.0 


169.2 


197.4 


225.6 


253.8 


281 ! 28.1 


56.2 


84.3 


112 4 


140.5 


168.6 


196.7 


224.8 


252.9 


280 28.0 


56.0 


84.0 


112.0 


140.0 


168.0 


196.0 


224.0 


252.0 


279 


27.9 


55.8 


83.7 


111.6 


139.5 


167.4 


195.3 223.2 


251.1 


278 


27.8 


55.6 


83.4 


111.2 


139.0 


166.8 


194.6 ! 222.4 


250.2 


277 


27.7 


55.4 


83.1 


110.8 


138.5 


166.2 


193.9 221.6 


249.3 


276 


27.6 


55.2 


82.8 


110.4 


138.0 


165.6 


193.2 


220.8 


248.4 


275 


27.5 


55.0 


82.5 


110.0 


137.5 


165.0 


192.5 


220.0 


247.5 


274 


27.4 


54.8 


82.2 


109.6 


137.0 


164.4 


191.8 


219.2 


246.6 


273 


27.3 


54.6 


81.9 


109.2 


136.5 


163.8 


191.1 


218.4 


245.7 


272 


27.2 


54.4 


81.6 


108.8 


136.0 


163.2 


190.4 


217.6 


244.8 


271 


27.1 


54.2 


81.3 


108.4 


135.5 


162.6 


189.7 


216.8 


243.9 


270 


27.0 


54.0 


81.0 


108.0 


135.0 


162.0 


189.0 


216.0 


243.0 


269 


26.9 


53.8 


80.7 


107.6 


134.5 


161.4 


188.3 


215.2 


242.1 


268 


26.8 


53.6 


80.4 


107.2 


134.0 


160.8 


187.6 


214.4 


241.2 


267 


26.7 


53.4 


80.1 


106.8 


133.5 


i 160.2 


186.9 


213.6 


240.3 


266 


26.6 


53.2 


79.8 


106.4 


133.0 


159.6 


186.2 


212.8 


239.4 


265 


26.5 


53.0 


79.5 


106.0 


132.5 


159.0 


185.5 212.0 


238.5 


264 


26.4 


52.8 


79.2 


105.6 


132.0 


158.4 


184.8 I 211.2 


237.6 


263 


26.3 


52.6 


78.9 


105.2 


131.5 


157.8 


184.1 : 210.4 


236.7 


262 


26.2 


52.4 


78.6 


104.8 


131.0 


157.2 


183.4 ! 209.6 


235.8 


261 


26.1 


52.2 


78.3 


104.4 


130.5 


156.6 


182.7 j 208.8 


234.9 


260 


26.0 


52.0 


78.0 


104.0 


130.0 


156.0 


182.0 : 208.0 


234.0 


259 


25.9 


51.8 


77.7 


103.6 


129.5 


155.4 


181.3 207.2 


233.1 


258 


25.8 


51.6 


77.4 


103.2 


129.0 


1 154.8 


180.6 i 206.4 


232.2 


257 


25.7 


51.4 


77.1 


102.8 


128.5 


154.2 


179.9 205.6 


231.3 


256 


25.6 


51.2 


76.8 


102.4 


128.0 


1 153.6 


179.2 j 204.8 


230.4 


255 


25.5 


51.0 


76.5 


102.0 


1&7.5 


153.0 


178.5 j 204.0 


229.5 



LOGARITHMS OP NUMBERS. 



135 



No. 


170 L. 230.] 














[N 


o. 189 L. 278. 


N. 





1 


2 


3 


4 


6 


6 


7 


8 


9 


Diff. 


170 
1 

3 


230449 
2996 

5528 
8046 


0704 
3250 

5781 
8297 


0960 

3504 
6033 
8548 


1215 

3757 
6285 
8799 


1470 
4011 
6537 
9049 


1724 
4264 
6789 
9299 


1979 
4517 
7041 
9550 


2234 

4770 
7292 
9800 


24S8 
5023 
7544 


2742 
5276 
7795 


255 
253 

252 


0050 
2541 
5019 
7482 
9932 


0300 
2790 
5266 
7728 


250 
249 
248 
246 


4 

5 
C 


240549 
3038 
5513 
7973 


0799 
3286 
5759 
8219 


1048 
3534 
6006 
8464 


1297 
3782 
6252 
8709 


1546 
4030 
6499 
8954 


1795 
4277 
6745 
9198 


2044 
4525 
6991 
9443 


2293 
4772 
7237 
9687 


' 


0176 
2610 
5031 

7439 
9833 


245 
243 
242 

241 
339 


8 
9 

ISO 
1 


250420 
2853 

5273 

7679 


0664 
3096 

5514 
7918 


0908 
3338 

5755 
8158 


1151 
3580 

5996 
8398 


1395 

3822 

6237 

8637 


1638 
4064 

6477 

8877 


1881 
4306 

6718 
9116 


2125 
4548 

6958 
9355 


2368 
4790 

7198 
9594 


2 
3 
4 

5 
6 


260071 
2451 
4818 
7172 
9513 


0310 

2688 
5054 
7406 
9746 


0548 
2925 
5290 
7641 
9980 


0787 
3162 
5525 
7875 


1025 
3399 
5761 
8110 


1263 
3636 
5996 
8344 


1501 

3873 
6232 

8578 


1739 
4109 
6467 

8812 


1976 
-1346 
6702 
9046 


2214 
4582 
6937 
9279 


238 
237 
235 

234 


0213 
2538 
4850 
7151 


0446 
2770 ! 
5081 

7380 


0679 
3001 
5311 
7609 


0912 
3233 
5542 

7838 


1144 
3464 
5772 

8067 


1377 
3696 
6002 
8296 


1609 
3927 
6232 

8525 


233 
232 
230 

229 


8 

9 


271842 
4158 
6462 


2074 
4389 
6692 


2306 
4620 
6921 



Proportional Parts. 



Diff. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


255 


25.5 


51.0 


76.5 


102.0 


127.5 


153.0 


175.5 


204.0 


229.5 


254 


25.4 


50.8 


76.2 


101.6 


127.0 


152.4 


177.8 


203.2 


228.6 


253 


25.3 


50.6 


75.9 


101.2 


126.5 


151.8 


177.1 


202.4 


227.7 


252 


25.2 


50.4 


75.6 


100.8 


126.0 


151.2 


176.4 


201.6 


226.8 


251 


25.1 


50.2 


75.3 


100.4 


125.5 


150.6 


175.7 


200.8 


225.9 


250 


25 


50.0 


75.0 


100.0 


125.0 


150.0 


175.0 


200.0 


225.0 


249 


24.9 


49.8 


74.7 


99.6 


124.5 


149.4 


174.3 


199.2 


224.1 


248 


24.8 


49.6 


74.4 


99.2 


124.0 


148.8 


173.6 


198.4 


223.2 


247 


24.7 


49.4 


74.1 


98.8 


123.5 


148.2 


172.9 


197.6 


222.3 


246 


24.6 


49.2 


73.8 


98.4 


123.0 


147.6 


172.2 


196.8 


221.4 


245 


24.5 


49.0 


73.5 


98.0 


122.5 


147.0 


171.5 


196.0 


220.5 


244 


24.4 


48.8 


73.2 


97.6 


122.0 


146.4 


170.8 


195.2 


219.6 


243 


24.3 


48.6 


72.9 


97.2 


121.5 


145.8 


170.1 


194.4 


218.7 


242 


24.2 


48.4 


72.6 


96.8 


121.0 


145.2 


169.4 


193.6 


217.8 


241 


24.1 


48.2 


72.3 


96.4 


120.5 


144.6 


168.7 


192.8 


216.9 


240 


24.0 


48.0 


72.0 


96.0 


120.0 


144.0 


168.0 


192.0 


216.0 


239 


23.9 


47.8 


71.7 


95.6 


119.5 


143.4 


167.3 


191.2 


215.1 


238 


23.8 


47.6 


71.4 


95.2 


119.0 


142.8 


166.6 


190.4 


214.2 


237 


23.7 


47.4 


71.1 


94.8 


118.5 


142.2 


165.9 


189.6 


213.3 


236 


23.6 


47.2 ~ 


70.8 


94.4 


118.0 


141.6 


165.2 


188.8 


212.4 


235 


23.5 


47.0 


70.5 


94.0 


117.5 


141.0 


164.5 


188.0 


211.5 


234 


23.4 


46.8 


70.2 


93.6 


117.0 


140.4 


163.8 


187.2 


210.6 


233 


23.3 


46.6 


69.9 


93.2 


116.5 


139.8 


163.1 


186.4 


209.7 


232 


23.2 


46.4 


69.6 


92.8 


116.0 


139.2 


162.4 


185.6 


208.8 


231 


23.1 


46.2 


69.3 


' 92.4 


115.5 


138.6 


161.7 


1&4.8 


207.9 


230 


23.0 


46.0 


69 


92.0 


115.0 


138.0 


161.0 


184.0 


207.0 


229 


22.9 


45.8 


68.7 


91.6 


114.5 


137.4 


160.3 


ias.2 


206.1 


228 


22.8 


45.6 


68.4 


91.2 


114.0 


136.8 


159.6 


182.4 


205.2 


227 


22.7 


45.4 


68.1 


90.8 


113.5 


136.2 


158.9 


181.6 


204.3 


226 


22.6 


45.2 


67.8 


90.4 


113.0 


135.6 


158 2 


180.8 


203.4 



136 








LOGABITHMS 


OF HUMBEKS. 












No. 190 L. 278.] 




[No. 214 


L. 332 


iu 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


Diff. 


Is, 


190 


278754 


8982 


9211 


9439 


9667 


9895 






' 




fi 




























0123 

2396 


0351 

2622 


0578 

2849 


0806 
3075 


228 . 
227 




1 


281033 


1261 


1488 


1715 


1942 


2169 


t 


2 


3301 


3527 


3753 


3979 


4205 


4431 


4656 


4882 


5107 


5332 


226 




8 


5557 


5782 


6007 


6232 


6456 


6681 


6905 


7130 


7354 


7578 


225 




4 
5 


7802 


8026 


8249 


8473 


8696 


8920 


9143 


9366 


9589 


9812 


223 
222 


.'0 


290035 


0257 


0480 


0702 


0925 


1147 


1369 


1591 


1813 


2034 


1 


6 


2256 


2478 


2699 


2920 


3141 


3363 


3584 


3804 


4025 


4246 


221 


:i 


7 


4466 


4687 


4907 


5127 


5347 


5567 


5787 


6007 


6226 


6446 


220 




8 


6665 


6884 


7104 


7323 


7542 


7761 


7979 


8198 


8416 


8635 


219 


•I 


9 


8853 


9071 


9289 


9507 


9725 


9943 














0161 
2331 


0378 
2547 


0595 

2764 


0813 
2980 


218 
217 




200 


301030 


1247 


1464 


1681 


1898 


2114 


i 


1 


3196 


3412 


3628 


3844 


4059 


4275 


4491 


4706 


4921 


5136 


216 




2 


5351 


5566 


5781 


5996 


6211 


6425 


6639 


6854 


7068 


7282 


215 


:l 


8 


7496 


7710 


7924 


8137 


8351 


8564 


8778 


8991 


9204 


9417 


213 




4 


9630 


9843 




















:)' 


0056 
2177 


0268 
2389 


0481 
2600 


0693 
2812 


0906 
3023 


1118 
3234 


1330 
3445 




212 
211 


5 


311754 


1966 


3656 


I 


fi 


3867 


4078 


4289 


4499 


4710 


4920 


5130 


5340 


5551 


5760 


210 




7 


5970 


6180 


6390 


6599 


6809 


7018 


7227 


7436 


7646 


7854 


209 


\ 


8 
9 


8063 


8272 


8481 


8689 


8898 


9106 


9314 


9522 


9730 


9938 


208 
207 




320146 


0354 


0562 


0769 


0977 


1184 


1391 


1598 


1805 


2012 


fi 


210 


2219 


2426 


2633 


2839 


3046 


3252 


3458 


3665 


3871 


4077 


206 




1 


4282 


4488 


4694 


4899 


5105 


5310 


5516 


5721 


5926 


6131 


205 


: 


2 


6336 


6541 


6745 


6950 


7155 


7359 


7563 


7767 


7972 


8176 


204 


SI 








8787 


8991 


9194 


9398 


9601 


9805 
















0008 
2034 


0211 
2236 


203 

202 




4 


330414 


0617 


0819 


1022 


1225 


1427 


1630 


1832 




Proportional Parts. 




Diff. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


Dil 


225 


22.5 


45.0 


67.5 


90.0 


112.5 


135.0 


157.5 


180.0 


202. f 


;i 


224 


22.4 


44.8 


67.2 


89 


fi 


112.0 


134.4 


156.8 


179.2 


201J 




223 


22.3 


44.6 


66.9 


89 


2 


111.5 


133.8 


156.1 


178.4 


200." 




222 


22.2 


44.4 


66.6 


88 


8 


111.0 


133.2 


155.4 


177.6 


199.* 




221 


22.1 


44.2 


66.3 


88 


4 


110.5 


132.6 


154.7 


176.8 


198. 1 




220 


22.0 


44.0 


66.0 


88 





110.0 


132.0 


154.0 


176.0 


198.1 




219 


21.9 


43.8 


65.7 


87 


6 


109.5 


131.4 


153.3 


175.2 


197.1 




218 


21.8 


43.6 


65.4 


87 


2 


109.0 


130.8 


152.6 


174.4 


196.J 


i. 


217 


21.7 


43.4 


65.1 


86 


8 


108.5 


130.2 


151.9 


173.6 


195.,' 




216 


21.6 


43.2 


64.8 


HIS 


4 


108.0 


129.6 


151.2 


172.8 


194 ^ 


I 1 ;, 


215 


21.5 


43.0 


64.5 


86 





107.5 


129.0 


150.5 


172.0 


193.f 


1 


214 


21.4 


42.8 


64.2 


85 


6 


107.0 


128.4 


149.8 


171.2 


193.f 


1 


213 


21.3 


42.6 


63.9 


85 


2 


106.5 


127.8 


149.1 


170.4 


191. 1 


1: 


212 


21.2 


42.4 


63.6 


84 


8 


106.0 


127.2 


148.4 


169.6 


190.* 


II 


211 


21.1 


42.2 


63.3 


84 


4 


105.5 


126.6 


147.7 


168.8 


189,t 


\< 


210 


21.0 


42.0 


63.0 


84 





105.0 


126.0 


147.0 


168.0 


189.1 


li. 


209 


20.9 


41.8 


62.7 


83 


6 


104.5 


125.4 


146.3 


167.2 


188.1 


1 


208 


20.8 


41.6 


62.4 


83 


2 


104.0 


124.8 


145.6 


166 4 


187.2 


11 


207 


20.7 


41.4 


62.1 


82 


8 


103.5 


124.2 


144.9 


165.6 


1864 


1: 


206 


20.6 


41.2 


61.8 


82 


4 


103.0 


123.6 


144.2 


164.8 


185.4 


l;>: 


205 


20.5 


4d.O 


C1.5 


82 





102.5 


123 


143.5 


164.0 


184.5 


llv 


204 


20.4 


40.8 


61.2 


81 


H 


102.0 


122.4 


142.8 


163.2 


183. b 


1M 


203 


20.3 


40.6 


60.9 


81 


2 


101.5 


121.8 


142.1 


162.4 


182.7 
181.8 


|:~ 


202 


20.2 


40.4 


60.6 


,0 


8 


101.0 


121.2 


141.4 


161.6 


1,: 

















































LOGARITHMS 


OF NUMBERS. 






m 


fro. 215 L. 332.] 




[No. 239 L. 380. 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


Diff. 


315 


332438 


2640 


2842 


3044 


3246 


3447 


3649 


3850 


4051 


4253 


202 


6 


4454 


4055 


4856 


50&7 


5257 


5458 


5658 


5859 


6059 


6260 


201 


r 


6460 


6600 


6860 


7060 


7260 


7459 


7659 


7858 


8058 


8257 


200 


8 


8456 


8656 


8855 


9054 


9253 


9451 


9650 


9849 
















0047 
2028 


0246 
2225 


199 

198 


9 


340444 


0642 


0841 


1039 


1237 


1435 


1632 


1830 


220 


2423 


2620 


2817 


3014 


3212 


3409 


3606 


3802 


3999 


4196 


197 


1 


4392 


4589 


4785 


4981 


5178 


5374 


5570 


5766 


5962 


6157 


196 


2 


6353 


6549 


6744 


6939 


7135 


7330 


7525 


7720 


7915 


8110 


195 


| 3 
4 


8305 


8500 


8694 


8889 


9083 


9278 
1216 


9472 
1410 


9666 


9860 
1796 




194 
193 


0054 

1989 


350248 


0442 


0636 


0829 


1023 


1603 


5 


2183 


2375 


2568 


2761 


2954 


3147 


3339 


3532 


3724 


3916 


193 


6 


4108 


4301 


4493 


4685 


4876 


5068 


5260 


5452 


5643 


5834 


192 


7 


6026 


6217 


6408 


6599 


6790 


6981 


7172 


7363 


7554 


7744 


191 


8 


7935 


8125 


8316 


8506 


8696 


8886 


9076 


9266 


9456 


9646 


190 


9 


9835 






















0025 
1917 


0215 
2105 


0404 
2294 


0593 

2482 


0783 
2671 


0972 

2859 


1161 

3048 


1350 
3236 


1539 
3424 


189 
188 


230 


361728 


1 


3612 


3800 


3988 


4176 


4363 


4551 


4739 


4926 


5113 


5301 


188 


2 


5488 


5675 


5862 


6049 


6236 


6423 


6610 


6796 


6983 


7169 


187 


3 


7356 


7542 


7729 


7915 


8101 


8287 


8473 


8659 


8845 


9030 


186 


4 


9216 


9401 


9587 


9772 


9958 
























0143 
1991 


0328 
2175 


0513 
2360 


0698 
2544 


0883 
2728 


185 
184 


5 


371068 


1253 


1437 


1622 


1806 


6 


2912 


3096 


3280 


3464 


3647 


3831 


4015 


4198 


4382 


4565 


184 


7 


4748 


4932 


5115 


5298 


5481 


5664 


5846 


6029 


6212 


6394 


183 


8 


6577 


6759 


6942 


7124 


7306 


7488 


7670 


7852 


8034 


8216 


182 


9 


8398 


8580 


8761 


8943 


9124 


9306 


9487 


9668 


9849 








38 




0030 


181 


Proportio 


NAL Pi 


RTS. 




Diff. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


! oqo 


20.2 


40.4 


60.6 


80.8 


101.0 


121.2 


141.4 


161.6 


181.8 


201 


20.1 


40.2 


60.3 


80.4 


100.5 


120.6 


140.7 


160.8 


180.9 


200 


20.0 


40.0 


60.0 


80.0 


100.0 


120.0 


140.0 


160.0 


180.0 


199 


19.9 


39.8 


59.7 


79.6 


99.5 


119.4 


139.3 


159.2 


179.1 


198 


19.8 


39.6 


59.4 


79.2 


99.0 


118.8 


138.6 


158.4 


178.2 


197 


19.7 


S9.4 


59.1 


78.8 


98.5 


118.2 


137.9 


157.6 


177.3 


196 


19.6 


39.2 


58.8 


78.4 


98.0 


117.6 


137.2 


156.8 


176.4 


195 


19.5 


39.0 


58.5 


78.0 


97.5 


117.0 


136.5 


156.0 


175.5 


194 


19.4 


38.8 


58.2 


77.6 


97.0 


116.4 


135.8 


155.2 


174.6 


193 


19.3 


38.6 


57.9 


77.2 


96.5 


115.8 


135.1 


154.4 


173.7 


192 


19.2 


38.4 


57.6 


76.8 


96.0 


115.2 


134.4 


153.6 


172.8 


191 


19.1 


38.2 


57.3 


76.4 


95.5 


114.6 


133.7 


152.8 


171.9 


190 


19.0 


38.0 


57.0 


76.0 


95.0 


114.0 


133.0 


152.0 


171.0 


189 1 18.9 


37.8 


56.7 


75.6 


94.5 


113.4 


132.3 


151.2 


170.1 


188 18.8 


37.6 


56.4 


75.2 


94.0 


112.8 


131.6 


150.4 


169.2 


187 


is. r 


37 4 


56.1 


74.8 


93.5 


112.2 


130.9 


149.6 


168.3 


186 


18.6 


37.3 


55.8 


74.4 


93.0 


111.6 


130.2 


148.8 


167.4 


185 


18.5 


37.0 


55.5 


74.0 


92.5 


111.0 


129.5 


148.0 


166.5 


184 


18.4 


36.8 


55.2 


73.6 


92.0 


110.4 


128.8 


147.2 


165.6 


183 


18.3 


36.6 


54.9 


73.2 


91.5 


109.8 


128.1 


146.4 


164.7 


182 


18.2 


36.4 


54.6 


72.8 


91.0 


109.2 


127.4 


145.6 


163.8 


181 


18.1 


36.2 


54.3 


72.4 


90.5 


108.6 


126.7 


144.8 


162.9 


180 


18.0 


36.0 


54.0 


72.0 


90.0 


108.0 


126.0 


144.0 


162.0 


179 


17.9 


35.8 


53.7 71.6 


89.5 


107.4 


125.3 


143.2 


161.1 



138 






LOGAKITHMS OP J5fuMBEft& 










r 


No. 240 L. 380.] ^ [No. 269 L. 431. 


:; 


N. 





1 


2 


3 


4 


5 


6 


J 


8 


9 


Diff. 


i 


240 


380211 


0392 


0573 


0754 


0934 


1115 


1296 


1476 


1656 


1837 


181 


u 


1 


2017 


2:97 


2377 


2557 


2737 


2917 


3097 


3277 


3456 


3636 


180 




2 


3815 


3995 


4174 


4353 


4533 


4712 


4891 


5070 


5249 


5428 


179 




3 


5606 


5785 


5964 


6142 


6321 


6499 


6677 


6856 


7034 


7212 


178 ! 




4 


7390 


7568 


7746 


7924 


8101 


8279 


8456 


8634 


8811 


8989 


178 ; 




5 


9166 


9343 


9520 


9698 


9875 
















0051 
1817 


0228 
1993 


0405 
2169 


0582 
2345 


0759 
2521 


177 
176 




6 


390935 


1112 


1288 


1464 


1641 




7 


2697 


2873 


3048 


3224 


3400 


3575 


3751 


3926 


4101 


4277 


176 




8 


4452 


4627 


4802 


4977 


5152 


5326 


5501 


5676 


5850 


6025 


175 




9 


6199 


6374 


6548 


6722 


6896 


7071 


7245 


7419 


7592 


7766 


174 




250 


7940 


8114 


8287 


8461 


8634 


8808 


8981 


9154 


9328 


9501 


173 


i 


1 


9674 


9847 






















0020 
1745 


0192 
1917 


0365 
2089 


0538 
2261 


0711 
2433 


0883 
2605 


1056 

2777 


1228 
2949 






2 


401401 


1573 


172 




3 


3121 


3292 


3464 


3635 


.3807 


3978 


4149 


4320 


4492 


4663 


171 




4 


4834 


5005 


5176 


5346 


5517 


5688 


f>858 


6029 


6199 


6370 


171 




5 


6540 


6710 


6881 7051 


7221 


7391 


7561 


7731 


7901 


8070 


170 i 




6 


8240 


8410 


8579 , 8749 


8918 


9087 


9257 


9426 


9595 


9764 


169 : 






9933 
























0102 
1788 




0609 
2293 


0777 
2461 


0946 
2629 


1114 

2796 


1283 
2964 


1451 
3132 






8 


411620 


1956 


2124 


168 




9 


3300 


3467 


3635 


3803 


3970 


4137 


4305 


4472 


4639 


4806 


167 




260 


4973 


5140 


5307 


5474 


5641 


5808 


5974 


6141 


6308 


6474 


167 




1 


6641 


6807 


6973 


7139 


7306 


7472 


7638 


7804 


7970 


8135 


166 




2 
3 


8301 
9956 


8467 
0121 


8633 
0286 


8798 


8964 
0616 


91*29 


9295 


9460 


9625 


9791 


165 : 

165 




0451 


0781 


0945 


1110 


1275 


1439 




4 


421604 


1768 


1933 


2097 


2261 


2426 


2590 


2754 


2918 


3082 


164 




5 


3246 


3410 


3574 3737 


3901 


4065 


4228 


4392 


4555 


4718 


164 




6 


4882 


5045 


5208 


5371 


5534 


5697 


5S60 


6023 


6186 


6349 


163 




7 


6511 


6674 


6836 


6999 


7161 


7324 


7486 


7648 


7811 


7973 


162 




8 


8135 


8297 


8459 


8621 


8783 


8944 


9106 


9268 


9429 


9591 


162 




9 


9752 
43 


9914 






















0075 | 0236 


0398 


1 0559 


0720 


0881 


1042 


1203 


161 




Proportional Parts. 


I 


Diff 


1 


2 


3 


4 


5 


6 


7 


8 


9 




178 


17.8 


35.6 


53.4 


71.2 


89.0 


106.8 


124.6 


142.4 


160.2 




177 


17.7 


35.4 


53.1 


70.8 


88.5 


106.2 


123.9 


141.6 


159.3 




176 


17.6 


35.2 


52.8 


70.4 


88.0 


105.6 


123.2 


140.8 


158.4 




175 


17.5 


35.0 


52.5 


70.0 


87.5 


105.0 


122.5 


140.0 


157.5 




174 


17.4 


34.8 


52.2 


69.6 


87.0 


104.4 


121.8 


139.2 


156.6 




173 


17.3 


34.6 


51.9 


69.2 


86.5 


103.8 


121.1 


138.4 


155.7: 




172 


17.2 


34.4 


51.6 


68.8 


86.0 


103.2 


120.4 


137.6 


154.8 




171 


17.1 


34.2 


51.3 


68.4 


85.5 


102.6 


119.7 


136.8 


153.9 




170 


17.0 


34.0 


51.0 


68.0 


85.0 


102.0 


119.0 


136.0 


153.0 




169 


16.9 


33.8 


50.7 


67.6 


84.5 


101.4 


118.3 


135.2 


152.1 




168 


16.8 


33.6 


50.4 


67.2 


84.0 


100.8 


117.6 


134.4 


151.2 




167 


16.7 


33.4 


50.1 


66.8 


83.5 


100.2 


116.9 


133.6 


150.3 




166 


16.6 


33.2 


49.8 


66.4 


83.0 


99.6 


116.2 


132.8 


149.4 




165 


16.5 


33.0 


49.5 


66.0 


82.5 


99.0 


115.5 


132.0 


148.5 




164 


16.4 


32.8 


49.2 


65.6 


82.0 


98.4 


114.8 


131.2 


147.6 




163 


16.3 


32.6 


48.9 


65.2 


81.5 


97.8 


114.1 


130.4 


146.7 




162 


16.2 


32.4 


48.5 


64.8 


81.0 


97.2 


113.4 


129.6 


145.8 




161 


16.1 


32.2 


48.3 


64.4 


80.5 


96.6 


112.7 


128.8 144.9 









LOGARITHMS OF NUMBERS. 



139 



No. 270 L. 431.] #> 



[No. 299 L. 476. 



6163 
7751 



4045 
5604 

7158 
8706 



4845 
6366 

7882 



471292 
2756 
4216 
5671 



1525 1685 
3130 I 3290 

47" 

6481 

8067 



9491 



1066 
2637 
4201 
5760 
7313 



! I ! 



1846 
3450 

5048 
6640 

8226 



4357 
5915 



0403 
1940 
3471 
4997 
6518 
8033 
9543 



1048 

2548 
4042 
5532 

7016 
8495 



5150 
6670 
8184 



0116 
1585 



9170 



4:340 
5829 
7312 
8790 



4653 
6107 



2007 
3610 
5207 
6799 
8384 
.9964 

1538 
3106 
4669 
6226 

7778 
9324 



0865 
2400 



5454 
6973 

8487 



1499 

2997 
4490 
5977 
7460 



0410 
1878 
3341 
4799 
6252 



2167 I 
3770 I 
5367 i 
6957 
8542 ! 

0122 
1695 
3263 
4825 



947t 



1018 
2553 
4082 
5606 
7125 



0146 
1649 

3146 
4639 
6126 



1852 
3419 
4981 
6537 



1172 
2706 
4235 

5758 
7276 



1799 

3296 

4788 
6274 
7756 



0557 
2025 
3487 
4944 
6397 



0704 

2171 



7 


8 


9 


2488 


2649 


2809 


4090 


4219 


4409 


5685 


5844 


6004 


7275 


7433 


7592 


8859 
0437 


9017 


9175 


0594 


0752 


2009 


2166 


2323 


3576 


3732 


3889 


5137 


5293 


5449 


6692 


6848 


7003 


8242 


8397 


8552 


9787 


9941 




0095 
1633 


1326 


1479 


2859 


3012 


3165 


4387 


4540 


4692 


5910 


6062 


6214 


7428 


7579 


'7731 


8940 


9091 


9242 


0447 


0597 


0748 


1948 


2098 


2248 


3445 


3594 


3744 


4936 


5085 


52:34 


6423 


6571 


6719 


7904 


8052 


8200 


9380 


9527 


9675 


0851 


0998 


1145 


2318 


2464 


2610 


3779 


3925 


4071 


5235 


5381 


5526 


6687 


6832 


6976 



Proportional Parts. 



Diff. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


161 


16.1 


32.2 


48.3 


64.4 


80.5 


96.6 


112.7 


128.8 


144.9 


160 


16.0 


32.0 


48.0 


64.0 


80.0 


96.0 


112.0 


128.0 


144.0 


159 


15.9 


31.8 


47.7 


63.6 


79.5 


95.4 


111.3 


127.2 


143.1 


158 


15.8 


31.6 


47.4 


63.2 


79.0 


94.8 


110.6 


126.4 


142.2 


157 


15.7 


31.4 


47.1 


62.8 


78.5 


94.2 


109.9 


125.6 


141.3 


156 


15.6 


31.2 


46.8 


62.4 


78.0 


93.6 


109.2 


124.8 


140.4 


155 


15.5 


31.0 


46.5 


62.0 


77.5 


93.0 


108.5 


124.0 


139.5 


154 


15.4 


30.8 


46.2 


61.6 


77.0 


92.4 


107.8 


123.2 


138.6 


153 


15.3 


30.6 


45.9 


61.2 


76.5 


91.8 


107.1 


122.4 


137.7 


152 


15.2 


30.4 


45.6 


60.8 


76.0 


91.2 


106.4 


121.6 


136.8 


151 


15.1 


30.2 


45.3 


60.4 


75.5 


90.6 


105.7 


120.8 


135.9 


150 


15.0 


30.0 


45.0 


60.0 


75.0 


90.0 


105.0 


120.0 


135.0 


149 


14.9 


29.8 


44.7 


59.6 


74.5 


89.4 


104.3 


119.2 


134.1 


148 


14.8 


29.6 


44.4 


59.2 


74.0 


88.8 


103.6 


118.4 


133.2 


147 


14.7 


29.4 


44.1 


58.8 


73.5 


88.2 


102.9 


117.6 


132.3 


146 


14.6 


29.2 


43.8 


58.4 


73.0 


87.6 


102.2 


116.8 


131.4 


145 


14.5 


29.0 


43.5 


58.0 


72.5 


87.0 


101.5 


116.0 


130.5 


144 


14.4 


28.8 


43.2 


57.6 


72.0 


86.4 


100.8 


115.2 


129.6 


143 


14.3 


28.6 


42.9 


57.2 


71.5 


85.8 


100.1 


114.4 


128.7 


142 


14.2 


28.4 


42.6 


56.8 


71.0 


85.2 


99.4 


113.6 


127.8 


141 


14.1 


28.2 


42.3 


56.4 


70.5 


84.6 


98.7 


112.8 


126.9 


140 


14.0 


28.0 


42.0 


56.0 


70.0 


84.0 


98.0 


112.0 


126.0 



140 



LOGARITHMS OF NUMBERS. 



No. 300 L. 477.] 



[No. 339 L. 531. 



477121 
8566 



480007 
1443 
2874 
4300 
5721 
7138 
8551 



2760 
4155 
5544 
6930 
8311 
9687 



501059 
2427 
3791 

5150 
6505 
7856 



3218 
4548 
5874 
7196 

8514 

9828 



521138 
2444 
3746 
5045 



8917 
530200 



(711 



0151 

1586 
3016 
4442 

5863 



1502 
2900 
4294 

5(583 
7068 
8448 
9824 



5286 
6640 
7991 



0679 
2017 
3351 

4681 



8646 
9959 



2575 

3876 
5174 
6469 
7759 

9015 



0294 
1729 
3159 
4585 

01 )().■} 
7421 
8833 



0239 

1642 
3040 
4433 

5*22 
7206 



1872 
3302 
4727 
6147 
7563 
8974 



0380 

1782 
3179 
4572 
5960 
7344 
8724 

0099 
1470 
2837 
4199 

5557 
6911 



0947 

2284 
3617 
4946 
6271 
7592 



0221 
1530 

2835 
4136 
5434 

6727 



0520 

1922 
3319 
4711 
6099 

7483 



1740 



1081 
2418 
3750 
5079 
6403 
7724 
9040 

0353 
1661 



8145 



0456 0584 0712 0840 0968 



7844 7989 
9287 9431 



0725 
2159 
3587 
5011 
6430 
7845 



3458 
4850 



0374 

1744 
3109 
4471 

5828 
7181 
8530 

9874 



1215 
2551 

5211 
6535 

7855 



0484 

17! 12 



8274 
9559 



8133 8278 8422 
9575 9719 



3730 
5153 
6572 



1012 
2445 
3872 
5295 
6714 
8127 
9537 



4015 

5437 
6855 



0801 
2201 



77'59 
9137 



3246 
4607 

5964 

7316 



0009 

2684 
4016 
5344 

6668 
7987 



0941 
2341 



5128 
6515 



0615 



7114 
8402 
96b'7 



0143 
1482 
2818 
4149 
5476 
6800 
8119 

9434 

0745 
2053 
3356 
4656 
5951 
7243 
8531 
9815 



5267 
6653 



4878 
6234 



0277 
1616 
2951 
4282 



0876 
2183 



1096 



5014 

6370 
7721 



0411 
1750 
3084 
4415 
5741 
7064 
8382 



1007 
2314 
3616 
4915 
6210 
7501 
8788 



0072 
1351 



Proportional Parts. 



Diff. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


139 


13.9 


27.8 


41.7 


55.6 


69.5 


83.4 


97.3 


111.2 


125.1 


138 


13.8 


27.6 


41.4 


55.2 


69.0 


82.8 


96.6 


110.4 


124.2 


137 


13.7 


27.4 


41.1 


54.8 


68.5 


82.2 


95.9 


109.6 


123.3 


136 


13.6 


27.2 


40.8 


54.4 


68.0 


81.6 


95.2 


108.8 


122.4 


135 


13.5 


27.0 


40.5 


54.0 


67.5 


81.0 


94.5 


108.0 


121.5 


134 


13.4 


26.8 


40.2 


53.6 


67.0 


80.4 


93.8 


107.2 


120.6 


133 


13.3 


26.6 


39.9 


53.2 


66.5 


79.8 


93.1 


106.4 


119.7 


132 


13.2 


26.4 


39.6 


52.8 


66.0 


79.2 


92.4 


105.6 


118.8 


131 


13.1 


26.2 


89.3 


52.4 


65.5 


78.6 


91.7 


104.8 


117.9 


130 


13.0 


26.0 


39.0 


52.0 


65.0 


78.0 


91.0 


104.0 


117.0 


129 


12.9 


25.8 


38.7 


51.6 


64.5 


77.4 


90.3 


103.2 


116.1 


128 


12.8 


25.6 


38.4 


51.2 


64.0 


76.8 


89.6 


102.4 


115.2 


127 


12 7 


25.4 


38.1 


50.8 


63.5 


76.2 


88.9 


101.6 


114.3 



LOGARITHMS OF NUMBERS. 



141 



I No. 340 L. 531.] 



[No. 379 L. 579. 



531479 
2754 
4026 
5294 
6558 
7819 
9076 



540329 
1579 



5307 
6543 

7775 



550228 
1450 



5094 
6303 

7507 
8709 
9907 



3481 
4666 
5848 
7026 

8202 
9374 



6341 

7492 






0351 

1572 
2790 
4004 
5215 

6423 

7627 
8829 



0026 
1221 
2412 
3600 
4784 
5966 
7144 

8319 

9491 



0660 



4147 
5303 
6457 
7607 

8754 



4280 
5547 
6811 



0473 
1694 
2911 
4126 
5336 

6544 

7748 



0146 
1340 
2531 
3718 
4903 
6084 
7262 
8436 



0776 
1942 
3104 
4263 
5419 
6572 
7722 



3 



1862 
3136 
4407 
5674 
6937 
8197 
9452 



0595 
1816 
3033 
4247 
5457 
6664 
7868 
9068 



0265 
1459 
2650 
3837 
5021 
6202 
7379 

8554 
9725 



2058 
3220 
4379 
5534 
6687 



1990 2117 

3264 3391 

4534 4661 

5800 5927 

7063 j 7189 

8322 : 8448 

9578 I 9703 



0955 
2203 
3447 



2078 
3323 

4564 j 

5802 

7036 

8267 
9494 



0717 
1938 
3155 



6785 
7988 
9188 



0385 
1578 
2769 
3955 
5139 
6320 
7497 
8671 
9842 



5925 
7159 



0840 
2060 
3276 
4489 
5699 
6905 
8108 



0504 
1698 
2887 
4074 
5257 
6437 
7614 



1010 


1126 


2174 


2291 


3336 


3452 


4494 


4610 


5650 


5765 


6802 


6917 


7951 


8086 


9097 


9212 



2245 
3518 
4787 
6053 
7315 
8574 



1080 
2327 
3571 

4812 
6049 

8512 
9739 



0962 
2181 



9428 



0624 
1817 
3006 
4192 
5376 
6555 



0076 
1243 

2407 



8181 
9326 



4914 



3519 
4731 
5940 

7146 
8349 
9548 



0743 
1936 
3125 
4311 
5494 
6673 
7849 
9023 



0193 
1359 



7147 
8295 
9441 



2500 
3772 
5041 



1206 
2425 
3640 



: 

3244 
4429 
5612 
6791 
7967 
9140 

0309 
1476 
2639 
3800 
4957 
6111 
7262 
8410 
9555 



2627 
3899 
5167 

7693 



0982 

2174 



4548 
5730 



1592 
2755 
3915 
5072 
6226 
7377 



Proportional Parts. 



Diff. 


» 


2 


3 


4 


5 


6 


7 


8 


9 


128 


12.8 


25.6 


38.4 


51.2 


64.0 


76.8 


89.6 


102.4 


115.2 


127 


12.7 


25.4 


38.1 


50.8 


63.5 


76.2 


88.9 


101.6 


114.3 


126 


12.6 


25.2 


37.8 


50.4 


63.0 


75.6 


88.2 


100.8 


113.4 


125 


12.5 


25.0 


37.5 


50.0 


62.5 


75.0 


87.5 


100.0 


112.5 


124 


12.4 


24.8 


37.2 


49.6 


62.0 


74.4 


86.8 


99.2 


111.6 


123 


12.3 


24.6 


36.9 


49.2 


61.5 


73.8 


86.1 


98.4 


110.7 


122 


12.2 


24.4 


36.6 


48.8 


61.0 


73.2 


85.4 


97.6 


109.8 


121 


12.1 


24.2 


36.3 


48.4 


60.5 


72.6 


84.7 


96.8 


108.9 


120 


12.0 


24.0 


36.0 


48.0 


60.0 


72.0 


84.0 


96.0 


108.0 


119 


11.9 


23.8 


35.7 


47.6 


59.5 


71.4 


83.3 


95.2 


107.1 



142 



LOGARITHMS OF NUMBERS. 



No. 380. L. 579.]^ 


[No. 414 L. 617. 


N. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


Diff. 


380 
1 


579784 


9898 


0012 

1153 


, 












0126 
1267 


0241 

1381 


j 0355 


0469 ' 0583 | 0697 


0811 
1950 


114 


580925 


1039 


1495 1 


1608 '< 1722 I 1836 


2 


2063 


2177 


2291 


2404 


2518 


2631 


2745 2858 


2972 


3085 




3 


3199 


3312 


3426 


3539 


3652 


3765 


3879 


3992 


4105 


4218 




4 


4331 


4444 


4557 


4670 


4783 


4896 


5009 


5122 


5235 


5348 


113 


5 


5461 


5574 


5686 


5799 


5912 


6024 


6137 


6250 


6362 


6475 




fi 


6587 


6700 


6812 


6925 


7037 


! 7149 


7262 


7374 


7486 


7599 




7 


7711 


7823 


7935 


8047 


8160 


! 8272 


8384 


8496 


8608 


8720 


112 


8 


8832 


8944 


9056 


9167 


'9279 


9391 


9503 


9615 


9726 


9838 




9 


9950 






0284 
1399 








0842 






0061 
1176 


0173 

1287 


0396 
1510 




0619 1 0730 


390 


591065 


i 1621 


1732 1843 


1955 


2066 




1 


2177 


2288 


2399 


2510 


2621 


2732 


2843 i 2954 


3064 


3175 


111 


2 


3286 


3397 


3508 


3618 


3729 


3840 


3950 1 4061 


4171 


4282 




3 


4393 


4503 


4614 


4724 


4834 


I 4945 


5055 j 5165 


5276 


5386 




4 


5496 


5606 


5717 


5827 


5937 


! 6047 


6157 6267 


6377 


6487 




5 


6597 


6707 


6817 


6927 


7037 


| 7146 


7256 j 7366 


7476 


7586 


110 


6 


7695 


7805 


7914 


8024 


8134 


8243 


8353 8462 


8572 


8681 




7 


8791 


8900 


9009 


9119 


9228 


! 9337 


9446 . 9556 


9665 


9774 




8 


9883 


9992 


















0101 


0210 


0319 


i 0428 


0537 


0646 


0755 


0864 




9 


600973 


1082 


1191 


1299 


1408 


1517 


1625 


1734 


1843 


1951 




400 


2060 


2169 


2277 


2386 


2494 


2603 


2711 


2819 


2928 


3036 




1 


3144 


3253 


3361 


3469 


3577 


3686 


3794 


3902 


4010 


4118 


108 


2 


4226 


4334 


4442 


4550 


4658 


4766 


4874 


4982 


5089 


5197 




3 


, 5305 


5413 


5521 


5628 


5736 


5844 


5951 


6059 


6166 


6274 




4 


6381 


6489 


6596 


6704 


6811 


6919 


7026 


7133 


7241 


7348 




5 


7455 


7562 


7669 


7777 


7884 


7991 


8098 


8205 


8312 


8419 


107 


6 


8526 


8633 


8740 


8847 


8954 


9061 


9167 


9274 


9381 


9488 




7 


9594 


9701 


9808 


9914 


















0128 
1192 




0341 
1405 


0447 
1511 






8 


610660 


0767 


0873 


0979 


1086 


1298 


1617 




9 


1723 


1829 


1936 


2042 


2148 


2254 


2360 


2466 


2572 


2678 


106 


410 


2784 


2890 


2996 


3102 


3207 


3313 


3419 


• 3525 


3630 


3736 




1 


3842 


3947 


4053 


4159 


4264 


4370 


4475 


4581 


4686 


4792 




2 


4897 


5003 


5108 


5213 


5319 


5424 


5529 


5634 


5740 


5845 




3 


5950 


6055 


6160 


6265 


6370 


6476 


6581 


6686 


6790 


6895 


105 


4 i 


7000 


7105 


7210 


7315 | 7420 


7525 


7629 


7734 


7839 


7943 




Proportional. Parts. 


Diff 


. 1 


2 


3 


4 


5 


6 


7 


8 


9 


118 


11.8 


23.6 


35.4 


47.2 


59.0 


70.8 


82.6 


94.4 


106.2 


117 


11.7 


23.4 


35.1 


46.8 


58.5 


70.2 


81.9 


93.6 


105.3 


116 


11.6 


23.2 


34.8 


46.4 


58.0 


69 6 


81.2 


92.8 


104.4 


115 


11.5 


23.0 


34.5 


46.0 


57.5 


69.0 


80.5 


92.0 


103.5 


114 


11.4 


22.8 


34.2 


45.6 


57.0 


68.4 


79.8 


91.2 


102.6 


113 


11.3 


22.6 


33.9 


45.2 


56.5 


67.8 


79.1 


90.4 


101.7 


112 


11.2 


22.4 


33.6 


44.8 


56.0 


67.2 


78.4 


89.6 


100.8 


111 


11.1 


22.2 


33.3 


44.4 


55.5 


66.6 


77.7 


88.8 


99.9 


110 


11.0 


22.0 


33.0 


44.0 


55.0 


66.0 


77.0 


88.0 


99.0 


109 


10.9 


21.8 


32.7 


43.6 


54.5 


65.4 


76.3 


87.2 


98.1 


108 


10.8 


21.6 


32.4 


43.2 


54.0 


64.8 


75.6 


86.4 


97.2 


107 


10.7 


21.4 


32.1 


42.8 


53.5 


64.2 


74.9 


85.6 


96.3 


106 


10.6 


21.2 


31.8 


42.4 


53.0 


63.6 


74.2 


84.8 


95.4 


105 


10.5 


21.0 


31.5 


42.0 


52.5 


63.0 


73.5 


84.0 


94.5 


105 


10.5 


21.0 


31.5 


42.0 


52.5 


63.0 


73.5 


84.0 


94.5 


104 


10.4 


20.8 


31.2 


41.6 


52.0 


62.4 


72.8 


83.2 


93.6 



LOGARITHMS OF KtJMBERS. 



143 



No. 415 L. 618.] 



[No. 459 L. 662 



N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


Diff. 


415 
6 


618048 
9093 


8153 
9198 


8257 
9302 


8362 
9406 


8466 
9511 


8571 
9615 


8676 
9719 

0760 
1799 
2835 

3869 
4901 
5929 
6956 
7980 
9002 


8780 
9824 


8884 
9928 

0968 
2007 
3042 

4076 
5107 
6135 
7161 
8185 
9206 


8989 

0032 
1072 
2110 
3146 
4179 
5210 
6238 
7263 
8287 
9308 


105 


r 

8 

9 

420 

1 

2 

4 
5 
6 


620136 
1176 
2214 

3249 

4282 
5312 
6340 
7366 
8389 
9410 


0240 
1280 
2318 
3353 
4385 
5415 
6443 
7468 
8491 
9512 


0344 
"1384 
2421 

3456 
4488 
5518 
6546 
7571 
8593 
9613 


0448 
1488 
2525 

3559 
4591 
5621 
6648 
7673 
8695 
9715 


0552 
1592 
2628 
3663 
4695 
5724 
6751 

8797 
9817 


0656 
1695 
2732 

3766 

4798 
5827 
6853 
7878 
8900 
9919 

0936 
1951 
2963 

3973 
4981 
5986 
6989 
7990 
8988 
9984 

0978 
1970 
2959 

3946 
4931 
5913 

6894 
7872 
8848 
9821 


0864 
1903 
2939 

3973 
5004 
6032 

7058 
8082 
9104 


104 

103 
102 


0021 
1038 
2052 
3064 

4074 

5081 
6087 
7089 
8090 
9088 


0123 
1139 
2153 
3165 

4175 
5182 
6187 
7189 
8190 
9188 

0183 
1177 
2168 
3156 

4143 
5127 
6110 
7089 
8067 
9043 


0224 
1241 
2255 
3266 

4276 
5283 
6287 
7290 
8290 
9287 

0283 
1276 
2267 
3255 

4242 
5226 
6208 
7187 
8165 
9140 


0326 
1342 
2356 
3367 

4376 

5383 
6388 
7390 
8389 
9387 




8 
9 
430 
1 
2 
3 
4 
5 
6 


630428 
1444 
2457 

3468 

4477 
5484 
6488 
7490 
8489 
9486 


0530 
1545 
2559 

3569 

4578 
5584 
6588 
7590 
8589 
9586 


0631 
1647 
2660 

3670 

4679 
5685 
6688 
7690 
8689 
9686 


0733 
1748 
2761 

3771 

4779 
5785 
6789 
7790 
8789 
9785 


0835 
1849 
2862 
3872 
4880 
5886 
6889 
7890 
8888 
9885 


101 
100 


0084 
1077 
2069 
3058 

4044 
5029 
6011 
6992 
7969 
8945 
9919 

~0890 
1859 
2826 

3791 
4754 
5715 
6673 
7629 
8584 
9536 


0382 
1375 
2366 
3354 

4340 

5324 
6306 
7285 
8262 
9237 

0210 
1181 
2150 
3116 

4080 
5042 
6002 
6960 
7916 
8870 
9821 




7 
8 
9 

440 
1 
2 
3 
4 
5 
6 


640481 
1474 
2465 

3453 

4439 
5422 
6404 
7383 
8360 
9335 


0581 
1573 
2563 
3551 
4537 
5521 
6502 
7481 
8458 
9432 


0680 
1672 
2662 

3650 
4636 
5619 
6600 
7579 
8555 
9530 


0779 
1771 
2761 

3749 
4734 
5717 
6698 
7676 
8653 
9627 


0879 
1871 
2860 

3847 

4832 
5815 

8750 
9724 


99 
98 


0016 
0987 
1956 
2923 

3888 
4850 
5810 
6769 
7725 
8679 
9631 


0113 
1084 
2053 
3019 

3984 
4946 
5906 
6864 
7820 
8774 
9726 




7 
8 
9 
450 
1 
2 
3 
4 
5 
6 


650308 
1278 
2246 

3213 
4177 
5138 
6098 
7056 
8011 
8965 
9916 


0405 
1375 
2343 

3309 

4273 
5235 
6194 
7152 
8107 
9060 


0502 
1472 
2440 

3405 
4369 
5331 
6290 
7247 
8202 
9155 


0599 
1569 
2536 

3502 
4465 
5427 
6386 
7343 
8298 
9250 


0696 
1666 
2633 

3598 
4562 
5523 
6482 
7438 
8393 
9346 


0793 
1762 
2730 

3695 
4658 
5619 
6577 
7534 
8488 
9441 


97 
96 




0011 
0960 
1907 


0106 
1055 
2002 


0201 
1150 
2096 


0296 
1245 
2191 


0391 
1339 
2286 


0486 
1434 
2380 


0581 
1529 
2475 


0676 
1623 
2569 


0771 
1718 
2663 




8 

9 


660865 
1813 





Proportional Parts. 



Diff. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


105 


10.5 


21.0 


31.5 


42.0 


52.5 


63.0 


73.5 


84.0 


: 94.5 


104 


10.4 


20.8 


31.2 


41.6 


52.0 


62.4 


72 8 


83.2 


93.6 


103 


10.3 


20.6 


30.9 


41.2 


51.5 


61.8 


72 1 


82.4 


92.7 


102 


10.2 


20.4 


30.6 


40.8 


51.0 


61.2 


71.4 


81.6 


91.8 


101 


10.1 


20.2 


30.3 


40.4 


50.5 


60.6 


70 7 


80.8 


90.9 


100 


10.0 


20.0 


30.0 


40.0 


50.0 


60.0 


70 


80.0 


90.0 


99 


9.9 


19.8 


29.7 


39.6 


49.5 


59.4 


69.3 


79.2 


89.1 



144 



LOGARITHMS OF NUMBERS. 





No. 460 L. 662.] fNo. 499 T. 698 




N 
460 





1 


2 


8 


4 


5 


6 

3324 


7 8 


9 


Diff. 




662758 


2852 


2947 


3041 


3135 


3230 


3418 


3512 








1 


3701 


3795 


3889 


3983 


4078 


4172 


4266 


4360 


4454 








2 


4642 


4736 


4830 


4924 


5018 


5112 


5206 


5299 


5393 


5487 


94 




3 


5581 


5675 


5769 


5862 


5956 


6050 


6143 


6237 


6331 


6424 




4 


6518 


6612 


6705 


6799 


6892 


6986 


7079 


7173 


7266 








5 


7453 


7546 


7640 


7733 


7826 


7920 


8013 


8106 


8199 








6 


8386 


8479 


8572 


8665 


8759 


8852 


8945 


9038 


9131 


9224 






7 
8 


9317 


9410 


9503 


9596 


9689 


9782 


9875 


9967 
0895 










0060 
0988 


0153 






670246 


0339 


0431 


0524 


0617 


0710 


0802 


93 




9 


1173 


1265 


1358 


1451 


1543 


1636 


1728 


1821 


1913 


2005 






470 


2098 


2190 


2283 


2375 


2467 


2560 


2652 


2744 


2836 








1 


3021 


3113 


3205 


3297 


3390 


3482 


3574 


3666 


3758 








2 


3942 


4034 


4126 


4218 


4310 


4402 


4494 


4586 


4677 


4769 


92 




4 


4861 

5778 


i 4953 
5870 


5045 
5962 


5137 
6053 


5228 
6145 


5320 
6236 


5412 
6328 


5503 
6419 


5595 
6511 


5687 
6602 








6694 


6785 


6876 


6968 


7059 


7151 


7242 


7333 


7424 








6 


7607 


7698 


7789 


7881 


7972 


8063 


8154 


8245 


8336 


8427 
9337 






7 
8 

9 


8518 
9428 


8609 
9519 


8700 
9610 


8791 
9700 


8882 
9791 


8973 
9882 


9064 
9973 


9155 


9246 


91 




680336 


0426 


0517 


0607 


0698 


0789 


0879 


0063 
0970 


0154 
1060 


0245 
1151 






480 
1 
2 


1241 
2145 
3047 


1332 

2235 
3137 


1422 
2326 
3227 


1513 

2416 
3317 


1603 
2506 
3407 


1693 

2596 
3497 


1784 
2686 
3587 


1874 

2777 
3677 


1964 

2867 
3767 


2055 
2957 
3857 
4756 
5652 
6547 


90 




3 

4 
5 
6 


3947 
4845 
5742 
6636 


4037 
4935 
5831 
6726 


4127 
5025 
5921 
6815 


4217 
5114 
6010 
6904 


4307 1 
5204 
6100 
6994 


4396 
5294 
6189 

7083 


4486 
5383 
6279 

7172 


4576 
5473 
6368 
7261 


4666 
5563 
6458 
7351 




7 
8 
9 


7529 
8420 
9309 


7618 
8509 
9398 


7707 
8598 
9486 


7796 
8687 
9575 


.7886 
8776 
9664 


7975 
8865 
9753 


8064 
8953 
9841 


8153 
9042 
9930 - 


8242 
9131 


8331 
9220 


89 




















[)()19 


0107 




490 
1 
2 
3 

4 
5 
6 

8 
9 


690196 
1081 
1965 
2847 
3727 
4605 
5482 
6356 
7229 
8100 


0285 
1170 

2053 


0373 

1258 
2142 


0462 
1347 
2230 


0550 : 

1435 
2318 ; 


0639 
1524 
2406 


0728 
1612 
2494 


0816 
1700 
2583 


0905 
1789 
^671 


0993 
1877 
2759 






2935 
3815 
4693 
5569 
6444 
7317 
8188 


3023 
3903 
4731 
5657 
6531 
7404 
8275 


3111 
3991 
4868 
5744 
6618 
7491 
3362 


3199 

4078 
4956 
5832 ,' 
6706 
7578 1 
8449 


3287 
4166 
5044 
5919 
6793 
7665 
8535 


3375 
4254 
5131 
6007 

6880 
7752 
8622 


3463 

4342 
5219 
6094 
6968 
7839 ' 
8709 I 


3551 
4430 
5307 
3182 
?055 
"926 
S796 


3639 
4517 
5394 
6269 
7142 
8014 
8883 


88 
87 




Proportional, Parts. 


Diff. 


1 


2 


3 


4 


5 


6 1 7 


8 


9 


1 


98 

97 

96 

95 

94 

93 

92 

91 

90 

89 

88 

87 i 

86 


1 9.8 
9.7 
9.6 
9.5 
9.4 
9.3 
9.2 
9.1 
9.0 
8.9 
8.8 
8.7 [ 
8.6 | 


19.6 
19.4 
19.2 
19.0 
18.8 
18.6 
18.4 
18.2 
18.0 
17.8 
17:6 
17.4 
17.2 


29.4 
29.1 
28.8 
28.5 
28.2 
27.9 
27.6 
27.3 
27.0 
26.7 
26.4 
26.1 
25.8 


39.2 
38.8 
38.4 
38.0 
37.6 
37.2 
36.8 
36.4 
36.0 
35.6 
35.2 

t 34.-8 I 

1 34.4 


49.0 

48.5 

48.0 

47.5 

47.0 

46.5 

46.0 

45.5 

45.0 

44.5 

44.0 1 

43 /5 I 

43.0 1 


58.8 
58.2 
57.6 
57.0 
56.4 
55.8 
55.2 
54.6 
54.0 
53.4 
52.8 
"52.2 
51.6 


68.6 
67.9 
67.2 
66.5 
65.8 
65.1 
64.4 
63.7 
63.0 
62.3 
61.6 
| -60. "9 
60.2 


78.4 
77.6 
76.8 
76.0 
75.2 
74.4 
73.6 
72.8 
72.0 
71.2 
70.4 
69.6 
68.8 


88.2 

87.3 

86.4 

85.5 

84.6 

83.7 

82.8 

81.9 

81.0 

80.1 

79.2 1 

78."3 i 

77.4 | 



LOGARITHMS OF LUMBERS. 



145 



No. 500 L. 698.] ^ 






[No. 544 L. 736. 


N. 





1 


2 


3 


4 


6 


6 


7 


8 


9 


Diff. 


500 698970 


9057 


9144 


9231 


9317 


9404 


9491 


9578 


9664 


9751 




1 9838 


9924 




















0011 

0877 


0098 
0963 


0184 
1050 






0444 
1309 


0531 
1395 


0617 

1482 




f 2 


700704 


0790 


1136 


1222 




3 


1568 


1654 


1741 


1827 


1913 


1999 


2086 


2172 


2258 


2344 




4 


2431 


2517 


2603 


2689 


2775 


2861 


2947 


3033 


3119 


3205 




5 


3291 


3377 


3463 


3549 


3635 


3721 


3807 


3893 


3979 


4065 


86 


6 


4151 


4236 


4322 


4408 


4494 


4579 


4665 


4751 


4837 


4922 




7 


5008 


5094 


5179 


5265 


5350 


5436 


5522 


5607 


5693 


5778 




8 


5864 


5949 


6035 


6120 


6206 


6291 


6376 


6462 


6547 


6632 




9 


6718 


6803 


6888 


6974 


7059 


7144 


7229 


7315 


7400 


7485 




510 


7570 


7655 


7740 


7826 


7911 


7996 


8081 


8166 


8251 


8336 


85 


1 


8421 


8506 


8591 


8676 


8761 


8846 


8931 


9015 


9100 


9185 














































0033 

0879 




3 


710117 


0202 


0287 


0371 


0456 


0540 


0625 


0710 


0794 




4 


0963 


1048 


1132 


1217 


1301 


1385 


1470 


1554 


1639 


1723 




5 


1807 


1892 


1976 


2060 


2144 


2229 


2313 


2397 


2481 


2566 




6 


2650 


2734 


2818 


2902 


2986 


3070 


3154 


3238 


3323 


3407 


84 


7 


3491 


3575 


3659 


3742 


3826 


3910 


3994 


4078 


4162 


4246 


8 


4330 


4414 


4497 


4581 


4665 


4749 


4833 


4916 


5000 


5084 




9 


5167 


5251 


5335 


5418 


5502 


5586 


5669 


5753 


5836 


5920 




520 


6003 


6087 


6170 


6254 


6337 


6421 


6504 


6588 


6671 


6754 




1 1 


6838 


6921 


7004 


7088 


7171 


7254 


7338 


7421 


7504 


7587 




2 7671 


7754 


7837 


7920 


8003 


8086 


8169 


8253 


8336 


8419 


83 


3 


8502 


8585 


8668 


8751 


8834 


8917 


9000 


9083 


9165 


9248 


4 


9331 


9414 


9497 


9580 


9663 


9745 


9828 


9911 


9994 


0077 
0903 




5 


720159 


0242 


0325 


0407 


0490 


0573 


0655 


0738 


0821 




6 


0986 


1068 


1151 


1233 


1316 


1398 


1481 


1563 


1646 


1728 




7 


1811 


1893 


1975 


2058 


2140 


2222 


2305 


2387 


2469 


2552 




8 


2634 


2716 


2798 


2881 


2963 


3045 


3127 


3209 


3291 


3374 




9 


3456 


3538 


3620 


3702 


3784 


3866 


3948 


4030 


4112 


4194 


82 


530 


4276 


4358 


4440 


4522 


4604 


4685 


4767 


4849 


4931 


5013 




1 


5095 


5176 


5258 


5340 


5422 


5503 


5585 


5667 


5748 


5830 




2 


5912 


5993 


6075 


6156 


6238 


6320 


6401 


6483 


6564 


6646 




3 


6727 


6809 


6890 


6972 


7053 


7134 


7216 


7297 


7379 


7460 




4 


7541 


7623 


7704 


7785 


7866 


7948 


8029 


8110 


8191 


8273 




5 


8354 


8435 


8516 


8597 


8678 


8759 


8841 


8922 


9003 


9084 




6 


9165 


9246 


9327 


9408 


9489 


9570 


9651 


9732 


9813 


9893 


81 


7 


9974 






















0055 
0863 


0136 
0944 


0217 
1024 


0298 
1105 


0378 
1186 


0459 
1266 


0540 
1347 


0621 
1428 


0702 
1508 




8 


730782 




y 


1589 


1669 


1750 


1830 


1911 


1991 


2072 


2152 


2233 


2313 




540 


2394 


2474 


2555 


2635 


2715 


2796 


2876 


2956 


3037 


3117 




1 


8197 


3278 


3358 


3438 


3518 


3598 


3679 


3759 


3839 


3919 




2 


3999 


4079 


4160 


4240 


4320 


4400 


4480 


4560 


4640 


4720 


80 


3 


4800 


4880 


4960 


5040 


5120 


5200 


5279 


5359 


5439 


5519 


4 


5599 


5679 


5759 


5838 


5918 


5998 


6078 


6157 


6237 


6317 




Proportional Parts. 


Diff. 


1 

8.7 


2 


3 


4 


5 


e J r 


8 


9 


87 


17.4 


26 


1 




34.8 


43.5 


52.2 


60.9 


69.6 


78.3 


86 


8.6 


17.2 


25 


8 




34.4 ' 


43.0 


51.6 


60.2 


68.8 


77.4 


85 


8.5 


17.0 


25 


5 




14.0 


42.5 


51.0 


59.5 


68.0 


76.5 


84 


8.4 


16.8 


25 


2 




33.6 


42.0 


50.4 58.8 


67.2 


75.6 





































146 








LOGARITHMS OF NUMBEEg, 








No. 545 L. 736.] 


[No. 584 L. 787. 


N. | 1 


2 3 


4 


5 


6 


7 


8 j 9 


Diff. 


545 


736397 6476 


6556 


6635 


6715 


6795 


6874 


6954 


7034 


7113 




6 


7193 


7272 


7352 


7431 


7511 


7590 


7670 


7749 


7829 


7908 




7 


7987 


8067 


8146 


8225 


8305 


8384 


8463 


8543 


8622 


8701 




8 


8781 


8860 


8939 


9018 


9097 


9177 


9256 


9335 


9414 


9493 




9 


9572 


9651 


9731 


9810 


9889 


9968 












0047 
0836 


0126 
0915 


0205 
0994 


0284 
1073 


79 


550 


740363 


0442 


0521 


0600 


0678 


0757 


1 


1152 


1230 


1309 


1388 


1467 


1546 


1624 


1703 


1782 


1860 




2 


1939 


2018 


2096 


2175 


2254 


2332 


2411 


2489 


2568 


2647 




3 


2725 


2804 


2882 


2961 


3039 


3118 


3196 


3275 


3353 


3431 




4 


3510 


3588 


3667 


3745 


3823 


3902 


3980 


4058 


4136 


4215 




5 


4293 


4371 


4449 


4528 


4606 


4684 


4762 


4840 


4919 


4997 




6 


5075 


5153 


5231 


5309 


5387 


5465 


5543 


5621 


5699 


5777 


78 ' 


7 


5855 


5933 6011 


6089 


6167 


6245 


6323 


6401 


6479 


6556 




8 


6634 


6712 


6790 


6868 


6945 


7023 


7101 


7179 


7256 


7334 




9 


7412 


7489 


7567 


7645 


7722 


7800 


7878 


7955 


8033 


8110 




560 


8188 


8266 


8343 


8421 


8498 


8576 


8653 


8731 


8808 


8885 




1 


8963 


9040 


9118 


9195 


9272 


9350 


9427 


9504 


9582 


9659 




2 


9736 


9814 


9891 


9968 
















0045 
0817 


0123 

0894 


0200 
0971 


0277 
1048 


0354 
1125 


0431 
1202 




3 


750508 


0586 


0663 


0740 




4 


1279 


1356 


1433 


1510 


1587 


1604 


1741 


1818 


1895 


1972 


77 ; 


5 


2048 


2125 


2202 


2279 


2356 


2433 


2509 


2586 


2663 


2740 


6 


2816 


2893 


2970 


3047 


3123 


3200 


3277 


3353 


3430 


3506 




7 


3583 


3660 


3736 


3813 


3889 


3966 


4042 


4119 


4195 


4272 




8 


4348 


4425 


4501 


4578 


4654 


4730 


4807 


4883 


4960 


5036 




9 


5112 


5189 


5265 


5341 


5417 


5494 


5570 


5646 


. 5722 


5799 




570 


5875 


5951 


6027 


6103 


6180 


6256 


6332 


6408 


6484 


6560 




1 


6636 


6712 


6788 


6864 


6940 


| 7016 


7092 


7168 


7244 


7320 


76 


2 


7396 


7472 


7548 


7624 


7700 


i 7r?5 


7851 


7927 


8003 


8079 




3 


8155 


8230 


8306 


8382 


8458 


8533 


8609 


8685 


8761 


8836 




4 


8912 


8988 


9063 


9139 


9214 


| 9290 


9366 


9441 


9517 


9592 




5 


9668 


9743 


9819 


9894 


9970 














1 0045 
0799 


0121 

0875 


0196 
0950 


0272 
1025 


0347 
1101 




6 


760422 


0498 


0573 


0649 


0724 




7 


1176 


1251 


1326 


1402 


1477 


1552 


1627 


1702 


1778 


1853 




8 


1928 


2003 


2078 


2153 


2228 


2303 


2378 


2453 


2529 


2604 


75 


9 


2679 


2754 


2829 


2904 


2978 


! 3053 


3128 


3203 


3278 


3353 


580 


3428 


3503 


3578 


3653 


3727 


3802 


3877 


3952 


4027 


4101 




1 


4176 


4251 


4326 


4400 


4475 


j 4550 


4624 


4699 


4774 


4848 




2 


4923 


4998 


5072 


5147 


5221 


5296 


5370 


5445 


5520 


5594 




3 


5669 


5743 


5818 


5892 


5966 


6041 


6115 


6190 


6264 


6338 




4 


6413 


6487 


6562 


6636 


6710 


j 6785 


6859 


6933 


7007 


7082 




Proportional Parts. 


Difl. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


83 


8.3 


16.6 


24.9 


33.2 


41.5 


49.8 


58.1 


66.4 


74.7 


82 


8.2 


16.4 


24.6 


32.8 


41.0 


49.2 


57.4 


65.6 


73.8 


81 


8.1 


16.2 


24.3 


32.4 


40.5 


48.6 


56.7 


64.8 


72.9 


80 


8.0 


16.0 


24.0 


32.0 


40.0 


48.0 


56.0 


64.0 


72.0 


79 


7.9 


15.8 


23.7 


31.6 


39.5 


47.4 


55.3 


63.2 


7i.i : 


78 


7.8 


15.6 


23.4 


31.2 


39.0 


46.8 


54.6 


62.4 


70.2 1 


77 


7.7 


15.4 


23.1 


30.8 


38.5 


46.2 


53.9 


61.6 


69.3 


76 


7.6 


15.2 


• 22.8 


30.4 


38.0 


45.6 


53.2 


60.8 


68.4 


75 


7.5 


15.0 


22.5 


30.0 


37.5 


45.0 


52.5 


60.0 


67.5 


74 


7.4 


14.8 


22.2 


29.6 


37.0 


44.4 


51.8 


59.2 


63.6 



































LOGARITHMS OF NUMBERS. 



147 



No. 585 L. 767.] 



[No. 629 L. ' 



770115 

0852 

1587 



4517 
5246 
5974 
6701 

7427 
8151 
8874 
9596 

7803lf 
1037 
1755 
2473 
3189 
3904 
4617 



6751 
7460 



3790 
4488 
5185 
5880 
6574 
7268 
7960 
8651 



7230 ! 7304 

7972 ! 8046 

8712 I 8786 

9451 ; 9525 

0189 | 0263 

0926 0999 
1661 1734 
2395 i 2468 
3128 3201 
3860 ■ 3933 
4590 ; 4663 
5319 i 5392 
6047 I 6120 
6774 ! 6S46 



7379 
8120 



7499 
8224 



1109 

1827 
2544 



5401 
6112 



- 



9019 
9741 



2462 
3162 
3860 
4558 
5254 
5949 
6644 
7337 
8029 
8720 



0461 
1181 
1899 
2616 
3332 
4046 
4760 

5472 
6183 
6893 
7602 
8310 
9016 
9722 



1129 
1831 

2532 
3231 
3930 
4627 
5324 
6019 



1073 

1808 
2542 
3274 
4006 
4736 
5465 
6193 
6919 
7644 



3403 
4118 
4831 

5543 

6254 



1199 
1901 



7475 
8167 




8564 
9303 



1514 
2248 
2981 
3713 
4444 
5173 
5902 
6629 
7354 



9524 



9245 
0965 
1684 
2401 
3117 
3832 
4546 
5259 

5970 



0215 
0918 
1620 
2322 
3022 
3721 
4418 
5115 
5811 
6505 



8582 
9272 



Proportional Parts. 



Diff. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


75 


7.5 


15.0 


22.5 


30.0 


37.5 . 


45.0 


52.5 


60.0 


67.5 


74 


7.4 


14.8 


22.2 


29.6 


37.0 


44.4 


51.8 


59.2 


66.6 


73 


7.3 


14.6 


21.9 


29.2 


36.5 


43.8 


51.1 


58.4 


65.7 


72 


7.2 


14.4 


21.6 


28.8 


36.0 


43.2 


50.4 


57.6 


64.8 


71 


7.1 


14.2 


21.3 


28.4 


.35.5 


42.6 


49.7 


56.8 


63.9 


70 


7.0 


14.0 


21.0 


28.0 


35.0 


42.0 


49.0 
48.3 


56.0 


63.0 


69 


6.9 


13.8 


20.7 


27.6 


34.5 


41.4 


55.2 


62.1 



148 



LOGARITHMS OF NUMBERS. 



No. 680 L. 799.] 




[No. 674 L. 829. 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


Diff. 


630 

1 


799341 


9409 


9478 


9547 


9616 


9685 


9754 


9823 


9892 


9961 


1 


800029 


0098 


0167 


0236 


0305 


0373 


0442 


0511 


0580 


0648 


2 


0717 


0786 


0854 


0923 


0992 


1061 


1129 


1198 


1266 


1335 




3 


1404 


1472 


1541 


1609 


1678 


1747 


1815 


1884 


1952 


2021 




4 


2089 


2158 


2226 


2295 


2363 


2432 


2500 


2568 


2637 


2705 




5 


2774 


2842 


2910 


2979 


3047 


3116 


3184 


3252 


3321 


3389 




6 


3457 


3525 


3594 


3662 


3730 


3798 


3867 


3935 


4003 


4071 




7 


4139 


4208 


4276 


4344 


4412 


4480 


4548 


4616 


4685 


4753 




8 


4821 


4889 


4957 


5025 


5093 


5161 


5229 


5297 


5365 


5433 


68 


9 


5501 


5569 


5637 


5705 


5773 


5841 


5908 


5976 


6044 


6112 




640 


806180 


6248 


6316 


6384 


6451 


6519 


6587 


6655 


6723 


6790 




1 


6858 


6926 


6994 


7061 


7129 


7197 


7264 


7332 


7400 


7467 




2 


7535 


7603 


7670 


7738 


7806 


7873 


7941 


8008 


8076 


8143 




3 


8211 


8279 


8346 


8414 


8481 


8549 


8616 


8684 


8751 


8818 




4 


8886 


8953 


9021 


9088 


9156 


9223 


9290 


9358 


9425 


9492 




5 


9560 










9896 






















0031 
0703 


0098 
0770 


0165 
0837 




6 


810233 


0300 


0367 


0434 


0501 


0569 


0636 




7 


0904 


0971 


1039 


1106 


1173 


1240 


1307 


1374 


1441 


1508 


67 


8 


1575 


1642 


1709 


1776 


1843 


1910 


1977 


2044 


2111 


2178 




9 


2245 


2312 


2379 


2445 


2512 


2579 


2646 


2713 


2780 


2847 




650 


2913 


2980 


3047 


3114 


3181 


3247 


3314 


3381 


3448 


3514 




1 


3581 


3648 


3714 


3781 


3848 


3914 


3981 


4048 


4114 


4181 




2 


4248 


4314 


4381 


4447 


4514 


4581 


4647 


4714 


4780 


4847 




3 


4913 


4980 


5046 


5113 


5179 


5246 


5312 


5378 


5445 


5511 




4 


5578 


5644 


5711 


5777 


5843 


5910 


5976 


6042 


6109 


6175 




5 


6241 


6308 


6374 


6440 


6506 


6573 


6639 


6705 


6771 


6838 




6 


6904 


6970 


7036 


7102 


7169 


7235 


7301 


7367 


7433 


7499 




7 


7565 


7631 


7698 


7764 


7830 


7896 


7962 


8028 


8094 


8160 




8 


8226 


8292 


8358 


8424 


8490 


8556 


8622 


8688 


8754 


8820 


66 


9 
660 


8885 
9544 


8951 
9610 


9017 
9676 


9083 
9741 


9149 
9807 


9215 
9873 


9281 
9939 


9346 


9412 


9478 
























0004 
0661 


0070 

0727 


0136 

0792 




1 


820201 


0267 


0333 


0399 


0464 


0530 


0595 




2 


0858 


0924 


0989 


1055 


1120 


1186 


1251 


1317 


1382 


1448 




3 


1514 


1579 


1645 


1710 


1775 


1841 


1906 


1972 


2037 


2103 




4 


2168 


2233 


2299 


2364 


2430 


2495 


2560 


2626 


2691 


2756 




5 


2822 


2887 


2952 


3018 


3083 


3148 


3213 


3279 


3344 


3409 




6 


3474 


3539 


3605 


3670 


3735 


3800 


3865 


3930 


3996 


4061 




7 


4126 


4191 


4256 


4321 


4386 


4451 


4516 


4581 


4646 


4711 


65 


8 


4776 


4841 


4906 


4971 


5036 


5101 


5166 


5231 


5296 


5361 


9 


5426 


5491 


5556 


5621 


5686 


5751 


5815 


5880 


5945 


6010 




670 


6075 


6140 


6204 


6269 


6334 


6399 


6464 


6528 


6593 


6658 




1 


6723 


6787 


6852 


6917 


6981 


7046 


7111 


7175 


7240 


7305 




2 


7369 


7434 


7499 


7563 


7628 


7692 


7757 


7821 


7886 


7951 




3 


8015 


8080 


8144 


8209 


8273 


8338 


8402 


8467 


8531 


8595 




4 


8660 


8724 


8789 


8853 


8918 


8982 


9046 


9111 


9175 


9239 




Proportional Parts. 


Diff. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


68 


6.8 


13.6 


20 


4 




27.2 


34.0 


40.8 


47.6 


54.4 


61.2 


67 


6.7 


13.4 


20 


1 




26.8 


33.5 


40.2 


46.9 


53.6 


60.3 


66 


6.6 


13.2 


19 


8 




26.4 


33.0 


39.6 


46.2 


52.8 


59 4 


65 


6.5 


13.0 


19 


5 




26.0 


32.5 


39.0 


45.5 


52.0 


58.5 


64 


6.4 


IS. 8 


19 


2 




25.6 


32.0 


3S.4 


44.8 


51.2 


57. G 



LOGARITHMS OF NUMBERS. 



149 



No. 675 L. 829.] [No. 719 L. 857. 


N. 





1 i 2 


'i 


4 


$ 


6 


7 


8 

9818 


9 


Diff. 


675 


829304 


9368 


9432 


9497 


9561 


9625 


9690 


9754 


9882 




6 


9947 


























0011 


0075 


0139 


0204 


0268 


0332 


0396 


0460 


0525 




7 


830589 


0653 


0717 


0781 


0845 


, 0909 


0973 


1037 


1102 


1166 




8 


1230 


1294 


1358 


1422 


1486 


j 1550 


1614 


1678 


1742 


1806 


64 


9 


1870 


1934 


1998 


2062 


2126 


! 2189 


2253 


2317 


2381 


2445 




680 


2509 


2573 


2637 


2700 


2764 


2828 


2892 


2956 


3020 


3083 




1 


3147 


3211 


3275 


3338 


3402 


3466 


3530 


3593 


3657 


3721 




2 


' 3784 


3848 


3912 


3975 


4039 


4103 


4166 


4230 


4294 


4357 




3 


4421 


4484 


4548 


4611 


4675 


4739 


4802 


4866 


4929 


4993 




4 


5056 


5120 


5183 


5247 


5310 


5373 


5437 


5500 


5564 


5627 




5 


5691 


5754 


5817 


5881 


5944 


6007 


6071 


6134 


6197 


6261 




6 


6324 


6387 


6451 


6514 


6577 


6641 


6704 


6767 


6830 


6894 




7 


6957 


7020 


7083 


7146 


7210 


7273 


7336 


7399 


7462 


7525 




8 


7588 


7652 


7715 


7778 


7841 


7904 


7967 


8030 


8093 


8156 




9 1 8219 


8282 


8345 


8408 


8471 


8534 


8597 


8660 


8723 


8786 


63 


690 


8849 


8912 


8975 


9038 


9101 


9164 


9227 


9289 


9352 


9415 




1 


9478 


9541 


9604 


9667 


9729 


9792 


9855 


9918 


9981 






0043 
0671 




2 


840106 


0169 


0232 


0294 


0357 


0420 


0482 


0545 


0608 




3 


0733 


0796 


0859 


0921 


0984 


1046 


1109 


1172 


1234 


1297 




4 


1359 


1422 


1485 


1547 


1610 


1672 


1735 


1797 


1860 


1922 




5 


1985 


2047 


2110- 


2172 


2235 


2297 


2360 


2422 


2484 


2547 




6 


2609 


2672 


2734 


2796 


2859 


2921 


2983 


3046 


3108 


3170 




7 


3233 


3295 


3357 


3420 


3482 


3544 


3606 


3669 


3731 


3793 




8 


3855 


3918 


3980 


4042 


4104 


4166 


4229 


4291 


4353 


4415 




9 


4477 


4539 


4601 


4664 


4726 


4788 


4850 


4912 


4974 


5036 




700 


5098 


5160 


5222 


5284 


5346 


5408 


5470 


5532 


5594 


5656 


62 


1 


5718 


5780 


5842 


5904 


5966 


6028 


6090 


6151 


6213 


6275 




2 


6337 


6399 


6461 


6523 


6585 


6646 


6708 


6770 


6832 


6894 




3 


6955 


7017 


7079 


7141 


7202 


7264 


7326 


7388 


7449 


7511 




4 


7573 


7634 


7696 


7758 


7819 


7881 


7943 


8004 


8066 


8128 




5 


8189 


8251 


8312 


8374 


8435 


8497 


8559 


8620 


8682 


8743 




6 


8805 


8866 


8928 


8989 


9051 


9112 


9174 


9235 


9297 


9358 




7 
8 


9419 


9481 


9542 


9604 


9665 


9726 


9788 


9849 
0462 


9911 

0524 


9972 
0585 




850033 


0095 


0156 


0217 


0279 


0340 


0401 


9 


0646 


0707 


0769 


0830 


0891 


0952 


1014 


1075 


1136 


1197 




710 


1258 


1320 


1381 


1442 


1503 


1564 


1625 


1686 


1747 


1809 




1 


1870 


1931 


1992 


2053 


2114 


2175 


2236 


2297 


2358 


2419 




2 


2480 


2541 


2602 


2663 


2724 


2785 


2846 


2907 


2968 


3029 


61 


3 


3090 


3150 


3211 


3272 


3333 


3394 


3455 


3516 


3577 


3637 




4 


3698 


3759 


3820 


3881 


3941 


4002 


4063 


4124 


4185 


4245 




5 


4306 


4367 


4428 


4488 


4549 


4610 


4670 


4731 


4792 


4852 




6 | 4913 


4974 


5034 


5095 


5156 


5216 


5277 


5337 


5398 


5459 




7 5519 


5580 


5640 


5701 


5761 


5822 


5882 


5943 


6003 


6064 




8 i 6124 


6185 


6245 


6306 


6366 


6427 


6487 


6548 


6608 


6668 




9 i 6729 


6789 


6850 


6910 


6970 


7031 7091 


7152 


7212 


7272 




Proportional Parts. 


Diff. 


1 


2 


3 


4 


5 6 


7 


8 


9 


65 


6.5 


13.0 


19.5 


26.0 


32.5 39.0 


45.5 


52.0 


58.5 


64 


6.4 


12.8 


19.2 


25.6 


32.0 38.4 


44.8 


51.2 


57.6 


63 


6.3 


12.6 


18.9 


25.2 


31.5 37.8 


44.1 


50.4 


56.7 


62 


6.2 


12.4 


18.6 


24.8 


31.0 37.2 


43.4 


49.6 


55.8 


61 


6.1 


12.2 


18.3 


24.4 


30.5 36.6 


42.7 


48.8 


54 9 


60 


6.0 


12.0 


18.0 


24.0 


30.0 ! 36.0 42.0 


48.0 


54.0 



150 



LOGARITHMS OF NUMBERS. 



No. 720 L. 857.] 


[No. 764 L. 883. 


N. 





> I * 


3 


4 


6 


6 


7 


8 


9 


Diff. 


720 


857332 


7393 7453 


7513 


7574 


7634 


7694 


7755 


7815 


7875 




1 


7935 


7995 8056 


8116 


8176 


1 8236 


8297 


8357 


8417 


8477 




2 


8537 


8597 


8657 


8718 


8778 


! 8838 


8898 


8958 


9018 


9078 




3 


9138 


9198 


9258 


9318 


9379 


i 9439 


9499 


9559 


9619 


9679 


60 


4 


9739 


9799 


9859 


9918 


9978 














0038 
0637 


0098 
0697 


0158 
0757 


0218 
0817 


0278 

0877 




5 


860338 


0398 


0458 


0518 


0578 




i 6 


0937 


0996 


1056 


1116 


1176 


1236 


1295 


1355 


1415 


1475 




i 7 


1534 


1594 


1654 


1714 


1773 


1833 


1893 


1952 


2012 


2072 




8 


2131 


2191 


2251 


2310 


2370 


2430 


2489 


2549 


2608 


2668 




9 


2728 


2787 


2847 


2906 


2966 


3025 


3085 


3114 


3204 


3263 




730 


3323 


3382 


3442 


3501 


3561 


3620 


3680 


3739 


3799 


3858 




1 


3917 


3977 


4036 


4096 


4155 


4214 


4274 


4333 


4392 


4452 






4511 


4570 


4630 


4689 


4748 


4808 


4867 


4926 


4985 


5045 




3 


5104 


5163 


5222 


5282 


5341 


5400 


5459 


5519 


5578 


5637 




4 


5696 


5755 


5814 


5874 


5933 


5992 


6051 


6110 


6169 


6228 




5 


6287 


6346 


6405 


6465 


6524 


6583 


C642 


6701 


6760 


6819 


59 


6 


6878 


6937 


6996 


7055 


7114 


7173 


7232 


7291 


7350 


7409 


7 


7467 


7526 


7585 


7644 


7703 


7762 


7821 


7880 


7939 


7998 




8 


8056 


8115 


8174 


8233 


8292 


8350 


8409 


8468 


8527 


8586 




9 


8644 


8703 


8762 


8821 


8879 


8938 


8997 


9056 


9114 


91,3 




740 


9232 


9290 


9349 


9408 


9466 


9525 


9584 


9642 


9701 


9760 




1 


9818 


9877 


9935 


9994 
















0053 
0638 


0111 
0696 


0170 
0755 


0228 
0813 


0287 

0872 


0345 
0930 




2 


870404 


0462 


0521 


0579 




3 


0989 


1047 


1106 


1164 


1223 


1281 


1339 


1398 


1456 


1515 




4 1573 


1631 


1690 


1748 


1806 


1865 


1923 


1981 


2040 


2008 




5 2156 


2215 


2273 


2331 


2389 


2448 


2506 


2564 


2622 


2681 




6 2739 


2797 


2855 


2913 


2972 


3030 


3088 


8146 


3204 


3262 




7 3321 


3379 


3437 


3495 


3553 


3611 


3669 


3727 


3785 


3844 




8 | 3902 


3960 


4018 


4076 


4134 


4192 


4250 


4308 


4366 


4424 


58 


9 ! 4482 


4540 


4598 


4656 


4714 


4772 


4830 


4888 


4945 


5003 




750 5061 


5119 


5177 


5235 


5293 


5351 


5409 


5466 


5524 


5582 




1 5640 


5698 


5756 


5813 


5871 


5929 


5987 


6045 


6102 


6160 




2 6218 


6276 


6333 


6391 


6449 


6507 


6564 


6622 


6680 


6737 




3 


6795 


6853 


6910 


6968 


7026 


7083 


7141 


7199 


7256 


7314 




4 


7371 


7429 


7487 


7544 


7602 


7659 


7717 


7774 


7832 


7889 




5 


7947 


8004 


8062 


8119 


8177 


8234 


8292 


8349 


8407 


8464 




6 


8522 


8579 


8637 


8694 


8752 


8809 


8866 


8924 


8981 


9039 




7 


9096 


9153 


9211 


9268 


9325 


9383 


9440 


9497 


9555 


9612 




8 9669 


9726 


9784 


9841 


9898 


9956 












0013 
0585 


0070 
0642 


0127 
0699 


0185 
0756 




9 


880242 


0299 


0356 


0413 


0471 


0528 




760 


0814 


0871 


0928 


0985 


1042 


1099 


1156 


1213 


1271 


1328 




1 


1385 


1442 


1499 


1556 


1613 


1670 


1727 


1784 


1841 


1898 


57 


2 


1955 


2012 


2069 


2126 


2183 


2240 


2297 


2354 


2411 


2468 


3 


2525 


2581 


2638 


2695 


2752 


2809 


2866 


2923 


2980 


3037 




4 


3093 


3150 


3207 


3264 


3321 


3377 


3434 


3491 


3548 


3605 




Proportional Parts. 


Diff. 


1 


2 




3 


4 


5 


6 


7 


8 


9 


59 


5.9 


11.8 




17.7 


23.6 


29.5 


35.4 


41.3 


47.2 


53.1 


58 


5.8 


11.6 


17.4 


23.2 


29.0 


34.8 


40.6 


46.4 


52.2 


57 


5.7 


11.4 


17.1 


22.8 


28.5 


34.2 


39.9 


45.6 


51.3 


56 


5.6 


11.2 | 16.8 


22.4 


28.0 


33.6 


39.2 


44.8 


50.4 



LOGARITHMS OF NUMBERS. 



151 



No. 765 L. 883.] 






[No. 809 L. 908. 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


Diff. 


765 


883661 


3718 


3775 


3832 


3888 


3945 


4002 


4059 


4115 


4172 


6 


4229 


4285 


4342 


4399 


4455 


4512 


4569 


4625 


4682 


4739 




7 


4^95 


4852 


4909 


4965 


5022 


5078 


5135 


5192 


5248 


5305 




8 


5361 


5418 


5474 


5531 


5587 


5644 


5700 


5757 


5813 i 5870 




9 


5926 


5983 


6039 


6096 


6152 


6209' 


6265 


6321 


6378 


6434 




770 


6491 


6547 


6604 


6660 


6716 


6773 


6829 


6885 


6942 


6998 




1 


7054 


7111 


7167 


7223 


7280 


7336 


7392 


7449 


7505 


7561 




2 


7617 


7674 


7730 


7786 


7842 


7898 


7955 


8011 


8067 


8123 




3 


8179 


8236 


8292 


8348 


8404 


8460 


8516 


8573 


8629 


8685 




4 


8741 


8797 


8853 


8909 


8965 


9021 


9077 


9134 


9190 


9246 




5 


9302 


9358 


9414 


9470 


9526 


9582 


9638 


9694 


9750 


9806 


56 


6 


9862 


9918 


9974 


















0030 
0589 


0086 
0645 


0141 

0700 


0197 
0756 


0253 

0812 


0309 
0868 


0365 
0924 




7 


890421 


0477 


0533 




8 


0980 


1035 


1091 


1147 


1203 


1259 


1314 


1370 


1426 


1482 




9 


1537 


1593 


1649 


1705 


1760 


1816 


1872 


1928 


1983 


2039 




780 


2095 


2150 


2206 


2262 


2317 


2373 


2429 


2484 


2540 


2595 




1 


2651 


2707 


2762 


2818 


2873 


2929 


2985 


3040 


3096 


3151 




2 


3207 


3262 


3318 


3373 


3429 


3484 


3540 


3595 


3651 


3706 




3 


3762 


3817 


3873 


3928 


3984 


4039 


4094 


4150 


4205 


4261 




4 


4316 


4371 


4427 


4482 


4538 


4593 


4648 


4704 


4759 


4814 




5 


4870 


4925 


4980 


5036 


5091 


5146 


5201 


5257 


5312 


5367 




6 


5423 


5478 


5533 


5588 


5644 


5699 


5754 


5809 


5864 


5920 




7 


5975 


6030 


6085 


6140 


6195 


6251 


6306 


6361 


6416 


6471 




8 


6526 


6581 


6636 


6692 


6747 


6802 


6857 


6912 


6967 


7022 




9 


7077 


7132 


7187 


7242 


7297 


7352 


7407 


7462 


7517 


7572 


55 


790 


7627 


7682 


7737 


7792 


7847 


7902 


7957 


8012 


8067 


8122 


1 


8176 


8231 


8286 


8341 


8396 


8451 


8506 


8561 


8615 


8670 




2 


8725 


8780 


8835 


8890 


8944 


8999 


9054 


9109 


9164 


9218 




3 


9273 


9328 


9383 


9437 


9492 


9547 


9602 


9656 


9711 


9766 




4 


9821 


9875 


9930 


9985 
















0039 
0586 


0094 
0640 


0149 
0695 


0203 
0749 


0258 
0804 


0312 
0859 




5 


900367 


0422 


0476 


0531 




6 


0913 


0968 


1022 


1077 


1131 


1186 


1240 


1295 


1349 


1404 




7 


1458 


1513 


1567 


1622 


1676 


1731 


1785 


1840 


1894 


1948 




8 


2003 


2057 


2112 


2166 


2221 


2275 


2329 


2384 


2438 


2492 




9 


2547 


2601 


2655 


2710 


2764 


2818 


2873 


2927 


2981 


3036 




800 


3090 


3144 


3199 


3253 


3307 


3361 


3416 


3470 


3524 


3578 




1 


3633 


3687 


3741 


3795 


3849 


3904 


3958 


4012 


4066 


4120 




2 


4174 


4229 


4283 


4337 


4391 


4445 


4499 


4553 


4607 


4661 




3 


4716 


4770 


4824 


4878 


4932 


4986 


5040 


5094 


5148 


5202 


54 


4 


5256 


5310 


5364 


5418 


5472 


5526 


5580 


5634 


5688 


5742 


5 


5796 


5850 


5904 


5958 


6012 


6066 


6119 


6173 


6227 


6281 




6 


6335 


6389 


6443 


6497 


6551 


6604 


6658 


6712 


6766 


6820 




7 


6874 


6927 


6981 


7035 


7089 


7143 


7196 


7250 


7304 


7358 




8 


7411 


7465 


7519 


7573 


7626 


7680 


7734 


7787 


7841 


7895 




9 


7949 


8002 


8056 


8110 


8163 


8217 


8270 


8324 


8378 


8431 




Proportional Parts. 


Diff. 


1 


2 


3 


4 


5 J 6 


7 


8 


9 


57 


5.7 


11.4 




17.1 




22.8 


28.5 


34.2 


39.9 


45.6 


51.3 


56 


5.6 


11.2 




16.8 




22.4 


28.0 


33.6 


39.2 


44.8 


50.4 


55 


5.5 


11.0 




16.5 




22.0 


27.5 


33.0 


38.5 


44.0 


49.5 


54 


5.4 


10.8 




16.2 




21.6 


27.0 


32.4 


37.8 


43.2 


48.6 



LOGARITHMS OF NUMBERS. 



No. 810 L. 908.] 


[No. 854 L. 931. 


N. 





1 


2 


3 


4 I 


5 

8753 


"• ! 


7 


8 ! 


9 

8967 


Diff. 


810 908485 


8539 


8592 


8646 


8699 


8807 


8860 


8914 


1 1 9021 


9074 


9128 


9181 


9235 | 


9289 


9342 


9396 


9449 


9503 




2 


9556 


9610 


9663 


9716 


9770 ! 


9823 


9877 


9930 


9984 


0037 
0571 




3 


910091 


0144 


0197 


0251 


0304 


0358 


0411 


0464 


0518 


4 


0624 


0678 


0731 


0784 


0838 


0891 


0944 


0998 


1051 


1104 




5 


1158 


1211 


1264 


1317 


1371 ! 


1424 


1477 


1530 


1584 


163? 




6 


1690 


1743 


1797 


1850 


1903 


1956 


2009 


2063 


2116 


2169 




7 


2222 


2275 


2328 


2381 


2435 


2488 


2541 


2594 


2647 


2700 




8 


2753 


2806 


2859 


2913 


2966 


31)19 


3072 


3125 


3178 


3231 




9 


3284 


3337 


3390 


3443 


3496 | 


3549 


3602 


3655 


3708 


3761 


53 


820 


3814 


3867 


3920 


3973 


4026 


4079 


4132 


4184 


4237 


4290 




1 


4343 


4396 


4449 


4502 


4555 


4608 


4660 


4713 


4766 


4819 




2 


4872 


4925 


4977 


5030 


5083 


5136 


5189 


5241 


5294 


5347 




3 


5400 


5453 


5505 


5558 


5611 


5664 


5716 


5769 


5822 


5875 




4 


5927 


5980 


6033 


6085 


6138 


6191 


6243 


6296 


6349 


6401 




5 


6454 


6507 


6559 


6612 


6664 


6717 


6770 


1822 


6875 


6927 




6 


6980 


7033 


7085 


7138 


7190 


7243 


7295 


7348 


7400 


7453 




7 


7506 


7558 


7611 


7663 


7716 


7768 


7820 


7873 


7925 


7978 




8 


8030 


8083 


8135 


8188 


8240 


8293 


8345 


8397 


8450 


8502 




9 


8555 


8607 


8659 


8712 


8764 


8816 


8869 


8921 


8973 


9026 




830 


9078 


9130 


9183 


9235 


9287 


i 9340 


9392 


9444 


9496 


9549 




1 


9601 


9653 


9706 


9758 


9810 


9862 


9914 


9967 














0019 
0541 


0071 
0593 




2 


920123 


0176 


0228 


0280 


0332 


0384 


0436 


0489 




3 


0645 


0697 


0749 


0801 


0853 


0906 


0958 


1010 


1062 


1114 


52 


4 


1166 


1218 


1270 


1322 


1374 


1426 


1478 


1530 


1582 


1634 


5 


1686 


1738 


1790 


1842 


1894 


1946 


1998 


2050 


2102 


2154 




6 


2206 


2258 


2310 


2362 


2414 


2466 


2518 


2570 


2622 


2674 




7 


2725 


2777 


2829 


2881 


2933 


2985 


3037 


3089 


3140 


3192 




8 


3244 


3296 


3348 


3399 


3451 


3503 


3555 


3607 


3658 


3710 




9 


3762 


3814 


3865 


3917 


3969 


4021 


4072 


4124 


4176 


4228 




840 


4279 


4331 


4383 


4434 


4486 


4538 


4589 


4641 


4693 


4744 




1 


4796 


4848 


4899 


4951 


5003 


5054 


5106 


5157 


5209 


5261 




2 


5312 


5364 


5415 


5467 


5518 


5570 


5621 


5673 


5725 


5776 




3 


5828 


5879 


5931 


5982 


6034 


6085 


6137 


6188 


6240 


6291 




4 


6342 


6394 


6445 


6497 


6548 


6600 


6651 


6702 


6754 


6805 




5 


6857 


6908 


6959 


7011 


7062 


7114 


7165 


7216 


7268 


7319 




6 


7370 


7422 


7473 


7524 


7576 


7627 


7678 


7730 


7781 


7832 




7 


7883 


7935 


7986 


8037 


8088 


8140 


8191 


8242 


8293 


8345 




8 


8396 


8447 


8498 


8549 


8601 


8652 


8703 


8754 


8805 


8857 




9 


8908 


8959 


9010 


9061 


9112 


9163 


9215 


9266 


9317 


9368 




850 


9419 


9470 


9521 


9572 


9623 


i 9674 


9725 


9776 


9827 


9879 


51 


1 


9930 


9981 


















0032 
0542 


0083 
0592 


0134 
0643 


i 0185 
1 0694 


0236 
0745 


0287 
0796 


0338 


0389 




2 


930440 


0491 


0847 


0898 




3 


0949 


1000 


1051 


1102 


1153 


! 1204 


1254 


1305 


1356 


1407 




4 


1458 


1509 


1560 


1610 


1661 


1712 


1763 


1814 


1865 


1915 




Proportional Parts. 


Diff. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


53 


5.3 


10.6 


15.9 


21.2 


26.5 


31.8 


37.1 


42.4 


47.7 


52 


5.2 


10.4 


15.6 


20.8 


26.0 


31.2 


36.4 


41.6 


46.8 


51 


5.1 


10.2 


15.3 


20.4 


25.5 


30.6. 


35.7 


40.8 


45.9 


50 


5.0 10.0 


15.0 


20.0 


25.0 


30.0 


35.0 


40.0 


45.0 



LOGAEITHMS OF NUMBEKS. 



153 



No. 855 L. 931.1 

N. 



[No. 899 L. 954. 



4498 
5003 
5507 
6011 
6514 
7016 
7518 
8019 



9519 



940018 
0516 
1014 
1511 
2008 
2504 
3000 
3495 



4976 
5469 
5961 
6452 
6943 
7434 
7924 
8413 



0851 
1338 



2792 
3276 
3760 



3538 
4044 

4549 
5054 
5558 
6061 
6564 
7066 
7568 
8069 
8570 
9070 



1064 
1561 

2058 
2&54 
3049 
3544 
4038 

4532 
5025 
5518 
6010 
6501 



4094 

4599 
5104 
5608 
6111 
6614 
7116 
7618 
8119 
8620 
9120 

9619 



0414 
0900 

1386 

1872 
2356 

2841 



9975 
0462 



1435 
1920 



2118 
2626 
3133 
3639 
4145 
4650 
5154 
5658 



8670 
9170 



2677 . , 
3183 | 3234 
3690 i 3740 
4195 | 4246 
' 4751 
5255 
5759 
6262 
6765 



^roo 

5205 
5709 

6212 
6715 
7217 
7718 
8219 
8720 



0168 


0218 


0666 


0716 


1163 


1213 


1660 


1710 


2157 


2207 


2653 


2702 


3148 


3198 


3643 


3692 


4137 


4186 


4631 


4680 


5124 


5173 


5616 


5665 


6108 


6157 


6600 


6649 


7090 


7139 


7581 


7630 


8070 


8119 


8560 


8608 


9048 


9097 


9536 


9585 


0024 


0073 


0511 


0560 


0997 


1046 


1483 


1532 


1969 


2017 


2453 


2502 


2938 


2986 


3421 


3470 


3905 


3953 



8770 
9270 



0267 
0765 
1263 
1760 
2256 
2752 
3247 
3742 
4236 

4729 

5222 
5715 
6207 



8657 
9146 



0121 
0608 
1095 
1580 
2066 
2550 
3034 
3518 
4001 



2271 

2778 
3285 
3791 
4296 

4801 
5306 



6815 
7317 
7819 



0317 
0815 
1313 



2801 
3297 
3791 
4285 

4?79 

5272 
5764 
6256 
6747 
7238 
7728 
8217 
8706 
9195 



3841 
4347 



5356 

5860 



8370 

8870 



1362 
1859 
2355 



5321 
5813 



8755 
9244 



0170 
0657 
1143 
1629 
2114 
2599' 



0219 
0706 
1192 
1677 
2163 
2647 
3131 
3615 



4902 
5406 
5910 
6413 
6916 
7418 
7919 
8420 



0417 
0915 
1412 
1909 
2405 
2901 



4877 
5370 



0267 
0754 
1240 



4049 4098 ' 4146 



3437 
3943 
4448 



6966 
7468 



8470 
8970 



0467 
0964 
1462' 
1958 
2455 
2950 
3445 



0316 



3711 
4194 









Proportional Parts. 








Diff. 


1 

5.1 
5.0 

! 4.9 

! 4.8 


2 


3 


4 


5 


6 


7 


8 


9 


51 
50 
49 
48 


10.2 
10.0 
9.8 
9.6 


15.3 
15.0 

14.7 
14.4 

— -r 


20.4 
20.0 
19.6 
19.2 


25.5 
25.0 
24.5 
24.0 


30.6 
30.0 
29.4 

28.8 


35.7 
35.0 
34.3 
33.6 


40.8 
40.0 
39.2 
38.4 


45.9 
45.0 
44.1 
43.2 



154 



LOGARITHMS OF NUMBERS. 



No 


900 L. 954.1 






[No. 944 L. 975. 


N. 





1 


2 


3 


4 


5 


6 


7 
4580 


8 


9 


Diff. 


900 


954243 


4291 


4339 


4387 


4435 


4484 


4532 


4628 


4677 




1 


4725 


4773 


4821 


4869 


4918 


4966 


5014 


5062 


5110 


5158 




2 


5207 


5255 


5303 


5351 


5399 


5447 


5495 


5543 


5592 


5640 




3 


5688 


5736 


5784 


5832 


5880 


5928 


5976 


6024 


6072 


6120 




4 


6168 


6216 


6265 


6313 


6361 


6409 


6457 


6505 


6553 


6601 


48 


5 


6649 


6697 


6745 


6793 


6840 


6888 


6936 


6984 


7032 


7080 


6 


7128 


7176 


7224 


7272 


7320 


7368 


7416 


7464 


7512 


7559 




7 


7607 


7655 


7703 


7751 


7799 


7847 


7894 


7942 


7990 


8038 




8 


8086 


8134 


8181 


8229 


8277 


8325 


8373 


8421 


8468 


8516 




9 


8564 


8612 


8659 


.8707 


8755 


8803 


8850 


8898 


8946 


8994 




910 


9041 


9089 


9137 


9185 


9232 


9280 


9328 


9375 


9423 


9471 




1 


9518 


9566 


9614 


9661 


9709 


9757 


9804 


9852 


9900 


9947 




2 


9995 






















0042 


0090 
0566 


0138 
0613 


0185 
0661 


0233 


0280 
0756 


0328 


0376 


0423 




3 


960471 


0518 


0709 


0804 


0851 


0899 




4 


0946 


0994 


1041 


1089 


1136 


1184 


1231 


1279 


1326 


1374 




5 


1421 


1469 


1516 


1563 


1611 


1658 


1706 


1753 


1801 


1848 




6 


1895 


1943 


1990 


2038 


2085 


2132 


2180 


2227 


2275 


2322 




7 


2369 


2417 


2464 


2511 


2559 


2606 


2653 


2701 


2748 


2795 




8 


2843 


2890 


2937 


2985 


3032 


3079 


3126 


3174 


3221 


3268 




9 


* 3316 


3363 


3410 


3457 


3504 


3552 


3599 


3646 


3693 


3741 




920 


3788 


3835 


3882 


3929 


3977 


4024 


4071 


4118 


4165 


4212 




1 


4260 


4307 


4354 


4401 


4448 


4495 


4542 


4590 


4637 


4684 




2 


4731 


4778 


4825 


4872 


4919 


4966 


5013 


5061 


5108 


5155 




3 


5202 


5249 


5296 


5343 


5390 


5437 


5484 


5531 


5578 


5625 




4 


5672 


5719 


5766 


5813 


5860 


5907 


5954 


6001 


6048 


6095 


47 


5 


6142 


6189 


6236 


6283 


6329 


6376 


6423 


6470 


6517 


6564 




6 


6611 


6658 


6705 


6752 


6799 


6845 


6892 


6939 


6986 


7033 




7 


7080 


7127 


7173 


7220 


7267 


7314 


7361 


7408 


7454 


7501 




8 


7548 


7595 


7642 


7688 


7735 


7782 


7829 


7875 


7922 


7969 




9 


8016 


8062 


8109 


8156 


8203 


8249 


8296 


8343 


8390 


8436 




930 


8483 


8530 


8576 


8623 


8670 


8716 


8763 


8810 


8856 


8903 




1 


8950 


8996 


9043 


9090 


9136 


9183 


9229 


9276 


9323 


9369 




2 


9416 


9463 


9509 


9556 


9602 


9649 


9695 


9742 


9789 


9835 




3 


9882 


9928 


9975 




















0021 
0486 


0068 
0533 


0114 
0579 


0161 
0626 


0207 
0672 


0254 
0719 


0300 




4 


970347 


0393 


0440 


0765 




5 


0812 


0858 


0904 


0951 


0997 


1044 


1090 


1137 


1183 


1229 




6 


1276 


1322 


1369 


1415 


1461 


1508 


1554 


1601 


1647 


1693 




7 


1740 


1786 


1832 


1879 


1925 


1971 


2018 


2064 


2110 


2157 




8 


2203 


2249 


2295 


2342 


2388 


2434 


2481 


2527 


2573 


2619 




9 


2666 


2712 


2758 


2804 


2851 


2897 


2943 


2989 


3035 


3082 




940 


3128 


3174 


3220 


3266 


3313 


3359 


3405 


3451 


3497 


3543 




1 


3590 


3636 


3682 


3728 


3774 


3820 


3866 


3913 


3959 


4005 




2 


4051 


4097 


4143 


4189 


4235 


4281 


4327 


4374 


4420 


4466 




3 


4512 


4558 


4604 


4650 


4696 


4742 


4788 


4834 


4880 


4926 




4 


4972 


5018 


5064 


5110 


5156 


5202 


5248 


5294 


5340 


5386 


46 






• 


Pro 


PORTIO 


i 
*al Parts. 


DifE 


1 


2 


3 




4 


5 


6 


7 


8 


9 


47 


4.7 


9.4 14 


1 




8.8 


23.5 


28.2 


32.9 


37.6 


42.3 I 


46 


4.6 


9.2 | ]3 


8 




8.4 


23.0 


27.6 


32.2 


36.8 


41.4 (. 



LOGARITHMS OF KUMBERS. 



155 



{ No. 945 L. 975.] 






1 
[No. 989 L. 995. 


N. 





1 


i 


1 

55 




4 


5 


6 


r 


8 


9 


Diff. 


945 


975432 


5478 


5524 


'0 


5616 


5662 


5707 


5753 


5799 


5845 




6 


5891 


5937 


5983 


6029 


6075 


6121 


6167 


6212 


6258 


6304 




7 


6350 


6396 


6442 


6488 


6533 


6579 


6625 


6671 


6717 


6763 




8 


6808 


6854 


6900 


6946 


6992 


7037 


7083 


7129 


7175 


7220 




9 


7266 


7312 


7358 


7403 


7449 


7495 


7541 


7586 


7632 


7678 




950 


7724 


7769 


7815 


7861 


7906 


7952 


7998 


8043 


8089 


8135 




1 


8181 


8226 


8272 


8317 


8363 


8409 


8454 


8500 


8546 


8591 




2 


8637 


8683 


8728 


8774 


8819 


8865 


8911 


8956 


9002 


9047 




3 


9093 


9138 


9184 


9230 


9275 


9321 


9366 


9412 


9457 


9503 




4 
5 


9548 
980003 


9594 
0049 


9639 


9685 


9730 


9776 


9821 


9867 


9912 


9958 




0094 


0140 


0185 


0231 


0276 


0322 


0367 


0412 


6 


(1458 


0503 


0549 


0594 


0640 


0685 


0730 


0776 


0821 


0867 




7 


0912 


0957 


1003 


1048 


1093 


1139 


1184 


1229 


1275 


1320 




8 


1366 


1411 


1456 


1501 


1547 


1592 


1637 


1683 


1728 


1773 




9 


1819 


1864 


1909 


1954 


2000 


2045 


2090 


2135 


2181 


2226 




960 


2271 


2316 


2362 


2407 


2452 


2497 


2543 


2588 


2633 


2678 




1 


2723 


2769 


2814 


2859 


2904 


2949 


2994 


3040 


3085 


3130 




2 


3175 


3220 


3265 


3310 


3356 


3401 


3446 


3491 


3536 


3581 




3 


3626 


3671 


3716 


3762 


3807 


3852 


3897 


3942 


3987 


4032 




4 


4077 


4122 


4167 


4212 


4257 


4302 


4347 


4392 


4437 


4482 


45 


5 


4527 


4572 


4617 


4662 


4707 


4752 


4797 


4842 


4887 


4932 


6 


4977 


5022 


5067 


5112 


5157 


5202 


5247 


5292 


5337 


5382 




7 


5426 


5471 


5516 


5561 


5606 


5651 


5696 


5741 


5786 


5830 




8 


5875 


5920 


5965 


6010 


6055 


6100 


6144 


6189 


6234 


6279 




9 


6324 


6369 


6413 


6458 


6503 


6548 


6593 


6637 


6682 


6727 




970 


6772 


6817 


6861 


6906 


6951 


6996 


7040 


7085 


7130 


7175 




1 


7219 


7264 


7309 


7353 


7398 


7443 


7488 


7532 


7577 


7622 




2 


7666 


7711 


7756 


7800 


7845 


7890 


7934 


7979 


8024 


8068 




3 


8113 


8157 


8202 


8247 


8291 


8336 


8381 


8425 


8470 


8514 




4 


8559 


8604 


8648 


8693 


8737 


8782 


8826 


8871 


8916 


8960 




5 


9005 


9049 


9094 


9138 


9183 


9227 


9272 


9316 


9361 


9405 




6 


9450 


9494 


9539 


9583 


9628 


9672 


9717 


9761 


9806 


9850 




7 


9895 


9939 


9983 


















0028 
0472 


0072 
0516 


0117 
0561 


0161 
0605 


0206 
0650 


0250 
0694 


0294 
0738 




8 


990339 


0383 


0428 




9 


0783 


0827 


0871 


0916 


0960 


1004 


1049 


1093 


1137 


1182 




980 


1226 


1270 


1315 


1359 


1403 


1448^ 


1492 


1536 


1580 


1625 




1 


1669 


1713 


1758 


1802 


1846 


1890 


1935 


1979 


2023 


2067 




2 


2111 


2156 


2200 


2244 


2288 


2333 


2377 


2421 


2465 


2509 




3 


2554 


2598 


2642 


2686 


2730 


2774 


2819 


2863 


2907 


2951 




4 


2995 


3039 


3083 


3127 


3172 


3216 


3260 


3304 


3348 


3392 




5 


3436 


3480 


3524 


3568 


3613 


3657 


3701 


3745 


3789 


3833 




6 


3877 


3921 


3965 


4009 


4053 


4097 


4141 


4185 


4229 


4273 




7 


4317 


4361 


4405 


4449 


4493 


4537 


4581 


4625 


4669 


4713 


44 


8 


4757 


4801 


4845 


4889 


4i)33 


4977 


5021 


5065 


5108 


5152 




9 


5196 


5240 


5284 


5328 


5372 


5416 


5460 


5504 


5547 


5591 




Proportional, Parts. 


Diff. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


46 


4.6 


9.2 


'■ 13.8 


] 


8.4 


23.0 


27.6 


32 


.2 


36.8 


41.4 


45 


4.5 


9.0 


j 13.5 


] 


8.0 


22.5 


27.0 


31 


.5 


36.0 


40.5 


'| 44 


4.4 


8.8 


13.2 




7.6 


22.0 


26.4 


30 


.8 


35.2 


39. e 


1 43 
k, 


4.3 


8.6 


I 12.9 


] 


7.2 


21.5 


25.8 


30 


.1 


34.4 


38.7 



J56 



MATHEMATICAL TABLES. 



No. 


}90 L. 995.] 














[N 


o. 999 L. 999. 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


Diff. 


990 


995635 


5679 


5723 


5767 


5811 


5854 


5898 


5942 


5986 


6030 




1 


6074 


6117 


6161 


6205 


6249 


6293 


6337 


6380 


6424 


6468 


44 


2 


6512 


6555 


6599 


6643 


6687 


6731 


6774 


6818 


6862 


6906 




3 


6949 


6993 


7037 


7080 


7124 


7168 


7212 


7255 


7299 


7343 




4 


7386 


7430 


7474 


7517 


7561 


7605 


7648 


7692 


7736 


7779 




5 


7823 


7867 


7910 


7954 


7998 


8041 


8085 


8129 


8172 


8216 




6 


8259 


8303 


8347 


8390 


8434 


8477 


8521 


8564 


8608 


8652 




7 


8695 


8739 


8782 


8826 


8869 


8913 


8956 


9000 


9043 


9087 




8 


9131 


9174 


9218 


9261 


9305 


9348 


9392 


9435 


9479 


9522 




9 


9565 


9609 


9652 


9696 


9739 


9783 


9826 


9870 


9913 


9957 


43 







HYPERBOLIC 


LOGARITHMS. 






No. 


Log. 


No. 


Log. 


No. 


Log. 


No. 


Log. 


No. 


Log. 


1.01 


.0099 


1.45 


.3716 


1.89 


.6366 


2.33 


.8458 


2.77 


1.0188 


1.02 


.0198 


1.46 


.3784 


1.90 


.6419 


2.34 


.8502 


2.78 


1.0225 


1.03 


.0296 


1.47 


.3853 


1.91 


.6471 


2.35 


.8544 


2.79 


1.0260 


1.04 


.0392 


1.48 


.3920 


1.92 


.6523 


2.36 


.8587 


2.80 


1.0296 


1.05 


.0488 


1.49 


.3988 


1.93 


.6575 


2.37 


.8629 


2.81 


1.0332 


1.06 


.0583 


1.50 


.4055 


1.94 


.6627 


2.38 


.8671 


2.82 


1.0367 


1.07 


.0677 


1.51 


.4121 


1.95 


.6678 


2.39 


.8713 


2.83 


1.0403 


1.08 


.0770 


1.52 


.4187 


1.96 


.6729 


2.40 


.8755 


2.S4 


1.0438 


1.09 


.0862 


1.53 


.4253 


1.97 


.6780 


2.41 


.8796 


2.85 


1.0473 


1.10 


.0953 


1.54 


.4318 


1.98 


.6831 


2.42 


.8838 


2.86 


1.0508 


1.11 


.1044 


1.55 


.4383 


1.99 


.6881 


2.43 


.8879 


2.87 


1.0543 


1.12 


.1133 


1.56 


.4447 


2.00 


.6931 


2.44 


.8920 


2.88 


1.0578 


1.13 


.1222 


1.57 


.4511 


2.01 


.6981 


2.45 


.8961 


2.89 


1.0613 


1.14 


.1310 


1.58 


.4574 


2.02 


.7031 


2.46 


.9002 


2.90 


1.0647 


1.15 


.1398 


1.59 


.4637 


2.03 


.7080 


2.47 


.9042 


2.91 


1.0682 


1.16 


.1484 


1.60 


.4700 


2.04 


.7129 


2.48 


.9083 


2.92 


1.0716 


1.17 


.1570 


1.61 


.4762 


2.05 


.7178 


2.49 


.9123 


2.93 


1.0750 


1.18 


.1655 


1.62 


.4824 


2.06 


.7227 


2.50 


.9163 


2.94 


1.0784 


1.19 


.1740 


1.63 


.4886 


2.07 


.7275 


2.51 


.9203 


2.95 


1.0813 


1.20 


.1823 


1.64 


.4947 


2.08 


.7324 


2.52 


.9243 


2.96 


1.0852 


1.21 


.1906 


1.65 


.5008 


2.09 


.7372 


2.53 


.9282 


2.97 


1.0886 


1.22 


.1988 


1.66 


.5068 


2.10 


.7419 


2.54 


.9322 


2.98 


1.0919 


1.23 


.2070 


1.67 


.5128 


2.11 


.7467 


2.55 


.9361 


2.99 


1.0953 


1.24 


.2151 


1.68 


.5188 


2.12 


.7514 


2.56 


.9400 


3.00 


1.0986 


1.25 


.2231 


1.69 


.5247 


2.13 


.7561 


2.57 


.9439 


3.01 


1.1019 


1.26 


.2311 


1.70 


.5306 


2.14 


.7608 


2.58 


.9478 


3.02 


1.1053 


1.27 


.2390 


1.71 


.5365 


2.15 


.7655 


2.59 


.9517 


3.03 


1.1086 


1.28 


.2469 


1.72 


.5423 


2.13 


.7701 


2.60 


.9555 


3.04 


1.1119 


1.29 


.2546 


1.73 


.5481 


2.17 


.7747 


2.61 


.9594 


3.05 


1.1151 


1.30 


.2624 


1.74 


.5539 


2.18 


.7793 


2.62 


.9632 


3.06 


1.1184 


1.31 


.2700 


1.75 


.5596 


2.19 


.7839 


2.63 


.9670 


3.07 


1.1217 


1.32 


.2776 


1.76 


.5653 


2.20 


.7885 


2.64 


.9708 


3.08 


1.1249 


1.33 


.2852 


1.77 


.5710 


2.21 


.7930 


2.65 


.9746 


3.09 


1.1282 


1.34 


.2927 


1.78 


.5766 


2.22 


.7975 


2.66 


.9783 


3.10 


1.1314 


1.35 


.3001 


1.79 


.5822 


2^23 


.8020 


2.67 


.9821 


3.11 


1.1346 


1.36 


.3075 


1.80 


.5878 


2.24 


.8065 


2.68 


.9858 


3.12 


1.1378 


1.37 


.3148 


1.81 


.5933 


2.25 


.8109 


2.69 


.9895 


3.13 


1.1410 


1.38 


.3221 


1.82 


.5988 


2.26 


.8154 


2.70 


.9933 


3.14 


1.1442 


1.39 


.3293 


1.83 


.6043 


2.27 


.8198 


2.71 


.9969 


3.15 


1.1474 


1.40 


.3365 


1.84 


.6098 


2.28 


.8242 


2.72 


1.0006 


3 16 


1.1506 


1.41 


.3436 


1.85 


.6152 


2.29 


.8286 


2.73 


1.0043 


3.17 


1.1537 


1.42 


.3507 


1.86 


.U206 


2.30 


.8329 


2.74 


1.0080 


3.18 


1.1569 


1.43 


.3577 


1.87 


.6259 


2.31 


.8372 


2.75 


1.0116 


3.19 


1.1600 


1.44 


.3646 


1.88 


.6313 


2.32 


.8416 


2.76 


1.0152 


3.20 


1.1632 



HYPERBOLIC LOGARITHMS. 



157 



No. 


Log. 


No. 


Log. 


No. 


Log. 


No. 


Log. 


No. 


Log. 


3.21 


1 . 1663 


3.87 


1.3533 


4.53 


1.5107 


5.19 


1.6467 


5.85 


1.7664 


3.22 


1.1694 


3.88 


1.3558 


4.54 


1.5129 


5.20 


1.6487 


5.86 


1 .7681 


3.23 


1.1725 


3.89 


1.3584 


4.55 


1.5151 


5.21 


1.6506 


5.87 


1.7699 


3.24 


1.1756 


3.90 


1.3610 


4.56 


1.5173 


5.22 


1.6525 


5.88 


1.7716 


3.25 


1.1787 


3.91 


1.3635 


4.57 


1.5195 


5.23 


1.6514 


5.89 


1.7733 


3.26 


1.1817 


3.92 


1.3661 


4.58 


1.5217 


5.24 


1.6563 


5.90 


1.7750 


3.27 


1.1848 


3.93 


1.3686 


4.59 


1.5239 


5.25 


1.6582 


5.91 


1.7766 


3.28 


1.1878 


3.94 


1.3712 


4.60 


1.5261 


5.26 


1.6601 


5.92 


1.7783 


3.29 


1.1909 


3.95 


1.3737 


4.61 


1.5282 


5.27 


1.6620 


5.93 


1.7800 


* 3.30 


1.1939 


3.96 


1.3762 


4.62 


1.5304 


5.28 


1.6639 


5.94 


1.7817 


3.31 


1 1969 


3.97 


1.3788 


4.63 


1.5326 


5.29 


1.6658 


5.95 


1.7834 


3.32 


1.1999 


3.98 


1.3813 


4.64 


1.5347 


5.30 


1.6677 


5.96 


1.7851 


3.33 


1.2030 


3.99 


1.3838 


4.65 


1.5369 


5.31 


1.6696 


5.97 


1.7867 


3.34 


1.2060 


4.00 


1.3863 


4.66 


1.5390 


5.32 


1.6715 


5.98 


1.7884 


3.35 


1.2090 


4.01 


1.3888 


4.67 


1.5412 


5.33 


1.6734 


5.99 


1.7901 


3.36 


1.2119 


4.02 


1.3913 


4.68 


1.5433 


5.34 


1.6752 


6.00 


1.7918 


3.37 


1.2149 


4.03 


1.3938 


4.69 


1.5454 


5.35 


1.6771 


6.01 


1.7934 


3! 38 


1.2179 


4.04 


1.3962 


4.70 


1.5476 


5.36 


1.6790 


6.02 


1.7951 


3.39 


1.2208 


4.05 


1.3987 


4.71 


1.5497 


5.37 


1.6808 


6.03 


1.7967 


3.40 


1.2238 


4.06 


1.4012 


4.72 


1.5518 


5.38 


1.6827 


6.04 


1.7984 


3.41 


1.2267 


4.07 


1.4036 


4.73 


1.5539 


5.39 


1.6845 


6.05 


1.8001 


3.42 


1.2296 


4.08 


1.4061 


4.74 


1.5560 


5.40 


1.6864 


6.06 


1.8017 


3.43 


1.2326 


4.09 


1.4085 


4.75 


1.5581 


5.41 


1.6882 


6.07 


1.8034 


3.44 


1.2355 


4.10 


1.4110 


4.76 


1.5602 


5.42 


1.6901 


6.08 


1.8050 


3.45 


1.2384 


4.11 


1.4134 


4.77 


1.5623 


5.43 


1.6919 


6.09 


1.8066 


3.46 


1.2413 


4.12 


1 .4159 


4.78 


1.5644 


5.44 


1.6938 


6.10 


1.8083 


3.47 


1.2442 


4.13 


1.4183 


4.79 


1.5665 


5.45 


1.6956 


6.11 


1.8099 


3.48 


1.2470 


4.14 


1.4207 


4.80 


1.5686 


5.46 


1.6974 


6.12 


1.8116 


3.49 


1.2499 


4.15 


1.4231 


4.81 


1.5707 


5.47 


1.6993 


6.13 


1.8132 


3.50 


1.2528 


4.16 


1.4255 


4.82 


1.5728 


5.48 


1.7011 


6.14 


1.8148 


3.51 


1.2556 


4.17 


1.4279 


4.83 


1.5748 


5.49 


1.7029 


6.15 


1.8165 


3.52 


1.2585 


4.18 


1.4303 


4.84 


1.5769 


5.50 


1.7047 


6.16 


1.8181 


3.53 


1.2613 


4.19 


1.4327 


1 4.85 


1.5790 


5.51 


1.7066 


6.17 


1.8197 


3.54 


1.2641 


4.20 


1.4351 


4.86 


1.5810 


5.52 


1.7084 


6.18 


1.8213 


3.55 


1.2669 


4.21 


1.4375 


. 4.87 


1.5831 


5.53 


1.7102 


6.19 


1 .8229 


3.56 


1.2698 


4.22 


1.4398 


4.88 


1.5851 


5.54 


1.7120 


6.20 


1.8245 


3.57 


1.2726 


4.23 


1.4422 


4.89 


1.5872 


5.55 


1.7138 


6.21 


1.8262 


3.58 


1.2754 


4.24 


1.4446 


4.90 


1.5892 


5.56 


1.7156 


6.22 


1.8278 


3.59 


1.2782 


4.25 


1.4469 


4.91 


1.5913 


5.57 


1.7174 


6.23 


1.8294 


3.60 


1.2809 


4.26 


1.4493 


4.92 


1.5933 


5.58 


1.7192 


6.24 


1.8310 


3.61 


1.2837 


4.27 


1.4516 


4.93 


1.5953 


5.59 


1.7210 


6.25 


1.8326 


3.62 


1.2865 


4.28 


1.4540 


4.94 


1.5974 


5.60 


1.7228 


6.26 


1.8342 


3.63 


1.2892 


4.29 


1.4563 


4.95 


1.5994 


5.61 


1.7246 


6.27 


1.8358 


3.64 


1.2920 


4.30 


1.4586 


4.96 


1.6014 


5.62 


1.7263 


6.28 


1.8374 


3.65 


1.2917 


4.31 


1.4609 


4.97 


1.6034 


5.63 


1.7281 1 


6.29 


1.8390 


3.66 


1.2975 


4.32 


1.4633 


4.98 


1.6054 


5.64 


1.7299 ! 


6.30 


1.8405 


3.67 


1.3002 


4.33 


1 .4656 


4.99 


1.6074 


5.65 


1.7317 ! 


6.31 


1.8421 


3.68 


1.3029 


4.34 


1.4679 


5.00 


1.6094 


5.66 


1.7334 


6.32 


1.8437 


3.69 


1.3056 


4.35 


1.4702 


5.01 


1.6114 


5.67 


1.7352 ! 


6.33 


1.8453 


3.70 


1.3083 


4.36 


1.4725 


5.02 


1.6134 


5.68 


1.7370 


6.34 


1.8469 


3.71 


1.3110 


4.37 


1.4748 


5.03 


1.6154 


5.69 


1.7387 


6.35 


1.8485 


3.72 


1.3137 


4.38 


1.4770 


5.04 


1.6174 


5.70 


1.7405 


6.36 


1.8500 


3.73 


1.3164 


4.39 


1.4793 


5.05 


1.6194 


5.71 


1.7422 | 


637 


1.8516 


3.74 


1.3191 


4.40 


1.4816 


5.06 


1.6214 


5.72 


1.7440 


6.38 


1.8532 


3.75 


1.3218 


4.41 


1.4839 


5.07 


1.6233 


5.73 


1.7457 


6.39 


1.8547 


3.76 


1.3244 


4.42 


1.4861 


5.08 


1.6253 


5.74 


1.7475 ! 


6.40 


1.8563 


3.77 


1 .3271 


4.43 


1.4884 


5.09 


1.6273 


5.75 


1.7492 


6.41 


1 8579 


3.78 


L3297 


4.44 


1.4907 


5.10 


1.6292 


5.76 


1.7509 


6.42 


1.8594 


3.79 


1.3324 


4.45 


1.4929 


5.11 


1.6312 


5.77 


1.7527 


6.43 


1.8610 


3.80 


1.3350 


4.46 


1.4951 


5.12 


1.6332 


5.78 


1.7544 


6.44 


1.8625 


3.81 


1.3376 


4.47 


1.4974 


5.13 


1.6351 


5.79 


1.7561 


6.45 


1.8641 


3.82 


1.3403 


4.48 


1.4996 


5.14 


1.6371 


5.80 


1.7579 


6.46 


1.8(156 


3.83 


1.3429 


4.49 


1.5019 1 


5.15 


1.6390 


5.81 


1.7596 


6.47 


1.8672 


3.84 


1.3455 


4.50 


1.5041 1 


5.16 


1.6409 


5.82 


1.7613 


6.48 


1.8687 


3.85 


1.3481 


4.51 


1.5063 j 


5.17 


1.6429 


5.83 


1.7630 


6.49 


1.8703 


3.86 


1.3507 


4.52 


1.5085 1 


5.18 


1.6148 


5.84 


1.7647 1 


6.50 


1.8718 



158 



MATHEMATICAL TABLES. 



No. 


Log. 


No. 


Log. 


No. 


Log. 


No. Log. 


No. 


Log. 


6.51 


1.8733 


7.15 


1.9671 


7.79 


2.0528 


8.66 


2.1587 


9.94 


2.2966 


6.52 


1.8749 


7.16 


1.9685 


7.80 


2.0541 


8.68 


2.1610 


9.96 


2.2986 


6.53 


1.8764 


7.17 


1.9699 


7.81 


2.0554 


8.70 


2.1633 


9.98 


2.3006 


6.54 


1.8779 


7.18 


1.9713 


7.82 


2.0567 


8.72 


2.1656 


10.00 


2.3026 


6.55 


1.8795 


7.19 


1.9727 


7.83 


2.0580 


8.74 


2.1679 


10.25 


2.3279 


6.56 


1.8810 


7.20 


1.9741 


7.84 


2.0592 


8.76 


2.1702 


10.50 


2.3513 


6.57 


1.8825 


7.21 


1.9754 


7.85 


2.0605 


8.78 


2.1725 


10.75 


2.3749 


6.58 


1.8840 


7.22 


1.9769 


7.86 


2.0618 


8.80 


2.1748 


11.00 


2.3979 


6.59 


1.8856 


7.23 


1.9782 


7.87 


2.0631 


8.82 


2.1770 


11.25 


2.4201 


6.60 


1.8871 


7.24 


1.9796 


7.88 


2.0643 


8.84 


2.1793 


11.50 


2.4430 


6.61 


1.8886 


7.25 


1.9810 


7.89 


2.0656 


8.86 


2.1815 


11.75 


2.4636 


6.62 


1.8901 


7.26 


1.9824 


7.90 


2.0669 


8.88 


2.1838 


12.00 


2.4849 


6.63 


1.8916 


7.27 


1.9838 


7.91 


2.0681 


8.90 


2.1861 


12.25 


2.5052 


6.64 


1.8931 


7.28 


1.9851 


7.92 


2.0694 


8.92 


2.1883 


12.50 


2.5262 


6.65 


1.8946 


7.29 


1.9865 


7.93 


2.0707 


8.94 


2.1905 


12.75 


2.5455 


6.66 


1.8961 


7.30 


1.9879 


7.94 


2.0719 


8.96 


2.1928 


13.00 


2.5649 


6.67 


1.8976 


7.31 


1.9892 


7.95 


2.0732 


8.98 


2.1950 


13.25 


2.5840 


6.68 


1.8991 


7.32 


1.9906 


7.96 


2.0744 


9.00 


2.1972 


13.50 


2.6027 


6.69 


1.9006 


7.33 


1.9920 


7.97 


2.0757 


9.02 


2.1994 


13.75 


2.6211 


6.70 


1.9021 


7.34 


1.9933 


7.98 


2.0769 


9.04 


2.2017 


14.00 


2.6391 


6.71 


1.9036 


7.35 


1.9947 


7.99 


2.0782 


9.06 


2.2039 


14.25 


2.6567 


6.72 


1.9051 


7.36 


1.9961 


8.00 


2.0794 


9.08 


2.2061 


14.50 


2.6740 


6.73 


1.9066 


7.37 


1.9974 


8.01 


2.0807 


9 10 


2.2083 


14.75 


2.6913 


6.74 


1.9081 


7.38 


1.9988 


8.02 


2.0819 


9.12 


2.2105 


15.00 


2.7081 


6.75 


1.9095 


7.39 


2.0001 


8.03 


2.0832 


9.14 


2.2127 


15.50 


2.7408 


6.76 


1.9110 


7.40 


2.0015 


8.04 


2.0844 


9.16 


2.2148 


16.00 


2.7726 


6.77 


1.9125 


7.41 


2-0028 


8.05 


2.0857 


9.18 


2.2170 


16.50 


2.8034 


6.78 


1.9140 


7.42 


2.0041 


8.06 


2.0869 


9.20 


2.2192 


17.00 


2.8332 


6.79 


1.9155 


7.43 


2.0055 


8-07 


2.0882 


9.22 


2.2214 


17.50 


2.8621 


6.80 


1.9169 


7.44 


2.0069 


8-08 


2.0894 


9.24 


2.2235 


18.00 


2.8904 


6.81 


1.9184 


7.45 


2.0082 


8-09 


2.0906 


9.26 


2.2257 


18.50 


2.9173 


6.82 


1.9199 


7.46 


2-0096 


8-10 


2.0919 


9.28 


2.2279 


19.00 


2.9444 


6.83 


1.9213 


7.47 


2.0108 


8-11 


2.0931 


9.30 


2.2300 


19.50 


2.9703 


6.84 


1.9228 


7.48 


2.0122 


8-12 


2.0943 


9.32 


2.2322 


20.00 


2.9957 


6.85 


1.9242 


7.49 


2.0136 


8.13 


2.0956 


9.34 


2.2343 


21 


3.0445 


6.86 


1.9257 


7.50 


2.0149 


8.14 


2.0968 


9.36 


2.2364 


22 


3.0910 


6.87 


1.9272 


7.51 


2.0162 


8.15 


2.0980 


9.38 


2.2386 


23 


3.1355 


6.88 


1.9286 


7.52 


2.0176 


8.16 


2.0992 


9.40 


2.2407 


24 


3.1781 


6.89 


1.9301 


7.53 


2.0189 


8.17 


2.1005 


9.42 


2.2428 


25 


3.2189 


6.90 


1.9315 


7.54 


2.0202 


8.18 


2.1017 


9.44 


2.2450 


26 


3.2581 


6.91 


1.9330 


7.55 


2.0215 


8.19 


2.1029 


9.46 


2.2471 


27 


3.2958 


6.92 


1.9314 


7.56 


2.0229 


8-20 


2.1041 


9.48 


2.2492 


28 


3.3322 


6.93 


1.9359 


7.57 


2.0242 


8-22 


2.1066 


9.50 


2.2513 


29 


3.3673 


6.94 


1.9373 


7.58 


2.0255 


8.24 


2.1090 


9.52 


2.2534 


30 


3.4012 


8.95 


1.9387 


7.59 


2.0268 


8.26 


2.1114 


9.54 


2.2555 


31 


3.4340 


6.96 


1.9102 


7.60 


2.0281 


8.28 


2.1138 


9.56 


2.2576 


32 


3.4657 


6.97 


1.9416 


7.61 


2.0295 


8.30 


2.1163 


9.58 


2.2597 


33 


3.4965 


6.98 


1.9430 


7.62 


2.0308 


8.32 


2.1187 


9.60 


2.2618 


34 


3.5263 


6.99 


1.9445 


7.63 


2.0321 


8.34 


2.1211 


9.62 


2.2638 


35 


3.5553 


7.00 


1.9459 


7.64 


2.0334 


8.36 


2.1235 


9.64 


2.2659 


36 


3.5835 


7.01 


1.9473 


7.65 


2.0347 


8.38 


2.1258 


9.66 


2.2680 


37 


3.6109 


7.02 


1.9488 


7.66 


2.0360 


8.40 


2.1282 


9.68 


2.2701 


38 


3.6376 


7.03 


1.9502 


7.67 


2.0373 


8.42 


2.1306 


9.70 


2.2721 


39 


3.6636 


7.04 


1.9516 


7.68 


2.0386 


8.44 


2.1330 


9.72 


2.2742 


40 


3.6889 


7.05 


1.9530 


7.69 


2.0399 


8.46 


2.1353 


9.74 


2.2762 


41 


3.7136 


7.06 


1.9544 


7.70 


2.0412 


8.48 


2.1377 


9.76 


2.2783 


42 


3.7377 


7.07 


1.9559 


7.71 


2.0425 


8.50 


2.1401 


9.78 


2.2803 


43 


3.7612 


7.08 


1.9573 


7.72 


2.0438 


8.52 


2.1424 


9.80 


2.2824 


44 


3.7842 


7.09 


1.9587 


7.73 


2.0451 


8.54 


2.1448 


9.82 


2.2844 


45 


3.8067 


7.10 


1.9601 


7.74 


2.0464 


8.56 


2.1471 


9.84 


2.2865 


46 


3.8286 


7.11 


1.9615 


7.75 


2.0477 


8.58 


2.1494 


9.86 


2.2885 


47 


3.8501 


7.12 


1.9629 


7.76 


2.0490 


8.60 


2.1518 


9.88 


2.2905 


48 


3.8712 


7.13 


1.9643 


7.77 


2.0503 


8.62 


2.1541 


9.90 


2.2925 


49 


3.8918 


7.14 


1.9657 


7.78 


2.0516 


8.64 


2.1564 


9.92 


2.2946 


50 


3.9120 



NATURAL- TRIGONOMETRICAL FUNCTIONS. 



159 



NATURAL TRIGONOMETRICAL FUNCTIONS. 



o 


M. 


Sine. 


Co-Vers. 


Cosec. | Tang. 


Cotan. | 


Secant. Ver. Sin.j 


Cosine. 












00000 


L.0O0O 


Infinite 


00000 


Infinite ] 


1.0000 .00000 1.0000 90 







15 


00436 


.99564 


229.18 


00136 


229.18 ! 


1.0000 


.00001 


.99999 


45 




30 


00873 


.99127 


114.59 


00K73 


114.59 I 


1.0000 


.00001 


.99996 s 


30 




45 


01309 


.98691 


76.397 


01309 


76.390 


1.0001 


.00009 


. 99991 J 


15 


1 





01745 


.98255 


57.299 


01745 


57.290 


1.0001 


.00015 


.99985 89 







15 


.02181 


.97819 


15.840 


02182 


45.829 


1.0002 


.00024 


.99976' 


45 




.30 


.02618 


.973S2 


38.202 


02618 


38.188 


1.0003 


.00034 


.99966 


30 




45 


.03051 


.96946 


32.746 


03055 


32.730 


1.0005 


.00047 


.99953 


15 


2 





.03190 


.96510 


28.654 


03492 


28.636 


1.0006 


.00061 


.99939 88 







15 


.03926 


.96074 


25.471 


03929 


25.452 


1.0008 


.00077 


.99923 


45 




30 


.04362 


.95638 


22.926 


04366 


22.904 


1.0009 


.00095 


.99905, 


30 




45 


.04798 


.95202 


20.843 


04803 


20.819 


1.0011 


.00115 


.99885! 


15 


3 





.05234 


.94766 


19.107 


05211 


19.081 


1.0014 


.00137 


.99863 87 







15 


.05669 


.94331 


17.639 


05678 


17.611 


1.0016 


.00161 


.99839 




45 




30 


.06105 


.93895 


16.380 


06116 


16.350 


1.0019 


.00187 


.99813 




30 




45 


.06510 


.93460 


15.290 


06551 


15.257 • 


1.0021 


.00214 


.99786 




15 


4 





.06976 


.93024 


14.336 


06993 


14.301 j 


1.0024 


.00244 


.99756 


86 







15 


.07411 


.92589 


13.494 


07131 


13.457 ! 


1.0028 


.00275 


.99725 




45 




30 


.07846 


.92154 


12.745 


07870 


12.706 ! 


1.0031 


.00308 


.99692 




30 




45 


.08281 


.91719 


12.076 


08309 


12.035 


1.0034 


.00313 


.99656 




15 


5 





.08716 


.91284 


11.174 


08719 


11.430 


1.0038 


.00381 


.99619 


85 







15 


.09150 


.90850 


10.929 


09189 


10.883 i 


1.0012 


.00420 


.99580 




45 




30 


.09585 


.90415 


10.433 


09629 


10.385 


1.0046 


.00460 


.99540 




30 




45 


.10019 


.89981 


9.9812 


10069 


9.9310 


1.0051 


.00503 


.99497 




15 


6 





.10153 


.89547 


9.5668 


10510 


9.5144 


1.0055 


.00548 


.99152 


84 







15 


.10887 


.89113 


9.1855 


10952 


9.1309 


1.0060 


.00594 


.99406 




45 




30 


.11320 


.88680 


8.8337 


11393 


8.7769 


1.0065 


.00643 


.99357 




30 




45 


.11754 


.88246 


8.5079 


11836 


8.4190 


1.0070 


.00693 


.93307 




15 


1 





.12187 


.87813 


8.2055 


12278 


8.1443 


1.0075 


.00745 


.99255 


83 







15 


.12620 


.87380 


7.9240 


12722 


7.8606 


1.0081 


.00800 


.99200 




45 




30 


. 13053 


.86947 


7.6613 


.13165 


7.5958 


1.0086 


.00856 


.99144 




30 




45 


.13185 


.86515 


7.4156 


.13609 


7.3479 


1.0092 


.00913 


.99086 




15 


8 





.13917 


.86083 


7.1853 


.14054 


7.1154 


1.0098 


.00973 


.99027 


82 







15 


.14349 


.85651 


6.9690 


.11199 


6.S969 


1.0105 


.01035 


.98965 




45 




30 


.14781 


.85219 


6.7655 


.141)15 


6.6912 


1.0111 


.01098 


.98902 




30 




45 


.15212 


.84788 


6.5736 


.15391 


6.4971 


1.0118 


.01164 


.98836 




15 


9 





.15613 


.84357 


6.3924 


.15838 


6.3138 


1.0125 


.01231 


.98769 


81 







15 


.16074 


.83926 


6.2211 


.16286 


6.1402 


1.0132 


.01300 


.98700 




45 




30 


.16505 


.83495 


6.0589 


.16731 


5.9758 


1.0139 


.01371 


.98629 




30 




45 


.16935 


.83065 


5.9049 


•17183 


5.8197 


1.0147 


.01444 


.98556 




15 


10 





!•■'. ' 


.82635 


5.7588 


•17633 


5.6713 


1.0154 


.01519 


.98481 


80 







15 


.17794 


.82206 


5.6198 


.18083 


5.5301 


1.0162 


.01596 


.98104 




45 




30 


.18224 


.81776 


5.4874 


•18534 


5.3955 


1.0170 


.01675 


.98325 




30 




45 


.18652 


.81348 


5.3612 


.18986 


5.2672 


1.0179 


.01755 


.98245 




15 


11 





.19081 


.80919 


5.2408 


•19438 


5.1446 


1.0187 


.01837 


.98163 


79 







15 


.19509 


.80491 


5.1258 


•19891 


5.0273 


1.0196 


.01921 


.98079 




45 




30 


.19937 


.80063 


5.0158 


.20315 


4.9152 


1.0205 


.02008 


.97992 




30 




45 


.20364 


.79636 


4.9106 


•20800 


4.8077 


1.0214 


.02095 


.97905 




15 


12 





.20791 


.7920P 


4.8097 


.21256 


4.7046 


1.0223 


.02185 


.97815 


7S 







15 


.21218 


.7878; 


4.7130 


•21712 


4.6057 


1.0233 


.02277 


.97723 




45 




30 


.21644 


.78356 


4.6202 


•22169 


4.5107 


1.0243 


.02370 


.97630 




30 




45 


.2207C 


.7793C 


4.5311 


•22628 


4.4191 


1.0253 


.02466 


.97534 




15 


13 





.22495 


.7750c 


4.4154 


•23087 


4.3315 


1.0263 


.02563 


.97137 


77 







15 


.2292C 


.7708( 


) 4.3630 


•23547 


4.2468 


1.0273 


.02662 


.97338 




45 




30 


.2334c 


.7665? 


) 4.2837 


•2400S 


4.1653 


1.0284 


.02763 


.97237 




30 




45 


.2376C 


.7623 


4.2072 


•24171 


4.0867 


1.0295 


.02866 


.97134 




15 


14 





.2419; 


.7580* 


] 4.1336 


.2493? 


4.010E 


1.0306 


.0297C 


.97030 


76 







15 


.2461f 


.7538 


5 4.0625 


. 


3.937c 


1.0317 


.03077 


.96923 




45 




30 


.2503* 


.7496 


I 3.9939 


.2586- 


3.8667 


1.032S 


.0318c 


.96815 




30 




45 


.2546( 


) .7454 


) 3.9277 


.26325 


3.7981 


1.034 


. 0329c 


.98705 




15 


15 





.2588; 


5 .7411 


* 3.8637 


.2679." 


1 3.732C 


1.0353 


.03407 


.96593 


75 









Cosine 


. Ver. Sin 


. Secant. 


Cotan 


| Tang. 


Cosec. 


Co-Vers 


Sine. 


° 


M. 



From 75° to SO° read from pottom of table upwards, 



160 



MATHEMATICAL TABLES. 



• 


M. 


Sine. 


Co-Vers. 


c™,. 


Tang. 


Cotan. 


Secant. 


Ver. Sin. 


c.„„. 






15 





.25882 


.74118 


3.8637 


.26795 


3.7320 


1.0353 


.03407 


.96693 


75 







15 


.26303 


.73697 


3.8018 


.27263 


3.6680 


1.0365 


.03521 


.96479 




45 




30 


.26724 


.73276 


3.7420 


.27732 


3.6059 


1.0377 


.03637 


.96363 




30 




45 


.27144 


.72856 


3.6840 


.28203 


3.5457 


1.0390 


.03754 


.96246 




15 


16 





.27564 


.72436 


3.6280 


.28674 


3.4874 


1.0403 


.03874 


.96126 


74 







15 


.27983 


.72017 


3.5736 


.29147 


3.4308 


1.0416 


.03995 


.96005 




45 




30 


.28402 


.71598 


3.5209 


.29621 


3.3759 


1 .0429 


.04118 


.95882 




30 




45 


.28820 


.71180 


3.4699 


.30096 


3.3226 


1.0443 


.04243 


.95757 




15 


17 





.29237 


.70763 


3.4203 


.30573 


3.2709 


1.0457 


.04370 


.95630 


73 







15 


.21)654 


.70346 


3.3722 


.31051 


3.2205 


1.0471 


.04498 


.95502 




45 




30 


.30070 


.69929 


3.3255 


.31530 


3.1716 


1.0485 


.04628 


.95372 




30 




45 


.30486 


.69514 


3.2801 


.32010 


3.1240 


1.0500 


.04760 


.95240 




15 


18 





.30902 


.69098 


3.2361 


.32492 


3.0777 


1.0515 


.04894 


.95106 


72 







15 


.31316 


.68684 


3.1932 


.32975 


3.0326 


1.0530 


.05030 


.94970 




45 




30 


.31730 


.68270 


3.1515 


.33459 


2.9887 


1.0545 


.05168 


.94832 




30 




45 


.32144 


.67856 


3.1110 


.33945 


2.9459 


1.0560 


.05307 


.94693 




15 


19 





.32557 


.67443 


3.0715 


.34433 


2.9042 


1.0576 


.05448 


.94552 


71 







15 


.32969 


.67031 


3.0331 


.34921 


2.8636 


1.0592 


.05591 


.94409 




45 




30 


.33381 


.66619 


2.9957 


.35412 


2.8239 


1.0608 


.05736 


.94264 




30 




45 


.33792 


.66208 


2.9593 


.35904 


2.7852 


1.0625 


.05882 


.94118 




15 


20 





.34202 


.65798 


2.9238 


.36397 


2-7475 


1 .0642 


.06031 


.93969 


70 







15 


.34612 


.65388 


2.8892 


.36892 


2.7106 


1.0659 


.06181 


.93819 




45 




30 


.35021 


.64979 


2.8554 


.37388 


2-6746 


1.0676 


.06333 


.93667 




30 




45 


.35429 


.64571 


2.8225 


.37887 


2-6395 


1.0694 


.06486 


.93514 




15 


21 





.35837 


.64163 


2.7904 


.38386 


2.6051 


1.0711 


.06642 


.93358 


69 







15 


.36244 


.63756 


2.7591 


.38888 


2-5715 


1.0729 


.06799 


.93201 




45 




30 


.36650 


.63350 


2.7285 


.39391 


2-5386 


1.0748 


.06958 


.93042 




30 




45 


.37056 


.62944 


2.6986 


.39896 


2-5065 


1.0766 


.07119 


.92881 




15 


22 





.37461 


.62539 


2.6695 


.40403 


2-4751 


1.0785 


.07282 


.92718 


68 







15 


.37865 


.62135 


2.6410 


.40911 


2-4443 


1.0804 


.07446 


.92554 




45 




30 


.38268 


.61732 


2.6131 


.41421 


2-4142 


1.0824 


.07612 


.92388 




30 




45 


.38671 


.61329 


2.5859 


.41933 


2-3847 


1.0844 


.07780 


.92220 




15 


23 





.39073 


.60927 


2.5593 


.42447 


2-3559 


1.0864 


.07950 


.92050 


67 







15 


.39474 


.60526 


2.5333 


.42963 


2-3276 


1.0884 


.08121 


.91879 




45 




30 


.39875 


.60125 


2.5078 


.43481 


2-2998 


1.0904 


.08294 


.91706 




30 




45 


.40275 


.59725 


2.4829 


.44001 


2 2727 


1.0925 


.08469 


.91531 




15 


24 





.40674 


.59326 


2.4586 


.44523 


2-2460 


1.0946 


.08645 


.91355 


66 







15 


.41072 


.58928 


2.4348 


.45047 


2.2199 


1.0968 


.08824 


.91176 




45 




30 


.41469 


.58531 


2.4114 


.45573 


2.1943 


1.0989 


.09004 


.90996 




30 




45 


41866 


.58134 


2.3886 


.46101 


2.1692 


1.1011 


.09186 


.90814 




15 


25 





.-12262 


.57738 


2.3662 


.46631 


2.1445 


1.1034 


.09369 


.90631 


65 







15 


.42657 


.57343 


2.3443 


.47163 


2.1203 


1.1056 


.09554 


.90446 




45 




30 


.43051 


.56949 


2.3228 


.47697 


2.0965 


1.1079 


.09741 


.90259 




30 




45 


.43445 


.56555 


2.3018 


.48234 


2.0732 


1.1102 


.09930 


.90070 




15 


26 





.43837 


.56163 


2.2812 


.48773 


2.0503 


1.1126 


. 10121 


.89879 


64 







15 


.44229 


.55771 


2.2610 


.49314 


2.0278 


1.1150 


.10313 


.89687 




45 




30 


.44620 


.55380 


2.2412 


.49858 


2.0057 


1.1174 


. 10507 


.89493 




30 




45 


.45010 


.54990 


2.2217 


.50404 


1.9840 


1.1198 


.10702 


.89298 




15 


27 





.45399 


.54601 


2.2027 


.50952 


1.9626 


1.1223 


.10899 


.89101 


63 







15 


.45787 


.54213 


2.1840 


.51503 


1.9416 


1.1248 


.11098 


.88902 




45 




30 


.46175 


.53825 


2.1657 


.52057 


1.9210 


1.1274 


.11299 


.88701 




30 




45 


.46561 


.53439 


2.1477 


.52612 


1.9007 


1.1300 


.11501 


.88499 




15 


28 





.46947 


.53053 


2.1300 


.53171 


1.8807 


1.1326 


.11705 


.88295 


62 







15 


.47332 


.52668 


2.1127 


; 


1.8611 


1.1352 


.11911 


.88089 




45 




30 


.47716 


.52284 


2.0957 


.54295 


1.8418 


1.1379 


.12118 


.87882 




30 




45 


.48099 


.51901 


2.0790 


.54862 


1.8228 


1.1406 


.12327 


.87673 




15 


29 





.48481 


.51519 


2.0627 


.55431 


1.8040 


1.1433 


.12538 


.87462 


61 







15 


.48862 


.51138 


2.0466 


.56003 


1.7856 


1.1461 


.12750 


.87250 




45 




30 


.49242 


.50758 


2.0308 


.56577 


1.7675 


1 . 1490 


.12964 


.87036 




30 




45 


.49622 


.50378 


2.0152 


.57155 


1.7496 


1.1518 


.13180 


.868:20 




15 


30 





.50000 


.50000 


2.0000 


.57735 


1.7320 


1.1547 


.13397 


.86603 


60 









Cosine. 


Ver. Sin. 


Sec,.,. 


Cotan. 


Tang. 


o«. 


Co-Vers. 


Sine. 


• 


M. 



From 60° to 75° read from bottom of table upwards. 



NATURAL TRIGONOMETRICAL FUNCTIONS. 



161 



■• 


M. 



Sine. 


Co-Vers. 


C.,.c. 


Tang. 


Cotan. 


Secant. 


Ver. Sin. 


Cosine. 






30 


.50000 


.50000 


2.0000 


.57735 


1.7320 1.1547 


.13397 


.86603 


GO 







15 


.50377 


.49623 


1.9850 


.58318 


1.7147 


1.1576 


.13616 


.86384 




45 




30 


.50754 


.49246 


1.9703 


.58904 


1.6977 


1.1606 


.13837 


.86163 




30 




45 


.51129 


.48871 


1.9558 


.59494 


1.6808 


1.1636 


. 14059 


.85941 




15 


31 





.51504 


.48496 


1.9416 


.60086 


1.6643 


1.1666 


.14283 


.85717 


59 







15 


.51877 


.48123 


1.9276 


.60681 


1.6479 


1.1697 


.14509 


.85491 




45 




30 


.52250 


.47750 


1.9139 


.61280 


1.6319 


1.1728 


.14736 


.85264 




30 




45 


.52621 


.47379 


1.9004 


.61882 


1.6160 


1.1760 


.14965 


.85035 




15 


32 





.52992 


.47008 


1.8871 


.62487 


1.6003 


1.1792 


.15195 


.84805 


58 







15 


.53361 


.46639 


1.8740 


.63095 


1.5849 


1.1824 


.15427 


.84573 




45 




30 


.53730 


.46270 


1.8612 


.63707 


1.5697 


1.1857 


.15661 


.84339 




30 




45 


.54097 


.45903 


1.8485 


.64322 


1.5547 


1.1890 


.15896 


. 84104 




15 


33 





.54464 


.45536 


1.8361 


.64941 


1.5399 


1 1924 


. 16133 


.83867 


57 







15 


.54829 


.45171 


1.8238 


.65563 


1.5253 


1.1958 


.16371 


.83629 




45 




30 


.55194 


.44806 


1.8118 


.66188 


1.5108 


1.1992 


.16611 


.83389 




30 




45 


.55557 


.44443 


1.7999 


.66818 


1.4966 


1.2027 


.16853 


.83147 




15 


34 





.55919 


.44081 


1.7883 


.67451 


1.4826 


1.2062 


.17096 


.82904 


56 







15 


.56280 


.43720 


1.7768 


.68087 


1.4687 


1.2098 


.17341 


.82659 




45 




30 


.56641 


.43359 


1.7655 


.68728 


1.4550 


1.2134 


.17587 


.82413 




30 




45 


.57000 


.43000 


1.7544 


.69372 


1.4415 


1.2171 


.17835 


.82165 




15 


35 





.57358 


.42642 


1.7434 


.70021 


1.4281 


1.2208 


.18085 


.81915 


55 







15 


.57715 


.42285 


1.7327 


.70673 


1.4150 


1.2245 


.18336 


.81664 




45 




30 


.58070 


.41930 


t.7220 


.71329 


1.4019 


1.2283 


.18588 


.81412 




30 




45 


.58425 


.41575 


1.7116 


.71990 


1.3891 


1.2322 


.18843 


.81157 




15 


36 





.58779 


.41221 


1.7013 


.72654 


1.3764 


1.2361 


.19098 


.80902 


54 







15 


.59131 


.40869 


1.6912 




1.3638 


1.2400 


.19356 


.80644 




45 




30 


.59482 


.40518 


1.6812 




1.3514 


1.2440 


.19614 


.80386 




30 




45 


.59832 


.40168 


1.6713 


.74673 


1.3392 


1.2480 


.19875 


.80125 




15 


37 





.60181 


.39819 


1.6616 


.75355 


1.3270 


1.2521 


.20136 


.79864 


53 







15 


.60529 


.39471 


1.G521 


.76042 


1.3151 


1.2563 


.20400 


.79600 




45 




30 


.60876 


.39124 


1.G427 


.76733 


1.3032 


1.2605 


.20665 


.79335 




30 




45 


.61222 


.38778 


1.G334 


.77428 


1.2915 


1.2647 


.20931 


.79069 




15 


38 





.61566 


.38434 


1.G243 


.78129 


1.2799 


1.2690 


.21199 


.78801 


52 







15 


.61909 


.38091 


1.0153 


.78834 


1.2685 


1.2734 


.21468 


.78532 




45 




30 


.62251 


.37749 


1.6064 


.79543 


1.2572 


1.2778 


.21739 


.78261 




3" 




45 


- 


.37408 


1.5976 


.80258 


1.2460 


1.2822 


.22012 


.77988 




15 


39 







.37068 


1. 589.1 


.80978 


1.2349 


1.2868 


.22285 


.77715 


51 







15 


.63271 


.36729 


1.5805 


.81703 


1.2239 


1.2913 


.22561 


.77439 




45 




30 


.63608 


.36392 


1.5721 


.82434 


1.2131 


1.2960 


.22838 


.77162 




30 




45 


.63944 


.36056 


1.5639 


.83169 


1.2024 


1.3007 


.23116 


.76884 




15 


40 





.64279 


.35721 


1.5557 


.83910 


1.1918 


1.3054 


.23396 


.76604 


50 







15 


.64612 


.353S8 


1.5477 


.84656 


1.1812 


1.3102 


.23677 


.76323 




45 




30 


.64945 


. 35055 


1.5398 


.85408 


1.1708 


1.3151 


.23959 


.76041 




30 




45 


.65276 


.34724 


1.5320 


.86165 


1.1606 


1.3200 


.24244 


.75756 




15 


41 





.65606 


.34394 


1 5242 


.86929 


1.1504 


1.3250 


.24529 


.75471 


49 







15 


.65935 


.34065 


1.5166 


.87698 


1.1403 


1.3301 


.24816 


.75184 




45 




30 


.66262 


.33738 


1 5092 


.88472 


1.1303 


1.3352 


.25104 


.74896 




30 




45 


.66588 


.33412 


1.5018 




1.1204 


1.3404 


.25394 


.74606 




15 


42 





.66913 


.33087 


1.4945 


.90040 


1.1106 


1.3456 


.25686 


.74314 


48 







15 


.67237 


.32763 


1.4873 


.90834 


1.1009 


1.3509 


.25978 


.74022 




45 




30 


.67559 


.32441 


1.4802 


.91633 


1 0913 


1.3563 


.26272 


.73728 




30 




45 


.67880 


.32120 


1.4732 


.92439 


1.0818 


1.3618 


.26568 


.73432 




15 


43 





.68200 


.31800 


1.4663 


93251 


1.0724 


1.3673 


.26865 


.73135 


47 







15 


.68518 


.31482 


1.4595 


.94071 


1.0630 


1.3729 


.27163 


.72837 




45 




30 


.68835 


.31165 


1.4527 


.94896 


1.0538 


1 .3786 


.27463 


.72537 




30 




45 


.69151 


.30849 


1.4461 


.95729 


1.0446 


1.3843 


.27764 


.72236 




15 


44 





.69466 


.30534 


1.4396 


.96569 


1.0355 


1.3902 


.28066 


.71934 


46 







15 


.69779 


.30221 


1.4331 


.97416 


1 .0265 


1.3961 


.28370 


.71630 




45 




30 


.70091 


.29909 


1.4267 


.98270 


1.0176 


1.4020 


.28675 


.71325 




30 




45 


.70401 


.29599 


1.4204 


.99131 


1.0088 


1.4081 


.28981 


.71019 




15 


45 





.70711 


.29289 

Ver. Sin. 


1.4142 


1.0000 


1 .0000 


1.4142 


.29289 


.70711 

Sine. 


45 





c„,«. 


Secant. 


Cotan. 


Tang. 


Cosec. 


Co-Vers. 


M. 


! 


Fro 


m45 


° to 6 


J° read 


fron 


i ootttc 


in of i 


aole i 


ipwa 


rds. 





162 



MATHEMATICAL TABLES. 
LOGARITHMIC SINES, ETC. 



Deg. 


Sine. 


Cosec. 


Versin. 


Tangent. 


Cotan. 


Covers. 


Secant. 


Cosine. 


Deg. 





In.Neg. 
8.24186 


Infinite. 


In.Neg. 


In.Nea:. 


Infinite. 


10.00000 


10.00000 


10.00000 


90 


1 


11.75814 


6.18271 


8.24192 


11.75808 


9.99235 


10.00007 


9.99993 


89 


2 


8.54282 


11.45718 


6.78474 


8.54308 


11.45692 


9.98457 


10.00026 


9.99974 


88 ' 


3 


8.71880 


11.28120 


7.13687 


8.71940 


11.28060 


9.97665 


10.00060 


9.99940 


87 


4 


8.84358 


11.15642 


7.38667 


8.84464 


11.15536 


9.96860 


10.00106 


9.99894 


86 


5 


8.94030 


11.05970 


7.58039 


8.94195 


11.05805 


9.96040 


10.00166 


9.99834 


85 


6 


9.01923 


10.98077 


7.73863 


9.02162 


10.97838 


9.95205 


10.00239 


9.99761 


84 


7 


9.08589 


10.91411 


7.87238 


9.08914 


10.91086 


9.94356 


10.00325 


9.99675 


83 


8 


9.14356 


10.85644 


7.98820 


9.14780 


10.85220 


9.93492 


10.00425 


9.99575 


82 i 


9 


9.19433 


10.80567 


8.09032 


9.19971 


10.80029 


9.92612 


10.00538 


9.99462 


81 


10 


9.23967 


10.76033 


8.18162 


9.24632 


10.75368 


9.91717 


10.00665 


9.99335 


80 


11 


9.28060 


10.71940 


8.26418 


9.28865 


10.71135 


9.90805 


10.00805 


9.99195 


79 ! 


12 


9.31788 


10.68212 


8.33950 


9.32747 


10.67253 


9.89877 


10.00960 


9.99040 


78 


13 


9.35209 


10.64791 


8.40875 


9.36336 


10.63664 


9.88933 


10.01128 


9.98872 


77 


14 


9.38368 


10.61632 


8.47282 


9.39677 


10.60323 


9.87971 


10.01310 


9.98690 


76 


15 


9.41300 


10.58700 


8.53243 


9.42805 


10.57195 


9.86992 


10.01506 


9.98494 


75 


16 


9.44034 


10.55966 


8.58814 


9.45750 


10.54250 


9.85996 


10.01716 


9.98284 


74 


17 


9.46594 


10.53406 


8.64043 


9.48534 


10.51466 


9.84981 


10.01940 


9.98060 


73 


18 


9.48998 


10.51002 


8.68969 


9.51178 


10.48822 


9.83947 


10.02179 


9.97821 


72 


19 


9.51264 


10.48736 


8.73625 


9.53697 


10.46303 


9.82894 


10.02433 


9.97567 


71 


20 


9.53405 


10.46595 


8.78037 


9.56107 


10.43893 


9.81821 


10.02701 


9.97299 


70 


21 


9.55433 


10.44567 


8.82230 


9.58418 


10.41582 


9.80729 


10.02985 


9.97015 


69 


22 


9.57358 


10.42642 


8.86223 


9.60641 


10.39359 


9.79615 


10.03283 


9.96717 


68 


23 


9.59188 


10.40812 


8.90034 


9.62785 


10.37215 


9.78481 


10.03597 


9.96403 


67 


24 


9.60931 


10.39069 


8.93679 


9.64858 


10.35142 


9.77325 


10.03927 


9.96073 


66 


25 


9.62595 


10.37405 


8.97170 


9.66867 


10.33133 


9.7614t' 


10.04272 


9.95728 


65 


26 


9.64184 


10.35816 


9.00521 


9.68818 


10.31182 


9.74945 


10.04634 


9.95366 


64 


27 


9.65705 


10.34295 


9.03740 


9.70717 


10.29283 


9.73720 


10.05012 


9.94988 


63 


28 


9.67161 


10.32839 


9.06838 


9.72567 


10.27433 


9.72471 


10.05407 


9.94593 


62 


29 


9.68557 


10.31443 


9.09823 


9.74375 


10.25625 


9.71197 


10.05818 


9.94182 


61 


30 


9.69897 


10.30103 


9.12702 


9.76144 


10.23856 


9.69897 


10.06247 


9.93753 


60 


31 


9.71184 


10.28S16 


9.15483 


9.77877 


10.22123 


9.68571 


10.06693 


9.93307 


59 


32 


9.72421 


10.27579 


9.18171 


9.79579 


10.20421 


9.67217 


10.07158 


9.92842 


58 


33 


9.73611 


10.26389 


9.20771 


9.81252 


10.18748 


9.65836 


10.07641 


9.92359 


57 i 


34 


9.74756 


10.25244 


9.23290 


9.82899 


10.17101 


9.64425 


10.08143 


9.91857 


56 


35 


9.75859 


10.24141 


9.25731 


9.84523 


10.15477 


9.62984 


10.08664 


9.91336 


55 


36 


9.76922 


10.23078 


9.28099 


9.86126 


10.13874 


9.61512 


10.09204 


9.90796 


54 


37 


9.77946 


10.22054 


9.30398 


9.87711 


10.12289 


9.60008 


10.09765 


9.90235 


53 


38 


9.78934 


10.21066 


9.32631 


9.89281 


10.10719 


9.58471 


10.10347 


9.89653 


52 


39 


9.79887 


10.20113 


9.34802 


9.90837 


10.09163 


9.56900 


10.10950 


9.89050 


51 


40 


9.80807 


10.19193 


9.36913 


9.92381 


10.07619 


9.55293 


10.11575 


9.88425 


50 


41 


9.81694 


10.18306 


9.38968 


9.93916 


10.06084 


9.53648 


10.12222 


9.87778 


49 


42 


9.82551 


10.17449 


9.40969 


9.95444 


10.04556 


9.51966 


10.12893 


9.87107 


48 


43 


9.88378 


10.16622 


9.42918 


9.96966 


10.03034 


9.50243 


10.13587 


9.86413 


47 


44 


9.84177 


10.15823 


9.44818 


9.98484 


10.01516 


9.48479 


10.14307 


9.85693 


46 


45 


9.84949 


10.15052 


9.46671 


10.00000 


10.00000 


9.46671 


10.15052 


9.84949 


t 
45 




Cosine. 


Secant. 


Covers. 


Cotan. 


Tangent. 


Versin. 


Cosec. 


Sine. 


t 


From 45° to 90° rea< 


1 from 


bottoi 


a of tal 


Me npi 


vards. 


i 

i 1 

• 1 



SPECIFIC GRAVITY. 



163 



MATERIALS. 



THE CHEMICAL ELEMENTS. 
Tl»e Common Elements (42). 



o o 


Name. 


sit 
0-53 



11 


Name. 


-i_j 




§ s 


Name. 


053 






<£ 






% 






<£ 


Al 


Aluminum 


27.1 


F 


Fluorine 


19. 


Pd 


Palladium 


106. 


Sb 


Antimony 


120. 


Au 


Gold 


196.2 


P 


Phosphorus 


30.96 


As 


Arsenic 


75. 


H 


Hydrogen 


1. 


Pt 


Platinum 


195. 


Ba 


Barium 


137. 


I 


Iodine 


126.6 


K 


Potassium 


39.03 


Bi 


Bismuth 


208. 


Ir 


Iridium 


193. 


Si 


Silicon 


28.4 


B 


Boron 


10.9 


Fe 


Iron 


56. 


Ag 


Silver 


107.7 


Br 


Bromine 


79.8 


Pb 


Lead 


206.4 


Na 


Sodium 


23. 


Cd 


Cadmium 


111.8 


Li 


Lithium 


7.01 


Sr 


Strontium 


87.4 


Ca 


Calcium 


40. 


Mg 


Magnesium 


24. 


S 


Sulphur 


32. 


C 


Carbon 


12. 


Mn 


Manganese 


55. 


Sn 


Tin 


118. 


CI 


Chlorine 


35.4 


Hg 


Mercury 


199.8 


Ti 


Titanium 


50. 


Cr 


Chromium 


52.3 


Ni 


Nickel 


58.3 


W 


Tungsten 


184. 


Co 


Cobalt 


59. 


N 


Nitrogen 


14. 


Va 


Vanadium 


51.2 


Cu 


Copper 


63.2 





Oxygen 


15.96 


Zn 


Zinc 


65. 



The atomic weights of many of the elements vary in the decimal place as 
given by different authorities. 



Tlie Rare Elements (27). 



Beryllium, Be. 
Caesium, Cs. 
Cerium, Ce. 
Didymium, D. 
Erbium, E. 
Gallium, Ga. 



Glucinum, G. 
Indium, In. 
Lanthanum, La. 
Molybdenum, Mo. 
Niobium, Nb. 
Osmium, Os. 



Germanium, Ge. Rhodium, R. 



Rubidium, Rb. 
Ruthenium, Ru. 
Samarium, Sm. 
Scandium, Sc. 
Selenium, Se. 
Tantalum, Ta. 
Tellurium, Te. 



Thallium, Tl. 
Thorium, Th. 
Uranium, U. 
Ytterbium, Yr. 
Yttrium, Y. 
Zirconium, Zr. 



SPECIFIC GRAVITY. 



The specific gravity of a substance is its weight as compared with the 
weight of an equal bulk of pure water. 
To find, the specific gravity of a substance. 

W = weight of body in air; 10 = weight of body submerged in water. 

W 

Specific gravity = — . 

W — io 

If the substance be lighter than the water, sink it by means of a heavier 
substance, and deduct the weight of the heavier substance. 

Specific-gravity determinations are usually referred to the standard of the 
weight of water at 62° F., 62.355 lbs. per cubic foot. Some experimenters 
have used 60° F. as the standard, and others 32° and 39.1° F. There is no 
general agreement. 

Given sp. gr. referred to water at 39.1° F., to reduce it to the standard of 
62° F. multiply it by 1.00112. 

Given sp. gr. referred to water at 62° F.. to find weight per cubic foot mul- 
tiply by 62.355. Given weight per cubic foot, to find sp. gr. multiply by 
0.016037. Given sp. gr., to rind weight per cubic inch multiply by .036085, 



164 



MATERIALS. 



Weight and Specific Gravity of Metals. 



Aluminum 

Antimony 

Bismuth 

Brass: Copper + Zinc 1 
80 20 i 

70 30 }-.. 

60 40 | 

50 50 J 

Bionze-j Tinj 5 to 20| 

Cadmium 

Calcium 

Chromium 

Cobalt 

Gold, pure 

Copper , 

Iridium 

Iron, Cast 

" Wrought 

Lead 

Manganese 

Magnesium 

( 32< 
Mercury •< 60' 

(212- 

Nickel 

Platinum 

Potassium 

Silver 

Sodium 

Steel 

Tin : 

Titanium 

Tungsten 

Zinc 



Specific Gravity. 
Range accord- 
ing to 
several 
Authorities. 



2.56 
6.66 
9.74 



to 2.71 
to 6.86 
to 9.90 



8.52 to 8.96 

8.6 to 8.7 
1.58 
5.0 
8.5 to 8.6 
19.245 to 19.361 
8.69 to 8.92 
to 23. 
to 7.48 
to 7.9 
to 11.44 






7.4 
11.07 

7. 



to 8. 



13.60 



to 1.75 
to 13.62 
13.58 
13.37 to 13.38 
8.279 to 8.93 
20.33 to 22.07 

0.865 
10.474 to 10.511 

0.97 
7.69* to 7.932t 
7.291 to 7." 

5.3 
17. to 17. 
6.86 to 7. 



Specific Grav- 
ity. Approx. 


Weight 


Mean Value, 

used in 
Calculation of 


Cubic 
Foot, 
lbs. 


Weight. 


2.67 


166.5 


6.76 


421.6 


9.82 


612.4 


f8.60 


536.3 


J 8.40 
|8.36 


523.8 


521.3 


L8.20 


511.4 


8.853 


552. 


8.65 


539. 


19.258 


1200.9 


8.853 


552. 




1396. 


7.218 


450. . 


7.70 


480. 


11.38 


709.7 


8. 


499. 


1.75 


109. 


13.62 


849.3 


13.58 


846.8 


13.38 


834.4 


8.8 


548.7 


21.5 


1347.0 


10.505 


655.1 


7.854 


489.6 


7.350 


458.3 


7.00 


436.5 



Weight 

per 
Cubic 
Inch, 

lbs. 



.0963 



.3103 
.3031 

.3017 



.3195 
.3121 



.6949 
.3195 

.8076 
.2601 
.2779 
.4106 
.2887 
.0641 
.4915 
.4900 
.4828 
.3175 
.7758 

.3791 

.2834 

.2652 



* Hard and burned. 

t Very pure and soft. The sp. gr. decreases as the carbon is increased. 

In the first column of figures the lowest are usually those of cast metals. 
which are more or less porous; the highest are of metals finely rolled or 
drawn into wire. 

Specific Gravity of Liquids at 60° F. 



Acid, Muriatic 1.200 

" Nitric 1.217 

" Sulphuric 1.849 

Alcohol, pure 794 

95 per cent 816 

50 " " 934 

Ammonia, 27.9 per cent 891 

Bromine 2.97 

Carbon disulphide 1.26 

Ether, Sulphuric 72 

Oil, Linseed 94 



Oil, Olive 92 

" Palm 97 

" Petroleum 78 to .88 

" Rape 92 

" Turpentine 87 



Whale.. 



Tar 

Vinegar.. 

Water 



.92 



Compression of the following Fluids under a Pressure of 
15 lbs. per Square Inch. 

Water 00004663 I Ether 00006158 

Alcohol 0000216 I Mercury 00000265 



SPECIFIC GRAVITY. 



165 



The Hydrometer. 

The hydrometer is an instrument for determining the density of liquids. 
It is usually made of glass, and consists of three parts: (1) the upper part, 
a graduated stem or fine tube of uniform diameter; (2) a bulb, or enlarge- 
ment of the tube, containing air ; and (3) a small bulb at the bottom, con- 
taining shot or mercury which causes the instrument to float in a vertical 
position. The graduations are figures representing either specific gravities, 
or the numbers of an arbitrary scale, as in Beaume's, Tvvaddell"s, Beck's, 
and other hydrometers. . 

There is a tendency to discard all hydrometers with arbitrary scales and 
to use only those which read in terms of the specific gravity directly. 

Beaume's Hydrometer and Specific Gravities Compared, 





Liquids 
Heavier 

than 
Water, 
sp. gr. 


Liquids 
Lighter 

than 
Water, 
sp. gr. 


60 § 


Liquids Li 
Heavier Li 
than t 
Water, W 
sp. gr. si 


quids 
ghter 
nan 
ater, 
). gr. 




Liquids 
Heavier 

than 
Water, 
sp. gr. 


Liquids 
Lighter 

than 
Water, 
sp. gr. 





1.000 
1.007 
1.013 
1.020 
1.027 
1.034 
1.041 
1.048 
1.056 
1.063 
1.070 
1.078 
1.086 
1.094 
1.101 
1.109 
1.118 
1.126 
1.134 




19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 


1.143 
1.152 
1.160 
1.169 
1.178 
1.188 
1.197 
1.206 
1.216 
1.226 
1.236 
1.246 
1.256 
1.267 
1.277 
1.288 
1.299 
1.310 
1.322 


942 
936 
930 
924 
918 
913 
907 
901 
896 
890 
885 
880 
874 
869 
864 
859 
854 
849 
844 


38 
39 
40 
41 
42 
44 
46 
48 
50 
52 
54 
56 
58 
60 
65 
70 
75 
76 


1.333 
1.345 
1.357 
1.369 
1.382 
1.407 
1.434 
1.462 
1.490 
1.520 
1.551 
1.583 
1.617 
1.652 
1.747 
1.854 
1.974 
2.000 


.839 


1 




.834 
.830 


3 

4 




.825 
.820 


5 




.811 


6 

7 




.802 
.794 


8 




.785 


9 

10 
11 
12 
13 
14 
15 


'"l'.bbb" 

.993 
.986 
.980 
.973 
.967 
.960 
.954 
.948 


;768 
.760 
.753 
.745 


16 




17 




18 





Specific Gravity and "Weight of Wood, 



Alder 

Apple 

Ash 

Bamboo.. . 

Beech 

Birch 

Box, 

Cedar 

Cherry 

Chestnut . . . 

Cork 

Cypress 

Dogwood . . . 

Ebony 

Elm .. 

Fir. 



Gum 

Hackmatack 
Hemlock . . 

Hickory 

Holly 



.60 to .1 

.31 to .< 

.62 to .1 

.56 to .' 
.91tol.: 

.49 to .' 

.61 to .' 

.46 to J 
.24 



Avge. 
).80 .68 
.76 



.13tol 
1.55 to 
.48 to 
.84 to 1 
.59 
.36 to 
.69 to , 



.65 
1.12 
.62 
.66 
.56 
.24 
.53 
.76 
1.23 
.61 



Weight 



Hornbeam. . . 

Juniper 

Larch....' 

Lignum vitse 

Linden 

Locust 

Mahogany. .. 

Maple 

Mulberry — 
Oak, Live 

" White.. 

" Red.... 
Pine, White . 

" Yellow. 

Poplar 

Spruce 

Sycamore 

Teak 

Walnut 

Willow 



Avge, 

.76 

.56 

.56 

3 1.00 



.76 

.56 

.56 

.65 to 1. 

.604 



.56 to 1.06 .81 

.57 to .79 .68 

.56 to .90 .73 

.96 to 1.26 1.11 

.69 to .86 .77 

.73 to .75 .74 

.35 to .55 .45 

.46 to .76 .61 

.38 to .58 .48 

.40 to .50 

.59 to .62 .60 

.66 to .98 .82 

.50 to .67 .58 

.49 to .59 



166 



MATERIALS. 



Weight and Specific Gravity of Stones, Brick, 
Cement, etc. 





Pounds per 
Cubic Foot. 


Specific 
Gravity. 


Asphaltum 

Brick, Soft 


87 
100 
112 
125 
135 

140 to 150 
100 
112 

60 

78 

120 to 150 
120 to 140 

72 to 80 

90 to 110 
250 

156 to 172 
180 to 196 
160 to 170 
100 to 120 
130 to 150 
200 to 220 

50 to 55 
170 to 200 
150 

160 to 180 
140 to 160 
140 to 180 

90 to 100 

72 

74 to 80 
165 

90 to 110 
140 to 150 
170 to 180 
135 to 200 
170 to 200 
110 to 120 
166 to 175 


1.39 
1.6 


Common 

Hard 


1.79 
2.0 


" Pressed 

" Fire 


2.16 

2.24 to 2.4 




1.6 




1.79 




.96 




1.25 


Clay 


1.92 to 2.4 
1.92 to 2.24 




1.15 to 1.28 


rammed 

Emery 


1.44 to 1.76 

4. 

2.5 to 2.75 


" flint 

Gneiss 1 


2.88 to 3.14 
2.56 to 2.72 


Granite j 


1.6 to 1.92 


Gypsum 


2.08 to 2.4 
3.2 to 3.52 


Lime, quick, in bulk 


.8 to .88 
2.72 to 3.2 




2.4 




2.56 to 2.88 




2.24 to 2.56 




2.24 to 2.88 


Mortar 

Pitch 


1.44 to 1.6 

1.15 

1.18 to 1.28 




2.64 


Sand 


1.44 to 1.76 




2.24 to 2.4 


Slate 


2.72 to 2.88 
2.16 to 3.4 


Trap 


2.72 to 3.4 


Tile 


1.76 to 1.92 
2.65 to 2.8 







PROPERTIES OF THE USEFUL. METALS. 

Aluminum, Al.— Atomic weight 27.1. Specific gravity 2.6 to 2.7. 
The lightest of all the useful metals except magnesium. A soft, ductile, 
malleable metal, of a white color, approaching silver, but with a bluish cast. 
Very non-corrosive. Tenacity about one third that of wrought-iron. For- 
merly a rare metal, but since 1890 its production and use have greatly in- 
creased on account of the discovery of cheap processes for reducing it from 
the ore. Melts at about 1160° F. For further description see Aluminum, 
under Strength of Materials. 

Antimony (Stibium), Sb.— At. wt. 120. Sp. gr. 6.7 to 6.8. A brittle 
metal of a bluish-white color and highly crystalline or laminated structure. 
Melts at 842° F. Heated in the open air it burns with a bluish-white flame. 
Its chief use is for the manufacture of certain alloys, as type-metal (anti- 
mony 1, lead 4), britannia (antimony 1, tin 9), and various anti-friction 
metals (see Alloys). Cubical expansion by heat from 32° to 212° F., 0.0070. 
Specific heat .050. 

Bismuth, Bi.— At. wt. 20S. Bismuth is of a peculiar light reddish 
color, highly crystalline, and so brittle that it can readily be pulverized. It 
melts at 510° F., and boils at about 2300° F. Sp. gr. 9.823 at 54° F., and 
10.055 just above the melting-point. Specific heat about .0301 at ordinary 



PROPERTIES OF THE USEFUL METALS. 167 

temperatures. Coefficient of cubical expansion from 32° to 212° , 0.0040. Con- 
ductivity for heat about 1/56 and for electricity only about 1/80 of that of 
silver. Its tensile strength is about 6400 lbs. per square inch. Bismuth ex- 
pands in cooling, and Tribe has shown that this expansion does not take 
place until after solidification. Bismuth is the most diamagnetic element 
known, a sphere of it being repelled by a magnet; and on account of its 
marked thermo-electric properties it is much used in laboratories in the 
construction of delicate thermopiles. 

In the arts bismuth is used chiefly in the preparation of alloys. 

Cadmium, Cd.— At. wt. 112. Sp. gr. 8.6 to 8.7. A bluish-white metal, 
lustrous, with a fibrous fracture. Melts below 500° F. and volatilizes at 
about 680° F. It is used as an ingredient in some fusible alloys with lead, 
tin, and bismuth. Cubical expansion from 32° to 212° F., 0.0094. 

Copper, Cu.— At. wt. 63.2. Sp. gr. 8.81 to 8.95. Fuses at about 1930° 
F. Distinguished from all other metals by its reddish color. Very ductile 
and malleable, and its tenacity is next to iron. Tensile strength 20.000 to 
30,000 lbs. per square inch. Heat conductivity 73.6$ of that of silver, and su- 
perior to that of other metals. Electric conductivity equal to that of gold 
and silver. Expansion by heat from 32° to 212° F., 0.0051 of its volume. 
Specific heat .093. (See Copper under Strength of Materials; also Alloys.) 

Gold (Aurum), An..— At. wt. 197. Sp. gr., when pure and pressed in a 
die, 19.34. Melts at about 1915° F. The most malleable and ductile of all 
metals. One ounce Troy may be beaten so as to cover 160 sq. ft. of surface. 
The average thickness of gold-leaf is 1/282000 of an inch, or 100 sq. ft. per 
ounce. One grain may be drawn into a wire 500 ft. in length. The ductil- 
ity is destroyed by the presence of 1/2000 part of lead, bismuth, or antimony. 
Gold is hardened by the addition of silver or of copper. In U. S. gold coin 
there are 90 parts gold and 10 parts of alloy, which is chiefly copper with a 
little silver. By jewelers the fineness of gold is expressed in carats, pure 
gold being 24 carats, three fourths fine 18 carats, etc. 

Iridium.— Iridium is one of the rarer metals. It has a white lustre, re- 
sembling that of steel; its hardness is about equal to that of the ruby; in 
the cold it is quite brittle, but at a white heat it is somewhat malleable. It 
is one of the heaviest of metals, having a specific gravity of 22.38. When 
heated in the air to a red heat the metal is very slowly oxidized. It is insol- 
uble in all single acids, but is very slightly soluble in aqua regia after being 
heated in the state of fine powder for many hours. In a massive state, how- 
ever, aqua regia does not attack it. 

Iridium is extremely infusible. With the heat of the oxyhydrogen or 
electric furnaces, a globule of very small size may be melted. Mr. John 
Holland found that by heating the ore in a Hessian crucible to a white heat 
and adding to it phosphorus, and continuing the heating for a few minutes, 
he could obtain a perfect fusion of the metal, which could be poured out 
and cast into almost any desired shape. This material was about as hard 
as the natural grains of iridium, and contained, according co two determina- 
tions, 7.52$ and 7.74$ of phosphorus. By heating the metal in a bed of lime 
the phosphorus could be completely removed. In this operation the metal 
is first heated in an ordinary furnace at a white heat, and finally, after no 
more phosphorus makes it's appearance, it is removed and placed in an 
electric furnace with a lime crucible, and there heated until the last traces 
of phosphorus are removed; the metal which then remains will resist as 
much heat without fusion as the native metal. 

For uses of iridium, methods of manufacturing it, etc., see paper by W. D. 
Dudley on the "Iridium Industry," Trans. A. I. M. E. 1884. 

Iron (Ferrum), Fe.— At. wt. 56. Sp. gr.: Cast, 6.85 to 7.48; Wrought, 
7.4 to 7.9. Pure iron is extremely infusible, its melting point being above 
3000° F.,but its fusibility increases with the addition of carbon, cast iron fus- 
ing about 2500° F. Conductivity for heat 11.9, and for electricity 12 to 14.8, 
silver being 100. Expansion in bulk by heat: cast iron .0033, and wrought iron 
.0035, from 32° to 212° F. Specific heat: cast iron .1298, wrought iron .1138, 
steel .1165. Cast iron exposed to continued heat becomes permanently ex- 
panded 1)4, to -3 per cent of its length. Grate-bars should therefore be 
allowed about 4 per cent play. (For other properties see Iron and Steel 
under Strength of Materials.) 

Lead (Plumbum), Pb.— At. wt. 206.4. Sp. gr. 11.07 to 11.44 by different 
authorities. Melts at about 625° F., softens and becomes pasty at about 
617° F. If broken by a sudden blow when just below the melting-point it is 
quite brittle and the fracture appears crystalline. Lead is very malleable 
and ductile, but its tenacity js such that it can be drawn into wire with great 



168 MATERIALS. 

difficulty. Tensile strength, 1600 to 2400 lbs. per square inch. Its elasticity is 
very low, and the metal flows under very slight strain. Lead dissolves to 
some extent in pure water, but water containing- carbonates or sulphates 
forms over it a film of insoluble salt which prevents further action. (For 
alloys of lead see Alloys.) 

Magnesium, Mg<- At. wt. 24. Sp. gr. 1.69 to 1.75. Silver-white, 
brilliant, malleable, and ductile. It is one of the lightest of metals, weighing 
only about two thirds as much as aluminum. In the form of filings, wire, 
or thin ribbons it is highly combustible, burning with a light of dazzling 
brilliancy, useful for signal-lights and for flash-lights for photographers. It 
is nearly non-corrosive, a thin film of carbonate of magnesia forming on ex- 
posure to damp air, which protects it from further corrosion. It may be 
alloyed with aluminum, 5 per cent Mg added to Al giving about as much in- 
crease of strength and hardness as 10 per cent of copper. Cubical expansion 
by heat 0.0083, from 32° to 212° F. Melts at 1200° F. Specific heat .25. 

Manganese, Mn.-At. wt. 55. Sp. gr. 7 to 8. The pure metal is not 
used in tne arts, but alloys of manganese and iron, called spiegeleisen when 
containing below 25 per cent of manganese, and ferro-manganese when con- 
taining from 25 to 90 per cent, are used in the manufacture of steel. Metallic 
manganese oxidizes rapidly in the air, and its function in steel manufacture 
is to remove the oxygen from the bath of steel whether it exists as oxide of 
iron or as occluded gas. 

Mercury (Hydrargyrum), Hg,- At. wt. 199.8. A silver- white metal, 
liquid at temperatures above— 39° F., and boils at 680° F. Unchangeable as 
gold, silver, and platinum in the atmosphere at ordinary temperatures, but 
oxidizes to the red oxide when near its boiling-point. Sp.gr.: when liquid 
13.58 to 13.59, when frozen 14.4 to 14.5. Easily tarnished by sulphur fumes, 
also by dust, from which it may be freed by straining through a cloth. No 
metal except iron or platinum should be allowed to touch mercury. The 
smallest portions of tin, lead, zinc, and even copper to a less extent, cause it 
to tarnish and lose its perfect liquidity. Coefficient of cubical expansion 
from 32° to 212° F. .0182; per deg. .000101. 

Nickel, Ni.— At. wt. 58.3. Sp. gr. 8.27 to 8.93. A silvery-white metal 
with a strong lustre, not tarnishing on exposure to the air. Ductile, hard, 
and as tenacious as iron. It is attracted to the magnet and may be made 
magnetic like iron. Nickel is very difficult of fusion, melting at about 
3000° F. Chiefly used in alloys with copper, as german-silver, nickel-silver, 
etc., and recently in the manufacture of steel to increase its hardness and 
strength, also for nickel-plating. Cubical expansion from 32° to 212° F., 
0.0038. Specific heat .109. 

Platinum, Pt.— At. wt. 195. A whitish steel-gray metal, malleable, 
very ductile, and as unalterable by ordinary agencies as gold. When fused 
and refined it is as soft as copper. Sp. gr. 21.15. It is fusible only by the 
oxyhydrogen blowpipe or in strong electric currents. When combined with 
iridium it forms an alloy of great hardness, which has been used for gun- 
vents and for standard weights and measures. The most important uses of 
platinum in the arts are for vessels for chemical laboratories and manufac- 
tories, and" for the connecting wires in incandescent electric lamps. Cubical 
expansion from 32° to 212° F., 0.0027, less than that of any other metal ex- 
cept the rare metals, and almost the same as glass. 

Silver (Argentum), Ag.- At. wt. 107.7. Sp. gr. 10.1 to 11.1, according to 
condition and purity. It is the whitest of the metals, very malleable and 
ductile, and in hardness intermediate between gold and copper. Melts at 
about 1750° F. Specific heat .056. Cubical expansion from 32° to 212° F., 
0.0058. As a conductor of electricity it is equal to copper. As a conductor 
of heat it is superior to all other metals. 

Tin (Stannum) Sn.— At. wt. 118. Sp. gr. 7.293. White, lustrous, soft, 
malleable, of little strength, tenacity about 3500 lbs. per square inch. Fuses 
at 442° F. Not sensibly volatile when melted at ordinary heats. Heat con- 
ductivity 14.5, electric conductivitj r 12.4; silver being 100 in each case. 
Expansion of volume by heat .0069 from 32° to 212° F. Specific heat .055. Its 
chief uses are for coating of sheet-iron (called tin plate) aDd for making 
alloys with copper and other metals. 

Zinc, Zn,- At. wt. 65. Sp. gr. 7.14. Melts at 780° F. Volatilizes and 
burns in the air when melted, with bluish-white fumes of zinc oxide. It is 
ductile and malleable, but to a much less extent than copper, and its tenacity, 
about 5000 to 6000 lbs. per square inch, is about one tenth that of wrought 
iron. It is practically non-corrosive in the atmosphere, a thin film of car- 
bonate of zinc forming upon it. Cubical expansion between 32° and 212° F., 



MEASURES AND WEIGHTS OF VARIOUS MATERIALS. 169 



0.0088. Specific heat .096. Electric conductivity 29, heat conductivity 36, 
silver being 100. Its principal uses are for coating iron surfaces, called 
" galvanizing," and for making brass and other alloys. 

Table Showing the Order of 



Malleability. 


Ductility. 


Tenacity. 


Infusibility 


Gold 


Platinum 


Iron 


Platinum 


Silver 


Silver 


Copper 


Iron 


Aluminum 


Iron 


Aluminum 


Copper 
Gold 


Copper 


Copper 
Gold 


Platinum 


Tin 


Silver 


Silver 


Lead 


Aluminum 


Zinc 


Aluminum 


Zinc 


Zinc 


Gold 


Zinc 


Platinum 


Tin 


Tin 


Lead 


Iron 


Lead 


Lead 


Tin 



FORMULA AND TABLE FOR CALCULATING THE 
WEIGHT OF RODS, BARS, PLATES, TUBES, AND 
SPHERES OF DIFFERENT MATERIALS. 

Notation : b = breadth, t = thickness, s = side of square, d = external 
diameter, d 1 = internal diameter, all in inches. 

Sectional areas : of square bars — s 2 ; of flat bars = bt ; of round rods = 
.7854d 2 ; f tubes = .7854(d 2 - dj 2 ) = 3.1416(d*- * 2 ). 

Volume of 1 foot in length : of square bars = 12s 2 ; of flat bars = 126* ; of 
round bars = 9.4248d 2 ; of tubes = 9.4248(d 2 - d 3 2 ) = 37.6992(d* - * 2 ), in cubic 
inches. 

Weight per foot length = volume x weight per cubic inch of the material. 



Weight of a sphere 


= diam. 3 X .5236 X weight per cubic incl 






Material. 


1 
1 


3 


If J 


Us 

°ftfl 




t per cubic 
, lbs. 

re Weights, 
ught Iron 


■d 

n 
gf . 

285 






1 

m 


0, O 


Us C 


fit 










Cast iron 


7.218 


450. 


37.5 


3^s 2 


3%bt 


.260415-16 


2.454d 2 


.1363d 3 


Wrought Iron. . . . 


7.7 


480. 


40. 


3& 2 


3Hbt 


.2779|1. 


2.618d 2 


.1455d s 


Steel 


7.854 
8.855 


489.6 
552. 


40.8 
46. 


3.4s 2 
3.833s 2 


SAbt 
3.8336* 


.2833 1.02 
.31951.15 


2.670d 2 
3. Olid 2 


.1484d 3 


Copper & Bronze | 
(copper and tin) j 


.1673d 3 


** rass \ 35 Zinc 

Lead 


8.393 


523.2 


43.6 


3.633s 2 


3.6336* 


.30291.09 


2.854d 2 


.1586d 3 


11.38 


709.6 


59.1 


4.93s 2 


4.936* 


.4106 1.48 


3.870d 2 


.2150d s 


Aluminum 


2.67 


166.5 


13.9 


1.16s 2 


1.166* 


.09630.347 


0.908d 2 


.0504d 3 


Glass 


2.62 


163.4 


13.6 


1.13s 2 


1.136t 


.09450.34 


0.891d 2 


.0495d 3 


Pine Wood, dry . . . 


0.481 


30.0 


2.5 


0.21s 2 


0.216* 


.01741-16 


0.164d 2 


.0091d 3 


For tubes use the coefficient of d 2 in ninth column, as for rods, and 


multiply it into (d 2 — dj 2 ); or take four times this coefficient and multiply it 


into (dt - * 2 ). 


For hollow spheres use the coefficient of d 3 in the last column and 


multiply ic into (d 3 — d l 3 ). 


MEASURES AND WEIGHTS OF VARIOUS 


MATERIALS (APPROXIMATE). 


Brickwork.— Brickwork is estimated by the thousand, and for various 


thicknesses of wall runs as follows: 


8J4-in. wall, or 1 brick in thickness, 14 bricks per superficial foot. 

mi " " V \\i " " " 21 " 


17 " " " 2 " " " 28 " 
21^ « «-« «« 2% " " " 35 " 


An ordinary brick measures about 814. X 4 X 2 inches, which is equal to 60 


cubic inches, or 26 


2 bricl 


ts to < 


i cub 


c foot. 


The a 


perage wei 


?ht is 4} 


^lbs. 



m 



MATERIALS. 



Fuel.— A bushel of bituminous coal weighs 76 pounds and contains 2688 
cubic inches = 1.554 cubic feet. 29.47 bushels = 1 gross ton. 

A bushel of coke weighs 40 lbs. (35 to 42 lbs.). 

One acre of bituminous coal contains 1600 tons of 2240 lbs. per foot of 
thickness of coal worked. 15 to 25 per cent must be deducted for waste in 
mining. 

44.8 cubic feet bituminous coal when broken down = 1 ton, 2240 lbs. 

42.3 " " anthracite " " " " - 1 ton, 2240 lbs. 

123 " " of charcoal = 1 ton, 2240 lbs. 

70.9 " " "coke ... = 1 ton, 2240 lbs. 

cubic foot of anthracite coal =50 to 55 lbs. 

" " " bituminous" = 45 to 55 lbs. 

" " Cumberland coal = 53 lbs. 

" " Cannel coal = 50.3 lbs. 

" " charcoal (hardwood) = 18.5 lbs. 

(pine) =18 lbs. 

A bushel of charcoal.— In 1881 the American Charcoal-Iron Work- 
ers' Association adopted for use in its official publications for the standard 
bushel of charcoal 2748 cubic inches, or 20 pounds. A ton of charcoal is to 
be taken at 2000 pounds. This figure of 20 pounds to the bushel was taken 
as a fair average of different bushels used throughout the country, and it 
has since been established by law in some States. 

Ores, Earths, etc. 

13 cubic feet of ordinary gold or silver ore, in mine =1 ton = 2000 lbs. 

20 " " " broken quartz ,..= 1 ton = 2000 lbs. 

18 feet of gravel in bank =1 ton. 

27 cubic feet of gravel when dry,... , =1 ton. 

25 " " " sand = 1 ton. 

18 " " " earth in bank = 1 ton. 

27 " " " " when dry .= 1 ton. 

17 " " " clay = 1 ton. 

Cement.— English Portland, sp. gr. 1.25 to 1.51, per bbl 400 to 430 lbs. 

Rosendale, U. S., a struck bushel 62 to 70 lbs. 

Liime.— A struck bushel 72 to 75 lbs. 

Grain.— A struck bushel of wheat = 60 lbs.; of corn = 56 lbs. ; of oats = 
30 lbs. 

Salt.— A struck bushel of salt, coarse, Syracuse, N. Y. = 56 lbs.; Turk's 
Island = 76 to 80 lbs. 

Weight of Earth Filling. 
(From Howe's " Retaining Walls.") 

Average weight in 
lbs. per cubic foot. 

Earth, common loam, loose 72 to 80 

" shaken 82 to 92 

" " " rammed moderately 90 to 100 

Gravel 90 to 106 

Sand 90 to 106 

Soft flowing mud 104 to 120 

Sand, perfectly wet 118 to 129 





COMMERCIAL SIZES OF IRON BARS. 




Flats. 






Width. 


Thickness. 


Width. 


Thickness. 


Width. 


Thickness. 


% 


%to % 


m 


^tol^ 


4 


Mto2 


Vsto % 


2 


% to m 
V A to\% 

Htoiy 8 


Qi 


M to 2 
Mto2 


1 


Ys to 15/16 


m 


5 


m 


fctol 


2% 


5^ 


^to2 


i% 


^tol^ 


2% 


3/16 to W* 


6 


Mto2 


Mtol^ 


y> to iy 8 
H to 1^ 


w* 


J4to2 




y 6 to\y 4 


m 


7 


14 to 2 


i% 


^tolj^ 


3 


^to2 


Wz 


^to2 


m 


3/16 to V/% 


Wz 


14 to 2 







WEIGHTS OF WROUGHT IRON BARS. 



171 



Rounds : M to 1% inches, advancing by 16ths, and 1% to 5 inches by 
8ths. 

Squares : 5/16 to 1J4 inches, advancing by 16ths, and V/± to 3 inches by 
8th s. 

Half rounds: 7/16, J^, %, 11/16, %, 1, \%, 1*4, 1)4, 1%, 3 inches. 

Hexagons : ' 6 Ato\y% inches, advancing by 8ths. 

Ovals : % X U, % X 5/16, % X %, % X 7/16 inch. 

Half ovals : ^ X H, % X 5/33, % x 3/16, % x 7/32, \y a X ^, 1% X %, 
1% X % inch. 

Round-edge flats : 1% X ^, 1M X %, W& X % inch. 

Rands : >^ to \% inches, advancing by 8ths, 7 to 16 B. W. gauge. 

134 to 5 inches, advancing by 4ths, 7 to 16 gauge up to 3 inches, 4 to 14 
gauge, 3J4 to 5 inches. 

WEIGHTS OF SQUARE AND ROUND RARS OF 
WROUUHT IRON IN POUNDS PER LINEAL FOOT. 



Iron weighing 480 lbs. per cubic foot. 


For steel add £ 


per cent. 




htof 
are Bar 
Foot 

g- 


htof 
nd Bar 
Foot 


. . 

M © <Q 

gf-3 


htof 
are Bar 
Foot 
g- 


htof 
nd Bar 
Foot 


. 
111 


htof 
are Bar 
Foot 


htof 
nd Bar 
Foot 

g 




M3«S 


6J33 <D C 




ocs © a 


M30 = 




M3 o> a 


&C3 <D C 


3 5. a 

Eh 


"3 o* fl 


|£o3 


g 5 - 3 


'53 cf = 


|SS3 




'53 <y 5. 9 


■r a 









11/16 


24,08 


18.91 


% 


96.30 


75.64 


1/16 


.013 


.010 


M 


25.21 


19.80 


7/16 


98.55 


77.40 


% 


.053 


.041 


13/16 


26.37 


20.71 


H 


100.8 


79.19 


3/16 


.117 


.092 


% 


27.55 


21.64 


9/16 


103.1 


81.00 


M 


.208 


.164 


15/16 


28.76 


22.59 


% 


105.5 


82.83 


5/16 


.336 


.256 


3 


30.00 


23.56 


11/16 


107.8 


84.69 


% 


.469 


.368 


1/16 


31.26 


24.55 


M 


110.3 


86.56 


7/16 


.638 


.501 


Vs 


32.55 


25.57 


13/16 


113.6 


88.45 


M 


.833 


.654 


3/16 


33.87 


26.60 


Vs 


115.1 


90.36 


9/16 


1.055 


.828 


H 


35.21 


27.65 


15/16 


117.5 


92.29 


Vs. 


1.303 


1.023 


5/16 


36.58 


'28.73 


6 


130.0 


94.25 


11/16 


1.576 


1.237 


% 


37.97 


29.82 


n 


135.1 


98.22 


M 


1.875 


1.473 


7/16 


39.39 


30.94 


130.2 


102.3 


13/16 


2.201 


1.728 


Y2 


40.83 


32.07 


% 


135.5 


106.4 


7 A 


2.552 


2.004 


9/16 


42.30 


33.23 


a 


140.8 


110.6 


15/16 


2.930 


2.301 


% 


43.80 


34.40 


% 


146.3 


114.9 


1 


3.333 


2.618 


11/16 


45.33 


35.60 


Ji 


151.9 


119.3 


1/16 


3.763 


2.955 


3 A 


46.88 


36.82 


157.6 


123.7 


y 8 


4.219 


3.313 


13/16 


48.45 


38.05 




163.3 


128.3 


3/16 


4.701 


3.692 


% 


50.05 


39.31 


M 


169.3 


132.9 


M 


5.208 


4.091 


15/16 


51.68 


40.59 


% 


175.2 


137.6 


5/16 


5.742 


4.510 


4 


53.33 


41.89 


181.3 


142.4 


% 


6.302 


4.950 


•1/16 


55.01 


43.21 


y* 


187.5 


147.3 


7/16 


6.888 


5.410 


y& 


56.72 


44.55 


if 


193.8 


152.2 


% 


7.500 


5.890 


3/16 


58.45 


45.91 


200.2 


157.2 


9/16 


8.138 


6.392 


H 


60.21 


47.39 


% 


206.7 


162.4 


% 


8.802 


6.913 


5/16 


61.99 


48.69 


8 


213.3 


167.6 


11/16 


9.492 


7.455 


% 


63.80 


50.11 


M 


226.9 


178.2 


u 


10.21 


8.018 


7/16 


65.64 


51.55 


t 


240.8 


189.2 


13/16 


10.95 


8.601 


H 


67.50 


53.01 


255.2 


200.4 


% 


11.72 


9.204 


9/16 


69.39 


54.50 


9 


270.0 


213.1 


15/16 


12.51 


9.828 


H 


71.30 


56.00 


M 


285.2 


224.0 


2 


13.33 


10.47 


11/16 


73.24 


57.53 


8 


300.8 


236.3 


1/16 


14.18 


11.14 


k 


75.21 


59.07 


316.9 


248.9 


% 


15.05 


11.82 


18/16 


77.30 


60.63 


10 


333.3 


361.8 


3/16 


15.95 


12.53 


7 A 


79.32 


63.32 


M 


350.2 


275.1 


M 


16.88 


13.25 


15/16 


81.26 


63.83 


K 


367.5 


288.6 


5/16 


17.83 


14.00 


5 


83.33 


65.45 


% 


385.2 


302.5 


% 


18.80 


14.77 


1/16 


85.43 


67.10 


11 


403.3 


316.8 


7/16 


19.80 


15.55 


Vs 


87.55 


68.76 


M 


421.9 


331.3 


^ 


20.83 


16.36 


3/16 


89.70 


70.45 


y% 


440.8 


346.2 


9/16 


21.89 


17.19 


H 


91.88 


73.16 


u 


460.2 


361.4 


K 


22.97 


18.04 


5/16 


94.08 


73.89 


12 


480. 


377. 



172 



MATERIALS. 





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WEIGHTS OF FLAT WROUGHT IRON. 



173 






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174 



MATERIALS. 



WEIGHT OF IRON AND STEEL SHEETS. 
Weights per Square Foot. 

(For weights by new U. S. Standard Gauge, see page 31.) 



Thickness by Birmingham Gauge. 


Thickness by American (Brown and 
Sharpe's) Gauge. 




Thick- 








Thick- 






No. of 


ness in 


Iron. 


Steel. 


No. of 


ness in 


Iron. 


Steel. 


Gauge. 


Inches. 






Gauge. 


Inches. 






0000 


.454 


18.16 


18.52 


0000 


.46 


18.40 


18.77 


000 


.425 


17.00 


17.34 


000 


.4096 


16.38 


16.71 


00 


.38 


15.20 


15.30 


00 


.3648 


14.59 


14.88 





.34 


13.60 


13.87 





.3249 


13.00 


13.26 


1 


.3 


12.00 


12.24 


1 


.2893 


11.57 


11.80 


2 


.284 


11.36 


11.59 


2 


.2576 


10.30 


10.51 


3 


.259 


10.36 


10.57 


3 


.2294 


9.18 


9.36 


4 


.238 


9.52 


9.71 


4 


.2043 


8.17 


8.34 


5 


.22 


8.80 


8.98 


5 


.1819 


7.28 


7.42 


6 


.203 


8.12 


8.28 


6 


.1620 


6.48 


6.61 


7 


.18 


7.20 


7.34 


7 


.1443 


5.77 


5.89 


8 


.165 


6.60 


6.73 


8 


.1285 


5.14 


5.24 


9 


.148 


5.92 


6.04 


9 


.1144 


4.58 


4.67 


10 


.134 


5.36 


5.47 


10 


.1019 


4.08 


4.16 


11 


.12 


4.80 


4.90 


11 


.0907 


3.63 


3.70 


12 


.109 


4.36 


4.45 


12 


.0808 


3.23 


3.30 


13 


.095 


3.80 


3.88 


13 


.0720 


2.88 


2.94 


14 


.083 


3.32 


3.39 


14 


.0641 


2.56 


2.62 


15 


.072 


2.88 


2.94 


15 


.0571 


2.28 


2.33 


16 


.065 


2.60 


2.65 


16 


.0508 


2.03 


2.07 


17 


.058 


2.32 


2.37 


17 


.0453 


1.81 


1.85 


18 


.049 


1.96 


2.00 


18 


.0403 


1.61 


1.64 


19 


.042 


1.68 


1.71 


19 


.0359 


1.44 


1.46 


20 


.035 


1.40 


1.43 


20 


.0320 


1.28 


1.31 


21 


.032 


1.28 


1.31 


21 


.0285 


1.14 


1.16 


22 


.028 


1.12 


1.14 


22 


.0253 


1.01 


1.03 


23 


.025 


1.00 


1.02 


23 


.0226 


.904 


.922 


24 


.022 


.88 


.898 


24 


.0201 


.804 


.820 


25 


.02 


.80 


.816 


25 


.0179 


.716 


.730 


26 


.018 


.72 


.734 


26 


.0159 


.636 


.649 


27 


.016 


.64 


.653 


27. 


.0142 


.568 


.579 


28 


.014 


.56 


.571 


28 


.0126 


.504 


.514 


29 


.013 


.52 


.530 


29 


.0113 


.452 


.461 


30 


.012 


.48 


.490 


30 


.0100 


.400 


.408 


31 


.01 


.40 


.408 


31 


.0089 


.356 


.363 


32 


.009 


.36 


.367 


32 


.0080 


.320 


.326 


33 


.008 


.32 


.326 


33 


.0071 


.284 


.290 


34 


.007 


.28 


.286 


34 


.0063 


.252 


.257 


35 


.005 


.20 


.204 


35 


.0056 


.224 


.228 



Iron. Steel. 

Specific gravity 7.7 7.854 

Weight per cubic foot 480. 489 . 6 

" " inch .2778 .2833 

As there are many gauges in use differing from each other, and even the 
thicknesses of a certain specified gauge, as the Birmingham, are not assumed 
the same by all manufacturers, orders for sheets and wires should always 
state the weight per square foot, or the thickness in thousandths of an inch. 



WEIGHT OF PLATE IROiN". 



175 





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T 


27.50 
29.79 
32.08 
34.38 
36.67 
38.96 
41.25 
43.54 
45.83 
48.13 
50.42 
52.71 
55.00 
57.29 
59.58 
61.88 
64.17 
66.46 
68.75 
73.33 
77.91 
82.50 
87.09 
91.67 
96.25 
100.8 
105.4 
110.0 
114.6 
119.2 
123.8 
128.3 
132.9 
137.5 




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CO 


17.50 
18.96 
20.42 
21.88 
23.33 
24.79 
26.25 
27.71 
29.17 
30.63 
32.08 
33.54 
35.00 
36.46 
37.92 
39.38 
40.83 
42.29 
43.75 
46.67 
49.58 
52.50 
55.42 
58.33 
61.25 
64.17 
67.08 
70.00 
72.91 
75.83 
78.75 
81.66 
84.58 
87.50 




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12.50 
13.54 
14.58 
15.63 
16.67 
17.71 
18.75 
19.79 
20.83 
21.88 
22.92 
23.96 
25.00 
26.04 
27.08 
28.13 
29.17 
30.21 
31.25 
33.33 
35.42 
37.50 
39.59 
41.67 
43.75 
45.84 
47.92 
50.00 
52.08 
54.17 
56.25 
58.33 
60.42 
62.50 




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176 



MATERIALS. 



WEIGHTS OF STEEL. BLOOMS. 

Soft steel. 1 cubic inch = 0.284 lb. 1 cubic foot = 490.75 lbs. 



Sizes. 


Lengths. 




























1" 


6" 


12" 


18" 


24" 


30" 


36" 


42" 


48" 


54" 


60" 


66" 


12" x 4" 


13.63 


82 


164 


245 


327 


409 


491 


573 


654 


736 


818 


900 


11 x 6 


18.75 


113 


225 


338 


450 


563 


675 


788 


900 


1013 


1125 


1238 


x 5 


15.62 


94 


188 


281 


375 


469 


562 


656 


750 


843 


937 


1031 


x 4 


12.50 


75 


150 


225 


300 


375 


450 


525 


600 


675 


750 


825 


10 x 7 


19.88 


120 


239 


358 


477 


596 


715 


835 


955 


1074 


1193 


1312 


x 6 


17.04 


102 


204 


307 


409 


511 


613 


716 


818 


920 


1022 


1125 


x 5 


14.20 


85 


170 


256 


341 


426 


511 


596 


682 


767 


852 


937 


x 4 


11.36 


68 


136 


205 


273 


341 


409 


477 


516 


614 


682 


750 


x 3 


8.52 


51 


102 


153 


204 


255 


306 


358 


409 


460 


511 


562 


9 x 7 


17.89 


107 


215 


322 


430 


537 


644 


751 


859 


966 


1073 


1181 


x 6 


15.34 


92 


184 


276 


368 


460 


552 


644 


736 


828 


920 


1012 


x 5 


12.78 


77 


153 


230 


307 


383 


460 


537 


614 


690 


767 


844 


x 4 


10.22 


61 


123 


184 


245 


307 


368 


429 


490 


552 


613 


674 


8x8 


18.18 


109 


218 


327 


436 


545 


655 


764 


873 


982 


1091 


1200 


x 7 


15.9 


95 


191 


286 


382 


477 


572 


668 


763 


859 


954 


1049 


x 6 


13.63 


82 


164 


245 


327 


409 


491 


573 


654 


736 


818 


900 


x 5 


11.36 


68 


136 


205 


273 


341 


409 


477 


546 


614 


682 


750 


x 4 


9.09 


55 


109 


164 


218 


273 


327 


382 


436 


491 


545 


600 


7x7 


13.92 


83 


167 


251 


334 


418 


501 


585 


668 


752 


835 


919 


x 6 


11.93 


72 


143 


215 


286 


358 


430 


501 


573 


644 


716 


788 


x 5 


9.94 


60 


119 


179 


238 


298 


358 


417 


477 


536 


596 


656 


x 4 


7.95 


48 


96 


143 


191 


239 


286 


334 


382 


429 


477 


525 


x 3 


5.96 


36 


72 


107 


143 


179 


214 


250 


286 


322 


358 


393 


6^x6^ 


12. 


72 


144 


216 


388 


360 


432 


504 


576 


648 


720 


792 


x 4 


7.38 


44 


89 


133 


177 


221 


266 


310 


354 


399 


443 


487 


6x6 


10.22 


61 


123 


184 


245 


307 


368 


429 


490 


551 


613 


674 


x 5 


8.52 


51 


102 


153 


204 


255 


307 


358 


409 


460 


511 


562 


x 4 


6.82 


41 


82 


123 


164 


204 


245 


286 


327 


368 


409 


450 


x 3 


5.11 


31 


61 


92 


123 


153 


184 


214 


245 


276 


307 


337 


5^x 5y 2 


8.59 


52 


103 


155 


206 


258 


309 


361 


412 


464 


515 


567 


x 4 


6.25 


37 


75 


112 


150 


188 


225 


262 


300 


337 


375 


412 


5x5 


7.10 


43 


85 


128 


170 


213 


256 


298 


341 


383 


426 


469 


x 4 


5.68 


34 


68 


102 


136 


170 


205 


239 


273 


307 


341 


375 


4y a x 4K> 


5.75 


35 


69 


104 


138 


173 


207 


242 


276 


311 


345 


380 


x 4 


5.11 


31 


61 


92 


123 


153 


184 


215 


246 


276 


307 


338 


4 x 4 


4.54 


27 


55 


82 


109 


136 


164 


191 


218 


246 


272 


300 


x 3^ 


3.97 


24 


48 


72 


96 


119 


143 


167 


181 


215 


238 


262 


x 3 


3.40 


20 


41 


61 


82 


102 


122 


143 


163 


184 


204 


224 


3^x zy Q 


3.48 


21 


42 


63 


84 


104 


125 


146 


167 


188 


209 


230 


x 3 


2.98 


18 


36 


54 


72 


89 


107 


125 


143 


161 


179 


197 


3x3 


2.56 


15 


31 


46 


61 


77 


92 


108 


123 


138 


154 


169 



SIZES AKD WEIGHTS OF STKtJCTlJRAL SHAPES. 177 



SIZES AND WEIGHTS OF STRUCTURAL. SHAPES. 

Minimum and Maximum "Weights and Dimensions oi 
Carnegie I-Beams. 

STEEL BEAMS. 



Section 




Weight per 
Foot, in lbs. 


Flange 


Width. 


Web Thickness. 


Increase of 

Web and 
Flanges for 




a£g 


















03 03.5 

fipq 


Min. 


Max. 


Min. 


Max. 


Min. 


Max. 


crease of 
weight. 


B 1 


24 


80.00 


100.00 


6,95 


7.20 


.50 


.75 


.0123 


B 2 


20 


80.00 


100.00 


7 00 


7.30 


.60 


.90 


.015 


B 3 


20 


64.00 


75.00 


6.25 


6.41 


.50 


.66 


.015 


B 4 


15 


80.00 


100.00 


^.41 


6.79 


.77 


1.16 


.020 


B 5 


15 


60.00 


75.00 


6.04 


6.34 


.54 


.84 


.020 


B 6 


15 


50.00 


59.00 


5.75 


5.93 


.45 


.63 


.020 


*B 7 


15 


41.00 


49.00 


5.50 


5.66 


.40 


.56 


.020 


B 8 


12 


40.00 


56.70 


5.50 


5.91 


.39 


.80 


.025 


*B 9 


12 


32.00 


39.00 


5.25 


5.42 


.35 


.52 


.025 


BIO 


10 


33.00 


40.00 


5.00 


5.21 


.37 


.58 


.029 


Bll 


10 


25.50 


32.00 


4.75 


4.94 


.32 


.51 


.029 


B12 


9 


27.00 


33.00 


4.75 


4.95 


.31 


.51 


.033 


B13 


9 


21.00 


26.00 


4.50 


4.66 


.27 


.43 


.033 


B14 


8 


22.00 


27.00 


4.50 


4.68 


.27 


.45 


.037 


B15 


8 


18.00 


21.70 


4.25 


4.39 


.25 


.39 


.037 


B16 


7 


20.00 


22.00 


4.25 


4.33 


.27 


.35 


.042 


B17 


7 


15.50 


19.00 


4.00 


4.15 


.23 


.38 


.042 


B18 


6 


16.00 


20.00 


3.63 


3.83 


.26 


.46 


.049 


B19 


6 


13.00 


15.00 


3.50 


3.60 


.23 


.34 


.049 


B20 


5 


13.00 


16.00 


3.13 


3.31 


.26 


.44 


.059 


B21 


5 


10.00 


12.00 


3.00 


3.12 


.22 


.33 


.059 


B22 


4 


10.00 


13.00 


2.75 


2.97 


.24 


.46 


.074 


B23 


4 


7.50 


9.00 


2.63 


2.74 


.20 


.31 


.074 


B24 


4 


6.00 


8.00 


2.18 


2.33 


.18 


.33 


.074 



Iron. Steel. 
Given weight in pounds per foot, to find sectional area-=- 3J^ 3.4 

" " " " " x 0.3 .2941 

Given sectional area, to find weight in lbs. per foot x Zy§ 3.4 

" " " " " "lbs. per yard x 10 10.2 



Maximum and Minimum Weights and Dimensions of 
Carnegie Deck Beams. 



Section 
Index. 


Depth 

of 
Beam, 
inches. 


Weight per 
Foot, lbs. 


Flange Width. 


Web 

Thickness. 


Increase of 
Web and 
Flanges per 
lb. in- 
crease of 
weight. 


Min. 


Max. 


Min. 


Max. 


Min. 


Max. 


B100 
B101 
B102 
B103 
B105 


10 
9 
8 

7 
6 


27.23 
26.52 
20.15 
18.10 
15.30 


35.70 
30.60 
24.48 
23.46 
18.36 


5.25 
4.94 
5.00 
4.87 
4.38 


5.50 
5.07 
5.16 
5.10 
4.53 


.38 
.44 
.31 
.31 

.28 


.63 
.57 
.47 
.54 
.43 


.029 
.032 
.037 
.042 
.049 



178 



MATERIALS. 



Weights and Dimensions of Carnegie Steel Channels. 












Increase 




Depth 


Weight per 
Foot, in lbs. 


Flange Width. 


Web 

Thickness. 


of Web 
and 


Sec- 
tion 


of 
Chan- 
nel, in 








Flanges 
for each 
lb. in- 


Index 
















inches. 


Min. 


Max. 


Min. 


Max. 


Min. 


Max. 


crease of 
weight. 


CI 


15 


32.00 


51 00 


3.40 


3.78 


.40 


.78 


.020 


C2 


12 


20.00 


30.25 


2.90 


3.15 


.30 


.55 


.025 


C3 


10 


15.25 


23.75 


2.66 


2.91 


.26 


.51 


.029 


C4 


9 


12.75 


20.50 


2.44 


2.69 


.24 


.49 


.033 


C5 


8 


10.00 


17.25 


2.20 


2.47 


.20 


.47 


.037 


C6 


7 


8.50 


14.50 


2.00 


2.25 


.20 


.45 


.042 


C7 


6 


7.00 


12.00 


1.89 


2.14 


.19 


.44 


.049 


C8 


5 


6.00 


10.25 


1.78 


2.03 


.18 


.43 


.059 


C9 


4 


5.00 


8.25 


1.67 


1.91 


.17 


.41 


.074 



Weights and Dimensions of Carnegie 'Z-Bars. 









Size. 




Weight. 


Section 


Thickness 
of Metal. 












Index. 


Flange. 


Web. 


Flange. 


Iron. 


Steel. 


Z 1 


Va 


3 y 2 


6 


3 H 


15.3 


15.6 


** 


7-16 


3 9-16 


6 1-16 


3 9-16 


18.0 


18.3 


" 


V2 


3 Va 


6 H 


3 % 


20.6 


21.0 


Z 2 


9-16 


3 \b 


6 


3 y* 


22.3 


22.7 


'• 


Va 


3 9-16 


6 1-16 


3 9-16 


24.9 


25.4 


" 


11-16 


3 Va 


6 % 


3 V 8 


27.5 


28.0 


Z 3 


% 


3 M 


6 


3 H 
3 9-16 


28.8 


29.3 


" 


13-16 


3 9-16 


6 1-16 


31.3 


32.0 


" 


Va 


3 Va 


6 % 


3 % 


33.9 


34.6 


Z 4 


5-16 


3 y A 


5 


3 M 


11.3 


11.6 


" 


Va 


3 5-16 


5 1-16 


3 5-16 


13.7 


13.9 


" 


7-16 


3 Va 


5 y 8 


3 % 


16.0 


16.4 


Z 5 


14 


3 Va 


5 


3 H 


17.5 


17.8 


44 


9-16 


3 5-16 


5 1-16 


3 5-16 


19.8 


20.2 


" 


Va 


3 % 


5 y 8 


in 


22.1 


22.6 


Z 6 


11-16 


3 X 


5 


23.2 


23.7 


" 


H 


3 5-16 


5 1-16 


3 5-16 


25.5 


26.0 


>t 


13-16 


3 % 


5 y a 


3 % 


27.8 


28.3 


Z 7 


H 


3 1-16 


4 


3 1-16 


8.0 


8.2 




5-1 6 


3 H 


4 1-16 


3 H 


10.1 


10.3 


" 


% 


3 3-16 


4 % 


3 3-16 


12.2 


12.4 


Z 8 


r-i6 


3 1-16 


4 


3 1-16 


13.5 


13.8 


** 


M 


3 % 


4 1-16 


3 H 


15.5 


15.8 


•* 


9-16 


3 3-16 


4 H 


3 3-16 


17.6 


17.9 


Z 9 


Va 


3 1-16 


4 


3 1-16 


18.5 


18.9 


" 


11-16 


3 X 


4 1-16 


3 y 8 


20.5 


20.9 


" 


H 


3 3-16 


4 14 


3 3-16 


22.5 


22.9 


Z10 


H 


2 11-16 


3 


2 11-16 


6.6 


6.7 


" 


5-16 


2 % 


3 1-16 


2 % 


8.3 


8.4 


Zll 


% 


2 11-16 


3 


2 11-16 


9.5 


9.7 


" 


7-16 


2 % 


3 1-16 


2 % 


11.2 


11.4 


Z12 


Y* 


2 11-16 


3 


2 11-16 


12.3 


12.5 




9-16 


2 % 


3 1-16 


2 % 


13.9 


14.2 



SIZES AND WEIGHTS OF STRUCTURAL SHAPES. 179 

Pencoyd Steel Angles. 

EVEN LEGS. 



t 


Size 

in 

Inches. 


Approximate Weight in Pounds per Foot for Various 
Thicknesses in Inches. 


o o 


% 


3-16 


M 


5-16 


.375 


7-16 


Vo 


9-16 


% 


11-16 


% 


% 


1 


'A 




.125 


.1875 


.25 


.3125 


.4375 


.50 


.5625 


.625 


.6875 


.75 


.875 


1.00 


120 


6 x6 










14.8 


17.3 


19.9 


22.3 


24.9 


26.5 


29.1 


34.2 


39.3 


121 


5 x5 












14.3 


16.4 


18.5 20.7 


22.8 




29 2 


33 4 


122 


4 x4 








8.2 


9.8 


11.3 


13.0 


14.6 


16.1 


177 


19 3 






123 


3^x3^ 








7.1 


8.6 


10.0 


11.4 


12.8 


14.2 










124 


3 x3 






4.9 


6.0 


7.1 


8.3 


9.4 


10.5 


11.6 










125 


'2% x 2?4 






4.5 


5.6 


6.7 


7.8 


8.9 














126 


2^x2^ 




3.1 


4.1 


5.1 


6.1 


7.1 
















127 


•2U*m 




2.7 


3.6 


4.5 


5.4 


















128 


2 x2 




2.44 


3.3 


4.1 


4.9 


















12SJ 


iMxiM 




2.14 


2.9 


3.6 


4.4 


















130 


l^xl^ 


1.16 


1.80 


2.4 


3.0 


3.6 


















131 


m*m 


1.02 


1.53 


2.04 






















132 


1 xl 


0.82 


1.16 


1.53 























UNEVEN LEGS. 



6 




Approximate Weight in Pounds per Foot for Various 


cc 


Size 


Thicknesses in Inches. 


'°.2 


Inches. 


H 


3-16 


H 


5-16 


% 


7-16 


% 


9-16 


% 


11-16 


H 


7,4 


1 


£ 




.125 


.1875 


.25 


.3125 


.375 


.43/5 


.50 


.5625 


.625 


.6875 


.75 


.875 


1.00 


154 


7 x3^ 














17.0 


18.9 


20.9 


22.8 


24.8 


28 6 


32.5 


152 


6^x4 










12.9 


15.0 


17.1 


19.3 


21.4 


23.6 


25,7 


30.0 


34.3 


140 


6 x4 










12.2 


14.4 


16.4 


18.6 


21 


22.8 


24.9 


29.1 


33.3 


151 


6 xSy 2 










11.5 


13.6 


15.6 


17.6 


19.7 


21,7 


23 8 


27 8 


31.9 


153 


5^x3V6 










11.0 


12.8 


14.6 


16.4 












141 


5 x4 










11.0 


12.8 


14.6 


16.4 


18 2 


20 


21 8 






142 


5 x3^ 








8.7 


10.3 


12.0 


13.6 


15.2 


16.8 


18 5 








143 


5 x3 








8.2 


9.7 


11.2 


12.8 


14.3 


15.8 


17 3 


18 9 






144 


4^x3 








7.7 


9.2 


10.6 


12.1 


13.6 


15.0 


16 5 


18 






145 


4 x3K 3 








7.7 


9.2 


10.6 


12.1 


13.6 


15.0 


16.5 


18 






146 


4 x3 








7.1 


S.6 


10.0 


11.4 


12 8 


14 2 










147 


3^x3 








6.6 


7.9 


9.2 


10.5 


11.8 


13.1 










150 


3^x2^ 






4.9 


6.0 


7.1 


8.3 


9.4 














159 


3^x2 






4.5 


5.6 


6.7 


















148 


3 x2^ 






4.5 


5.6 


6.7 


7.8 


8 9 














149 


3 x2 






4.1 


5.1 


6.1 


7.1 


8 2 














155 


2^x2 




2.7 


3.6 


4.5 


5.4 


6.3 


7,2 














156 


2^x1^ 




2.24 


a ! 


3.8 


4.6 


















157 


2 xlM 




1.94 


2.7 


3.3 


4.0 



















180 



MATERIALS. 



Pencoyd Tees. 



EVEN TEES. 



UNEVEN TEES. 







Weight per 






Weight per 
Foot. 




Size 


Foot. 




Size 


Chart 








Chart 








Number. 








Number. 


Inches. 










Iron. 


Steel. 






Iron. 


Steel. 


70 


4 x4 


12.40 


12.65 


107 


5 x4 


14.70 


15.00 


71 


&A x Wz 


10.17 


10.37 


106 


5 x3i^ 


16.13 


16.46 


72 


3 x3 


8.33 


8.50 


93 


5 x2 9-16 


11.03 


11 25 


82 


3 x3 


6.43 


6.56 


92 


5 x2^ 


10.23 


10.44 


83 




7.53 


7.68 


90 


4^x314 


14.83 


15.13 


84 


2^x2^ 
2^x2^ 


4.83 


4.93 


109 


4 x4^ 


13.23 


13.50 


73 


6.50 


6.63 


91 


4 x3^ 


13.93 


14.21 


74 


2^x2^ 


5.73 


5.85 


94 


4 x3 


8.63 


8 81 




2U * Wa 


3.90 


3.98 


95 


4 x3 


8.37 


8.53 


76 


2J4x2J4 


3.93 


4.01 


96 


4 x2 


6.43 


6.56 


77 


2 x2 


3.47 


3.54 


97 


3 x3^ 


9.37 


9.55 


78 




2.37 


2.41 


98 


3 x2^ 


7.93 


8.09 


79 


2.00 


2.04 


110 


3 x2^ 


5.87 


5.98 


80 


ty\*ty± 


1.50 


1.53 


111 


3 x2^ 


6.87 


7.00 


81 


1 xl 


1.03 


1.05 


117 


3 *2y z 


5.00 


5.10 


85 


4 x4 


10.98 


11.19 


99 


3 xlK 2 


3.73 


3.81 










105 


2^| xl% 
2^x114 


7.13 


7.28 










104 


6.53 


6.66 










100 


3.03 


3.09 










108 


2M x 9-16 


2.20 


2.24 










101 


2 xl^ 


2.90 


2.96 










112 


2 xl 1-16 


2.07 


2.11 










102 


2 xl 


2.33 


2.38 










103 


2 x 9-16 


2.03 


2.07 










116 


iMxiM 


3.47 


3.54 










113 


lMxl 1-16 


1.87 


1.90 










114 


l^x 15-16 


1.37 


1.39 










115 


lJ4x 15-16 


1.13 


1.16 










118 


3 x2i/ 2 


5.92 


6.03 










119 


m**% 


5.63 


5.74 







Pencoyd 


Car-Builders' 


Channels 


, Iron. 
















CG.GrQ 










« s 






1 


CO 


he a; 


<°s 


[m"" 


Approximate Weight in Pounds per 


*-P-i 


-§ 


£$ 


*S 


Foot for Each Thickness of 


3 


a 




Web, in Inches. 


T3 J § O 


£ 


a 


12 




g»| 








3^ 


PI 


Safe 




§ S.-^fc 


CD 


5-16 


¥a 


7-16 


k 


9-16 


% 




55 


13 


H 


29.5 


29.5 


32.2 


34.9 


37.6 


40.3 


.023 


54 


12 


3 


9-32 


22.4 


23.6 


26.1 


28.6 


31.1 


33.6 




.025 


33U 


10^ 

10^ 




7-16 


23.6 






23.6 


25.8 






.029 


33 


5-16 


17.6 


17.6 


19.8 










.029 







Pencoyd 


Car- 


Builders' 


Channels 


, Steel. 




55 
33 


13 
12 

10^ 
10^ 


We, 
3 

\Wh 

l2i/ 2 


7-16 
5-16 


30.1 

22.8 
24.1 
17.9 


24.1 
17.9 


30.1 
26.6 

20.2 


32.9 

29.2 
24.1 


35.6 
31.7 
26.3 


38.4 
34.3 


41.1 


.022 
.024 
.028 
.028 



SIZES AND WEIGHTS OF ROOFING MATERIALS. 181 



SIZES AND WEIGHTS OF ROOFING MATERIALS. 
Corrugated Iron (Phoenix Iron Co.). 



BLACK IRON. 


GALVANIZED IRON. 


Thick- 
ness in 
Indies. 


Weight 
in Lbs. 

per 
Sq. Ft., 
Flat. 


Weight 

iu Lbs. 

per 

Sq. Ft. on 

Roof. 

Flat. 


Weight 

in Lbs. 

per 

Sq. Ft., on 

Roof. 
Corrugated 


Weight 
in Lbs. 

per 

Sq. Ft., 

Flat. 


Weight 
in Lbs. 

per 

Sq. Ft. 

on Roof. 

Flat. 


Weight 

in Lbs. 

per 

Sq. Ft., on 

Roof. 
Corrugated 


0.065 
0.049 
0.035 
0.028 
0.022 
0.018 


2.61 
1.97 
1.40 
1.12 

0.88 
0.72 


3.03 
2.29 
1.63 
1.31 
1.03 
0.84 


3.37 
2.54 
1.82 
1.45 
1.14 
0.93 


3.00 
2.37 
1.75 
1.31 
1.06 
0.94 


3.50 
2.76 
2.03 
1.53 
1.24 
1.09 


3.88 
3.07 
2.26 
1.71 
1.37 
1.21 



The above table is calculated for the ordinary size of sheet, which is from 
2 to 2J/2 feet wide, and from 6 to 8 feet long, allowing 4 inches lap in length 
and 2)/2 inches iu width of sheet. 

The galvanizing of sheet iron adds about one- third of a pound to its weight 
per square foot. 

In corrugated iron made by the Keystone Bridge Co., the corrugations are 
2.42V long, measured on the straight line; they require a length of iron of 
2.725'' to make one corrugation, and the depth of corrugation is 21-32". 
One corrugation is allowed for lap in the width of the sheet and 6" in the 
length, for the usual pitch of roof of two to one. Sheets can be corrugated 
of any length not exceeding ten feet. The most advantageous width is 
30^j", which (allowing \Q' for irregularities) will make eleven corrugations 
= 30", or, making allowance for laps, will cover 24J4" of the surface of the 
roof. 

By actual trial it was found that corrugated iron No. 20, spanning 6 feet, 
will begin to give a permanent deflection for a load of 30 lbs. per square foot, 
and that it will collapse with a load of 60 lbs. per square foot. The distance 
between centres of purlins should therefore not exceed 6 feet, and, prefer- 
ably, be less than this. 

Terra-Cotta. 

Porous terra-cotta roofing 3" thick weighs 16 lbs. per square foot and 2" 
thick, 12 lbs. per square foot. 
Ceiling made of the same material 2" thick weighs 11 lbs. per square foot. 

Tiles. 

Flat tiles 6J4" X 10^" X %" weigh from 1480 to 1850 lbs. per square of 
roof, the lap being one-half the length of the tile. 

Tiles tvith grooves and fillets weigh from 740 to 925 lbs. per square of roof. 
Pan-tiles U}£" X 10^" laid 10" to the weather, weigh 850 lbs. per square. 

Tin. 

The usual sizes for roofing tin are 14" X 20" and 20" X 28". Without 
allowance for lap or waste, tin roofing weighs from 50 to 62 lbs. per square. 

Tin on the roof weighs from 62 to 75 lbs. per square. 

Roofing plates or terne plates (steel plates coated with an alloy of tin 
and lead) are made only in IC and IX thicknesses (27 and 29 Birmingham 
gauge). "Coke" and "charcoal" tin plates, old names used when iron 
made with coke and charcoal was used for the tinned plate, are still used in 
the trade, although steel plates have been substituted for iron ; a coke plate 
now commonly meaning one made of Bessemer steel, and a charcoal plate 
one of open-hearth steel. The thickness of the tin coating on the plates 
varies with different " brands. 11 

For valuable information on Tin Roofing, see circulars of Merchant & Co., 
Philadelphia. 



182 



MATERIALS. 



TIN PLATES. (TINNED SHEET STEEL.) 

Standard Stock Sizes, with Number of Sheets and Net 
Weight per Box. 



B. W. 


Thickness. 


Size. 


Sheets. 


Net 

Weight 

lbs. 


B. W. 


Thickness. 


Size. 


Sheets. 


Net 
Weight 


29 


IC 


10x14 


225 


108 


29 


IC 


10x20 


225 


160 


27 


IX 


lOx 14 


225 


135 


27 


IX 


10x20 


225 


195 


26 


IXX 


10x14 


225 


160 


26 


IXX 


10x20 


225 


222 


29 


IC 


12x12 


225 


110 


29 


IC 


11x22 


225 


190 


27 


IX 


12x12 


225 


138 


27 


IX 


11 x22 


225 


235 


26 


IXX 


12x12 


225 


165 


26 


IXX 


11x22 


225 


275 


29 


IC 


14x20 


112 


108 


29 


IC 


12 x 24 


112 


110 


27 


IX 


14x20 


112 


135 


27 


IX 


12x24 


112 


138 


26 


IXX 


14 x 20 


112 


160 


26 


IXX 


12x24 


112 


165 


25 


IXXX 


14x20 


112 


180 


29 


IC 


13 x 26 


112 


132 


24^ 


IXXXX 


14x20 


112 


200 


27 


IX 


13x26 


112 


162 


29 


IC 


20x28 


112 


216 


26 


IXX 


13x26 


112 


192 


27 


IX 


20 x 28 


112 


270 


29 


IC 


14x22 


112 


120 


26 


IXX 


20 x 28 


112 


320 


27 


IX 


14x22 


112 


148 


25 


IXXX 


20x28 


56 


180 


26 


IXX 


14x22 


112 


174 


24^ 


IXXXX 


20x28 


56 


200 


29 


IC 


14x24 


112 


130 


29 


IC 


13x13 


225 


132 


27 


IX 


14x24 


112 


161 


27 


IX 


13x13 


225 


162 


26 


IXX 


14x24 


112 


190 


26 


IXX 


13x13 


225 


192 


29 


IC 


14x28 


112 


155 


29 


IC 


14x14 


225 


155 


27 


IX 


14 x 28 


112 


193 


27 


IX 


14x14 


225 


193 


26 


IXX 


14x28 


112 


230 


26 


IXX 


14x14 


225 


230 


29 


IC 


14x31 


112 


178 


29 


IC 


15x15 


225 


178 


27 


IX 


14x31 


112 


210 


27 


IX 


15x15 


225 


218 


26 


IXX 


14x31 


112 


240 


26 


IXX 


15x15 


225 


260 


27 


IX 


14x56 


56 


185 


29 


IC 


16x16 


225 


200 


26 


IXX 


14 x 56 


56 


220 


27 


IX 


16x16 


225 


248 


27 


IX 


14x60 


56 


200 


26 


IXX 


16x16 


225 


290 


26 


IXX 


14x60 


56 


240 


29 


IC 


17x17 


225 


230 


29 


IC 


15x21 


112 


120 


27 


IX 


17x17 


225 


289 


27 


IX 


15 x 21 


112 


152 


26 


IXX 


17x17 


225 


340 


26 


IXX 


15x21 


112 


176 


29 


IC 


18x18 


112 


138 


29 


IC 


16x19 


112 


120 


27 


IX 


18x18 


112 


158 


27 


IX 


16x19 


112 


147 


26 


IXX 


18x18 


112 


178 


26 


IXX 


16x19 


112 


170 


29 


IC 


20x20 


112 


160 


29 


IC 


16x20 


112 


127 


27 


IX 


20x20 


112 


195 


27 


IX 


16x20 


112 


154 


26 


IXX 


20x20 


112 


222 


26 


IXX 


16x20 


112 


180 


29 


IC 


22x22 


112 


190 


29 


IC 


16x22 


112 


138 


27 


IX 


22x22 


112 


235 


27 


IX 


16x22 


112 


170 


26 


IXX 


22x22 


112 


275 


26 


IXX 


16x22 


112 


200 


29 


IC 


24x24 


112 


220 












27 


IX 


24x24 


112 


276 












26 


IXX 


24x24 


112 


330 













B. W. 


Thickness. 


Size. 


Sheets. 


Net 

Weight 

lbs. 


B. W. 


Thickness. 


Size. 


Sheets. 


Net 

Weight 

lbs. 


28 


DC 


12^x17 


100 


94 


23 


DXXX 


15x21 


100 


244 


25 


DX 


12^x17 


100 


122 


22 


DXXXX 


15x21 


100 


275 


24 


DXX 


12^x17 


100 


143 


28 


DC 


17x25 


50 


94 


23 


DXXX 


121^x17 


100 


164 


25 


DX 


17x25 


50 


122 


22 


DXXXX 


12^x17 


100 


185 


24 


DXX 


17x25 


50 


143 


28 


DC 


15x21 


100 


130 


23 


DXXX 


17x25 


50 


164 


25 


DX 


15x21 


100 


180 


22 


DXXXX 


17x25 


50 


185 


24 


DXX 


15x21 


100 


213 













Terne Plates, 112J|2^J^ 801! 



i a box 1 
Tagger's Tin and Iron, 36 and 



. per box. 
12 lbs.; IX 140 " " " 
28, IC, 224 lbs., 1X280 " " " 
B. W. G., 10 x 14 and 14 x 20. 112 lbs. per box. 



SIZES AtfD WEIGHTS OF ROOEIXG MATERIALS. 183 

Slate. 

Number and superficial area of slate required for one square of roof. 
(1 square ~ 1U0 square feet.) 



Dimensions 

in 

Inches. 


Number 

per 
Square. 


Superficial 
Area in 
Sq. Ft. 


Dimensions 

in 

Inches. 


Number 

per 
Square. 


Superficial 
Area in 
Sq. Ft. 


6x12 
7x12 
8x12 


533 
457 
400 
355 
374 
327 
291 
261 

246 
221 
213 
192 


267 


12x18 
10x20 
11x20 
12x20 
14x20 
16x20 
12x22 
14x22 
12x24 
14x24 
16 x 24 
14x26 
16x26 


160 
169 
154 
141 
121 
137 
126 
108 
114 
98 
86 
89 
78 


240 
235 


9x12 
7x14 
8x14 
9x 14 


'""254" 


231 


10x14 






8x16 
9x16 


246 


228 


10x16 
9x18 
10x18 


240"' 


225 









As slate is usually laid, the number of square feet of roof covered by one 
slate can be obtained from the following formula : 



width X (length — 3 inches) _ 



the number of square feet of roof covered. 



Weight of slate of various lengths and thicknesses required for one square 
of roof : 



Length 




Weight 


in Pounds per Square for the Thickness. 




Inches. 


%" 


3-16" 


Ya!' 


%" 


H" 


%" 


H" 


1" 


12 


483 


724 


967 


1450 


1936 


2419 


2902 


3872 


14 


460 


688 


920 


1379 


1842 


2301 


2760 


3683 


16 


445 


667 


890 


1336 


1784 


2229 


2670 


3567 


18 


434 


650 


869 


1303 


1740 


2174 


2607 


3480 


20 


425 


637 


851 


1276 


1704 


2129 


2553 


3408 


22 


418 


626 


836 


1254 


1675 


2093 


2508 


3350 


24 


412 


61? 


825 


1238 


1653 


2066 


2478 


3306 


26 


407 


610 


815 


1222 


1631 


2039 


2445 


3263 



The weights given above are based on the number of slate required for one 
square of roof, taking the weight of a cubic foot of slate at 175 pounds. 

Pine Shingles. 

Number and weight of pine shingles required to cover one square of 



Number of 

Inches 

Exposed to 

Weather. 


Number of 
Shingles 

per Square 
of Roof. 


Weight in 
Pounds of 
Shingle on 
One-square 
of Roofs. 


Remarks. 


4 

ft 

ft 


900 
800 
720 
655 
600 


216 
192 
173 
157 

144 


The number of shingles per square is 
for common gable-roofs. For hip- 
roofs add five per cent, to these figures. 

The weights per square are based on 
the number per square. 



184 



MATERIALS. 



Skylight Glass. 

■ The weights of various sizes and thicknesses of fluted or rough plate-glass 
required for one square of roof. 



Dimensions in 
Inches. 


Thickness in 
Inches. 


Area 
in Square Feet. 


Weight in Lbs. per 
Square Of Roof. 


12x48 
15x60 
20x100 
94x156 


3-16 

% 


3.997 

6.246 

13.880 

101.768 


250 
350 
500 
700 



In the above table no allowance is made for lap. 
If ordinary window-glass is used, singrle thick glass (about 1-16") will weigh 
about 82 lbs. per square, and double thick glass (about %") will weigh about 
164 lbs. per square, no allowance being made for lap. A box of ordinary 
window-glass contains as nearly 50 square feet as the size of the panes will 
admit of. Panes of any size are made to order by the manufacturers, but a 
great variety of sizes are usually kept in stock, ranging from 6x8 inches to 
36 x 60 inches. 

APPROXIMATE WEIGHTS OF VARIOUS ROOF- 
COVERINGS. 

For preliminary estimates the weights of various roof coverings may be 
taken as tabulated below: 

■pj aTT ,„ Weight in Lbs. per 

JName> Square of Roof. 

Cast-iron plates (%" thick) 1500 

Copper 80-125 

Felt and asphalt , 100 

Felt and gravel 800-1000 

Iron, corrugated 100-375 

Iron, galvanized, flat 100- 350 

Lath and plaster 900-1000 

Sheathing, pine, 1" thick yellow, northern . . 300 

" •' " " southern.. 400 

Spruce, 1" thick 200 

Sheathing, chestnut or maple, 1 " thick 400 

" ash, hickory, or oak, 1" thick 500 

Sheet iron (1-16" thick) 300 

" " " and laths 500 

Shingles, pine 200 

Slates (W thick) 900 

Skylights (glass 3-16" to Yq" thick) 250-700 

Sheet lead 500- 800 

Thatch 650 

Tin 70-125 

Tiles, flat . . 1500-2000 

(grooves and fillets) 700-1000 

" pan 1000 

" with mortar 2000-3000 

Zinc 100-200 



WEIGHT OF CAST-IRON" PIPES OR COLUMNS. 



185 



WEIGHT OF CAST-IRON PIPES OR COLUMNS. 
In libs, per Lineal Foot. 

Cast iron = 450 lbs. per cubic foot. 



Bore. 


Thick. 

of 
Metal. 


Weight 
per Foot. 


Bore. 


Thick. 

of 
Metal. 


Weight 
per Foot. 


Bore. 


Thick. 

of 
Metal. 


Weight 
per Foot. 


Ins. 


Ins. 


Lbs. 


Ins. 


Ins. 


Lbs. 


Ins. 


Ins. 


Lbs. 


3 


% 


12.4 


10 


H 


79.2 


22 


H 


167.5 




H 


17.2 


10K 


H 


54.0 




% 


196.5 




22.2 




% 


68 2 


23 


S 


174.9 


®A 


I 


14.3 




H 


82.8 




205.1 




19.6 


11 


H 


56.5 




l 


235.6 




% 


25.3 




% 


71.3 


24 


S 


182.2 


4 


% 


16.1 




% 


86.5 




213.7 




% 


22.1 


11^2 


1 


58.9 




l 


245.4 




% 


28.4 




74.4 


25 


s 


189.6 


*H 


% 


17.9 




% 


90.2 




222.3 




H 


24.5 


12 


1 


61.3 




1 


255.3 




% 


31.5 




77.5 


26 


n 


197.0 


5 


% 


19.8 




*A 


93.9 




% 


230.9 




u 


27.0 


12^ 




63.8 




i 


265.1 




% 


34.4 




% 


80.5 


27 


H 


204.3 


5H 




21.6 
29.4 


13 


V 


97.6 
66.3 




l 


239.4 
274.9 




% 


37.6 




% 


83.6 


28 


S 


211.7 


6 


% 


23.5 




% 


101.2 




248.1 




14 


31.8 


14 


% 


71.2 




l 


284.7 




% 


40.7 




% 


89.7 


29 


s 


219.1 


6^ 


% 


25.3 




% 


108.6 




256.6 




Yz 


34.4 


15 


95.9 




1 


294.5 




% 


43.7 




% 


116.0 


30 


% 


265.2 


7 


% 


27.1 




% 


136.4 




1 


304.3 




M 


36.8 


16 


% 


102.0 




m 


343.7 




Vs 


46.8 




M 


123.3 


31 


% 


273.8 


Wz 


¥s 


29.0 




if 


145.0 




l 


314.2 




8 


39.3 


17 


108.2 




i^ 


354.8 




49.9 




% 


130.7 


32 


% 


282.4 


8 


% 


30.8 




% 


153.6 




i 


324.0 






41.7 


18 


% 


114.3 




m 


365.8 




% 


52.9 




H 


138.1 


33 


% 


291.0 


8^ 


Vz 


44.2 




% 


162.1 




i 


333.8 




% 


56.0 


19 


% 


120.4 




m 


376.9 




s 


68.1 




I 


145.4 


34 


% 


299.6 


9 


46.6 




170.7 




l 


343.7 




% 


59.1 


20 


ft 


126.6 




i/^j 


388.0 




% 


71.8 




152.8 


35 


% 


308.1 


Wz 


H 


49.1 




7 4 


179.3 




i 


353.4 




% 


62.1 


21 


132.7 




m 


399.0 




M 


75.5 




8 


160.1 


36 


% 


316.6 


10 




51.5 




187.9 




i 


363.1 




% 


65.2 


22 


% 


138.8 




Wh 


410.0 



The weight of the two flanges may be reckoned = weight of one foot. 



186 



MATERIALS. 



WEIGHTS OF CAST-IRON PIPE TO LA¥ 12 FEET 
LENGTH. 

Weights are Gross Weights, including Hub. 

(Calculated by F. H. Lewis.) 



Thickness. 








Inside Diameter. 








Inches. 


Equiv. 
Decimals. 


4" 


6" 


8" 


10'/ 


12" 


14" 


16" 


18" 


20" 


13^32 
7-16 
15-32 

17-32 
9-16 
19-32 

Vs 
11-16 


.375 

.40625 

.4375 

.4687 

.5 

.53125 

.5625 

.59375 

.625 

.6875 

.75 

.8125 

.875 

.9375 

1. 

1.125 

1.25 

1.375 


209 
228 
247 
266 
286 
306 
327 


304 
331 
358 
386 
414 
442 
470 
498 


400 
435 
470 
505 
541 
577 
613 
649 
686 


581 
624 
668 
712 
756 
801 
845 
935 
1026 


692 
: 744 
795 
846 
899 
951 
1003 
1110 
1216 
1324 
1432 


804 

863 
922 
983 
1043 
1103 
1163 
1285 
1408 
1531 
1656 
1783 
1909 


1050 
1118 
1186 
1254 
1322 
1460 
1598 
1738 
1879 
2021 
2163 


1177 
1253 
1329 
1405 
1481 
1635 
1789 
1945 
2101 
2259 
2418 
2738 
3062 








1640 


H 

13 16 








1810 








1980 


% 
15 16 










2152 
2324 












2498 


1% 












2672 












3024 
















3380 

















Thickness. 








Inside Diameter. 








Inches. 


Equiv. 
Decimals. 


22" 


24" 


27" 


30" 


33" 


36" 


42" 


48" 


60" 


% 


.625 


1799 


















11-16 


.6875 


1985 


2160 


2422 














% 


.75 


2171 


2362 


2648 


2934 


3221 


3507 








13-16 


.8125 


2359 


2565 


2875 


3186 


3496 


3806 


4426 






% 


.875 


2547 


27G9 


3103 


3437 


3771. 


4105 


4773 


5442 




15-16 


.9375 


2737 


2975 


3332 


3690 


4048 


4406 


5122 


5839 






1. 


2927 


3180 


3562 


3942 


4325 


4708 


5472 


6236 




m 


1 125 . 


3310 


3598 


4027 


4456 


4886 


5316 


6176 


7034 




m 


1.25 


3698 


4016 


4492 


4970 


5447 


5924 


6880 


7833 


9742 


1.375 




4439 


4964 


5491 


6015 


6540 


7591 


8640 


10740 




1.5 






5439 


6012 


6584 


7158 


8303 


9447 


11738 


i% 


1.625 

1.75 

1.875 

2. 

2.25 

2.5 

2.75 








6539 


7159 
7737 


7782 
8405 


9022 
9742 
10468 
11197 


10260 
11076 
11898 
12725 
14385 


12744 


M 

2 








13750 










14763 














15776 


2 l A 

2U 














17821 
















19880 


















21956 





















CAST-IEOK" PIPE FITTINGS. 



187 



CAST-IRON PIPE FITTINGS. 

Approximate Weight. 

Addyston Pipe and Steel Co., Cincinnati, Ohio. 



Size in 


Weight 


Size in 


Weight 


Size in 1 Weight 
Inches, | in Lbs. 


Size in 1 Weight 


Inches. 


in Lbs. 


Inches. 


in Lbs. 


Inches. | in Lbs. 


CROSSES. 


TEES. 


SLEEVES. 


REDUCERS. 


2 


40 


8x3 


220 


6 


65 


10x4 


128 


3 


104 


10 


390 


8 


86 


12x10 


278 


3x2 


90 


10x8 


330 


10 


140 


12x8 


254 


4 


150 


10x6 


312 


12 


176 


12x6 


250 


4x3 


114 


10x4 


292 


14 


208 


12x4 


250 


4x2 


110 


10x3 


290 


16 


340 


14x12 


475 


6 


200 


12 


565 


20 


500 


14x10 


430 


6x4 


150 


12x10 


510 


24 


710 


14x8 


340 


6x3 


150 


12x8 


492 


30 


965 


14x6 


285 


8 


325 


12x6 


484 


36 


1500 


16x12 


475 


8x6 


265 


12x4 


460 




16x10 


435 


8x4 


265 


14x12 


650 


90° ELBOWS. 


20x16 


690 


8x3 


225 

510 


14x10 
14x8 


650 
575 




20x14 
20x12 


575 


10 


2 


14 


540 


10x8 


415 


14x6 


545 


3 


34 


20x8 


300 


10x6 


388 


14x4 


525 


4 


48 


24x20 


745 


10x4 


338 


14x3 


490 


6 


110 


30x24 


1305 


10x3 


350 


16 


790 


8 


145 


30x18 


1385 


12 


700 


16x14 


850 


10 


225 


36x30 


1730 


12x10 


650 


16x12 


825 


12 


370 




12x8 


615 


16x10 


890 


14 


450 


ANGLE REDUC- 


12x6 


540 


16x8 


755 


16 


525 


ERS FOR GAS. 


12x4 


525 

495 


16x6 
16x4 


630 
655 


20 
24 


900 

1400 




12x3 


6x4 1 95 


14x10 


750 
635 


20 
20x16 


1375 
1115 


Y s or 45° BENDS. 


6x3 | 80 


14x8 




14x6 


570 
1025 
1070 


20x12 
20x10 
20x8 


1025 
1090 
900 


S PIPES. 


16 


3 
4 


30 
65 


16x14 


4 1 90 


16x12 


1025 


20x6 


875 


6 


85 


6 1 190 


16x10 


1010 


20x4 


845 


8 


160 




16x8 


825 


21x10 


1465 


10 


190 


PLUGS. 


16x6 


700 
650 


24 
24 x 12 


1875 
1425 


12 
16 


290 
510 




16x4 


2 


2 


20 


1790 


24x8 


1375 


20 


740 


3 


5 


20x12 


1370 


24x6 


1375 


24 


1425 


4 


8 


20x10 


1225 


30 


3025 


30 


2000 


6 


12 


20x8 


1000 


30x24 


2640 




8 


26 


20x6 


1000 


30x20 


2200 


1-16 or 22W> 
BENDS. 


10 


46 


20x4 


1000 


30x12 


2035 


12 


66 


24 


2190 
2020 


30 x 10 
30x6 


2050 
1825 




14 

16 


70 


24x20 


6 


150 


100 


24x6 


1340 


*36 


5140 


8 


155 


20 


150 


30x20 


2635 


36x30 


4200 


10 


165 


24 


185 


30x12 


2250 


36 x 12 


4050 


12 


260 


30 


370 


30x8 


1995 


45° BR^ 
PIP] 


lNCH 

:s. 


16 


500 




TEES. 


24 
30 


1280 
1735 


CAPS. 


2 


28 


3 


90 




3 


15 


3 


76 


6x6x4 


145 


REDUCERS. 


4 


25 


3x2 


76 
100 


8 
8x6 


300 
290 




6 

8 


60 


4 


3x2 


35 


75 


4x3 


90 


24 


2765 


4x3 


42 


10 


100 


4x2 


87 
150 


24 x 24 x 20 
30 


2145 
4170 


4x2 
6x4 


40 
95 


12 


120 


6 




6x4 


130 
125 
120 

266 


36 


10300 


6x3 
8x6 
8x4 
8x3 


80 
126 
116 
116 


DRIP BOXES. 


6x3 


SLEEVES. 




6x2 


4 

8 


235 


8 


2 


10 


355 


8x6 


252 


3 


20 


10x8 


212 


10 


760 


8x4 


222 


4 


44 


10x6 


150 


20 


1420 



188 



MATERIALS. 



WEIGHTS OF CAST-IRON WATER- AND GAS-PIPE. 

(Addyston Pipe and Steel Co., Cincinnati, Ohio.) 



.5 <8 


Standard Water-Pipe. 


.9 t 


Standard Gas-Pipe. 


02 hH 


Per Foot. 


Thick- 
ness. 


Per 
Length. 




Per Foot. 


Thick- 
ness. 


Per 

Length. 


2 


7 


5-16 


63 


2 


6 


u 


48 


3 
3 


15 
17 


8 


180 
204 


3 


12M 


5-16 


150 


4 


22 


y* 


264 


4 


17 


% 


204 


6 


33 


H 


396 


6 


30 


7-16 


360 


8 


42 


y a 


504 


8 


40 


7-16 


480 


10 


60 


9-16 


720 


10 


50 


7-16 


600 


12 


75 


9-16 


900 


12 


70 


Yz 


840 


14 


117 


¥4 


1400 


14 


84 


9-16 


1000 


16 


125 


H 


1500 


16 


100 


9-16 


1200 


18 


167 


Va 


2000 


18 


134 


11-16 


1600 


20 


200 


15-16 


2400 


20 


150 


11-16 


1800 


24 


250 


1 


3000 


24 


184 


% 


2200 


30 


350 


m 


4200 


30 


250 


8 


3000 


36 


475 


1% 


5700 


36 


350 


4200 


42 


600 


w& 


7200 


42 


383 


% 


4600 


48 


775 


V4 


9300 


48 


542 


Ws 


6500 


60 


1330 


2 


15960 


W 


900 


1% 


10800 



THICKNESS OF CAST-IRON PIPES. 

P. H. Baermann, in a paper read before the Engineers' Clnb of Phila- 
delphia in 1882, gave twenty different formulas for determining the thick- 
ness of cast-iron water-pipes under pressure. The formulas are of three 
classes! 

1. Depending upon the diameter only. 

2. Those depending upon the diameter and head, and which add a con- 
stant. 

3. Those depending upon the diameter and head, contain an additive or 
subtractive term depending upon the diameter, and add a constant. 

The more modern formulas are of the third class, and are as follows: 

t = .OOOOHhd 4- .Old + .36 Shedd, No. 1. 

t = .00006/id + .0133d + .296 Warren Foundry, No. 2. 

t = .000058/id + .0152d + .312 Francis, No. 3. 

t= .000048/id-f .013d + .32 Dupuit, No. 4. 

t = .00004M + .1 Vd + .15 Box, No. 5. 

t = .000135/id-f .4 - .OOlld Whitman, No. 6. 

t = .00006(/i 4- 2S0d) 4- .333 - .0033d Fanning, No. 7. 

t = .00015/id + .25 - '.0052d Meggs, No. 8. 

In which t = thickness in inches, h = head in feet, d =» diameter in inches. 

Rankine, "Civil Engineering," p. 721, says: "Cast-iron pipes should be 
made of a soft and tough quality of iron. Great attention should be made 
to moulding them correctly, so that the thickness may be exactly uniform all 
round. Each pipe should be tested for air-bubbles and flaws by ringing it 
with a hammer, and for strength by exposing it to double the intended 
greatest working pressure." The rule for computing the thickness of a pipe 

to resist a given working pressure is t = -|-, where r is the radius in inches, 

p the pressure in pounds per square inch, and / the tenacity of the iron per 
square inch. When/ = 18000, and a factor of safety of 5 is used, the above 
expressed in terms of d and h becomes 

"There are limitations, however, arising from difficulties in casting, and 
by the strain produced by shocks, which cause the thickness to be made 
greater than that given by the above formula." 



THICKNESS OF CAST-IROX PIPE. 



189 



Thickness of Metal and Weight per Length for Different 
Sizes of Cast-iron Pipes under Various Heads of Water. 

(Warren Foundry and Machine Co.) 





50 


100 


150 


200 


250 


300 


I 


"t. Head. 


Ft. Head. 


Ft. Head. 


Ft. Head. 


Ft. Head. 


Ft. Head. 


Size. 1 

5 




^ 


W . 


« 


w . 


A 


w . 


ia 


80 _J 


& 


05 


A 


j'cS 


S"S> 


<V cS 


S"Sc 


% 


5"Sd 


0) eg 


S"So 


0>$ 


S'Sd 


o) ce 


S-!p 


: 




tefl 


c-^ 


bio 




B 1n 


tc = 


a 


tea 




ted 


1 


5^ 


£^ 




' S J 


%& 


& ^ 


2^ 


1^ 




^J 


"ota 


•s^ 


* 


3«H 


^fc 




&u 




& - 


,£3«M 


t^j- 


Iq^ 


&u 




l*n 


b 


H O 


P. 


^o 


ft 


HO 


ft 


H<-> 


& 


H° 


ft 


H<-> 


ft 


3 


344 


144 


.858 


149 


.362 


153 


.371 


157 


.380 


161 


.390 


166 


4 


361 


197 


.878 


204 


.385 


211 


.397 


218 


.409 


226 


.421 


235 


5 




254 


.398 


265 


.408 


275 


.423 


286 


.438 


298 


.453 


309 


6 


:m 


815 


.411 


330 


.42S 


345 


.447 


361 


.465 


377 


.483 


393 


8 


422 


445 


.450 


475 


.474 


502 


498 


529 


.522 


557 


.546 


584 


10 


459 


600 


.489 


641 


.519 


682 


.549 


723 


.579 


766 


.609 


808 


12 


491 


768 


.527 


826 


.563 


885 


.599 


944 


.635 


1004 


.671 


1064 


14 




952 


.566 


1031 


.608 


1111 


.650 


1191 


.692 


1272 


.734 


1352 


16 


557 


1152 


.604 


1253 


.652 


1360 


.700 


1463 


.748 


1568 


.796 


1673 


18 




1870 


.643 


1500 


.697 


1630 


.751 


1761 


.805 


1894 


.859 


2026 


20 




1608 


.682 


1763 


.742 


1924 


.802 


2086 


.862 


2248 


.922 


2412 


24 


: 


2120 


.759 


2349 


.831 


2580 


.903 


2811 


.975 


3045 


1.047 


3279 


30 


v 


3020 


.875 


3376 


.965 


3735 


1.055 


4095 


1.145 


4458 


1.235 


4822 


36 


Hi 


4070 


.990 


4581 


1.098 


5096 


1.206 


5613 


1.314 


6133 


1.422 


6656 


42 


ISO 


5265 


1.106 


5958 


1.232 


6657 


1 358 


7360 


1.484 


8070 


1.610 


8804 


48 1 


078 


6616 


1.222 


7521 


1.366 


8431 


1.510 


9340 


1.654 


10269 


1.798 


11195 



All pipe cast vertically in dry sand; the 3 to 12 inch in lengths of 12 feet, 
all larger sizes in lengths of 12 feet 4 inches. 

Safe Pressures and Equivalent Heads of Water for Cast- 
iron Pipe of Different Sizes and Thicknesses. 

(Calculated by F. H. Lewis, from Fanning's Formula.) 



















Size of 


Pipe. 
















Thick- 
ness. 


4" 


6" 


8" 


10" 


12" 


14" 


16" 


18" 


20" 


: = 

-- 


.2 


5 5 
5 5 


— T 


u 


— T 
Si 




11 






11 

^.5 








11 
ii 


•a 

si 
w 


II 


a 


7-16 
1-2 
9-16 
5-8 


112 

330 


516 

774 


49 

199 
X74 


112 
2v, 
458 
631 


18 

74 
130 

186 


42 
171 
300 
429 


a 

89 

132 

177 
224 


101 
205 
304 
408 
516 


24 

02 
99 
187 
174 
212 
249 


55 

148 
228 
310 
401 
488 
574 


42 
74 

100 
138 
170 
202 
234 
200 


97 
170 
244 
310 
392 
405 
538 
012 


56 

84 
112 
140 
108 
190 
224 


129 
194 

387 

452 
510 


41 
00 
91 
110 
141 
160 
191 
210 


95 
152 

210 
207 

382 
440 
497 


51 
74 
90 
119 
141 
104 
209 
250 


118 


3 4 














170 


13 16 














<?•>] 


7-8 


















"74 




















3"5 
























878 


118 






















481 






























580 





































190 



MATERIALS. 



Safe Pressures 


, etc., 


for Cast-iron 


Pipe. 


-(Continued.) 






Size of Pipe. 




22" 


24" 


27" 


30" 


33" 


36" 


42 




48" 


60" 


Thick- 
ness. 






















9$, 


a 


- ■ 


a 


a,* 


fl 




a 


I = 


d 


■ 


- 


ii 


PS 


■±£ 


a 


9% 


a 




- 


ffl 


: 

i,'- 




'■■ z 




u 


H 










{■ c 

Ph n 


s* 


- 


1| 




¥ 








































11-16 


40 


9?! 


::o 


r>9 


19 


64 


























3-4 


60 




49 


113 


36 


S3 


24 


55 






















13-16 


so 




6S 


157 


JW 


120 




90 






















7-8 


101 




si; 




(ill 


159 


54 


124 


42 


97 


32 


74 














15-16 


1",1 




III;-) 


'-{42 


85 


196 


69 


159 


55 


127 


44 


101 














1 


149! 


327 




286 


10" 


235 


S4 


194 


69 


159 


57 


131 


3S 


ss 


'24 


55 






1 1-8 




41 y 




MY 1 


135 


311 


114 


263 


96 


"21 


S" 


IS! 


5J) 


ii 




99 






1 1-4 


224 


511; 




>i 


11)9 




144 


332 


124 


:: 


107 


247 


si 


187 


' 


143 


34 


78 


1 3-8 






237 


.-in; 


202 


465 


174 


401 


151 


34 S 


13" 


K 


103 


237 


.81 




49 


113 


1 1-2 










236 


544 


204 


470 


17S 


410 


i.v; 




124 


286 


99 




64 


147 


1 5-8 














"34 


53S 


"(if, 


47" 


IS" 


4H 


14; 


ia 










1 3-4 


















"33 


537 


207 


477 


167 


385 










1 7-8 


























181 




"" 


" 






2 


























211 


484 


174 








2 1-8 






























193 












































355 
424 

482 




































2 3-4 


































214 































Note.— The absolute safe static pressure which may be 

put upon pipe is given by the formula P = -jr X -=-, in 

±J 5 
which formula P is the pressure per square inch; T, the 
thickness of the shell; S, the ultimate strength per square 
inch of the metal in tension; and D, the inside diameter of 
the pipe. In the tables S is taken as 18000 pounds per 
square inch, with a working strain of one fifth this amount 
or 3600 pounds per square inch. The formula for the 

7200 T 
absolute safe static pressure then is: P = — — . 

It is, however, usual to allow for "water-ram" by in- 
creasing the thickness enough to provide for 100 pounds 
additional static pressure, and, to insure sufficient metal for 
good casting and for wear and tear, a further increase 

equal to .333 yl — 7™)- 

The expression for the thickness then becomes: 

"Or I). 



T = [p±mD + 



100// 



100. 



7200 
and for safe working pressure 

P=^-Bsa(.- : , 

The additional section provided as above represents an 
increased value under static pressure for the different sizes 
of pipe as follows (see table in margin). So that to test 
the pipes up to one fifth of the ultimate strength of the 
material, the pressures in the marginal table should be 
added to the pressure-values given in the table above. 



Size 




of 


Lbs. 


Pipe. 




4" 


676 


6 


476 


8 


316 


10 


316 


12 


276 


14 


248 


16 


226 


18 


209 


20 


196 


22 


185 


24 


176 


27 


165 


30 


156 


33 


149 


36 


143 


42 


J 33 


48 


126 


60 


116 



SHEET-IRON" HYDRAULIC PIPE. 



191 



SHEET-IRON HYDRAULIC PIPE. 

(Pelt on Water-Wheel Co.) 
Weight per foot, with safe head for various sizes of double-riveted pipe. 



'o 




yjJ? 6X) 


-cS^ 


o £^ 


o 




w° to 




■"i-b 


u 


o • 


<8 c § 




^3 <V 1) 


"S <D 


?* 








S.I 




Jo.: 


£ .5 Etc 


tuoac 
©S3 


SB 


08 .& 


is! 




too-o 

|£3 


Q 


< 


H £ 


CO 


£ 


fi 


< 


EH £ 


w 


£ 


in. 


sq. in. 


B.W.G. 


feet. 


lbs. 


in. 


sq.in. 


B.W.G. 


feet. 


lbs. 


3 


7 


18 


400 


2 


18 


254 


16 


165 


16*3 


4 


12 


18 


350 


2*4 


18 


254 


14 


252 


20g 


4 


12 


16 


525 


3 


18 


254 


12 


385 


27*| 


5 


20 


18 


325 


3*3 


18 


254 


11 


424 


30 


5 


20 


16 


500 


4*4 


18 


254 


10 


505 


34 


5 


20 


14 


675 


5 


20 


314 


16 


148 


18 


6 


28 


18 


296 


4*4 


20 


314 


14 


227 


22*3 


6 


28 


16 


487 


I 


20 


314 


12 


346 


30 S 


6 


28 


14 


743 


20 


314 


11 


380 


32*3 


7 


38 


18 


254 


5*4 


20 


314 


10 


456 


36*13 


7 


38 


16 


419 


m 


22 


380 


16 


135 


20 


7 


38 


14 


640 


m 


C;2 


380 


14 


206 


24% 


8 


50 


16 


367 




22 


380 


12 


316 


32% 


8 


50 


14 


560 


9*f 


22 


380 


11 


347 


35% 


8 


50 


12 


854 


13 


22 


380 


10 


415 


40 


9 


63 


16 


327 


s*3 


24 


452 


14 


188 


27*4 


9 


63 


14 


499 


im 


24 


452 


12 


290 


35*3 


9 


63 


12 


761 


\m 


24 


452 


11 


318 


39 


10 


78 


16 


295 


9*4 


24 


452 


10 


379 


43*3 


10 


78 


14 


450 


ii% 


24 


/V2 


8 


466 


53 


10 


78 


12 


687 


15% 


26 


530 


14 


175 


29*4 
38*^ 


10 


78 


11 


754 


rn/ a 


Cj 


530 


12 


267 


10 


78 


10 


900 


19*4 

m 


26 


530 


11 


294 


42 


11 


95 


16 


C69 


26 


530 


10 


352 


47 


11 


95 


14 


412 


13 


20 


530 


8 


432 


57*4 


11 


95 


12 


026 


17M 

18% 


28 


615 


14 


162 


31*1 


11 


95 


11 


687 


28 


615 


12 


247 


41*4 


11 


95 


10 


820 


21. 


28 


015 


11 


273 


45 4 


12 


113 


16 


246 


n*4 


28 


615 


10 


327 


50*4 


12 


113 


14 


3^7 


14 


28 


C15 


8 


400 


61*| 


12 


113 


12 


574 


18*3 


30 


706 


12 


231 


44 /4 


12 


113 


11 


630 


19% 
22% 


30 


700 


11 


254 


48 


12 


113 


10 


753 


30 


706 


10 


304 


54 


13 


132 


16 


228 


12 


CO 


706 


8 


375 


65 


13 


132 


14 


348 


15 


30 


706 


7 




74 


13 


132 


12 


530 


20 


36 


1017 


11 




58 


13 


132 


11 


583 


22 


30 


1017 


10 




67 


13 


132 


10 


696 


24*3 


36 


1017 


8 




78 


14 


153 


16 


211 


13 


36 


1017 


7 




88 


14 


153 


14 


324 


16 


40 


1256 


10 




71 


14 


153 


12 


494 


*m 


40 


1256 


8 




86 


14 


153 


11 


543 


23*3 


40 


1256 


7 




97 


14 


153 


10 


648 


26 


40 


1256 


6 




108 


15 


176 


16 


197 


13% 


40 


1256 


4 




126 


15 


176 


14 


302 


17 


42 


1385 


10 




74*3 


15 


176 


12 


460 


23 


42 


1385 


8 




91 


15 


176 


11 


507 


24% 


42 


1385 


7 




102 


15 


176 


10 


606 


28 


42 


1385 


6 




114 


16 


201 


16 


185 


1.4*3 


42 


1385 


4 




133 


16 


201 


14 


283 


17M 


42 


1385 


H 




137 


16 


201 


12 


432 


24*4 


42 


1385 


3 




145 


16 


201 


11 


474 


26*3 


42 


1385 


5-16 




177 


16 


201 


10 


567 


29*3 


42 


1385 


% 




216 



192 



MATERIALS. 



STANDARD PIPE FLANGES. 

Adopted July 18, 1894, at a conference of committees of the American 
Society of Mechanical Engineers, and the Master Steam and Hot Water Fit- 
ters' Association, with representatives of leading manufacturers and users 
of pipe. 

The list is divided into two groups; for medium and high pressures, the 
first ranging up to 75 lbs. per square inch, and the second up to 200 lbs. 





^ - ^. 




P 




to 


02 
















'e j o 


S3 


3 




2 


2 












§5S 










2 


"o 















ao . 




| 


fa 








a 




u 




2 












5 


a 


w 




of 


<D 




.a 


Ji 


af o£ 








a ■ 








fa 


a 









m ^§ 


# a 


SB O 

a 

■8 - 

s ^ 

.24- 


3 

h 

3 


S3 

a — 
g-2 


: 

C— 1 

S8 
g| 





a 

5 

So 


a 


? 

6D 

a 


03 
60 

a 

p3 

fa • 

6 2 


s 

. 

.5" M 

a 


PQ 


s 

a 


| 

5 



-a" 
a 




03 a 1 ^ 
i3 cry 


S 


?\ 




1 


3Q"" 

460 


Ph 


5 


fa 


m' rt 


£ 


pq 


PQ 


2 


.409 


6 


% 


2 


4% 


4 


% 


2 


825 


% 


.429 


550 




7 


H 


2^ 


by A 


4 


! 




1050 


3 


.448 


IB 


690 




7^ 


% 


2M 


6 


4 




- 


1330 


3J^ 


.466 


H> 


700 




sy 2 


if 


m 


6J^ 


4 


P 




2530 


4 


.486 


^ 


800 




9 


it 


2¥s 


7*4 


4 


% 




2100 


*% 


.498 


H 


900 




9J4 


ii 


2% 


7% 


8 




3 


1430 


5 


.525 


¥ 


1000 


K 


10 


11 


2H 
2% 




8 




3 


1630 


6 


.563 




1060 


K 


11 




8 




3 


2360 


7 


.60 


J2 


1120 




12^ 


IPs 


2% 


10% 


8 




3200 


8 


.639 


ff 






1»H 


2% 


11% 


8 






4190 


9 


.678 


IS 


mo 




15 


m 


2% 


13 


12 






3610 


10 


.713 


M 


1380 




16 


ii 3 B 


3 


uy 4 


12 




2970 


12 


.79 


18 


1470 




19 


irf 


3J4 


uy 2 


12 






4280 


14 


.864 


% 




A 


21 




18% 


12 


1 




4280 


15 


.904 


if 


1600 




22M 


m 


35l 


20 


16 


1 




3660 


16 


.946 


l 


1600 




23^ 


l/s 


3% 


214 


16 


l 




4210 


18 


1.02 


1A 
IK 


1690 


A 


25 


lf f 


3^ 


22% 


16 




- 


4540 


20 


1.09 


1780 




27^ 




3*S 


25 






5 


4490 


22 


1.18 


h% 






29^ 


lie 


3% 


27M 








4320 


24 


1.25 


m 1920 


31^ 32 


1^ m 


3% 4 


29M 29^ 
31^ 31% 




U4 




5130 


26 


1.30 


l r 5 s 1980 
1% 2040 


H 


33% 34J4 


m 2 


3% 4^ 








5030 


28 


1.38 


!4 


36 36^ 


m 2A 


4 414 


33^ 34 






6 


5000 


30 


1.48 


1U 2000 




38 38% 


i 1 ^ 2^ 


4 4% 


35^ 36 


28 






4590 


36 


1.71 


1% '1920 




44^ 45% 


1% 2% 
1% 2% 


44 4v s 


42 42% 




1% 




5790 


42 


1.87 


2 12100 




51 52% 

*m 59^ 


4% 5% 


48^ 49y,\m 






5700 


48 


2.17 


2J4 '2130 




2 m 


434 5% 


54% 56 144 






6090 



Notes. — Sizes up to 24 inches are designed for 200 lbs. or less. 

Sizes from 24 to 48 inches are divided into two scale's, one for 200 lbs., the 
other for less. 

The sizes of bolts given are for high pressure. For medium pressures the 
diameters are J^-inch less for pipes 2 to 20 inches diameter inclusive, and J4 
inch less for larger sizes, except 48-inch pipe, for which the size of bolt is 1% 
inches. 

When two lines of figures occur under one heading, the single columns up 
to 24 inches are for both medium and high pressures. Beginning with 24 
inches, the left-hand columns are for medium and the right-hand lines are 
for high pressures. 

The sudden increase in diameters at 16 inches is due to the possible inser- 
tion of wrought-iron pipe, making with a nearly constant width of gasket a 
greater diameter desirable. 

When wrought-iron pipe is used, if thinner flanges than those given are 
sufficient, it is proposed that bosses be used to bring the nuts up to the 
standard lengths. This avoids the use of a reinforcement around the pipe. 

Figures in the third, fourth, fifth, and last columns refer only to pipe for 
200 lbs. pressure. 

In drilling valve flanges a vertical line parallel to the spindles should be 
midway between two holes on the upper side of the flanges. 



CAST-IRON PIPE AND PIPE FLANGES. 



193 



DIMENSIONS OF PIPE FLANGES AND CAST-IRON 
PIPES. 

(J. E. Codman, Engineers 1 Club of Philadelphia, 1889.) 



fe * 


f-> 


*-« 


u ^ 


'n 


GO ^ 


Thickness 


O3 <D 


oo-2 


2£ 


£ be 

d a 

rife 


rood 


S-3 

1? 


Mm 
3 




of Pipe. 




+3 &JOO 

•5 2m 




Frac. 


Dec. 


£ 03 


5 


ft^ 


ft 


ft 


fc 


HO 






2 


Wi 


4% 


M 


4 


% 


% 


.373 


6.96 


4.41 


3 


m 


5% 


9 
S 


4 


i 


13-33 


.396 


11.16 


5.93 


4 


9 




6 


11-16 


7-16 


.420 


15.84 


7.66 


5 


9% 

10% 


8 


6 


H 


7-16 


.443 


21.00 


9.63 


6 


m 


8 


i 


15-32 


.466 


26.64 


11.82 


8 


m, 


n% 


8 


13-16 


^ 


.511 


39.36 


16.91 


10 


15J4 


1314 


% 


10 


% 


9-16 


.557 


54.00 


23.00 


12 


17% 


15% 


% 


12 


15-16 


19-32 


.603 


70.56 


30.13 


14 


20 


18 


% 


14 


1 


21-32 


.649 


89.04 


38.34 


16 


22 


20 


% 


16 


1 1-16 


11-16 


.695 


109.44 


47.70 


18 


24 


2214 


% 


16 


1^ 


n 


.741 


131.76 


58.23 


20 


27 


24^ 


l 


18 


1 3-16 


25-32 


.787 


156.00 


70.00 


22 


28% 

3iy 4 
3334 


26^ 


l 


20 


M 


27-32 


.833 


182.16 


83.05 


24 


28% 


1 - 


22 


1 5-16 


% 


.879 


210.24 


97.42 


26 


31 


1 


24 


Ws 


15-16 


.925 


240.24 


113.18 


28 


35^ 


33^ 
35^ 
37^ 


l 


24 


1 7-16 


31-32 


.971 


272.16 


130.35 


30 


38 


l 


26 


1 9-16 


1 


1.017 


306.00 


149.00 


32 


40 


lfc 


28 


W% 


1 1-16 


1.063 


341.76 


169.17 


34 


42M 


40 


1^ 


30 


1 11-16 


m 


1.109 


379.44 


190.90 


36 


45 


42 


Ws 


32 


1% 


1 5-32 


1.155 


419.04 


214.26 


38 


47 


44 


iH 


32 


1 13-16 


1 3-16 


1.201 


460.56 


239.27 


40 


49 


46 


Ws 


34 


Ws 


m 


1.247 


504.00 


266.00 


42 


51M 


48^ 


Wb 


34 


1 15-16 


1 5-16 


1.293 


549.36 


294.49 


44 


5sy 2 


50^ 


m 


36 


2 


111-32 


1.339 


596.64 


324.78 


46 


55% 


52% 


m 


38 


2 1-16 


Ws 


1.385 


645.84 


356.94 


48 


58 


55 


M 


40 


2^ 


Ws 


1.431 


696.96 


391.00 



D = Diameter of pipe. All dimensions in inches. 
Formulae.— Thickness of flange = 0.033D + 0.56. 
Thickness of pipe = 0.023D + 0.327. 
Weight of pipe per foot = 0.24D 2 -f 3D. 
Weight of flange = .001D 3 + 0.1D 2 + D + 2. 
Diameter of flange = 1.125D + 4.25> 
Diameter of bolt-circle = 1.092D + 2.566. 
Diameter of bolt = 0.011D -f 0.73. 
Number of bolts = 0.78D + 2.56. 



PIPE 


FLANGES FOR HIGH STEAM-PRESSURE. 






(Chapman Valve Mfg. Co.) 




Size of 


Diameter 


Number of 


Diameter 


Diameter of 


Length of 


Pipe. 


of Flange. 


Bolts. 


of Bolts. 


Bolt Circle. 


Pipe-Thread. 


Inches. 


Inches. 


Inches. 


Inches. 


Inches. 


Inches. 


2^ 


7^ 


6 


Vs 


5V 8 


1% 


3 


9 


6 




6% 


Ws 


3^ 


9 


7 


% 


7J4 


1 7-16 


4 


10 


8 


% 


m 


1 9-16 


' 4)* 


10^ 


8 


i 
8 


m 


1 11-16 


5 


11 


9 


m 


1 13-16 


6 


13 


10 


10% 


m 


7 


14 


12 


Vs 


n% 


1 15-16 


8 


15 


12 


% 


13 




9 


16 


13 


Vs 


14 


2 


10 


17^ 


15 


Vs 


1534 

i«M 


2V S 


12 


20 


18 


% 


m \ 


14 


23 


18 


1 


2014 


2v 2 


1 15 


23^ 


18 


1 


21M 


2% 















194 



MATERIALS. 



STANDARD SIZES, ETC., OF WROUGHT-IRON PIPE. 
For Water, Gas, or Steam, 

(Briggs Standard.) 



Diameter of Tube. 


02 "cS 

a| 

H 

Ins. 


£ a a 

SocS 


« A o5 
a o v 

£ 9 v 


Length of 
Pipe per 
Sq. Ft. of 
Inside Sur- 
face. 


Length of 
Pipe per 
Sq. Ft. of 
Outside 
Surface. 


* . 
a eg 


OS . 
a aj 


1*1 

° a 52 


o 2 
Ins. 


$+6 

30* 




Ins. 


Ins. 


Ins. 


Ins. 


Feet. 


Feet. 


Ins. 


Ins. 


Ys 


.270 


.405 


.068 


.848 


1.272 


14.15 


9.44 


.0572 


.129 


Vi 


.364 


.540 


.088 


1.144 


1.696 


10.50 


7.075 


.1041 


.229 


% 


.494 


.675 


.091 


1.552 


2.121 


7.67 


5.657 


.1916 


.358 


y* 


.623 


.840 


.109 


1.957 


2.652 


6.13 


4.502 


.3048 


.554 


S 


.824 


1.050 


.113 


2.589 


3.299 


4.635 


3.637 


.5333 


.866 


l 


1.048 


1.315 


.134 


3.292 


4.134 


3.679 


2.903 


.8627 


1.357 


im 


1.380 


1.660 


.140 


4.335 


5.215 


2.768 


2.301 


1.496 


2.164 


m 


1.610 


1.900 


.145 


5.061 


5.969 


2.371 


2.01 


2.038 


2.835 


2 


2.067 


2.375 


.154 


6.494 


7.461 


1.848 


1.611 


3.355 


4.430 


%H 


2.468 


2.875 


.204 


7.754 


9.032 


1.547 


1.328 


4.783 


6.491 


3 


3.067 


3.500 


.217 


9.636 


10.996 


1.245 


1.091 


7.388 


9.621 


Wz 


3.548 


4.000 


.226 


11.146 


12.566 


1.077 


.955 


9.887 


12.566 


4 


4.026 


4.500 


.237 


12.648 


14.137 


.949 


.849 


12.730 


15.904 


4^ 


4.508 


5.000 


.246 


14.153 


15.708 


.848 


.765 


15.939 


19.635 


5 


5.045 


5.563 


.259 


15.849 


17.475 


.757 


.629 


19.990 


24.299 


6 


6.065 


6.625 


.2S0 


19.054 


20 813 


.63 


.577 


28.889 


34.471 


7 


7.023 


7.6?5 


.301 


22.063 


23.954 


.544 


.505 


38.737 


45.663 


8 


7.982 


8.625 


.322 


25.076 


27.096 


.478 


.444 


50.039 


58.426 


*9 


9.000 


9.688 


.344 


28.277 


30.433 


.425 


.394 


63.633 


73.715 


10 


10.019 


10.750 


.366 


31.475 


33.772 


.381 


.355 


78.838 90.762 



* By the action of the Manufacturers of Wrought-iron Pipe and Boiler 
Tubes, at a meeting held in New York, May 9, 1889, a change in size of actual 
outside diameter of 9-inch pipe was adopted, making the latter 9.625 instead 
of 9.688 inches, as given in the table of Briggs' standard pipe diameters. 

For discussion of the Briggs Standard of Wrought-iron Pipe Dimensions, 
see Report of the Committee of the A. S. M. E. in " Standard Pipe and Pipe 
Threads," 1886. Trans., Vol. VIII, p. 29. The figures in the next to the last 
column are derived from the formula 



Z>-(0.05Z> + 1.9) X 



1 



in which D = outside diameter of the tubes, and n the number of threads to 
the inch. The figures in the last column are derived from the formula 

0.8— x 2 -f d, or 1.6 \- d, in which d is the diameter at the bottom of the 

ti n 

thread at the end of the pipe. 

Having the taper, length of full-threaded portion, and the sizes at bottom 
and top of thread at the end of the pipe, as given in the table, taps and dies 
can be made to secure these points correctly, the length of the imperfect 
threaded portions on the pipe, and the length the tap is run into the fittings 
beyond the point at which the size is as given, or, in other words, beyond 
the end of the pipe, having no effect upon the standard. The angle of the 
thread is 60°, and it is slightly rounded off at top and bottom, so that, instead 
of its depth being equal to its pitch, as is the case with a full V-thread, it is 

4/5 the pitch, or equal to 0.8— , n being the number of threads per inch. 



WROUGHT-IRON PIPE. 



195 



Sizes, etc., of Wrought-iron Pipe— (Continued.) 





Sizes, etc. 






Screwed Ends. 




.5t3 CD 


«m cO o 


hi $ 


a 

i§ .so 

c 05O r O 




Oj tf G 


O-u . 

tit! £ 


eter of 
torn of 
ead at 
of Pipe. 


eter of 
of 

ead at 
of Pipe. 


o£tj 


i-5 


,Sfo5 


r*& 


H*s 


|e& 


EDO 

Inch. 


im 


Diarr 

Top 
Thr 
End 


Inch. 


Feet. 


Lbs. 




Lbs. 


No. 


Inches. 


Inches. 


M 


2500. 


.243 


.0006 


.005 


27 


.19 


.334 


.393 


*4 


1385. 


.422 


.0026 


.021 


18 


.29 


.433 


.522 


% 


751.5 


.561 


.0057 


.047 


18 


30 


.567 


.656 


u 


472.4 


.845 


.0102 


.085 


14 


.39 


.701 


.815 


3 4 


270. 


1.126 


.0230 


.190 


14 


.40 


.911 


1.025 


l 


166.9 


1.670 


.0408 


.349 


11^ 


.51 


1.144 


1.283 


J8 


96.25 


2.258 


.0638 


.527 


ll^S 


.54 


1.488 


1.627 


70.65 


2.694 


.0918 


.760 


11^ 


.55 


1.727 


1.866 




42.36 


3.667 


.1632 


l.:-56 


ny 2 


i£ 8 


2.2 


2.339 


2^ 


30.11 


5.773 


.2550 


2.116 


8 


?B9 


2.62 


2.82 


3 


19.49 


7.547 


.3673 


3.049 


8 


.95 


3.241 


3.441 


3*6 


14.56 


9.055 


.4998 


4.155 


8 


1.00 


3.738 


3.938 


4 


11.31 


10.728 


.6528 


5.405 


8 


1.05 


4.235 


4.435 


43^ 


9.03 


12.492 


.8263 


6.851 


8 


1.10 


4.732 


4.932 


5 


7.20 


14.564 


1.020 


8.500 


8 


1.16 


5.291 


5.491 


6 


4.98 


18.767 


1.469 


12.312 


8 


1.26 


6.346 


6.546 


7 


3.72 


23.410 


1.999 


16.662 


8 


1.36 


7.34 


7.54 


8 


2.88 


28.348 


2.611 


21.750 


8 


1.46 


8.334 


8.534 


9 


2.26 


34.077 


3.300 


27.500 


8 


1.57 


9.39 


9.59 


10 


1.80 


40.641 


4.081 


34.000 


8 


1.68 


10.445 


10.645 



Taper of conical tube ends, 1 in 32 to axis of tube = % inch to the foot 
total taper. 

1 inch and below are butt-welded, and proved to 300 pounds per square inch 
hydraulic pressure. 

V/± inch and above are lap-welded, and proved to 500 pounds per square 
inch hydraulic pressure. 



SIZES ABOVE 10 INCHES. 
(Morris, Tasker & Co., Limited.) 







^ 




aj 


. 6 


<S 


<S 


2* O e3 


8,-nS 




o 

o 




5 ,• 
11 

il 

©5 


3 <D 






OS 


< 


< 


Pn^-t 




^ c o 




"3 

a 


Oto 

11 

t>5 


a 


"3 ° 


cS O) 


3 
a 
53 
a 




O -CO 

"^53 "1 


O .02 

^o3'S 


o 3 ° 
S o o 




& 


< 


< 


H 




H 


H 


t^ M 


-1 O 


J 


in. 


in. 


in. 


in. 


in. 


in. 


i-q.in. 


sq. in. 


ft. 


ft. 


ft. 


lbs. 


11 


11.224 


12 


.388 


35.26 


37.70 


98.94 


113.10 


.340 


.318 


1.455 


47.73 


12 


12.180 


13 


.41 


38.26 


40.81 


116.54 


132.73 


.313 


.293 


1.235 


54.66 


13 


13.136 


14 


.432 


41.27 


43.98 


134.58 


153.94 


.290 


.273 


1.069 


61.94 


14 


14.092 


15 


.454 


44.27 


47.12 


155.97 


176.72 


.271 


.254 


.923 


70.01 


15 


15.048 


16 


.476 


47.27 


50.27 


177.87 


201.06 


.254 


.238 


.809 


78.27 


16 


16.004 


17 


.498 


50.28 


53.41 


201.16 


225.98 


.238 


.225 


.715 


87.12 


17 


16.960 


18 


.520 


53.28 


56.55 


225.91 


254.47 


.225 


.212 


.638 


96.38 


18 


17.916 


19 


.542 


56.28 


59.69 252.10 


283.53 


.213 


.201 


.571 


106.07 


19 


18.S72 


20 


.564 


59.29 


62.83 279.72 


314.16 


.202 


.191 


.515 


116.21 


20 


19.828 


21 


.586 


62.29 


65. 97 308.77 


346.36 


.192 


.183 


.466 


126.76 


i 























196 



MATERIALS. 



WROUGHT-IRON WELDED TUBES, EXTRA STRONG. 

Standard Dimensions. 









Thickness, 


Actual Inside 


Actual Inside 


Nominal 






Double 


Diameter, 


Diameter, 


Diameter. 


Diameter. 


Strong. 


Extra 


Extra 


Double Extra 








Strong. 


Strong. 


Strong. 


Inches. 


Inches. 


Inches. 


Inches. 


Inches. 


Inches. 


H 


0.405 


0.100 




0.205 




H 


0.54 


0.123 




0.294 




% 


0.675 


0.127 




0.421 




V*. 


0.84 


0.149 


0.298 


0.542 


0.244 


¥a 


1.05 


0.157 


0.314 


0.736 


0.422 


l 


1.315 


0.182 


0.364 


0.951 


0.587 


m 


1.66 


0.194 


0.388 


1.272 


0.884 


v& 


1.9 


0.203 


0.406 


1.494 


1.088 


2 


2.375 


0.221 


' 0.442 


1.933 


1.491 


&A 


2.875 


0.280 


0.560 


2.315 


1.755 


3 


3.5 


0.304 


0.608 


2.892 


2.284 


3^ 


4.0 


• 0.321 


0.642 


3.358 


2.716 


4 


4.5 


0.341 


0.682 


3.818 


3.136 



STANDARD SIZES, ETC., OF IiAP-WEL.DED CHAR- 
COAL-IRON BOILER-TUBES. 

(Morris, Tasker & Co., Limited). 



s 


s 
S 


2 


5g 


oa 


Internal 


External 




■So i 

t*0 


3®£ 


'4 


"3 


3 


V. to 


3=2 


3 e 


Area. 


Area. 


_ CD 

ill 


gaol 


ga§ 


"Sort 


H 


M 


m 


H° 


W 






a 


*A 


J 


Ins. 


Ins. 


Ins. 


Ins. 


Ins. 


Sq. In. 


Sq.Ft 


Sq. In. 


Sq.Ft 


Ft. . 


Ft. 


Ft. 


Lbs. 


1 


.856 


.072 


2.689 


3.142 


.575 


.004 


.785 


.0055 


4.460 


3.819 


4.139 


.708 


1 1-4 


1.106 


.072 


3.474 


3.927 


.960 


.0067 


1.227 


.0085 


3.455 


3.056 


3.255 


.9 


1 1-2 


1.334 


.083 


4.191 


4.712 


1.396 


.0097 


1.767 


.0123 


2.863 


2.547 


2.705 


1.25 


1 3-4 


1.560 


.095 


4.901 


5.498 


1.911 


.0133 


2.405 


.0167 


2.448 


2.183 


2.315 


1.665 


2 


1.804 


.098 


5.667 


0.888 


2.556 


.0177 


3.142 


.0218 


2.118 


1.909 


2.013 


1.981 


2 1-4 


2.054 


.098 


6.484 


7.069 


3.314 


.0280 


3.976 


.0276 


1.850 


1.698 


1.774 


2.238 


2 1-2 


2.283 


.109 


7.172 


7.854 


4.094 


.0284 


4.909 


.0311 


1.673 


1.528 


1.600 


2.755 


2 3-4 


2.538 


.109 


7.957 


8.639 


5.039 


.035 


5.940 


.0412 


1.508 


1.390 


1.449 


3.045 


3 


2.783 


.109 


8.743 


9.425 


6.083 


.0422 


7.069 


.0491 


1.373 


1.273 


1.323 


8 888 


3 1-4 


3.012 


.119 


9.462 10.210 


7.125 


.0495 


8.296 


.0576 


1.268 


1.175 


1.221 




3 1-2 




.119 


10.248 10.995 


8.357 


.058 


9.621 


.0668 


1.171 


1.091 


1.131 


4 8272 


3 3-4 


8.512 


.119 


11.033 11.781 


9.687 


.0673 


11.045 


.0767 


1.088 


1.018 


1.053 


4.590 


4 


3.741 


.130 


11.753 12.566 


10.1192 


.0763 


12.566 


.0872 


1.023 


.955 


.989 


5.82 


4 1-2 


4.241 


.130 


13.323 14.137 


14.126 


,0981 


15.904 


.1104 


.901 


.849 


.875 


0.01 


5 


4.7-1) 


.140 


14.818 15.708 


17.497 


. 1215 


19.035 


.1364 


.809 


.764 


.786 


7.226 




5.099 


.151 


17.904 18.849 


25.509 


.1771 


28.274 


.1963 


.670 


.637 


.653 


9.846 


7 


6.657 


.172 


20.914 21.991 


34.; Si )5 


.2417 


38.484 


.2678, 


.574 


.545 


.560 


12.485 


8 


7.636 


.182 


23.989 25.132 


45.795 


.318 


50.265 


.8491 


.500 


.478 


.489 


15.109 


9 


8.615 


.193 


27.055 28.274 


58.291 


.4048 


63.617 


.4418 


.444 


.424 


.434 


18.002 


10 


9.578 


.214 


30.074 31.416 


71.975 


.4998 


78.540 


.5454 


.399 


.382 


.391 


22.19 


11 


10.560 


.22 


33.175 34 557 


87.479 


.6075 


95.033 


.6001 


.361 


.347 


.354 


25.489 


12 


11.54.'; 


.229 


36.26 137.699 


103.749 


.7207 


113.097 


.7854 


.330 


.318 


.324 


28.516 


13 


12.524 


.238 


39.345 40.840 


123. 1N7 


.8554 


182.782 


.9213 


.305 


.293 


.299 




14 


13.504 


.248 


42.414 43.982 


143.189 


.9948 


153 938 


1.009 




.272 


.277 


20.271 


15 


14.1X2 


.259 


45.496 47.124 


164.718 1.1438 


176.715 


1.2272 


'.263 


.254 


.258 


40 012 


16 




.271 


48.562 50.265 


187.667 1.3032 


201.002 


1.188 


.247 


.238 


.242 


45.199 


17 


16.432 


.284 


51.662 53.407 


212.227 1.4738 


226.980 


1.5702 


.232 


.224 


.228 


49.908 


18 




.292 


54.714 56.548 


238.224 1.6543 


254.469 


1.7071 


.219 


.212 


.215 


54.816 


19 


" 


.3 


57.805 59.690 


265.903 1.8465 


283.529 


1.909 


.207 


.200 


.203 


59.479 


20 


19.360 


.32 


60.821 62.832 


294.373 2.0443 


314.159 


2.1817 


.197 


.190 


.193 


00.705 


21 


20.320 .34 


63.837 65.973 


324.311 2.2522 


346.361 


2.4053 


.188 


.181 


.184 


78.404 


In estimating tl 


e effective 


^team-heating 


or boiler sui 


face of 


tubes, t 


ne surface in 


contact with air o 
be taken. 
For heating- liq 


• gases of co 


mbustion (wh 


ether internal 


or exte 


•nal to t 


he tubes) is to 


uids by stea 


m, superheat 


ng- steam, or 


transft 


rring h 


eat from one 


liouid or aas to ai 


lother, the i 


lean surface c 


f the tubes is 


o be tal 


:en. 




1 



RIVETED IRON PIPE. 



197 



To find the square feet of surface, S, in a tube of a given length, L, in feet, 
and diameter, d, in inches, multiply the length in feet by the diameter in 

inches and by .2618. Or, S = 3 - 1416dL = ,2618dL. For the diameters in the 

table below, multiply the length in feet by the figures given opposite the 
diameter. 



Inches, 
Diameter. 


Square Feet 
per Foot 
Length. 


Inches, 
Diameter. 


Square Feet 
per Foot 
Length. 


Inches, 
Diameter. 


Square Feet 
per Foot 
Length. 


3 

m 
m 
m 

2 


.0654 
.1309 
.1963 
.2618 
.3272 
.3927 
.4581 
.5236 


m 

3 

m 

■ m 
m 

4 


1 


5890 
6545 
7199 
7854 
8508 
9163 
9817 
0472 


5 
6 
7 
8 
9 
10 
11 
12 


1.3090 
1.5708 
1.8326 
2.0944 
2.3562 
2.6180 
2.8798 
3.1416 



RIVETED IRON PIPE. 

(Abendroth & Root Mfg. Co.) 

Sheets punched and rolled, ready for riveting, are packed in convenient 
form for shipment. The following table shows the iron and rivets required 
for punched and formed sheets. 



Number Square Feet of Iron 


I'S'S'i'S'S 


Number Square Feet of Iron 


o'g'S'3'S'S 


required to make 100 Lineal 




required to make 100 Lineal 


fcsSg,Sg 


Feet Punched and Formed 


xiinate 
vets 1 1 
t requ 
100 Li 
Punc 
Fori 
ts. 


Feet Punched and Formed 




Sheets when put together. 


Sheets when put together. 














•R^«S $ 


Diam- 


Width of 


Square 
Feet. 


gtfa^-aS 


Diam- 


Width of 


Square 
Feet. 




eter in 
Inches. 


Lap in 
Inches. 


< 


eter in 
Inches. 


Lap in 
Inches. 


So tfcSfc 0JO2 

< 


3 


1 


90 


1,600 


14 


m 


397 


2,800 


4 


1 


116 


1,700 


15 


m 


423 


2,900 


5 


V4 


150 


1,800 


16 


m 


452 


3,000 


6 


m 


178 


1,900 


18 


m 


506 


3,200 


7 


m 


206 


2,000 


20 




562 


3,500 


8 


Wz 


234 


2,200 


22 


1/^2 


617 


3,700 


9 


m 


258 


2,300 


24 


m 


670 


3,900 


10 


VA 


289 


2,400 


26 


m 


725 


4,100 


11 


\y^ 


314 


2,500 


28 




779 


4,400 


12 


W* 


343 


2,600 


30 


iH 


836 


4,600 


13 


m 


369 


2,700 


36 


m 


998 


5,200 



WEIGHT OF ONE SQUARE FOOT OF SHEET-IRON 
FOR RIVETED PIPE. 





Thickness by the Rirmingham Wire 


-Gauge. 


No. of 
Gauge. 


Thick- 
ness in 
Decimals 
of an 
Inch. 


Weight 
in lbs , 
Black. 


Weight 
in lbs., 
Galvan- 
ized. 


No. of 
Gauge. 


Thick- 
ness in 
Decimals 
of an 
Inch. 


Weight 
in lbs., 
Black. 


Weight 
in lbs., 
Galvan- 
ized. 


26 
24 

20 


.018 
.022 
.028 
.035 


!S8 
1.12 
1.40 


.94 
1.13 
1.38 
1.69 


18 
16 
14 
12 


.049 
.065 
.083 
.109 


1.97 
2.61 
3.33 
4.37 


2.19 
2.82 
3.52 
4.50 

















198 



MATERIALS. 



SPIRAL RIVETED PIPE. 

(Abendroth & Root Mfg. Co.) 



Thickness. 


Diam- 


Approximate Weight 


Approximate Burst- 






eter, 


in lbs. per foot in 


ing Pressure in lbs. 


B. W. G. 

No. 


Inches. 


Inches. 


Length. 


per sq. in. 


26 


.018 


3 to 6 llbs.= 




24 


.022 


3 to 12 J " =y B of diam. in ins. 




22 


.028 


3 to 14 " =.4 




20 


.035 


3 to 24 " =.5 


2700 lbs.-4-diam. in ins. 


18 


.049 


3 to 24 " =.6 


8600 " h- " 


10 


.065 


6 to 24 " =.8 


4800 " -r- " 


14 


.083 


8 to 24 1 " =1.1 " 


(5400 " -- " " 



The above are black pipes. Galvanized weighs from 10 to 30 per cent 
heavier. Double Galvanized Spiral Riveted Flanged Pressure Pipe, tested, 
to 150 lbs. hydraulic pressure. 



Inside diameters, inches 

Thickness, B. W. G 

Nominal weight per foot, lbs. . . 


3 
20 
*-4 


4 

20 
3 


51 6 7 8, 9 10 11 12 13 14 
20,18 18 18118 16 16 16 16 14 
A\ 5 6 7| 8 11 12 14 15 20 


15 

14 

22 


16 IN 20 
14 14 14 
24 29 34 



DIMENSIONS OF SPIRAL, PIPE FITTINGS. 

Dimensions in Inches. 











Diameter 




Inside 
Diameter. 


Outside 
Diameter 
Flanges. 


Number 
Bolt Holes 


Diameter 
Bolt Holes. 


Circles on 

which Bolt 

Holes are 

Drilled. 


Sizes of 
Bolts. 


ins. 












3 


6 


4 


H 


m 


7-16x1% 


4 


7 


8 


y z 


5 15-16 


7-16x134 
7-16 x \% 


5 


8 


8 


H 


6 15-16 


6 


8% 


8 


% 


m 


7 


10 


8 


Va 


9 


y*xm 


8 


11 


8 


Va 


10 




9 


13 


8 


Va 




y 2 x2 


10 


14 


8 


Va 


}4x2 


11 


15 


12 


Va 


y**2 


12 


16 


12 


Va 


1414 


y 2 x2 


13 


17 


12 


Va 


15M 


y 2 x2 


14 


17% 


12 


Va 


16M 

17 7-16 


y^zy* 


15 


19 


12 


Va 


%*2}4 


16 


21 3-16 


12 


Va 


19^ 


Hx2*4 


18 


23 y A 


16 


11-16 


21J4 




20 


25 Ys 


1.6 


11-16 


2sy 8 


y*x2y 2 



SEAMLESS BRASS TUBE. IRON-PIPE SIZES. 

(Randolph & Clowes). 
(For actual dimensions see tables of Wrought-iron Pipe.) 





Weight 


Nom- 


Weight 


Nom- 


Weight 


Nom- 


Weight 




per 


inal 


per 


inal 


per 


inal 


per 




Foot, lbs. 


Size. 


Foot, lbs. 


Size. 


Foot, lbs. 


Size. 


Foot, lbs. 


ya 


.266 


H 


1.228 


2 


4. 


4 


11.719 


\ 


.461 


1 


1.837 


2^ 


6.323 


5 


15.935 


.617 


m 


2.468 


3 


8.266 


6 


20.690 


y* 


.925 


M 


3.045 


m 


9.878 


8 


26.286 
29.881 



BEASS TUBIKG ; COILED PIPES. 



199 



SEAMLESS' DRAWN BRASS-TUBING. 

(Randolph & Clowes, Waterbury, Conn.) 

Outside diameter 3-16 to 7% inches. Thickness of walls 8 to 25 Stubbs' 
Gauge, length 12 feet. The following are the standard sizes: 

SEAMLESS DRAWN BRASS-TUBING. 



Outside 
Diam- 


Length 
Feet. 


Stubbs' 
or Old 


Outside 
Diam- 


Length 
Feet. 


Stubbs' 
or Old 


Outside 
Diam- 


Length 
Feet. 


Stubbs' 
or Old 


eter. 


Gauge. 


eter. 


Gauge. 


eter. 


Gauge. 


Ya 


12 


20 


m 


12 


14 


2% 


12 


11 


5-16 


12 


19 


m 


12 


14 


2M 


12 


11 


¥s 


12 


19 


m 


12 


13 


3 


12 


11 


8 


12 


18 


m 


12 


13 


&A 


12 


11 


% 


12 


18 


1 13-16 


12 


13 


12 


11 


% 


12 


17 


m 


12 


12 


4 


10 to 12 


11 


13-16 


12 


17 


1 15-16 


12 


12 


5 


10 to 12 


11 


% 


12 


17 


2 


12 


12 


5H 


10 to 12 


11 


15-16 


12 


17 


m 


12 


12 


Wz 


10 to 12 


11 


1 


12 


16 


2U 


12 


12 


m 


10 to 12 


11 


% 


12 


16 


Ws 


12 


12 


6 


10 to 12 


11 


12 


15 


zyk 


12 


11 









COILED PIPES. 

(National Pipe-bending Co., New Haven, Conn.) 



COILS OF STEEL OR IRON PIPE 


; WELDED 


LENGTHS. 






Butt-welded Pipe. 


Lap- 
welded 
Pipe. 


Size of pipe Inches 

Least outside diameter of coil contain- 
ing 25 feet of pipe and less. . . Inches 
Least outside diameter of coils over 25 
feet and not over 200 feet Inches 


H 

2 

6 


% 

*A 
7 


V2 


% 

4 

SA 


1 
6 

9 

• 


1M 

8 
11 


1% 
12 

14 


2 

18 
18 



COILS OF SEAMLESS DRAWN BRASS AND COPPER TUBING. 



Size of tube, outside 
diameter Ins. 

Least outside diam- 
eter of coils — Ins. 



M 



1'4 



1A 



Welded solid drawn-steel tubes, imported by P. S. Justice & Co., Phila- 
delphia, are made in sizes from y% to 4J-£ inches external diameter, varying 
by J^ths, and with thickness of walls from 1-16 to 11-16 inches. The maxi- 
mum length is 15 feet, 



200 



MATERIALS. 



WEIGHT OF BRASS, COPPER, AND ZINC TUBING* 

Per Foot. 

Thickness by Brown & Sharped Gauge. 











Copper, 


Brass, 


No. 17. 


Brass, 


No. 20. 


Lightning-rod Tube, 
No. 23. 


Inch. 


Lbs. 


Inch. 


Lbs. 


Inch. 


Lbs. 


Va 


.107 


Vb 


.032 


y 2 


.162 


5-16 


.157 


3-16 


.039 


9-16 


.176 


% 


.185 


Va 


.063 


% 


.186 


7-16 


.234 


5-16 


.106 


11-16 


.211 


Vk 


.266 


% 


.126 


H 


.229 


9-11 


.318 


7-16 


.158 




% 
Va 


.3-33 

.377 


H 

9-16 


.189 
.208 


Zinc, No. 20. 


% 


.462 


% 


.220 






.542 
.675 


8 


.252 

.284 




W* 


^ 


.161 


m 


.740 


l 


.378 


% 


.185 


Wz 


.915 


m 
m 


.500 


Va 
Vs 


.234 


Wa 


.980 


.580 


.272 


2 


1.90 






1 


.311 


21^ 


1.506 






Wa 


.380 


3 


2.188 






m 


.452 



LEAD PIPE IN LENGTHS OF 10 FEET. 



In. 


3-8 Thick. 


5-16 Thick. 


Va Thick. 


3-16 Thick. 


2y a 

3 

m 

4 

M 

5 


lb. oz. 
17 
20 ' 
22 
25 

31 


lb. oz. 
14 

16 
18 
21 


lb. oz. 

11 

12 

15 

16 
18 
20 


lb. oz. 

8 

9 
9 8 

12 8 
14 



LEAD WASTE-PIPE. 

1}4 in., 2 lbs. per foot. I %% in., 4 lbs. per foot. 

2 " 3 and 4 lbs. per foot. 4 "5, 6, and 8 lbs. 

3 " 3^ and 5 lbs. per foot. | Ay 2 " 6 and 8 lbs. 

5 in. 8, 10, and 12 lbs. 

LEAD AND TIN TUBING. 

V% inch. 34 inch. 

SHEET LEAD. 

Weight per square foot, 2)4, 3, 3J4 4, i}4, 5, 6, 8, 9, 10 lbs. and upwards. 
Other weights rolled to order. 

BLOCK-TIN PIPE. 



% in , 4}4 6^, and 8 oz. per foot. 
}4 " 6, 73^, and 10 oz. " 
% " 8 and 10 oz. " 

Y± " 10 and 12 oz. " 



1 in., 15, and 18 oz. per foot. 
Wa " 1}4 and l^lbs. " 
1)4 " 2 and 2V£ lbs. 

2 il 2U and 3 lbs. " 



LEAD PIPE. 



201 





LEAD AND TIN-L.INED LEAD PIPE. 








(Tatham & Bros., New York.) 






, 




Weight per 
Foot and Rod. 


'IS 


6 




Weight per 


1^ 








£ 




Foot and Rod. 


II 


"3 


»3 




3^ 




s 




%in. 


E 


7 lbs. per rod 




1 in. 


E 


iy lbs. per foot 


10 


" 


D 


10 oz. per foot 


6 


" 


D 


2 " " 


11 


" 


C 


12 " 


8 


" 


C 


2y 2 " 


14 


" 


B 


1 lb. 


12 


" 


B 


sy 4 " 


17 


" 


A 


M " 


16 


" 


A 


4 « « 


21 


" 


AA 


\y « 


19 


" 


AA 


4% " 


24 


" 


AAA. 


m « 


27 


" 


AAA 


6 " 


30 


7-16 in. 




13 oz. 




1?4 in- 


E 


2 " 


10 


" 




1 lb. 






D 


2% " 


12 


y 2 m. 


E 


9 lbs. per rod 


7 


" 


C 


3 " 


14 




D 


% lb. per foot 


9 


" 


B 


4% " 


16 


" 


C 


1 " 


11 


'■ 


A 


19 


" 


B 


Wa " 


13 


" 


AA 


Wa " '• 

6% " 


25 


" 




\y " 

m " 




" 


AAA 




" 


A 


16 


l^in. 


E 


3 " 


12 


" 


AA 


2 " 


19 




D 


sy 2 " 


14 


" 




2^ <; 


23 


" 


C 


4M " 


17 




AAA 


3 " 


25 


" 


B 


5 " 


19 


%}n. 


. E 


12 " per rod 


8 


" 


A 


6^ " 


23 




D 


1 " per foot 


9 


" 


AA 


8 " 


27 


" 


C 


m ;; 


13 


" 


AAA 


9 " 




" 


B 




16 


l%in. 


C 


4 « 


13 


" 


A 


2y 2 " 


20 


B 


5 " " 


17 


" 


AA 


m « 


22 


" 


A 


6^ " 


21 


" 


AAA 


3^ " 


25 


" 


AA 


8^ " 


27 


%in.- 


E 


1 •' per foot 


8 


2 in. 


C 


4.% " 


15 




D 


1M " 


10 


" 


B 


6 " 


18 


" 


C 


2^ " 


12 


" 


A 


7 " 


22 


" 


B 


16 


" 


AA 


9 " 


27 


" 


A 


3 " 


20 


" 


AAA 


11% " 




" 


AA 


31^ " 


23 










" 


AAA 


4% " 


30 











WEIGHT OF L.EAD PIPE WHICH SHOULD BE USED 
FOR A GIVEN HEAD OF WATER. 

(Tatham & Bros., New York.) 



Head or 


Pressure 

per 
sq. inch. 




Calibre and Weight 


per Foot. 




of Feet 
Fall. 


Letter. 


%inch. 


y% inch. 


% inch. 


%inch. 


1 inch. 


Mm. 


30 ft, 
50 ft. 
75 ft. 

100 ft. 

150 ft. 

200 ft. 


15 lbs. 
25 lbs. 
38 lbs. 
50 lbs. 
75 lbs. 
100 lbs. 


D 

C 

B 

A 
AA 
AAA 


10 oz. 
12 oz. 

1 lb. 

1)4 lbs. 

1*6 lbs. 

\% lbs. 


JOS. 

1 lb. 

1% lbs. 
1M lbs. 

2 lbs. 

3 lbs. 


1 lb. 
\y 2 lbs. 

2 lbs. 
2}4 lbs. 
2% lbs. 
Sy lbs. 


\y± lbs. 

1H lbs. 
2J4 lbs. 
3 lbs. 
m lbs. 
4% lbs. 


2 lbs. 

2y lbs. 
zy 4 lbs. 

4 lbs. 
4% lbs. 
6 lbs. 


214 lbs. 
3 lbs. 
3% lbs. 
4% 1. s. 
6 lbs. 
Wa lbs. 



To find the thickness of lead pipe required when the 
head of water is given. (Chadwick Lead Works). 

Rule.— Multiply the head in feet by size of pipe wanted, expressed deci- 
mally, and divide by 750; the quotient will give thickness required, in one- 
hundredths of an inch. 

Example.— Required thickness of half -inch pipe for a head of 25 feet. 

25 X 0.50 -i- 750 = 0.16 inch. 



202 



MATERIALS. 



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BOLT COPPER — SHEET AND BAR BRASS. 



203 



WEIGHT OF ROUND BOUT COPPER. 

Per Foot. 



Inches. 


Pounds. 


Inches. 


Pounds. 


Inches. 


Pounds. 


% 


.425 




3.02 


1% 


7.99 


^2 


.756 


V/a 


3.83 


m 


9.27 


% 


1.18 


m 


4.72 


m 


10.64 


H 


1.70 


m 


5.72 




12.10 


% 


2.31 


m 


6.81 







WEIGHT OF SHEET AND BAR BRASS. 



Thick- 


Sheets 


Square 
Bars 


Round 


Thick- 


Sheets 


Square 
Bars 


Round 


ness. 


per 


Bars 


ness. 


per 


Bars 


Inches. 


sq. ft. 


1 ft. long. 


1 ft. long. 


Inches. 


sq. ft. 


1 ft. long. 


1 ft. long. 




lbs. 


lbs. 


lbs. 




lbs. 


lbs. 


lbs 


1-16 


2.7 


.015 


.011 


1 1-16 


45.95 


4.08 


3.20 


% 


5.41 


.055 


.045 


1^ 


48.69 


4.55 


3.57 


3-16 


8.12 


.125 


.1 


1 3-16 


51.4 


5.08 


3.97 


34 


10.76 


.225 


.175 


«4 


54.18 


5.65 


4.41 


5-16 


13.48 


.350 


.275 


1 5-16 


56.85 


6.22 


4.86 


% 


16.25 


.51 


.395 


m 


59 55 


6.81 


5.35 


7-16 


19. 


.69 


.54 


1 7-16 


62.25 


7.45 


5.85 


5 A 


21.65 


.905 


.71 


M 


65 


8.13 


6.37 


9-16 


24.3 


1.15 


.9 


1 9-16 


67.75 


8.83 


6.92 


% 


27.12 


1.4 


1.1 


m 


70.35 


9.55 


7.48 


11-16 


^9.77 


1.72 


1.35 


1 11-16 


73 


10.27 


8.05 


% 


32.46 


2.05 


1.66 


m 


75.86 


11 


8.65 


13-16 


35.18 


2.4 


1.85 


1 13-16 


78.55 


11.82 


9.29 


% 


37.85 


2.75 


2.15 


1% 


81.25 


12.68 


9.95 


15-16 


40.55 


3.15 


2.48 


1 15-16 


84 


13.5 


10.58 


1 


43.29 


3.65 


2.85 


2 


86.75 


14.35 


11.25 



COMPOSITION OF VARIOUS GRADES OF ROUUED 
BRASS, ETC. 



Trade Name. 


Copper 


Zinc. 


Tin. 


Lead. 


Nickel. 




61.5 

60 

66^ 

80 

60 

60 

Qiy 2 


38.5 

40 

33^ 

20 

40 

40 

33^ 

20^ 


































"iy 2 " 


Wz 
l^to2 




Drill rod 








18 per cent German silver 




IS 



The above table was furnished by the superintendent of a mill in Connec- 
ticut in 1894. He says: While each mill has its own proportions for various 
mixtures, depending upon the purposes for which the product is intended, 
the figures given are about the average standard. Thus, between cartridge 
brass with 33^g per cent zinc and common high brass with 38^ per cent 
zinc, there are any number of different mixtures known generally as "high 
brass," or specifically as "spinning brass, " "drawing brass," etc., wherein 
the amount of zinc is dependent upon the amount of scrap used in the mix- 
ture, the degree of working to which the metal is to be subjected, etc. 



204 



MATERIALS. 



AMERICAN STANDARD SIZES OF DROP-SHOT. 























^N 






o ^ 






9n 




Diameter. 






Diameter. 


6 o 




Diam- 
eter. 


6 o 


Fine Dust. 


3-100" 


10784 


No. 8 


Trap Shot 


472 


No. 2. . . . 


15-100" 


86 


Dust 


4-100 


4565 


" 8 


9-100" 


399 


" 1.. . 


16-100 


71 


No. 12 


5-100 


2326 


" 7 


Trap Shot 


338 


" B... 


17-100 


59 


" 11 


6-100 


1346 


" 7 


10-100" 


291 


" BB. 


18-100 


50 


" 10 


Trap Shot 


1056 


" 6 


11-100 


218 


" BBB 


19-100 


42 


" 10 


7-100" 


848 


" 5 


12-100 


168 


" T . . 


20-100 


36 


" 9 


Trap Shot 


688 


" 4 


13-100 


132 


" TT.. 


21-100 


31 


" 9 


8-100" 


568 


" 3 


14-100 


106 


" F.. 
" FF.. 


22-100 
23-100 


27 

24 





COMPRESSED RUCK-SHOT. 






Diameter. 


No. of Balls 
to the lb. 




Diameter. 


No. of Balls 
to the lb. 


No. 3 


25-100" 
27-100 
30-100 
32-100 


284 
232 
173 
140 


No. 00 

" 000 

Balls 


34-100" 
36-100 
38-100 
44-100 


115 


" 2 


98 


" 1 


85 


" . 


50 









SCREW-THREADS, SEINERS OR U. S. STANDARD. 

In 1864 a committee of the Franklin Institute recommended the adoption 
of the system of screw-threads and bolts which was devised by Mr. William 
Sellers, of Philadelphia. This same system was subsequemiy adopted as 
the standard by both the Army and Navy Departments of the United States, 
and by the Master Mechanics' and Master Car Builders' Associations, so 
that it may now be regarded, and in fact is called, the United States Stan- 
dard. 

The rule given by Mr. Sellers for proportioning the thread is as follows : 
Divide the pitch, or, what is the same thing, the side of the thread, into 
eight equal parts; take off one part from the top and fill in one part in the 
bottom of the thread; then the flat top and bottom will equal one eighth of 
the pitch, the wearing surface will be three quarters of the pitch, and the 
diameter of screw at bottom of the thread will be expressed by the for- 
mula 



diameter of bolt - 



" no. threads per inch 
For a sharp V thread with angle of 60° the formula is 

1.733 



diameter of bolt - 



no. of threads per inch 
The angle of the thread in the Sellers system is 60°. In the Whitworth or 
English system it is 55°, and the point and root of the thread are rounded. 
Screw-Threads, United States Standard. 



§ 




e3 




§ 


fl 


s 




s 

63 


,C 


Q 


Ph 


Q 


Ph 


P 


Ph 


• Q 


Ph 


ft 


Ph 


u 


20 


H 


10 


VA 


~ 


1 15-16 


5 


2 13-16 


3^ 
3V£ 


5-16 


18 


13-16 


10 


1 5-16 


6 


2 


4U 


3 


Va 


16 


% 


9 


1% 


6 


2^ 


4Y> 


m 


SV> 


7-16 


14 


15-16 


9 


VA 


6 


2 5-16 


3 5-16 


V/\ 


H 


13 


1 


8 


m 


5K 


m 


4 


m 


m 


9-16 


12 


1 1-16 


7 


m 


5 


m 


4 


m 


3 


% 


11 


m 


7 


Wa 


5 


2% 


4 


4 


3 


11-16 


11 



















U. S. OR SELLERS SYSTEM OF SCREW-THREADS. 205 



Screw-Tlireads, Whit worth (English) Standard. 



a 


,£3 


a 


o 


a" 


A 

o 


s 


"o 


i 





ft 


s 


Q 


S 


P 


Pn 


s 


Oh 


Q 


s 


4 


20 


96 


n 


1 


8 


m 


5 


3 


% 


5-16 


18 


11-16 


n 


XYa 


7 


Ws 


m, 


34 


m 


% 


16 


13-16 


10 


m 


7 


2 


4H 


3?4 


m 


7-16 


14 


10 


6 


24 


4 


m 


3 


% 


12 


Vs 


9 


Ws 


6 


zy* 


4 


4 


3 


9-16 


12 


15-16 


9 


5 


m 


3J/ 2 







u. 


S. 


OR SEINERS SYSTEM OF SCREW-THREADS. 


BOLTS AND THREADS. 


HEX. NUTS AND HEADS. 




^ 




« 


^ 




°,2 


.. 


, 








4a 



W 



g 

s 


ft 

5 2 

H 




o3 

5 
■g 


M.S rA 

< 


g.s 

aJEi 2 

< 


a . 

■£ ° 


a* 

in 


3 


oil 


C <» 
EH 




Ins. 




Ins. 


Ins. 






Ins. 


Ins. 


Ins. 


Ins. 


Ins. 


Ins. 


4 


20 


.185 


.0062 


.049 


.027 


^ 


7-16 


37-64 


4 


3-16 


■ 7-10 


5-16 


18 


.240 


.0074 


.077 


.045 


19-32 


17-32 


11-16 


5-16 


4 


10-12 


% 


16 


.294 


.0078 


.110 


.068 


11-16 


% 


51-64 


% 


5-16 


63-64 


7-16 


14 


.344 


.0089 


.150 


.093 


25-32 


23-32 


9-10 


7-16 


% 


17-64 


% 


13 


.400 


.0096 


.196 


.126 


% 


13-16 


1 


^ 


7-16 


1 15-64 


9-16 


12 


.454 


.0104 


.249 


.162 


31-32 


29-32 


M 


9-16 


y 2 


1 23-64 


% 


11 


.507 


.0113 


.307 


.202 


11-16 


1 


1 7-32 


% 


9-16 


m 


% 


10 


.620 


.0125 


.442 


.302 


1M 


13-16 


17-16 


M 


11-16 


1 49-64 


Vs 


9 


.731 


.0138 


.601 


.420 


17-16 


W* 


1 21-32 


% 


13-16 


2 1-32 


1 


8 


.837 


.0156 


.785 


.550 1% 


19-16 


W% 


1 


15-16 


2 19-64 


1^ 


7 


.940 


.0178 


.994 


.694 1 13-16 


194 


23-32 


1? 


^ 


1 1-16 


2 9-16 


14 


7 


1.065 


.0178 


1.227 


.893 


2 


1 15-16 


2 5-16 


l 1 


4 


13-16 


2 53-64 


1^ 


6 


1.160 


.0208 


1.485 


1.057 


2 3-16 


m 


2 17-32 \\ 




1 5-16 


3 3-32 


1^1 


6 


1.284 


.0208 


1.767 


1.295 


2% 


2 5-16 


m 13 


4 


17-16 


3 23-64 


5Vo 


1.3S9 


.0227 


2.074 


1.515 


2 9-16 


2^ 


2 31-32 If 


I 


19-16 


3% 


P 


5 


1.491 


.0250 


2.405 


1.746 


2^ 


2 11-16 


3 3-16 Is 


i 


1 11-16 


3 57-64 


5 


1.616 


.0250 


2.761 


2.051 


2 15-16 


2% 


3 13-32; r 


4 


1 13-16 


4 5-32 




1% 


1.712 


.0277 


3.142 


2.302 


3^ 


3 1-16 


3% 12 ' 


1 15-16 


4 27-64 


24 


4 l a 


1.962 


.0277 


3.976 


3.023 


Wz 


3 7-16 


4 1-16 2J4 


2 3-16 


4 61-64 


^ 


4 


2.176 


.0312 


4.909 


3.719 


3% 


3 13-16 


4y 2 Wz 


2 7-16 


5 31-C4 


2% 


4 


2.426 


.0312 


5.940 


4.620 


44 
4% 


4 3-16 


4 29-32 2% 


211-16 


6 


3 


3^3 


2.629 


.0357 


7.069 


5.428 


4 9-16 


5% 3 


2 15-16 6 17-32 


3^1 


3^ 


2.S71 


.0357 


8.296 


6.510 


5 


4 15-16 


5 13-16 314 


3 3-16 1 7 1-16 


34 


3.100 


.0384 


9.621 


7.548 


5% 


5 5-16 


6 7-64 3J^ 


3 7-16 7 39-61 


3% 


3 


3.317 


.0113 


11.045 


8.641 


Wa 


5 11-16 


6 21-32 3% 


3 11-161 8^ 


4 


3 


3.567 


.0413 


12.566 


9.963 


$8 


6 1-16 


7 3-32 4 


3 15-16 8 41-64 


44 


•- >r .s 


3.798 


.0435 


14.186 


11.329 


6^2 


6 7-16 


7 9-16 44 


4 3-16 1 9 3-16 


4 l A 


•-• ;i 4 


4.028 


.0454 


15.004 


12.753 


6^ 


6 13-16 


7 31-32 4^ 


4 7-16 1 9% 


4% 


2% 


4.256 


.0476 


17.721 


14.226 


-4 


7 3-16 


8 13-32 4% 


4 11-16 10J4 


5 


2>2 


4.480 


.0500 


19.635 


15.763 


7 9 16 


8 27-32 5 


4 15-16 10 49-64 


54 


'■ihi 


4.730 


.0500 


2l.l',4S 


17.572 


8 


7 15-16 


9 9-32 54 


5 3-16 11 23-64 


5^ 




4.953 


.0526 


23.758 


19.267 


8% 


8 5-16 


9 23-32 5^ 


5 7-16 uy 8 


5M 




5.203 


.0526 


25.967 


21.262 


8^ 


8 11-16 


10 5-32 5% 


5 11-16 12% 


6 


%Y4, 


5.423 .0555 
1 


28.274 


23.098 


W& 


9 1-16 


10 19-32 6 


5 15-16 12 15-16 



lilMIT GAUGES FOR IRON FOR SCREW THREADS. 

In adopting the Sellers, or Franklin Institute, or United States Standard. 
as it is variously called, a difficulty arose from tbe fact that it is the habit 
of iron manufacturers to make iron over- size, and as there are no over-size 



206 



MATERIALS. 



screws in the Sellers system, if iron is too large it is necessary to cut it away 
with the dies. So great is this difficulty, that the practice of making taps 
and dies over-size has become very general. If the Sellers system is adopted 
it is essential that iron should be obtained of the correct size, or very nearly 
so. Of course no high degree of precision is possible in rolling iron, and 
when exact sizes were demanded, the question arose how much allowable 
variationjthere should be from the true size. It was proposed to make limit- 
gauges for inspecting iron with two openings, one larger and the other 
smaller than the standard size, and then specify that the iron should enter 
the large end and not enter the small one. The following table of dimen- 
sions for the limit-gauges was recommended by the Master Car-Builders' 
Association and adopted by letter ballot in 1883. 





Size of 


Size of 






Size of 


Size of 




Size of 


Large 


Small 


Differ- 


Size of 


Large 


Small 


Differ- 


Iron. 


End of 


End of 


ence. 


Iron. 


End of 


End of 


ence. 




Gauge. 


Gauge. 






Gauge. 


Gauge. 




14 in. 


0.2550 


0.2450 


0.010 


96 in. 


0.6330 


0.6170 


0.016 


5-16 


0.3180 


0.3070 


0.011 


H 


0.7585 


0.7415 


0.017 


% 


0.3810 


0.3690 


.0.012 


% 


0.8840 


0.8660 


0.018 


7-16 


0.4440 


0.4310 


9.013 


1 


1.0095 


0.9905 


0.019 


% 


0.5070 


0.4930 


0.014 


m 


1.1350 


1.1150 


0.020 


9-16 


0.5700 


0.5550 


0.015 


m 


1.2605 


1.2395 


0.021 



Caliper gauges with the above dimensions, and standard reference gauges 
for testing them are made by the Pratt & Whitney Co. 

THE MAXIMUM VARIATION IN SIZE OF ROUGH 
IRON FOR U. S. STANUARD ROUTS. 

Am. Mach., May 12, 1892. 

By the adoption of the Sellers or U. S. Standard thread taps and dies keep 
their size much longer in use when flatted in accordance with this system 
than when sharp, though it has been found advisable in practice in most 
cases to make the taps of somewhat larger outside diameter than the nom- 
inal size, thus carrying the threads further towards the V-shape and giving 
corresponding clearance to the tops of the threads when in the nuts or 
tapped holes. 

Makers of taps and dies often have calls for taps and dies, U. S. Standard, 
" for rough iron." 

An examination of rough iron will show that much of it is rolled out of 
round to an amount exceeding the limit of variation in size allowed. 

In view of this it may be desirable to know what the extreme variation in 
iron may be, consistent with the maintenance of U. S. Standard threads, i.e., 
threads which are standard when measured upon the angles, the only place 
where it seems advisable to have them fit closely. Mr. Chas. A. Bauer, the 
general manager of the Warder. Bushnell & Glessner Co., at Springfield, 
Ohio, in 1884 adopted a plan which may be stated as follows : All bolts, 
whether cut from rough or finished stock, are standard size at the bottom 
and at the sides or angles of the threads, the variation for fit of the nut and 
allowance for wear of taps being made in the machine taps. Nuts are 
punched with holes of such size as to give 85 per cent of a full thread, expe- 
rience showing that the metal of wrought nuts will then crowd into the 
threads of the taps sufficiently to give practically a full thread, while if 
punched smaller some of the metal will be cut out by the tap at the bottom 
of the threads, which is of course undesirable. Machine taps are made 
enough larger than the nominal to bring the tops of the threads up sharp, 
plus the amount allowed for fit and wear of taps. This allows the iron to 
be enough above the nominal diameter to bring the threads up full (sharp) 
at top, while if it is small the only effect is to give a flat at top of tli reads ; 
neither condition affecting the actual size of the thread at the point at which 
it is intended to bear. Limit gauges are furnished to the mills, by which the 
iron is rolled, the maximum size being shown in the third column of the 
table. The minimum diameter is not given, the tendency in rolling being 
nearly always to exceed the nominal diameter. 

In making the taps the threaded portion is turned to the size given in the 
eighth column of the table, which gives 6 to 7 thousandths of an inch allow- 
ance for fit and wear of tap. Just above the threaded portion of the tap a 



SIZES OF SCREW-THREADS FOR BOLTS AND TAPS. 207 

place is turned to the size given in the ninth column, these sizes being the 
same as those of the regular U. S. Standard bolt, at the bottom of the 
thread, plus the amount allowed for fit and wear of tap ; or, in other words, 
d' = U. S. Standard d + (D' - D). Gauges like the one in the cut, Fig. 
72, are furnished for this sizing. In finishing the threads of the tap a tool 




Fig. 72. 
is used which has a removable cutter finished accurately to gauge by grind- 
ing, this tool being correct U. S. Standard as to angle, and flat at the point. 
It is fed in and the threads chased until the flat point just touches the por- 
tion of the tap which has been turned to size d'. Care having been taken 
with the form of the tool, with its grinding on the top face (a fixture being 
provided for this to insure its being ground properly), and also with the set- 
ting of the tool properly in the lathe, the result is that the threads of the tap 
are correctly sized without further attention. 

« It is evident that one of the points of advantage of the Sellers system is 
sacrificed, i.e., instead of the taps being flatted at the top of the threads 
they are sharp, and are consequently not so durable as they otherwise would 
be ; but practically this disadvantage is not found to be serious, and is far 
overbalanced by the greater ease of getting iron within the prescribed 
limits ; while any rough bolt when reduced in size at the top of the threads, 
by filing or otherwise, will fit a hole tapped with the U. S. Standard hand 
taps, thus affording proof that the two kinds of bolts or screws niade for the 
two different kind* of work are practically interchangeable. By this system 
\" iron can be .005'' smaller or .0108" larger than the nominal diameter, or, 
in other words, it may have a total variation of .0158", while \\" iron can be 
.0105" smaller or .0309" larger than nominal— a total variation of .0414"— 
and within these limits it is found practicable to procure the iron. 
STANDARD SIZES OF SCREW-THREADS FOR BOIiTS 
AND TAPS. 
(Chas. A. Bauer.) 



1 


2 


3 


4 


5 


6 


7 


8 


9 


10 


A 


u 


D 


d 


h 


/ 


D'-D 


D' 


d{ 


H 






Inches. 


Inches. 


Inches. 


Inches. 


Inches. 


Inches. 


Inches. 


Inches. 


5-16 


20 


.2608 


.1855 


.0379 


.0062 


.006 


.2668 


.1915 


.2024 


IS 


.3245 


.2403 


.0421 


.0070 


.006 


.3305 


.2463 


.2589 


% 


16 


.3885 


.2938 


.0474 


.0078 


.006 


.3945 


.2998 


.3139 


7-16 


14 


.4530 


.3447 


.0541 


.0089 


.006 


.4590 


.3507 


.3670 


% 


13 


.5166 


.4000 


.0582 


.0096 


.006 


.5226 


.4060 


.4236 


9-16 


12 


.5805 


.4543 


.0631 


.0104 


.007 


.5875 


.4613 


.4802 


3 A 


11 


.6447 


.5069 


.0689 


.0114 


.007 


.6517 


.5139 


.5346 


4 


10 


.7717 


.6201 


.0758 


.0125 


.007 


.7787 


.6271 


.6499 




9 


.8991 


.7307 


.0842 


.0139 


.007 


.9061 


.7377 


.7630 


i 


8 


1.0271 


.8376 


.0947 


.0156 


.007 


1.0341 


.8446 


.8731 


m 


7 


1.1559 


.9394 


.1083 


.0179 


.007 


1.1629 


.9464 


.9789 


7 


1.2809 


1.0644 


.1083 


.0179 


.007 


1.2879 


1.0714 


1.1039 



A = nominal diameter of bolt. 
D = actual diameter of bolt. 

d = diameter of bolt at bottom of 

thread. 
71 = number of threads per inch. 

/ = flat of bottom of thread. 

h = depth of thread. 
D' and d' = diameters of tap. 
H — hole in nut before tapping. 



D 

d 

h 

f=i 

H= D> 



' n 
1.29904 

.7577 D-d 



■ TV - .85(2/i.) 



208 



MATERIALS. 



STANDARD SET-SCREWS AND CAP-SCREWS. 

American, Hartford, and Worcester Machine-Screw Companies. 
(Compiled by W. S. Dix.) 



Diameter of Screw. . . . 

Threads per Inch 

Size of Tap Drill* 


(A) 
No. 43 


(B) 

3-16 

24 

No. 30 


(C) 

Va 

20 
No. 5 


(D) 

5-16 

18 

17-64 


(E) 

¥s 

16 

21-64 


(F) 
7-16 
14 
% 


(G) 

12 
27-64 


Diameter of Screw.... 

Threads per Inch 

Size of Tap Drill*.... 


(H) 

9-16 

12 

31-64 


(1) 

% 
11 
17-32 


(J) 

ft 

21-32 


(K) 

V i 
49-64 


(L) 

1 

8 
% 


(M) 

7 
63-64 


(N) 

m 



Set Screws. 


Hex. Head Cap-screws. 


Sq. Head Cap-screws, 


S sort 
Diam 


Long 
Diam. 


Lengths 
(under 


Short 
Diam. 

of 
Head. 


Long 
Diam, 

of 
Head. 


Lengths 
(under 


Short 
Diam. 

of 
Head. 


Long 
Diam. 

of 
Head. 


Lengths 
(under 


of Head 


of Head 


Head). 


Head). 


Head). 


(C) V A 


.35 


%to3 


7-16 


.51 


% to 3 

%to3M 


% 


.53 


%to 3 

%to3M 
M to 3^ 


(D) 5-16 


.44 


34 to 3J4 


y* 


.58 


7-16 




62 


(E) % 


.53 


%to3^ 


9-16 


.65 


% to 3y 2 


y 2 




71 


(F) 7-16 


.62 


34 to 3% 
9|to4 


n 


.72 


34 to Wa 


9-16 




so 


Mto3% 


(G) ^ 


.71 


% 


.87 


%to4 


% 




89 


U to 4 


(H) 9-16 


.80 


% to 4y 4 
%to4y 2 


13-16 


.94 


H to 4y 4 


11-16 




98 


¥a to 414 


(1) % 


.89 


Va 


1.01 


1 to 4% 


U 


1 


06 


1 to4V£ 
14 to 4Ya 


(J) Va 


1.06 


1 to 4% 


1 


1.15 


Wa to 4% 

\y 2 to 5 

Wa to 5 

2 to 5 


Vs 


1 


24 


(K) % 


1.24 


1M to 5 


V/a 


1.30 


m 


1 


00 


\y± to 5 


(L) 1 


1.42 


\\b to 5 
13^ to 5 


1M 


1.45 


m 


1 


77 


m to 5 


(M) iy 8 


1.60 




1.59 


1 


1)5 


2 to 5 


(N) m 


1.77 


2 to 5 


m 


1.73 


2 to 5 


m 


2 


13 


234 to 5 



Round and Filister Head 
Cap-screws. 


Flat Head Cap-screws. 


Button-head Cap- 
screws. 


Diam. of 
Head. 


Lengths 

(under 

Head). 


Diam. of 
Head. 


Lengths 

(including 

Head). 


Diam. of 
Head. 


Lengths 
(under 
Head). 


(A) 3-16 


Mto2J^ 


y± 


%-to 1% 


7-32 (.225) 


MtolM 


(B) H 


M to 234 


% 


34 to 2 


5-16 


34 to 2 


(O % 


Mto3 


15-32 


34 to 24 
%to2% 
% to 3 


7-16 


Mto2M 

M to 2^ 


(D) 7-16 


M to 314 


it 


9-16 


(E) 9-16 


%to3^ 
% to 3M 


% 


M to 2M 
% to 3 


(F) % 

(G) M 


13-16 


1 to 3 


¥a 


M to 4 


Va 


14 to 3 


13-16 


1 to 3 


(H) 13-16 


1 to 4J4 

Wa to 4^ 

Wz to 4% 


1 


l^to3 
l%to3 


15-16 


1J4 to 3 


0) % 


1% 


1 


l^to3 


(J) 1 


2 to 3 


1M 


1M to 3 


(K) iy 8 


1M to 5 
2 to 5 










(L) 14 











* For cast iron. 

Threads are U. S. Standard. Cap-screws are threaded 34 length up to and 
including 1" diam. x 4" long, and }/& length above. Lengths increase by 4" 
each regular size between the limits given. Lengths of heads, except flat 
and button, equal diam. of screws. 

The angle of the cone of the flat-head screw is 76°, the sides making angles 
of 52° with the top. 



STANDARD MACHINE SCREWS. 



209 



STANDARD MACHINE SCREWS. 

(Am. Screw Co.'s Catalogue, 1883, 1892.) 



No. 


Threads per 


Diam. of 


Diam. 
of Flat 
Head. 


Diam. of 
Round 
Head. 


Diam. of 
Filister 
Head. 


Lengths. 


Inch. 


Body. 


From 


To 


2 


56 


.0842 


.1631 


.1544 


.1332 


3-16 


k 


3 


48 


.0973 


.1894 


.1786 


.1545 


3-16 


% 


4 


32, 36, 40 


.1105 


.2158 


.2028 


.1747 


3-16 


% 


5 


32, 36, 40 


.1236 


.2421 


.2270 


.1985 


3-16 


Vs 


6 


30, 32 


.1368 


.2684 


.2512 


.2175 


3-16 




7 


30,32 


.1500 


.2947 


.2754 


.2392 


M 


m 


8 


30,32 


.1631 


.3210 


.2936 


.2610 


J4 


n 


9 


24, 30, 32 


■ .1763 


.3474 


.3238 


.2805 


H 


10 


24, 30, 32 


.1894 


.3737 


.3480 


.3035 


H 


M 


12 


20, 24 


.2158 


.4263 


.3922 


.3445 


% 


m 


14 


20, 24 


.2421 


.4790 


.4364 


.3885 


3 A 


2 


16 


16, 18, 20 


.2684 


.5316 


.4866 


.4300 


% 


m 


18 


16, 18 


.2947 


.5842 


.5248 


.4710 


Y2 


%\£ 


20 


16, 18 


.3210 


.63CS 


.5690 


.5200 


Vi 


m 


22 


16, 18 


.3474 


.6894 


.6106 


.5557 


% 


3 


24 


14, 16 


.3737 


.7420 


.6522 


.6005 


y* 


3 


26 


14, 16 


.4000 


.7420 


.6938 


.6425 




3 


28 


14, 16 


.4263 


.7946 


.7354 


.6920 


3 


30 


14, 16 


.4520 


- .8473 


.7770 


.7240 


i 


3 



Lengths vary by 16ths from 3-16 to y>, by Sths from y> to 1*4, by 4ths from 
1*4 to 3. 

SIZES AND WEIGHTS OF SQUARE AND 

HEXAGONAL NUTS. 

United States Standard Sizes. Chamfered and trimmed. 

Punched to suit U. S. Standard Taps. 



£ 








6< 


S-J8 


Square. 


Hexagon. 






a 

o 


M 
o 

a 
s 


is 

a 
o 


tCtJ 

2% 










o 

a 
s 


o 
o 

C to 




§ 

to 




% 


M 


V\ 


13-64 


11-16 


9-16 


7270 


.0138 


7615 


.0131 


5-16 


19-32 


5-16 


H 


13-16 


11-16 


4700 


.0231 


5200 


.0192 


% 


11-16 


% 


19-64 


1 


13-16 


2350 


.0426 


3000 


.0333 


7-16 


25-32 


7-16 


11-32 


Ws, 


% 


1630 


.0613 


2000 


.050 


H 


% 


y, 


25-64 




1 


1120 


.0893 


1430 


.070 


9-16 


31-32 


9-16 


29-64 


1^ 


890 


.1124 


1100 


.091 


% 


1 1-16 


% 


33-64 


i% 


m 


640 


.156 


740 


.135 


« 


\V± 


'*4 


39-64 


m 


1 7-16 


380 


.263 


450 


.222 


1 7-16 


% 


47-64 


2 1-16 


1 11-16 


280 


.357 


309 


.324 




i% 


1 


53-64 


2 5-16 


1% 


170 


.588 


216 


.463 


■?] 


4 


1 13-16 


1^ 


59-64 


2 9-16 


2 1-16 


130 


.769 


148 


.676 


1 


A 


2 


W A 


1 1-16 


2 13-16 


2 5-16 


96 


1.04 


111 


.901 


y. 


A 


2 3-16 


\% 


1 5-32 


W& 


2% 


70 


1.43 


85 


1.18 


1 


/» 


2% 


% 


1 9-32 


3% 


m 


58 


1.72 


68 


1.47 






2 9-16 


1 13-32 


3% 


2 15-16 


44 


2.27 


56 


1.79 






2% 


W A 


n 


Ws 


3 3-16 


34 


2.94 


40 


2.50 


y 


/r 


2 15-16 


m 


4y 8 


3% 


30 


3.33 


37 


2.70 


2 


%y& 


2 


1 23-32 


4 7-16 


3% 


23 


4.35 


29 


3.45 


2J4 


Wz 


2M 


1 15-16 


4 15-16 


4 1-16 


19 


5.26 ' 


21 


4.76 


Wh 


2Yo 


2 3-16 


5^ 


4^ 


12 


8.33 


15 


6.67 


2% 


4k 


23/j 


2 7-16 


6 


4 15-16 


9 


11.11 


11 


9.09 


3 


4% 


3 


2% 


Wz 


5 5-16 


m 


13.64 


Wfl 


11.76 



210 



MATERIALS, 



WEIGHTS OF 100 BOL.TS WITH SQUARE HEADS 
AND NUTS. 







(Hoopes & Townsend'. 


> List.) 








Length un- 


Diameter of Bolts. 






der Head 










to Point. 


J4 in. 5-16 in. 


%in. 


7-16 in. 


^in. 


%va. 


%in. 


%\n. 


lin. 




lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


M 


4.00 


7.00 


10.50 


15.20 


22.50 


39.50 


63.00 






m 


4.35 


7.50 


11.25 


16.30 


23.82 


41.62 


66.00 






2 


4.75 


8.00 


12.00 


17.40 


25.15 


43.75 


69.00 


109.00 


163 


&A 


5.15 


8.50 


12.75 


18.50 


26.47 


45.88 


72.00 


113.25 


169 


w/z 


5.50 


9.00 


13.50 


19.60 


27.80 


48.00 


75.00 


117.50 


174 


m 


5.75 


9.50 


14.25 


20.70 


29.12 


50.12 


78.00 


121.75 


180 


3 


6.25 


10.00 


15.00 


21.80 


30.45 


52.25 


81.00 


126.00 


185 


3^ 


7.00 


11.00 


16.50 


24.00 


33.10 


56.50 


87.00 


134.25 


196 


4 


7.75 


12.00 


18.00 


26.20 


35.75 


60.75 


93.10 


142.50 


207 


m 


8.50 


13.00 


19.50 


28.40 


38.40 


65.00 


99.05 


151.00 


218 


5 


9.25 


14.00 


21.00 


30.60 


41.05 


69.25 


105.20 


159.55 


229 


5^ 


10.00 


15.00 


22.50 


32.80 


43.70 


73.50 


111.25 


168.00 


240 


6 


10.75 


16.00 


24.00 


35.00 


46.35 


77.75 


117.30 


176.60 


251 


6^ 






25.50 


37.20 


49.00 


82.00 


123.35 


185.00 


262 


7 






27.00 


39.40 


51.65 


86.25 


129.40 


193.65 


WS 


V* 






28.50 
30.00 


41.60 
43.80 
46.00 


54.30 
59.60 
64.90 


90.50 
94.75 
103.25 


135.00 
141.50 
153.60 


202.00 
210.70 

227.75 


284 






295 


9 






317 


10 








48.20 
50.40 
52.60 


70.20 
75.50 
80.80 
86.10 
91.40 
96.70 
102.00 
107.30 
112.60 
117.90 
123.20 

5.45 


111.75 
120.25 
128.75 
137.25 
145.75 
154.25 
162.75 
171.00 
179.50 
188.00 
206.50 

8.52 


165.70 
177.80 
189.90 
202.00 
214.10 
226.20 
238.30 
250.40 
262.60 
274.70 
286.80 

12.27 


224.80 
261.85 
278.90 
295.95 
313.00 
330.05 
347.10 
364.15 
381.20 
398.25 
415.30 

16.70 


339 


11 








360 


12 








382 


13 








404 


14 










426 


15 










448 


16 










470 


17 










492 


18 










514 


19 










536 


20 










558 


Per inch 
additional. 


[■1.87 


2.13 


3.07 


4.18 


21.82 



TRACK BOLTS. 
Witli United States Standard Hexagon Nuts. 



Rails used. 


Bolts. 


Nuts. 


No. in Keg, 
200 lbs. 


Kegs per Mile. 


r 


%x4^£ 


m 


230 


6.3 




%x4 


m 


240 


6. 


45to851bs..J 


M*3^ 


m 

VA 


254 
260 


5.7 
5.5 


1 


m 


266 


5.4 


I 


Mx3 


1M 


283 


5.1 


r 


%x3^ 


1 1-16 


375 


4. 


30 to 40 lbs...-! 


%x3 

¥8*m 


1 1-16 
1 1-16 


410 
435 


3.7 
3.3 


1 


%*zy2 


1 1-16 


465 


3.1 


r 


^x3 


Vs 


715 


2. 


20to301bs..J 




Vs 
Vs 


760 
800 


2. 
2. 


I 




% 


820 


2. 



NUTS AND BOLT-HEADS — RIVETS. 



211 



WEIGHTS OF NUTS AND BOLT-HEADS, IN POUNDS. 
For Calculating the Weight of Longer Dolts. 



Diameter of Bolt, in Inches. 




X A 


% 


K 


% 


H 


% 


Weight of hexagon nut and head. 
Weight of square nut and head . . 




.017 
.021 


.057 
.069 


.128 
.164 


.267 
.320 


.43 

.55 


.13 

.88 


Diameter of Bolt, in Inches. 


1 


m 


m 


m 


2 


17 
21 


3 


Weight of hexagon nut and head. 
Weight of square nut and head . . 


1.10 
1.31 


2.14 

2.56 


3.78 
4.42 


5.6 

7.0 


8.75 
10.5 


28. S 
36.4 





NUMBER OF RIVETS IN 100 POUNDS. 




Lengths. 


96 in. 


7-16 in. 


^in. 


9-16 in. 


%iu. 


11-16 in. 


Mill. 


%m. 


% 


1965 

1848 
1692 


1419 
1335 
1222 


1092 
1027 
940 


944 
846 
763 


665 
597 
538 














l 


450 






!tf 


1512 
1437 


1092 
1036 


840 

797 


726 
691 


512 

487 


415 
389 






356 


„ 228 


Wb 


1368 


988 


760 


653 


460 


370 


329 


211 


m 
m 


1300 


949 


730 


624 


440 


357 


280 


180 


1260 


924 


711 


596 


420 


340 


271 


174 


m 
m 


1200 


900 


693 


553 


390 


325 


262 


169 


1156 


840 


648 


532 


375 


312 


257 


165 


2 


1100 


789 


608 


511 


360 


297 


243 


156 


m 


1031 


744 


573 


502 


354 


289 


237 


152 


2H 


999 


721 


555 


491 


347 


280 


232 


149 




945 


682 


525 


475 


335 


260 


220 


141 


hi 


900 


650 


500 


443 


312 


242 


208 


133 


3 


828 


598 


460 


411 


290 


224 


197 


127 




779 


562 


433 


379 


267 


212 


180 


115 


' 743 


536 


413 


352 


248 


201 


169 


108 


m 


715 


513 


395 


341 


241 


192 


160 


102 


4 








326 
312 


230 
220 


184 

177 


158 
150 


99 


4M 








96 








298 


210 


171 


146 


94 


4% 








284 


200 


166 


138 


89 


5 








270 
256 
244 
233 


190 
180 
172 
164 


161 
156 
151 
145 


135 
130 
124 
120 


87 










84 








80 










6 








223 


157 


140 


115 


74 


a 








213 


150 


138 


111 


71 








207 


146 


134 


107 


69 


6M 








203 


143 


129 


104 


67 


7 








198 


140 


125 


100 


64 













turnbuckli.es. 

Turnbuckles with right and left threads are made of standard sizes. B = 



H= length of 




Fig. 73. 
diameter of bolt, = 6 finches in all sizes of turnbuckle. 
tapped heads = 1}4B. L — length = 6 inches -f 3 B. 



212 



MATEKIALS. 



SIZES OF WASHERS. 



Diameter in 
inches. 


Size of Hole, in 
inches. 


Thickness, 
Birmingham 
Wire-gauge. 


Bolt in 
inches. 


No. in 100 lbs. 


1 


5-16 


No. 16 


M 


29,300 


% 




' 16 


5-16 


18,000 




7-16 




' 14 


% 


7,600 


Wz 


9-16 




' 11 


Mi 


3.300 


m 


y& 




' 11 


9-16 


2,180 


lit 


11-16 




' 11 


% 


2,350 


13-16 




' 11 


H 


1,680 


2 


31-32 




' 10 


% 


1,140 


M 


m 




' 8 


l 


580 


J H 




« 8 


m 


470 


3 


Wb 




' 7 


m 
m 


360 


3 


m 




' 6 


360 



TRACK SPIKES. 



Rails used. 


Spikes. 


Number in Keg, 
200 lbs. 


Kegs per Mile, 
Ties 24 in. 






between Centres. 


45 to 85 


5^x9-16 


380 


30 


40 " 52 


5 x 9-16 


400 


27 


35 " 40 


5 x}4 


490 


22 


24 " 35 


Qb^Y* 


550 


20 


24 " 30 


4^x7-16 


725 


15 


18 " 24 


4 x 7-16 


820 


13 


16 " 20 


3^x% 


1250 


9 


14 " 16 


3 x% 

2^x3^ 


1350 


8 


8 " 12 


1550 


7 


8 " 10 


2^x5-16 


22G0 


5 



STREET RAILWAY SPIKES. 



Spikes. 


Number in Keg, 200 lbs. 


Kegs per Mile, Ties 24 in. 
between Centres. 


5^x9-16 
5 xy 2 
4J^x7-16 


400 
575 
800 


30 
19 
13 



BOAT SPIKES. 

Number in Keg of 200 lbs. 



Length. 


H 


5-16 


% 


\k 




2375 
2050 
1825 








5 " 


1230 
1175 
990 

880 


940 

800 
650 
600 
525 
475 




6 " 

7 " 


450 
375 


8 " 




335 


9 " 




300 


10 " 






275 











spikes; cut nails. 



213 



WROUGHT SPIKES. 
Number of Nails in Keg of 1 50 Pounds. 



Size. 


Hin. 


5-16 in. 


36 in. 


7-16 in. 


y 2 in. 




2250 
1890 
1650 
1464 
1380 
1292 
1161 










VA I 

4 " 


1208 
1135 
1064 
930 
868 
662 
635 
573 














4y 2 ;; 

5 " 








742 
570 
482 
455 
424 
391 






6 " 






8 " '.'..'.'. 

9 " 

10 " .... 


445 
384 
300 
270 
249 
236 


306 
256 
240 
222 


11 " 






203 


12 " 






180 













WIRE SPIKES. 



Size. 


Approx. Size 
of Wire Nails. 


Ap. No. 
in 1 lb. 


Size. 


Approx. Size 
of Wire Nails. 


Ap. No. 
in 1 lb. 


lOd Spike 

16d " 

20d " 

30d " ...... 

40d " ...... 

50d " 


3 in. No. 7 
sy 2 " " 6 

4 " " 5 
4 y « « 4 

5 " " 3 

by, " » 2 


50 
35 
26 
20 
15 
12 


60d Spike 

6^ in/; 

8 " " '.'..'.'.'. 

9 " " ...... 


6 in. No. 1 
6^ " " 1 

7 " " 

8 " " 00 

9 " " 00 


10 
9 

5 
4^ 



LENGTH AND NUMRER OF CUT NAII.S TO THE 
POUND. 



Size. 


to 

a 

3 


a 

o 

s 
s 

o 

o 


5 


U 

a 


si 

'£, 
"a 


a 




si 

o 


*d 

m 


o 
o 

V 

O 


53 

o 


% 


Min. 

." 

M 

2 

VA 

m 

J* 

zy A 

4 

i*. 

5y 2 

6 












800 
500 
376 
224 

180 








% 




















2d 


800 
480 
288 
200 
168 
124 
88 
70 
58 
44 
34 
23 
18 
14 
10 
8 


95 

74 
62 
53 
46 
42 
3S 
33 
20 


'84' 
64 
48 
36 
30 
24 
20 
16 


iioo 

720 
523 
410 
268 
188 
146 
130 
102 
76 
62 
54 


1000 
760 
368 










3d 


"398 




130' 

96 

82 
68 




4d 

5d 




6d 






224 


126 
98 
75 
65 
55 
40 
27 




7d 








8d 






128 
110 
91 
71 
54 
40 
33 
27 




9d 








lOd 








oq 


12d 








16d 








oo 


20d 






oy 2 

8 


30d 










40d 














50d 












60d.. ....... 


















6 























214 



MATERIALS. 



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APPROXIMATE HUMBER OP WIRE KAILS PER POUHD. 215 



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216 



MATERIALS. 



SIZE, WEIGHT, LENGTH, AND STRENGTH OF IRON 
WIRE. 

(Trenton Iron Co.) 













Tensile Strength (Ap- 




Diam. 


Area of 






proximate) of Charcoal 


No. by- 


in Deci- 


Section in 
Decimals of 
One Inch. 


Feet to 


Weight of 


Iron Wire in Pounds. 


Wire 


mals of 


the 


One Mile 




Gauge. 


One 
Inch. 


Pound. 


in pounds. 
















Bright. 


Annealed. 


00000 


.450 


.15904 


1.863 


2833.248 


12598 


9449 


0000 


.400 


.12566 


2.358 


2238.878 


9955 


7466 


000 


.360 


.10179 


2.911 


1813.574 


8124 


6091 


00 


.330 


.08553 


3.465 


1523.861 


6880 


5160 





.305 


.07306 


4.057 


1301.678 


5926 


4445 


1 


.285 


.06379 


4.645 


1136.678 


5226 


3920 


2 


.265 


.05515 


5.374 


982 555 


4570 


3425 


3 


.245 


.04714 


6.286 


839.942 


3948 


2960 


4 


.225 


.03976 


7.454 


708.365 


3374 


2530 


5 


.205 


.03301 


8.976 


588.139 


2839 


2130 


6 


.190 


.02835 


10.453 


505.084 


2476 


1860 


7 


.175 


.02405 


12.322 


428.472 


2136 


1600 


8 


.160 


.02011 


14.736 


358.3008 


1813 


1360 


9 


.145 


.01651 


17.950 


294.1488 


1507 


1130 


10 


.130 


.01327 


22.333 


236.4384 


1233 


925 


11 


.1175 


.01084 


27.340 


193.1424 


1010 


758 


12 


.105 


.00866 


34.219 


154.2816 


810 


607 


13 


.0925 


.00672 


44 092 


119.7504 


631 


473 


14 


.080 


.00503 


58.916 


89.6016 


474 


356 


15 


.070 


.00385 


76.984 


68.5872 


372 


280 


16 


.061 


.00292 


101.488 


52.0080 


292 


220 


17 


.0525 


.00216 


137.174 


38.4912 


222 


165 


18 


.045 


.00159 


186.335 


28.3378 


169 


127 


19 


.040 


.0012566 


235.084 


22.3872 


137 


103 


20 


.035 


.0009621 


308.079 


17.1389 


107 


80 


21 


.031 


.0007547 


392.772 


13.4429 




22 


.028 


.0006157 


481.234 


10.9718 


5o 2 <a * 


23 


.025 


.0004909 


603.863 


8.7437 


£ ? § <u" ° z ' 


24 


.0225 


.0003976 


745.710 


7.0805 


«*|^ a 


25 


.020 


.0003142 


943.396 


5.5968 


■sS^^rtSS^ §* 


26 


.018 


.0002545 


1164.689 


4.5334 


dj'jS §"53 -S S ® • 


27 


.017 


.0002270 


1305.670 


4.0439 


^ mfl S °*='$ gg U 


28 


.016 


.0002011 


1476.869 


3.5819 


g-o 9 CD-** O * £ '% 


29 


.015 


.0001767 


1676.989 


3.1485 


^Sf<| §«•§** a 


30 


.014 


.0001539 


1925.321 


2.7424 


a " S^d sm®^ S 


31 


.013 


.0001327 


2232.653 


2.3649 




32 


.012 


.0001131 


2620.607 


2.0148 


33 


.011 


.0000950 


3119.092 


1.6928 


34 


.010 


.00007854 


3773.584 


1.3992 


•HgrS-OSs?! -g 


35 
36 


.0095 
.009 


.00007088 
.00006362 


4182.508 
4657.728 


1.2624 
1.1336 


035l5«C^Sog ^ 


37 


.0085 


.00005675 


5222.035 


1.0111 


e ab 
base 

coal- 
tensi 
ml r 
edis 

i 
?cial 

tha' 


38 


.008 


.00005027 


5896.147 


.89549 


39 


.0075 


.00004418 


6724.291 


.78672 


H 9 52 20tr Sf ~"-« < 


40 


.007 


.00003848 


7698.253 


.68587 


doH 


*" — ■ a 



TESTS OF TELEGRAPH WIRE. 



217 



GALVANIZED IRON WIRE FOR TELEGRAPH AN© 
TELEPHONE LINES. 

(Trenton lion Co.) 

Weight per Mile-Ohm.— This term is to be understood as distinguishing 
the resistance of material only, and means the weight of such material re- 
quired per mile to give the resistance of one ohm. To ascertain the mileage 
resistance of any wire, divide the "weight per mile-ohm " by the weight of 
the wire per mile. Thus in a grade of Extra Best Best, of winch the weight 
per mile-ohm is 5000, the mileage resistance of No. 6 (weight per mile 525 
lbs.) would be about 9^s ohms; and No. 14 steel wire. 6500 lbs. weight per 
mile-ohm (95 lbs. weight per mile), would show about 69 ohms. 

Sizes of Wire used in Telegraph and Telephone Lines. 

No. 4. Has not been much used until recently; is now used on important 
lines where the multiplex systems are applied. 

No. 5. Little used in the United States. 

No. 6. Used for important circuits between cities. 

No. 8. Medium size for circuits of 400 miles or less. 

No. 9. For similar locations to No. 8, but on somewhat shorter circuits ; 
until lately was the size most largely used in this country. 

Nos. 10, 11. For shorter circuits, railway telegraphs, private lines, police 
and fire-alarm lines, etc. 

No. 12. For telephone lines, police and fire-alarm lines, etc. 

Nos. 13, 14. For telephone lines and short private lines: steel wire is used 
most generally in these sizes. 

The coating of telegraph wire with zinc as a protection against oxidation 
is now generally admitted to be the most efficacious method. 

The grades of line wire are generally known to the trade as " Extra Best 
Best " (E. B. B.), " Best Best " (B. B.) ? and "Steel." 

" Extra Best Best " is made of the very best iron, as nearly pure as any 
commercial iron, soft, tough, uniform, and of very high conductivity, its 
weight per mile-ohm being about 5000 lbs. 

The " Best Best" is of iron, showing in mechanical tests almost as good 
results as the E. B. B., but not quite as soft, and being somewhat lower in 
conductivity; weight per mile-ohm about 5700 lbs. 

The Trenton " Steel " wire is well suited for telephone or short telegraph 
lines, and the weight per mile-ohm is about 6500 lbs. 

The following are (approximately) the weights per mile of various sizes of 
galvanized telegraph wire, drawn by Trenton Iron Co.'s gauge: 

No. 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14. 
Lbs. 720, 610, 525, 450, 375, 310, 250, 200, 160, 125, 95. 

TESTS OF TELEGRAPH WIRE. 

The following data are taken from a table given by Mr. Prescott relating 
to tests of E. B. B. galvanized wire furnished the Western Union Telegraph 



Size 
of 


Diam. 

Parts of 

One 


Weight. 


Length. 

Feet 

per 

pound. 


Resistance. 
Temp. 75.8° Fahr. 


Ratio of 
Breaking 
Weight to 

Weight 


Wire. 


Grains. 


Pounds 


Feet 


Ohms 






per foot. 


per mile. 


per ohm. 


per mile. 


per mile. 


4 


.238 


1043.2 


886.6 


6.00 


958 


5.51 




5 


.220 


891.3 


673.0 


7.85 


727 


7.26 




6 


.203 


758.9 


572.2 


9.20 


618 


8.54 


3.05 


7 


.180 


596.7 


449.9 


11.70 


578 


10.86 


3.40 


8 


.165 


501.4 


378.1 


14.00 


409 


12.92 


3.07 


9 


.148 


403.4 


304.2 


17.4 


328 


16.10 


3.38 


10 


.134 


330.7 


249.4 


21.2 


269 


19.60 


3 37 


11 


.120 


265.2 


200.0 


26.4 


216 


24.42 


2.97 


12 


.109 


218.8 


165.0 


32.0 


179 


29.60 


3.43 


14 


.083 


126.9 


95.7 


55.2 


104 


51.00 


3.05 



Joints in Telegraph Wires. — The fewer the joints in a line the better. 
All joints should be carefully made and well soldered over, for a bad joint 
may cause as much resistance to the electric current as several miles of 
wire. 



218 



MATEEIALS. 



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DIMENSIONS, WEIGHT, RESISTANCE OF COPPER WIRE. 219 



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220 



MATERIALS. 






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§§8«-« 



HARD-DRAWX COPPER WIRE; INSULATED WIRE. 221 



HARD-DRAWN COPPER TELEGRAPH WIRE. 

(J. A. Roebling's Sons Co.) 
Furnished in half-mile coils, either bare or insulated. 











Approximate 


Size, B. & S. 
Gauge. 


Resistance in 

Ohms 

per Mile. 


Breaking 
Strength. 


Weight 
per Mile. 


SizeofE.B.B. 
Iron Wire 
equal to 
Copper. 


9 


4.30 


625 


209 


2 W 


10 


5.40 


525 


166 


3 § 


11 


6.90 ^ 


< 420 


131 


° 1 


12 


8JZ0^" 


330 


104 


13 


^TOT90 
f 13.70 


270 


83 


6^3 


14 


213 


66 


8 Q 


15 


/ 17.40 


170 


52 


9 S 


16 


/ 22.10 


130 


41 


10 ft 



In handling this wire the greatest care should be observed to avoid kinks, 
bends, scratches, or cuts. Joints should be made only with Mclntire Con- 
nectors. 

On account of its conductivity being about five times that of Ex. B. B. 
Iron Wire, and its breaking strength over three times its weight per mile, 
copper may be used of which the section is smaller and the weight less than 
an equivalent iron wire, allowing a greater number of wires to be strung on 
the poles. 

Besides this advantage, the reduction of section materially decreases the 
electrostatic capacity, while its non-magnetic character lessens the self-in- 
duction of the line, both of which features tend to increase the possible 
speed of signalling in telegraphing, and to give greater clearness of enunci- 
ation over telephone lines, especially those of great length. 
INSULATED COPPER WIRES. 
Weight per 1000 feet. 





Weather- 


Under- 




Weather- 


Under- 




Weather- 


Under- 


°0 to 


proof 


writers 1 


« ft 


proof 


writers' 


was 


proof 


writers' 


^3 


Line 


Line 


**§ 


Line 


Line 


*JS 


Line 


Line 


rio 


Wire. 


Wire. 


m® 


Wire. 


Wire. 


wo 


Wire. 


Wire. 


0000 


671. 


701. 


5 


115. 


. 121. 


13 


26. 


26.5 


000 


537. 


565. 


6 


93. 


99. 


14 


20.5 


22. 


00 


426. 


447. 


7 


77. 


80. 


15 


17. 


20. 





342. 


364. 


8 


64. 


67. 


16 


14. 


15. 


1 


274. 


294. 


9 


53. 


54. 


17 


12. 


18. 


2 


220. 


241. 


10 


44. 


45. 


18 


10.75 


11. 


3 


178. 


185. 


11 


37. 


37. 


19 


9. 


10. 


4 


141. 


147. 


12 


30. 


31. 


20 


7.5 


8. 



LEAD-ENCASED ANTI-INDUCTION TELEPHONE AND 
TELEGRAPH CARLES. (Roebling's.) 



Plain Cables, Lead 
Encased. 



No. of Size Wire 
Wires. B. & S. Gauge. 



4 
7 
10 
50 
100 



For Metallic Circuit. 



No. of Size Wire 
Pairs. B. & S. Gauge. 



15 

25 
50 

75 



For Telegraph Cir- 
cuits. 



No. of 


Size Wire 


Wires. 


B. & S. Gauge. 


3 


14 


4 


14 


7 


14 


10 


14 


20 


14 


50 


14 


100 


14 



222 



MATERIALS. 



FLEXIBLE CABLES. 









5I-I+ 3 - 








SH-U - 








A o aP, 








o a £ 


Area 
Circ. 
Mils. 


No. of 
Wires. 


Size 
Wire 
B.&S. 
Gauge. 


ill? 

< 


Area 
Circ. 
Mils. 


No. of 
Wires. 


Size 
Wire 
B.&S. 
Gauge. 


Diameter 
Equivale 
Solid Wi 
Mils. 


15699.6 


49 


25 


8 B. & S. 


272410.6 


133 


17 


522. 


24963.0 


49 


23 


6 


433154.4 


133 


15 


658. 


39693.9 


49 


21 


4 


688727.2 


133 


13 


830. 


63116.9 


49 


19 


2 


868176.7 


133 


12 


932. 










1095135.3 


133 


11 


1046. 










210964.6 


103 


17 


459. 










420127.2 


129 


15 


649. 










657656.8 


127 


13 


811. 










835827.2 


128 


12 


914. 










1062198.9 


129 


11 


1035. 



WEATHERPROOF AERIAL CABLES. 





Weight per 




Weight per 




Weight per 


No. of Con- 


Conductor 


No. of Con- 


Conductor 


No. of Con- 


Conductor 


ductors. 


per 1000 


ductors. 


per 1000 


ductors. 


per 1000 




feet. 




feet. 




feet. 


1 


10.75 lbs. 


8 


9.25 lbs. 


15 


9.25 lbs. 


2 


18.00 " 


9 


9.25 " 


16 


9.25 " 


3 


13.00 " 


10 


9.25 " 


17 


9.25 " 


4 


10.75 " 


11 


9.25 " 


18 


9.25 " 


5 


10.00 " 


12 


9.25 " 


19 


9.25 " 


6 


9.50 " 


13 


• 9.25 " 


20 


9.25 " 


7 


9.25 " 


14 


9.25 " 







LEAD-ENCASED ELECTRIC-LIGHT CABLES. 

Single Wires. 

(J. A. Roebling's Sons Co.) 









Nearest Ap- 


Approxi- 


Approxi- 


Size, 
B.&S. 
Gauge. 


of Solid Cop- 
per Wire. 
Mils. 


Area. 

Circular 

Mils. 


proximate 
Birming- 
ham Wire- 
gauge No. 


Weight 

per Foot 

of Cable. 

Oz. 


mate 

Diameter 

of Cable. 

Mils. 


20 


31.96 


1021. 


21 


1.63 


170 


19 


35.39 


1252. 


20 


1.70 


175 


18 


40.30 


1624. 


19 


1.75 


180 


17 


45.25 


2048. 


18^ 


1.84 


185 


16 


50.82 


2583. 


18 


3.00 


245 


15 


57.07 


3257. 


17 


3.20 


250 


14 


64.08 


4107. 


16 


3.38 


255 


13 


71.96 


5178. 


15 


3.56 


265 


12 


80.80 


6530. 


14 


5.00 


310 


11 


90.74 


8234. 


13^ 


5.23 


320 


10 


101.89 


10381. 


12^ 


5.68 


335 


9 


114.23 


13094. 


11^ 


5.95 


345 


8 


128.49 


16509. 


10y 2 


6.35 


360 


7 


144.28 


20816. 


9 


6.90 


375 



As tested by the Bell Telephone Co. of Philadelphia, the insulation may 
be stated at 2000 megohms per mile, vs ith an electrostatic capacity of .14 
microfarad. 



STEEL WIRE CABLES. 



223 



GALVANIZED STEEL-WIRE STRAND. 
For Smokestack Guys, Signal Strand, etc. 

(J. A. Roebling's Sons Co.) 
This strand is composed of 7 wires, twisted together into a single strand. 





©' 




S&S 




33 


e<"5 


"S««^ 










O) 




+3 <D 




.Jj 


s 


f &- 


§1§ 


£i 


S 


1x fe 


£ 


eS 






£ 


.3 


"53 S 


tc £is 


J> 


ft 


£- 


£\&V2 


i> 


ft 


^ 


HPQw 




in. 


lbs. 


lbs. 




in. 


lbs. 


lbs. 


No. 8 


H 


52 


8,320 


No.15 


M 


10 


1,600 


9 


15-32 


42 


6,720 


16 


7-32 


8 


1,280 


10 


7-16 


36 


5,720 


17 


3-16 


6 


960 


11 


% 


29 


4,640 


18 


11-64 


4 3-10 


688 


12 


5-16 


21 


3,360 


19 


9-64 


3 3-10 


528 


13 


9^32 


16 


2,560 


20 


& 


2 4-10 


384 


14 


17-64 


12 


1,920 


21 


3-32 


2 - 


320 



For special purposes these strands can be made of 50 to 100 per cent 
greater tensile strength. When used to run over sheaves or pulleys the use 
of soft-iron stock is advisable. 

FLEXIBLE STEEL-WIRE CABLES FOR VESSELS. 
(Trenton Iron Co., 1886.) 

With numerous disadvantages, the system of working ships' anchors with 
chain cables is still in vogue. A heavy chain cable contributes to the hold- 
ing-power of the anchor, and the facility of increasing that resistance by 
paying out the cable is prized as an advantage. The requisite holding- 
power is obtained, however, by the combined action of a comparatively 
light anchor and a correspondingly great mass of chain of little service in 
proportion to its weight or to the weight of the anchor. If the weight and 
size of the anchor were increased so as to give the greatest holding-power 
required, and it were attached by means of a light wire cable, the combined 
weight of the cable and anchor would be much less than the total weight of 
the chain and anchor, and the facility of handling would be much greater. 
English shipbuilders have taken the initiative in this direction, and many of 
the largest and most serviceable vessels afloat are fitted with steel-wire 
cables. They have given complete satisfaction. 

The Trenton Iron Co.'s cables are made of crucible cast-steel wire, and 
guaranteed to fulfil Lloyd's requirements. They are composed of 72 wires 
subdivided into six strands of twelve wires each. In order to obtain great 
flexibility, hempen centres are introduced in the strands as well as in the 
completed cable. 

FLEXIBLE STEEL-WIRE HAWSERS. 

These hawsers are extensively used, They are made with six strands of 
twelve wires each, hemp centres being inserted in the individual strands as 
well as in the completed rope. The material employed is crucible cast steel, 
galvanized, and guaranteed to fulfil Lloyd's requirements. They are only 
one third the weight of hempen hawsers ; and are sufficiently pliable to work 
round any bitts to which hempen rope of equivalent strength can be applied. 

13-inch tarred Russian hemp hawser weighs about 39 lbs. per fathom. 

10-inch white manila hawser weighs about 20 lbs. per fathom. 

lj^-inch stud chain weighs about (58 lbs. per fathom. 

4-inch galvanized steel hawser weighs about 12 lbs. per fathom. 

Each of the above named has about the same tensile strength. 



224 



MATERIALS. 



SPECIFIC ATIONS FOR GALVANIZED IRON WIRE. 

Issued by the British Postal Telegraph Authorities. 



Weight per Mile. 


Diameter. 


Tests for Strength and 
Ductility. 


fe 




























73 
















2 




2 




2 


S^fa 


£<5 


V 

a 
a 

02 






-d 






bt- 


== a 


a 


'S d 


a 


S a 


ft §s> 


a a 


Allowed. 


eg 

a 

c3 


Allowed. 


m" 


r-* — 

SH 50 

© a 






H- 

•4-1° 

o a 
d 


bD i 


o a 

6 


•™ct_|.S 


.aw 


ts 






OQ 








fe 


SD~ 


K 


&C+- 3 


fe 


Ph 


ix 








53 















S 1 




























a 


a 

.a 
a 
a 


a 
a 

I a 

a 




' S 

a 

a 

'3 

a 


a 
1 

a 


a 

a 

a 

a 


a 

J 
a 

a 


M 
o 


S 
a 

a 

a 

a 


M 

o 
fa 


a 

a 

a 

a 

a 


a 
1 

a 


3.SP 


lbs. 


lbs. 


lbs. 


mils. 


mils. 


mils. 


lbs. 




lbs. 




lbs. 




ohms. 




800 


767 


833 


242 


237 


247 


2480 


15 


2550 


14 


2620 


13 


6.75 


5400 


600 


571 


«29 


209 


204 


214 




17 


1910 


16 


1960 


15 


9.00 


5400 


450 


424 


477 


181 


176 


186 


10 


19 




18 


1460 


17 


12.00 


5400 


400 


377 


424 


171 


166 


176 




21 


, 


20 


1300 


19 


13.50 


5400 


200 


190 


213 


121 


118 


125 


620 


30 


638 


28 


655 


26 


27.00 


5400 



STRENGTH OF PIANO-WIRE. 



The average strength of English piano- wire is given as follows by Web- 


ster, Horsfals & Lean : 










Numbers 


Equivalents 
in Fractions 


Ultimate 


Numbers 


Equivalents 
in Fractions 


Ultimate. 


in Music- 


Tensile 


in Music- 


Tensile 


wire 


of Inches in 


Strength in 


wire 


of inches in 


Strength in 


Gauge. 


Diameters. 


Pounds. 


Gauge. 


Diameters. 


Pounds. 


12 


.029 


225 


18 


.041 


395 


13 


.031 


250 


19 


.043 


425 


14 


.033 


285 


20 


.045 


500 


15 


.035 


305 


21 


.047 


540 


16 


.037 


340 


22 


.052 


650 


17 


.039 


360 









These strengths range from 300,000 to 340,000 lbs. per sq. in. The compo- 
sition of this wire is as follows: Carbon, 0.570; silicon, 0.090; sulphur, 0.011; 
phosphorus, 0.018; manganese, 0.425. 

"PIiOUGH"-STEEIi WIRE. 

The term "plough," given in England to steel wire of high quality, was 
derived from the fact that such wire is used for the construction of ropes 
used for ploughing purposes. It is to be hoped that the term will not be 
used in this country, as it tends to confusion of terms. Plough-steel is 
known here in some steel-works as the quality of plate steel used for the 
mould-boards of ploughs, for which a very ordinary grade is good enough. 

Experiments by Dr. Percy on the English plough-steel (so-called) gave the 
following results: Specific gravity, 7.814 ; carbon, 0.828 per cent; manga- 
nese, 0.587 per cent; silicon, 0.143 per cent; sulphur, 0.009 per cent; phos- 
phorus, nil; copper, 0.030 per cent. No traces of chromium, titanium, or 
tungsten were found. The breaking strains of the wire were as follows: 

Diameter, inch 093 .132 .159 .191 

Pounds per sq. inch 344,960 257,600 224,000 201,600 

The elongation was only from 0.75 to 1.1 per cent. 



SPECIFICATIONS FOR HARD-DRAWN" COPPER WIRE. 225 



WIRES OF DIFFERENT METALS AND AL.L.OYS. 

(J. Bucknall Smith's Treatise on Wire.) 

Brass Wire is commonly composed of an alloy of 1 3/4 to 2 parts of 
copper to 1 part of zinc. The tensile strength ranges from 20 to 40 tons per 
square inch, increasing with the percentage of zinc in the alloy. 

German or Nickel Silver, an alloy of copper, zinc, and nickel, is 
practically brass whitened by the addition of nickel. It has been drawn into 
wire as fine as .002" diam. 

Platinum wire may be drawn into the finest sizes. On account of its 
high price its use is practically confined to special scientific instruments and 
electrical appliances in which resistances to high temperature, oxygen, and 
acids are essential. It expands less than other metals when heated, which 
property permits its being sealed in glass without fear of cracking. It is 
therefore used in incandescent electric lamps. 

Phosphor-bronze Wire contains from 2 to 6 per cent of tin and 
from 2V to y% per cent of phosphorus. The presence of phosphorus is detri- 
mental to electric conductivity. 

" Delta-metal " wire is made from an alloy of copper, iron, and zinc. 
Its strength ranges from 45 to 62 tons per square inch. It is used for some 
kinds of wire rope, also for wire-gauze. It is not subject to deposits of ver- 
digris. It has great toughness, even when its tensile strength is over 60 
tons per square inch. 

Aluminum Wire. — Specific gravity .268. Tensile strength only 
about 10 tons per square inch. It has been drawn as fine as 11,400 yards to 
the ounce, or .042 grains per yard. 

Aluminum Bronze, 90 copper, 10 aluminum, has high strength and 
ductility; is inoxidizable, sonorous. Its electric conductivity is 12.6 per cent 
of that of pure copper. 

Silicon Bronze, patented in 1882 by L. Weiler of Paris, is made as 
follows: Fluosilicate of potash, pounded glass, chloride of sodium and cal- 
cium, carbonate of soda and lime, are heated in a plumbago crucible, and 
after the reaction takes place the contents are thrown into the molten 
bronze to be treated. Silicon-bronze wire has a conductivity of from 40 to 
98 per cent of that of copper wire and four times more than that of iron, 
while its tensile strength is nearly that of steel, or 28 to 55 tons per square 
inch of section. The conductivity decreases as the tensile strength in- 
creases. Wire whose conductivity equals 95 per cent of that of pure copper 
gives a tensile strength of 28 tons per square inch, but when its conductivity 
is 34 per cent of pure copper, its strength is 50 tons per square inch. It is 
being largely used for telegraph wires. It has great resistance to oxidation. 

Ordinary Drawn and Annealed Copper Wire has a strength 
of from 15 to 20 tons per square inch. 

SPECIFICATIONS FOR HARD-DRAWN COPPER 
WIRE. 

The British Post Office authorities require that hard-drawn copper wire 
supplied to them shall be of the lengths, sizes, weights, strengths, and con- 
ductivities as set forth in the annexed table. 



Weight per Statute 


Approximate Equiva- 


SB 


° J 


S°f 


BO 




Mile. 




len 


t Diameter. 




11 

a w 


a * £§ 


log 


p-a 


S 

p 

a 


s 
a 


id 




a 
p 

a 


5 


eg 

$02 


a 


9 


-2 




1 


a 


c.22 


|<S 


'So.. 


lbs. 


lbs. 


lbs. 


mils. 


mils. 


mils. 


lbs. 




ohms. 


lbs. 


100 


97^ 


102V6 

153% 


79 


78 


80 


330 


30 


9.10 


50 


150 


146J4 


97 


951^ 


98 


490 


25 


6.05 


50 


200 


195 


205 


112 


110i/> 


113J4 


650 


20 


4.53 


50 


400 


390 


410 


158 


155J^ 


160J4 


1300 


10 


2.27 


50 



226 



MATERIALS. 



WIRE ROPES. 

List adopted by manufacturers in 1892. See pamphlets of Trenton Iron 
Co., John A. Roebling's Sons Co., and other makers. 

Pliable Hoisting Rope. 

With 6 strands of 19 wires each. 

IRON. 









c 






«H 


a- 


S 






O o^ 


a «5 
S-© 


.S ° 
o o 


® "3 


0> 


& 


J 


0J 

si 


ft «5»S 


a o 


£s» 




g> 


g 


a 
s 




^ 03 


ftcco 






1 


2^ 


6% 


8.00 


74 


15 


14 


13 


2 


2 


6 


6.30 


65 


13 


13 


12 


3 


1% 


5^ 


5.25 


54 


11 


12 


10 


4 


1% 


5 


4.10 


44 


9 


11 


8^ 


5 


M 


4% 


3.65 


39 


8 


10 


7^ 


5% 


Ws 


4% 


3.00 


33 


6^ 


9^ 


7 


6 


ij| 


4 


2.50 


27 


5J^ 


8^3 


6V6 


7 


3^ 


2.00 


20 


4 




6 


8 


l 


m 


1.58 


16 


3 


6^ 


5M 
4^ 


9 


% 


2% 


1.20 


11.50 


2^ 


51^ 


10 


% 


2J4 


0.88 


8.64 


1% 


4% 


4 


10M 


% 


2 


0.60 


5.13 


■a 


3% 


m. 


10^ 


9-16 


1% 


0.44 


4.27 


3^ 


2H 


10% 


K 


m 


0.35 


3.48 




3 


m 


10a 


7-16 


m 


0.29 


3.00 


% 


2% 


2 


10^ 


% 


m 


0.26 


2.50 


Ya 


2^ 


i» 









CAST 


STEEL. 








1 


2M 
2 

M 


mi 

6 
®/ 2 


8.00 


155 


31 




sy 2 

8 


2 


6.30 
5.25 


125 
106 


25 
21 




3 




7^ 
6^4 
5% 


4 


m 


5 


4.10 


86 


17 


15 


5 


M 


4% 


3.65 


77 


15 


14 


5^ 




4% 


3.00 


63 


12 


13 


5^ 


6 


m 


4 


2.50 


52 


10 


12 


5 


7 


m 


3^ 


2.00 


42 


8 


11 


4y 9 


8 




3^ 


1.58 


33 


6 


9^ 


4 


9 


% 


2% 


1.20 


25 


5 


w 2 


3^ 


10 


H 


2y 4 


0.88 


18 


sy 2 


7 


3 


10J4 


% 


2 


0.60 


12 


zy 2 


5% 


2M 


10^ 


9-16 


1% 


0.44 


9 


w 2 


5 


18 


im 


X 


i« 


0.35 


7 


1 


4y 2 


10a 


7-16 


1% 


0.29 


sy 2 

*y 2 


% 


m 


IM 


10% 


% 


VA 


0.26 


H 


*y 2 


1 



Cable-Traction Ropes. 

According to English practice, cable-traction ropes, of about 3}4 in. in 
circumference, are commonly constructed with six strands of seven or fif- 
teen wires, the lays in the strands varying from, say, 3 in. to 3^ in., and the 
lays in the ropes from, say, 7^ in. to 9 in. In the United States, however, 
strands of nineteen wires are generally preferred as being more flexible; 
but, on the other hand, the smaller external wires wear out more rapidly. 
The Market street Street Railway Company, San Francisco, has used ropes 
1J4 in- hi diameter, composed of six strands of nineteen steel wires, weighing 
2^ lbs. per foot, the longest continuous length being 24,125 ft. The Chicago 
City Railroad Company has employed cables of identical construction, the 
longest length being 27,700 ft. On the New York and Brooktyn Bridge cable- 
railway steel ropes of 11,500 ft, long, containing 114 wires, have been used, 



WIRE EOPES. 



227 



Transmission and Standing; Rope. 

With 6 strands of 7 wires each. 











a 


&c«w 





N 


S 




a 


oo5 






® "eg 

= 08 2. 


© 


3 




a 




be 1 ** 




fig °M 


.2 a) 


t8 


s 

g3 


u 


'5o--£ 


go 


G.03O 


C ©S-C3 


■CO 


H 


s 


O 


> e.?*j 


PQ-" 


£^ w 


5 =tfw 


g 


11 


m 


4% 
4^ 


3.37 


36 


9 


10 


13 


12 


m 


2.77 


30 


Wz 


9 


12 


13 


m 


3% 


2.28 


25 


6J4 


8^ 


10% 
9v| 


14 


m 


3% 


1.82 


20 


5 


7*6 


15 


i 


3 


1.50 


16 


4 


6^ 


8^ 


16 


Vs 


2% 


1.12 


12.3 


3 


5% 


7^ 


17 


u 


2% 


0.88 


8.8 


m 


- 4% 
4^ 


63^ 


18 


11-16 


2^ 


0.70 


7.6 


2 


6 


19 


% 


1% 


0.57 


5.8 


1^ 


4 


514 


20 


9-16 


Ws 


0.41 


4.1 


1 


3J4 


4^ 


21 


M 


W& 


0.31 


2.83 


3 


2M 


4 


22 


7-16 


VA 


0.23 


2.13 


2^ 


3M 


23 


% 


Ui 


0.19 


1.65 




2^ 


2% 


24 


5-16 
9-32 


l 
% 


0.16 
0.125 


1.38 
1.03 




2 

1% 


2^ 
2M 


25 





CAST STEEL. 



11 


m 


m 


3.37 


62 


13 


13 


8^ 


12 


a* 


4U 


2.77 


52 


10 


12 


8 


13 


m 

3% 


2.28 


44 


9 


11 


7M 


14 


1H 


1.82 


36 


~y% 


10 


6M 


15 




3 


1.50 


30 


6 


9 


5% 


16 


% 


2% 


1.12 


22 


4^ 


8 


5 


17 


H 


m 


0.88 


17 


m 


7 


4V4 


18 


11-16 


2Y 8 


0.70 


14 


3 


6 


4 


19 


Vs 


m 


0.57 


11 


2M 


5M 

4M 


3^ 


20 


9-16 


1% 


0.41 


8 


m 


3 


21 


Y* 


m 


0.31 


6 


i« 


4 


2^ 


22 
23 


7-16 
% 


i*t 


0.23 
0.19 


4^ 
4 


JM 


ft 


f4 


24 


5-16 


1 


0.16 


3 


s 


2% 


1% 


25 


9-32 


% 


0.12 


2 


2J4 


1^ 



Plough-Steel Rope. 

Wire ropes of very high tensile strength, which are ordinarily called 
"Plough-steel Ropes," are made of a high grade of crucible steel, which, 
when put in the form of wire, will bear a strain of from 100 to 150 tons per 
square inch. 

Where it is necessary to use very Ions or very heavy ropes, a reduction of 
the dead weight of ropes becomes a matter of serious consideration. 

It is advisable to reduce all bends to a minimum, and to use somewhat 
larger drums or sheaves than are suitable for an ordinary crucible rope hav- 
ing a strength of 60 to 80 tons per square inch. Before using Plough-steel 
Ropes it is best to have advice on the subject of adaptability. 



228 



MATERIALS. 



Plough-Steel Rope. 

With 6 strands of 19 wires each. 



Trade 


Diameter in 


Weight per 
foot in 
pounds. 


Breaking 
Strain in 


Proper Work- 


Min. Size of 
Drum or 


Number. 


inches. 


tons of 
2000 lbs. 


ing Load. 


Sheave in 
feet. 


1 


2*4 


8.00 


240 


46 


9 


2 


2 


6.30 


189 


37 


8 


3 


m 


5.25 


157 


31 


7^ 


4 




4.10 


123 


25 


6 


5 


V4 


3.65 


110 


22 


&A 


5^ 


Ws 


3.00 


90 


18 


m 


6 


1M 


2.50 


75 


15 


5 


7 


V4 


2.00 


60 


12 


4*4 


8 




1.58 


47 


9 


414 

m 


9 


Va 


1.20 


37 


7 


10 


I 


0.88 


27 


5 


w% 


10J4 


0.60 


18 


m 


3 


ioy 2 


9-16 


0.44 


13 


2 


2Y2 


10% 


¥2 


0.35 


10 


M 


2 



With 7 Wires to the Strand. 



15 


1 


1.50 


45 


9 


% 


16 


1 


1.12 


33 


®A 


5 


17 


*0.88 


25 


qa 


4 


18 


11-16 


0.70 


21 


4 


3^ 


19 


% 


0.57 


16 


3M 


3 


20 


9-16 


0.41 


12 


2 




21 


y* 


0.31 


9 


IJUs 


22 


7-16 


0.23 


5 


a 


2 


23 


% 


0.19 


4 


i^ 



Galvanized Iron Wire Rope. 

For Ships' Rigging and Guys for Derricks. 
CHARCOAL ROPE. 







Cir. of 


Break- 




Weight 

per 
Fathom 

in 
pounds. 


Cir. of 


Break- 


Circum- 
ference 
in inches. 


Weight 
per Fath- 
om in 
pounds. 


new 

Manila 

Rope of 

equal 


ing 

Strain 
in tons 
of 2000 


Circum- 
ference 
in inches 


new 

Manila 

Rope of 

equal 


ing 
Strain 
in tons 
of 2000 






Strength. 


pounds 




Strength. 


pounds 


5^ 


26^ 


11 


43 


2% 


&A 


5 


9 


5^ 


24^ 


10^ 


40 


m 


*H 


m 


8 


5 


22 


10 


35 


2 


m 


*A 


7 


m 


21 


W2 


33 


m 


%A 


3M 


5 


*a 


19 


9 


30 


1^ 


2 


3 


m 


m 


16^ 


m 


26 


m 


m 


m 


za 


4 


14M 


8 


23 


m 


\ 


*A 


2J4 


m 


12% 


m 


20 


1 


&A 


2 


w* 


10-M 


&A 


16 


% 


a 


m 


1 


m 


W* 


6 


14 


3 A 


VA 


¥4 


3 


8 


5M 

514 


12 


¥s 


m 


% 


m 


m 


10 


A 


y* 


m 


% 



WIRE ROPES. 



229 



Galvanized Cast-steel Yacht Rigging. 



Circum- 
ference 
in inches. 


Weight 
per Fath- 
om in 
pounds. 


Cir. of 

new 
Manilla 
Rope of 

equal 
Strength. 


Break- 
ing 
Strain 
in tons 
of 2000 
pounds 


Circum- 
ference 
in inches 


Weight 

per 
Fathom 

in 
pounds. 


Cir. of 

new 

Manilla 

Rope of 

equal 

Strength. 


Break- 
ing 
Strain 
in tons 
of 2000 
pounds 


4 

3^ 

3 

m 

it 


10% 

8 

m 

4^ 


13 
11 
{$£ 

8 


66 
43 

32 
2? 

22 
18 


2 
Wa 

m 
i 


3^ 

2 

m 
m 


ey 2 
by A 
Wa 
414 
3% 
3 


14 

10 
8 

6^ 
5^ 
3^ 





Steel Hawsers, 

For Mooring, Sea, and Lake Towing. 




Circumfer- 
ence. 


Breaking 
Strength. 


Size of 
Manilla Haw- 
ser of equal 
Strength. 


Circumfer- 
ence. 


Breaking 
Strength. 


Size of 
Manilla Haw- 
ser of equal 
Strength. 


Inches. 

2y 2 


Tons. 
15 
18 
22 


Inches. 

7 


Inches. 
3^ 
4 


Tons. 
29 
35 


Inches. 
9 
10 



Steel Flat Ropes. 

(J. A. Roebling*s Sons Co.) 
Steel-wire Flat Ropes are composed of a number of strands, alternately 
twisted to the right and left, laid alongside of each other, and sewed together 
with soft iron wires, These ropes are used at times in place of round ropes 
in the shafts uf mines. They wind upon themselves on a narrow winding- 
drum, which takes up less room than one necessary for a round rope. The 
soft-iron sewing-wires wear out sooner than the steel strands, and then it 
becomes necessary to sew the rope with new iron wires. 



Width and 
Thickness 


Weight per 
foot in 


Strength in 
pounds. 


Width and 
Thickness 


Weight per 
foot in 


Strength in 
pounds. 


in inches. 


pounds. 


in inches. 


pounds. 


%x2 


1.19 


35,700 


^x3 


2.38 


71,400 


%x2^ 


1.86 


55.800 


%*3}4 


2.97 


89,000 


%x3 


2.00 


60,000 


3^x4 


3.30 


99,000 


%*m 


2.50 


75,000 


14x4^2 


4.00 


120,000 


%x4 


2.86 


85,800 


1^x5 


4.27 


128,000 


%x4V 2 


3.12 


93,600 


y^y* 


4.82 


144,600 


3 / 8 x5 


3.40 


100,000 


^x6 


5.10 


153,000 


9$x5« 


3.90 


110,000 


^x7 


5.90 


177,000 



For safe working load allow from one fifth to one seventh of the breaking 
stress. 

" Lang Lay" Rope. 

In wire rope, as ordinarily made, the component strands are laid up into 
rope in a direction opposite to that in which the wires are laid into strands; 
that is, if the wires in the strands are laid from right to left, the strands are 
laid into rope from left to right. In the " Lang Lay,' 1 sometimes known as 
"Universal Lay," the wires are laid into strands and the strands into rope 
in the same direction; that is, if the wire is laid in the strands from right to 
left, the strands are also laid into rope from right to left. Its use has beeu 
found desirable under certain conditions and for certain purposes, mostly 
for haulage plants, inclined planes, and street railway cables, although it 
has also been used for vertical hoists in mines, etc. Its advantages are that 



230 



MATERIALS. 



GALVANIZED STEEL CABLES. 
For Suspension Bridges, (Roebling's. 



* 


5 

^o 


o 


i 

A 


,3 


o 










58 

.go* 


o 


a 


&i 




a 


o 


a 




CD 




02 <" 


0) 




02 «m 




*£ 


^ 02 CO 


ft 


'£ 


®2« 


ft 


■J 




ft 


-2 
1 


+3 e O 


.bC 




'5 r- O 




S 


g c 3 
£.5 ft 


A 


5 


P-Sft 


£ 


S 


p .2ft 


£ 


5 


£ 


2% 


220 


13 


214 


155 


8.64 


1% 
1*6 


95 


5.6 


2W 


200 


11.3 


2 


110 


6.5 


75 


4 35 


m 


180 


10 


1% 


100 


5.8 


1W 


65 


3.7 



COMPARATIVE STRENGTHS OF FLEXIBLE GAL- 
VANIZED STEEL-WIRE HAWSERS, 

With Chain Cable, Tarred Russian Hemp, and White 
Manila Ropes. (Trenton Iron Co.) 



Patent Flexible 






Tarred Rus- 


White 


Steel-wire Hawsers 


Chain Cable 




sian Hemp 


Manilla 


and Cables. 






Rope. 


Ropes. 




a 


6fl 


oA . 

ft u 73 




£ 








R 






S 




c 




c3 


Pa 2 




£ 








S 







































2 

3 


fa 

o 

ft 


n 

9) 


3-n ° 

op 




fa 

s, 


1 


2 
So 

to 




fa 

J-l 
» 

ft 


o3 




« 
fa 
U 


si 

53 






"5.9 


P 0) b 






w 








a 






a 


U 


To 


c3 c3 


1>S 


o5 

N 


U 


2 

fa 




6 

33 


to 

'53 


ea 
£ 

fa 


6 


1 


ea 

fa 


1 


: M 


1% 


6 










2% 




1W 


2W 


1M 


2 


11.-, 


1 


214 


7W 


w 


14 


4W 


6 


n% 


3 


2U 


3 


1% 


2% 


1% 


4 


9 










4 


3W 


y. 


3V 4 


2 


*W 


i-'i 




51/, 


1(W 


9-16 


17 


5W 


TJ4 


5 


6 


5 


4 


8 


5 




2% 




12 










53/ ( 


8 


7 


5 


4W 


7% 


21 -j 


3% 


9 


18*6 


10-16 


21 


7 


9W 




10 


9 


5% 


6 


low 


2U 


4>., 


12 


15 












13 


nw 


pM 


7 


12^4 


2% 


51,, 


15 


16W 


11-16 


2b 


8W 


12% 




16 


14 


V 


t% 


15 


3 


7 


18 


18 


12-16 


30 


10W 


15W 


9 


19 


:6U 


Wo 




18 




8 


22 


19^ 


13-16 


35 


11% 
15 8-10 


17 8-10 


10 


23 


20 


m 


13 


22% 




9 


26 


SI 


15-16 


48 


23 7-10 


11 


28 




9 


!4Vo 


25 


4 


12 


33 


24 


1 


54 


18 


27 


12 


33 


29 


10 


18 


31W 

38W 




15 


39 


27 


1% 


68 


22% 


34W 


13 


39 


34 


11 


22 


5 




64 


30 


1 17-32 


113 


37W 


55W 


15 


56 


50 


123/, 




51 


>k 


es 


74 


33 


1% 




47W 


66W 


17 


67 


60 


13W 


8bW 


62 


6 


33 


88 


36 


m 


km; 


55W 


77W 


1!) 


84 


72 


15 


42 


',8W 


> l <> 


37 


102 


39 


1 15-16 




mi 


94W 


21 


106 


89 








7 


41 


116 


42 


2 1-16 




76W 


107 1-10 


23 


123 


106 








7W 


47 


130 


45 


2 3-16 




86W 


120W 


24 


134 


115 








8 


53 


150 


48 


2 5-16 




96J4 


134% 


25 


146 


125 









WIKE HOPES. 231 

it is somewhat more flexible than rope of the same diameter and composed 
of the same number of wires laid up in the ordinary manner; and (especi- 
ally) that owing to the fact that the wires are laid more axially in the rope, 
longer surfaces of the wire are exposed to wear, and the endurance of the 
rope is thereby increased. (Trenton Iron Co.) 

Notes on the Use of Wire Rope. 
(J. A. Roebling's Sons Co.) 

Two kinds of wire rope are manufactured. The most pliable variety con- 
tains nineteen wires in the strand, and is generally used for hoisting and 
running rope. The ropes with twelve wires and seven wires in the strand 
are stiffer, and are better adapted for standing rope, guys, and rigging. Or- 
ders should state the use of the rope, and advice will be given. Ropes are 
made up to three inches in diameter, upon application. 

For safe working load, allow one fifth to one seventh of the ultimate 
strength, according to speed, so as to get good wear from the rope. When 
substituting wire rope for hemp rope, it is good economy to allow for the 
former the same weight per foot which experience has approved for the 
latter. 

Wire rope is as pliable as new hemp rope of the same strength; the for- 
mer will therefore run over the same-sized sheaves and pulleys as the latter. 
But the greater the diameter of the sheaves, pulleys, or drums, the longer 
wire rope will last. The minimum size of drum is given in the table. 

Experience has demonstrated that the wear increases with the speed. It 
is, therefore, better to increase the load than the speed. 

Wire rope is manufactured either with a wire or a hemp centre. The lat- 
ter is more pliable than the former, and will wear better where there is 
short bending. Orders should specify what kind of centre is wanted. 

Wire rope must not be coiled or uncoiled like hemp rope. 

When mounted on a reel, the latter should be mounted on a spindle or flat 
turn-table to pay off the rope. When forwarded in a small coil, without reel, 
roll it over the ground like a wheel, and run off the rope in that way. All 
untwisting or kiukiug must be avoided. 

To preserve wire rope, apply raw linseed-oil with a piece of sheepskin, 
wool inside; or mix the oil with equal parts of Spanish brown or lamp-black. 

To preserve wire rope under water or under ground, take mineral or vege- 
table tar, and add one bushel of fresh-slacked lime to one barrel of tar, 
which will neutralize the acid. Boil it well, and saturate the rope with the- 
hot tar. To give the mixture body, add some sawdust. 

In no case should galvanized rope be used for running rope. One day's 
use scrapes off the coating of zinc, and rusting proceeds with twice the 
rapidity. 

The grooves of cast-iron pulleys and sheaves should be filled with well- 
seasoned blocks of hard wood, set on end, to be renewed when worn out. 
This end-wood -will save wear and increase adhesion. The smaller pulleys 
or rollers which support the ropes on inclined planes should be constructed 
on the same plan. When large sheaves run with very great velocity, the 
grooves should be lined with leather, set on end, or with India rubber. This 
is done in the case of sheaves used in the transmission of power between 
distant points by means of rope, which frequently runs at the rate of 4000 
feet per minute. 

Steel ropes are taking the place of iron ropes, where it is a special object 
to combine lightness with strength. 

But in substituting a steel rope for an iron running rope, the object in view 
should be to gain an increased wear from the rope rather than to reduce the 
size. 

Locked Wire Rope. 

Fig. 74 shows what is known as the Patent Locked Wire Rope, made by 
the Trenton Iron Co. It is claimed to wear two to three times as long as an 




ordinary wire rope of equal diameter and of like material. Sizes made are 
from %, to \y^ inches diameter- 



232 



MATERIALS. 



CRANE CHAINS. 

(Pencoycl Iron Works.) 





"D.B. G 


" Special Crane. 






Crane. 




© 


" 






© 


T3 


















be . 


ctf 






oi 






-^ >> 








O . 






O 


a 


S 


° 71 








J <u 






J dJ- 


O © 

c © 


x © 


o .2 

fee S 
*= g o 

- ft- 


3 
U 


si 


©1 
U O 

pq a 

© a 
hr.3 
oS o3 


CO o3 g 

3 § ^ 


If 
11 


O) 'CO 


$4 

r*>© o 
3 § & 


33 




© S* 


3 

o 


Pi 




.53 


h 


© 


l^ 5 




Oh 


^ 






«j 


o 




<J 


O 


M 


25-32 


% 


% 


1932 


3864 


1288 


1680 


3360 


1120 


5-16 


27-32 


1 


1 1-16 


2898 


5796 


1932 


2520 


5040 


1680 


% 


31-32 


1 7-10 


m 


4186 


8372 


2790 


3640 


7280 


2427 


7-16 


1 5-32 


2 


Ws 


5796 


11592 


3864 


5040 


10080 


3360 


^ 


1 11-32 


2^ 


1 11-16 


7728 


15456 


5182 


6720 


13440 


4480 


9-16 


1 15-32 


3 ",'-10 


m 


9660 


19320 


6440 


840C 


16800 


5600 


% 


1 23-32 


4^ 


2 1-16 


11914 


M3828 


7942 


1036C 


20720 


6907 


11-16 


1 27-32 


5 


2J4 


14490 


28980 


9660 


1260C 


25200 


8400 


U 


1 31-32 


5% 


173S8 


34776 


11592 


1512C 


30240 


10080 


12-16 


2 3-32 


6 7-10 


2 11-16 


20286 


40572 


13524 


1764C 


35280 


11760 


% 


2 7-32 


8 


Ws 


22484 


44968 


14989 


2044C 


40880 


13627 


15-16 


2 15-32 


9 


3 1-16 


25872 


51744 


17248 


2352C 


47040 


15680 


1 


2 19-32 


10 7-10 


3J4 


29568 


59136 


19712 


2688C 


53760 


17920 


1 1-16 


2 53-32 


11 2-10 


3 5-16 


33264 


66538 


22176 


3024C 


60480 


20160 


1*6 


2 27-32 


12J^ 


m 


37576 


75152 


25050 


3416C 


68320 


22773 


1 3-16 


3 5-32 


13 7-10 


m 


41888 


83776 


27925 


3808C 


76160 


25387 


VA 


3 7-32 


16 


4y 8 


46200 


92400 


30800 


4200C 


84000 


28000 


1 5-16 


3 15-32 


16J^ 


4% 


50512 


101024 


33074 


4592C 


91840 


30613 


1^ 


3% 


18 4-10 


4 9-16 


55748 


111496 


37165 


5068C 


101360 


33787 


1 7-16 


3 25-32 


19 7-10 


m 


60368 


120736 


40245 


5188C 


109760 


36587 


iH 


3 31-32 


21 7-10 


5 


66528 


133055 


41352 


60480 


120960 


40320 



The distance from centre of one link to centre of next is equal to the in- 
side length of link, but in practice 1/32 inch is allowed for weld. This is ap- 
proximate, and where exactness is required, chain should be made so. 

For Chain Sheaves. — The diameter, if possible, should be not less than 
twenty times the diameter of chain used. 

Example.— For 1-inch chain use 20-inch sheaves. 

WEIGHTS OF LOGS, LUMBER, ETC. 
Weight of Green Logs to Scale 1,000 Feet, Board Measure. 

Yellow pine (Southern) 8,000 to 10,000 lbs. 

Norway pine (Michigan) 7,000 to 8,000 " 

Whit«™-np(MiPhi M nJ oflofstum P 6,000 to 7,000 " 

V, hitepme (Michigan) -j outofwater 7,000 to 8,000 " 

White pine (Pennsylvania), bark off 5,000 to 6,000 " 

Hemlock (Pennsy lva nia), bark off 6,000 to 7,000 " 

Four acres of water are required to store 1,000,000 feet of logs. 

Weight of 1,000 Feet of Lumber, Board Measure. 

Yellow or Norway pine Dry, 3,000 lbs. Green, 5,000 lbs. 

White pine " 2,500 " " 4,000 " 

Weight of 1 Cord of Seasoned Wood, 128 Cubic Feet per 

Cord. 

Hickory or sugar maple 4,500 lbs. 

White oak 3,850 " 

Beech, red oak or black oak. 3,250 l ' 

Poplar, chestnut or elm 2,350 " 

Pine (white or Norway) 2,000 " 

Hemlock bark, dry ... 2,200 » 





SIZES OP EIRE-BRICK. 233 

SIZES OF FIRE-BRICK. 

9-iuch straight 9 x 4]4 x 2% inches. 

/ \ Soap 9x2^x2^ " 

Jamb \ Checker 9x3 x3 

\ 2-inch 9x4J^x2 

9x4^x2^ / S P lit 9x4^x1)4 " 

Jl^ll^L/ Jamb 9x4j|x2^ " 

No. 1 key 9x2i^thickx4^to4inches 

wide. 

113 bricks to circle 12 feet inside diam. 

Ke ^____J^ No. 2 key 9 x 2% thick x 4)4 to 3J^ 

f4V4-a^ inches wide. 

C3 bricks to circle 6 ft. inside diam. 

No. 3key 9x2^ thick x 4% to 3 

inches wide. 

~3 bricks to circle 3 ft. inside diam. 

Wedse \ No. 4key 9x2^ thick x 4\i to 2^ 

inches wide. 

25 bricks to circle 1^ ft. inside diam. 
fv**X*Wi\LX) No> j wedge (or bullhead). 9x4^ wide x 2J^ to 2 in. 
thick, tapering lengthwise. 

A . 98 bricks to circle 5 ft. inside diam. 

'\ Arch \ No. 2 wedge 9x4^x2^ to \y 2 in. thick. 

) 4 60 bricks to circle 2% ft. inside diam. 

/9x4>^x(2Xi%/ No. 1 arch 9x4J^x2^ to 2 in. thick, 

f " / tapering breadthwise. 

' 72 bricks to circle 4 ft. inside diam. 

No. 2 arch 9 x 4}4 x 2]4 to V& 

\ 42 bricks to circle 2 ft. inside diam. 

..iSkew\ No. 1 skew 9 to 7 x 4}4 to 2J^. 

\ Bevel on one end. 

„ } No. 2 skew 9x2j^x4^to2^. 

<§:7) x 4V^ x 2>^/ Equal bevel on both edges. 

' No.3skew.... 9 x 2% x 4% to iy 2 . 

Taper on one edge. 

— — \ 24 inch circle 8% to 5% x 4% x 2%. 

\____ \ Edges curved, 9 bricks line a 24-inch circle. 

f " ) 36-inch circle 8% to 6^ x 4}4 x 2^. 

/ 9 x 2kx av-^A 13 bricks line a 36-inch circle. 

J k X2 /2 j 48-inch circle 8^ to 714 x 4^ x 2%. 

* ' 17 bricks line a 48-inch circle. 

13J^-inch straight 13^ x 2% x 6. 

13^-inch key No. 1 13j| x 2% x 6 to 5 inch. 

Skew~ — \ 90 bricks turn a 12-ft. circle. 

1 133^-inch key No. 2 13^ x 2^ x 6 to 4% inch. 

, • , 52 bricks turn a 6-ft. circle. 

~ X2 l X2 x v Bridge wall, No. 1 13x6^x6. 

Bridge wall, No. 2 13X 6j| x 3. 

6 in. Ci rc i e Mill tile 18, 20, or 24 x 6 x 3. 

££ \ Stock-hole tiles 18, 20, or 24'x 9 x 4. 

18-inch block 18x9x6. 

Flat back 9x6x2^. 

Flat back arch 9 x 6 x 3J^ to 2%. 

22-inch radius, 56 bricks to circle. 

Locomotive tile 32 x 10 x 3. 

34x10x3. 
Cupola 

40x10x3. 
Tiles, slabs, and blocks, various sizes 12 to 30 inches 
long, 8 to 30 inches wide, 2 to 6 inches thick. 
Cupola brick, 4 and 6 inches high, 4 and 6 inches radial width, to line shells 
23 to 66 in diameter. 

A 9-inch straight brick weighs 7 lbs. and contains 100 cubic inches. (=120 
lbs. per cubic foot. Specific gravity 1.93.) 

One cubic foot of wall requires i7 9-inch bricks, one cubic yard requires 
460. Where keys, wedges, and other " shapes " are used, add 10 per cent iu 
estimating the number required. 





234 



MATERIALS. 



One ton of fire-clay should be sufficient to lay 3000 ordinary bricks. To 
secure the best results, fire-bricks should be laid in the same clay from which 
they are manufactured. It should be used as a thin paste, and not as mor- 
tar. The thinner the joint the better the furnace wall. In ordering bricks 
the service for which they are required should be stated. 



NUMBER OF FIRE-BRICK REQUIRED FOR 
VARIOUS CIRCL.ES. 





KEY BRICKS. 


ARCH BRICKS. 


WEDGE BRICKS. 


s s 


o 

to 

25 

17 
9 


d 
to 


6 

to 


6 

to 


1 
o 
Eh 


6 

to 


6 
to 


OS 


3 
o 
E-i 


6 

to 


d 

to 


OS 


O 

E-< 


ft. in. 
I 6 




25 
30 
34 
38 
42 
46 
51 
55 
59 
63 
67 
71 
76 
80 
84 
88 
92 
97 
101 
105 
109 
113 
117 










2 


13 
25 
38 
32 
25 
19 
13 
6 


10 
21 

32 
42 
53 
63 
58 
52 
47 
42 
37 
31 
2G 
21 
16 
11 
5 


9 
19 
29 
38 
47 
57 
66 
76 
85 
94 
104 
113 
113 


42 
31 

21 
10 






42 
49 
57 
64 
72 
80 
87 
95 
102 
110 
117 
125 
132 
140 
147 
155 
162 
170 
177 
185 
193 










2 6 


18 
3G 
54 

72 
72 

72 
72 

72 

72 
7:2 

72 

72 
72 
72 
72 

72 
72 


""8" 
15 
23 
30 
38 
45 
53 
60 
68 
75 
83 
90 
98 
105 
113 
121 


GO 
48 
3G 
24 
12 






60 


3 

3 6 

4 

4 6 

5 

5 6 

6 

6 6 

7 

7 6 

8 

8 6 

9 
9 6 

10 

10 G 

11 

11 6 

12 


20 
40 

59 
79 

98 
98 
98 
98 
98 
98 
98 
98 
98 
98 
98 
98 
98 
98 
98 


15 
23 
30 
38 
46 
53 
61 
68 
76 
83 
91 
98 
106 


68 
76 
83 
91 
98 
106 
113 
121 
128 
136 
144 
151 
159 
166 
174 
181 
189 
196 
304 


12 6 



































For larger circles than 12 feet use 113 No. 1 Key, and as many 9-inch brick 
as may be needed in addition. 



ANALYSES OF IT. SAVAGE FIRE-CLAY, 

(1) (2) (3) (4) 

1871 1877. 1878. 1885. 

Mass - ^TavJoT CteSfoScal (2 samples) 

Institute of N e^ ay T e r °i v Sm-v?v of D'"- Otto 

Technology. pr £ e £ g™*^ PemfsylJania. Wuth " 

50.457 56.80 Silica 44.395 56.15 

35.904 30.08 Alumina 33.558 33.295 

1.15 Titanic acid 1.530 

1.504 1.12 Peroxide iron . 1.080 0.59 

0.133 ... . Lime trace 0.17 

0.018 Magnesia 0.108 0.115 

trace 0.80 Potash (alkalies) 0.247 

12.744 10.50 Water and inorg. matter. 14.575 9.68 

100.760 100.450 100:493 100.000 



MAGNESIA BRICKS. 235 

MAGNESIA BRICKS. 

"Foreign Abstracts " of the Institution of Civil Engineers, 1893, gives a 
paper by C. Bischof on the production of magnesia bricks. The material 
most in favor at present is the magnesite of Styria, which, although less 
pure considered as a source of magnesia than the Greek, has the property 
of fritting at a high temperature without melting. The composition of the 
two substances, in the natural and burnt states, is as follows: 

Magnesite. Styrian. Greek. 

Carbonate of magnesia 90.0 to 96.0$ 94.46$ 

" lime 0.5 to 2.0 4.49 

" iron 3.0 to 6.0 FeO 0.08 

Silica 1.0 0.52 

Manganous oxide 0.5 Water 0.54 

Burnt Magnesite. 

Magnesia ... 77.6 82.46-95.36 

Lime 7.3 0.83—10.92 

Alumina and ferric oxide 13.0 0.56— 3.54 

Silica 1.2 0.73—7.98 

At a red heat magnesium carbonate is decomposed into carbonic acid and 
caustic magnesia, which resembles lime in becoming hydrated and recar- 
bonated when exposed to the air, and possesses a certain plasticity, so that 
it can be moulded when subjected to a heavy pressure. By long-continued 
or stronger heating the material becomes dead-burnt, giving a form of mag- 
nesia of high density, sp. gr. 3.8, as compared with 3.0 in the plastic form, 
which is unalterable in the air but devoid of plasticity. A mixture of two 
volumes of dead-burnt with one of plastic magnesia can be moulded into 
bricks which contract but little in firing. Other binding materials that have 
been used are: clay up to 10 or 15 per cent; gas -tar, perfectly freed from 
water, soda,, silica, vinegar as a solution of magnesium acetate which is 
readily decomposed by heat, and carbolates of alkalies or lime. Among . 
magnesium compounds a weak solution of magnesium chloride may also be 
used. For setting the bricks lightly burnt, caustic magnesia, with a small 
proportion of silica to render it less refractory, is recommended. The 
strength of the bricks may be increased by adding iron, either as oxide or 
silicate. If a porous product is required, sawdust or starch may be added 
to the mixture. When dead-burnt magnesia is used alone, soda is said to be 
the best binding material. 

See also papers by A. E. Hunt, Trans. A. I. M. E., xvi, 720, and by T. Egles- 
ton, Traus. A. I. M. E., xiv. 458. 

Asbestos.— J. T. Donald, Eng. and M. Jour., June 27, 1891. 

Analysis. 

Canadian. 

Italian. Broughton. Templeton. 

Silica.. 40.30$ 40.57$ 40.52?$ 

Magnesia 43.37 41.50 42.05 

Ferrous oxide 87 2.81 1.97 

Alumina 2.27 .90 2.10 

Water 13.72 13.55 13.46 

100.53 99.33 100.10 

Chemical analysis throws light upon an important point in connection 
with asbestos, i.e., the cause of the harshness of the fibre of some varieties. 
Asbestos is principally a hydrous silicate of magnesia, i.e.. silicate of mag- 
nesia combined with water. When harsh fibre is analyzed it is found to 
contain less water than the soft fibre. In fibre of very fine quality from 
Black Lake analysis showed 14.38$ of water, while a harsh-fibred sample 
gave only 11.70$. If soft fibre be heated to a temperature that will drive off 
a portion of the combined water, there results a substance so brittle that it 
may be crumbled between thumb and finger. There is evidently some con- 
nection between the consistency of the fibre and the amount of water in its 
composition. 



236 STRENGTH OF MATERIALS. 



STRENGTH OF MATERIALS. 

Stress and Strain.— There is much confusion among writers on 
strength of materials as to the definition of these terms. An external force 
applied to a body, so as to pull it apart, is resisted by an internal force, or 
resistance, and the action of these forces causes a displacement of the mole- 
cules, or deformation. By some writers the external force is called a stress, 
and the internal force a strain; others call the external force a strain, and 
the internal force a stress: this confusion of terms is not of importance, as 
the words stress and strain are quite commonly used synonymously, but the 
use of the word strain to mean molecular displacement, deformation, or dis- 
tortion, as is the custom of some, is a corruption of the language. See En- 
gineering NeioSy June 23, 1892. Definitions by leading authorities are given 
below. 

Stress.— A. stress is a force which acts in the interior of a body, and re- 
sists the external forces which tend to change its shape. A deformation is 
the amount of change of shape of a body caused by the stress. The word 
sti'ain is often used as synonymous with stress and sometimes it is also used 
to designate the deformation. (Merriman.) 

The force by which the molecules of a body resist a strain at any point is 
called the stress at that point. 

The summation of the displacements of the molecules of a body for a 
given point is called the distortion or strain at the point considered. (Burr). 

Stresses are the forces which are applied to bodies to bring into action 
their elastic and cohesive properties. These forces cause alterations of the 
forms of the bodies upon which they act. Strain is a name given to the 
kind of alteration produced by the stresses. The distinction between stress 
and strain is not always observed, one being used for the other. (Wood.) 

Stresses are of different kinds, viz. : tensile, compressive, transverse, tor- 
sional, and shearing stresses. 

A tensile stress, or pull, is a force tending to elougate a piece. A com- 
pressive stress, or push, is a force tending to shorten it. A travsverse stress 
tends to bend it. A torsional stress tends to twist it. A shearing stress 
tends to force one part of it to slide over the adjacent part. 

Tensile, compressive, and shearing stresses are called simple stresses. 
Transverse stress is compounded of tensile and compressive stresses, and 
torsional of tensile and shearing stresses. 

To these five varieties of stresses might be added tearing stress, which is 
either tensile or shearing, but in which the resistance of different portions 
of the material are brought into play in detail, or one after the other, in- 
stead of simultaneously, as in the simple stresses. 

Effects of Stresses.— The following general laws for cases of simple 
tension or compression have been established by experiment. (Merriman): 

1. When a small stress is applied to a body, a small deformation is pro- 
duced, and on the removal of the stress the body springs back to its original 
form. For small stresses, then, materials may be regarded as perfectly 
elastic. 

2. Under small stresses the deformations are approximately proportional 
to the forces or stresses which produce them, and also approximately pro- 
portional to the length of the bar or body. 

3. When the stress is great enough a deformation is produced which is 
partly permanent, that is, the body does not spring back entirely to its 
original form on removal of the stress. This permanent part is termed a 
set. In such cases the deformations are not proportional to the stress. 

4. When the stress is greater still the deformation rapidly increases and 
the body finally ruptures. 

5. A sudden stress, or shock, is more injurious than a steady stress or than 
a stress gradually applied. 

Elastic liiniit.— The elastic limit is defined as that point at which the 
deformations cease to be proportional to the stresses, or, the point at which 
the rate of stretch (or other, deformation) begins to increase. It is also 
defined as the point at which the first permanent set becomes visible. The 
last definition is not considered as good as the first, as it is found that with 
some materials a set occurs with any load, no matter how small, and that 
with others a set which might be called permanent vanishes with lapse of 
time, and as it is impossible to get the point of first set without removing 



STRESS AND STRAIN. 237 

the whole load after each increase of load, which is frequently inconvenient. 
The elastic limit, defined, however, as the point at which the extensions be- 
gin to increase at a higher ratio than the applied stresses, usually corresponds 
Very nearly with the point of first measurable permanent set. 

Yield-point.— The term yield-point has recently been introduced into 
the literature of the strength of materials. It is defined as that point at 
which the rate of stretch suddenly increases rapidly. The difference be- 
tween the elastic limit, strictly defined as the point at which the rate of 
stretch begins to increase, and the yield-point, at which the rate increases 
suddenly, may in some cases be considerable. This difference, however, will 
not be discovered in short test-pieces unless the readings of elongations are 

1 
OOOi 

of an inch. In using a coarser instrument, such as calipers reading to 1/100 
of an inch, the elastic limit and the yield-point will appear to be simultane- 
ous. Unfortunately for precision of language, the term yield-point was not 
introduced until long after the term elastic limit had been almost univer- 
sally adopted to signify the same physical fact which is now defined by the 
term yield-point, that is, not the point at which the first change in rate, 
observable only by a microscope, occurs, but that later point (more or less 
indefinite as to its precise position) at which the increase is great enough to 
be seen by the naked eye. A most convenient method of determining the 
point at which a sudden increase of rate of stretch occurs in short speci- 
mens, when a testing-machine in which the pulling is done by screws is 
used, is to note the weight on the beam at the instant that the beam " drops. " 
During the earlier portion of the test, as the extension is steadilj r increased 
by the uniform but slow rotation of the screws, the poise is moved steadily 
along the beam to keep it in equipoise; suddenly a point is reached at which 
the beam drops, and will not rise until the elongation has been considerably 
increased by the further rotation of the screws, the advancing of the poise 
meanwhile being suspended. This point corresponds practically to the point 
at which the rate of elongation suddenly increases, and to the point at 
which an appreciable permanent set is first found. It is also the point which 
has hitherto been called in practice and in text-books the elastic limit, and 
it will probably continue to be so called, although the use of the newer term 
"yield-point " for it, and the restriction of the term elastic limit to mean 
the earlier point at which the rate of stretch begins to increase, as determin- 
able only by micrometric measurements, is more precise and scientific. 

In tables of strength of materials hereafter given, the term elastic limit is 
used in its customary meaning, the point at which the rate of stress has be- 
gun to increase, as observable by ordinary instruments or by the drop of 
the beam. With this definition it is practically synonymous with yield- 
point. 

Coefficient (or Modulus) of Elasticity.— This is a term express- 
ing the relation between the amount of extension or compression of a mate- 
rial and the load producing that extension or compression. 

It may be defined as the load per unit of section divided by the extension 
per unit of length; or the reciprocal of the fraction expressing the elonga- 
tion per inch of length, divided by the pounds per square inch of section 
producing that elongation. 

Let P be the applied load, k the sectional area of the piece, I the length of 
the part extended, A the amount of the extension, and _£Jthe coefficient of 
elasticity. Then 

p 

— — the load on a unit of section ; 
k 

j = the elongation of a unit of length. 

k ' l~ kk' 

The coefficient of elasticity is sometimes defined as the figure expressing 
the load which would be necessary to elongate a piece of one square inch 
section to double its original length, provided the piece would not break, and 
the ratio of extension to the force producing it remained constant. This 
definition follows from the formula above given, thus: Iffc=one square 
inch, I and ^ each = one inch, then E — P. 

Within the elastic limit, when the deformations are proportional to the 



238 STRENGTH OF MATERIALS. 

stresses, the coefficient of elasticity is constant, but beyond the elastic limit 
it decreases rapidly. 

In cast iron there is generally no apparent limit of elasticity, the deforma- 
tions increasing at a faster rate than the stresses, and a permanent set being 
produced by small loads. The coefficient of elasticity therefore is not con- 
stant during any portion of a test, but grows smaller as the load increases. 
The same is true in the case of timber. In wrought iron and steel, however, 
there is a well-defined elastic limit, and the coefficient of elasticity within 
that limit is nearly constant. 

Resilience, or Work of Resistance of a Material.— Within 
the elastic limit, the resistance increasing uniformly from zero stress to the 
stress at the elastic limit, the work done by a load applied gradually is equal 
to one half the product of the final stress by the extension or other deforma- 
tion. Beyond the elastic limit, the extensions increasing more rapidly than 
the loads, and the strain diagram approximating a parabolic form, the work 
is approximately equal to two thirds the product of the maximum stress by 
the extension. 

The amount of work required to break a bar, measured usually in inch- 
pounds, is called its resilience; the work required to strain it to the elastic 
limit is called its elastic resilience. 

Under a load applied suddenly the momentary elastic distortion is equal 
to twice that caused by the same load applied gradually. 

When a solid material is exposed to percussive stress, as when a weight 
falls upon a beam transversely, the work of resistance is measured by the 
product of the weight into the total fall. 

Elevation of Ultimate Resistance and Elastic Limit.— It 
was first observed by Pruf. R. H. Thurston, and Commander L. A. Beards- 
lee, U. S. N., independently, in 1873, that if wrought iron be subjected to a 
stress beyond its elastic limit, but not beyond its ultimate resistance, and 
then allovvtd to "rest 1 ' for a definite interval of time, a considerable in- 
crease of elastic limit and ultimate resistance may be experienced. In other 
words, the application of stress and subsequent " rest " increases the resist- 
ance of wrought iron. 

This " rest " may be an entire release from stress or a simple holding the 
test-piece at a given intensity of stress. 

Commander Beardslee prepared twelve specimens and subjected them to 
an intensity of stress equal to the ultimate resistance of the material, with- 
out breaking the specimens. These were then allowed to rest, entirely free 
from stress, from 24 to 30 hours, after which period they were again stressed 
until broken. The gain in ultimate resistance by the rest was found to vary 
from 4.4 to 17 per cent. 

This elevation of elastic and ultimate resistance appears to be peculiar to 
iron and steel: it has not been found in other metals. 

Relation of the Elastic liimit to Endurance under Re» 
peated Stresses (condensed from Engineering, August 7, 1891). — 
When engineers first began to test materials, it was soon recognized that 
if a specimen was loaded beyond a certain point it did not recover its origi- 
nal dimensions on removing the load, but took a permanent set; this point 
was called the elastic limit. Since below this point a bar appeared to recover 
completely its original form and dimensions on removing the load, it ap- 
peared obvious that it had not been injured by the load, and hence the work- 
ing load might be deduced from the elastic limit by using a small factor of 
safety. 

Experience showed, however, that in many cases a bar would not carry 
safely a stress anywhere near the elastic limit of the material as determined 
by these experiments, and the whole theory of any connection between the 
elastic limit of a bar and its working load became almost discredited, and 
engineers employed the ultimate strength only in deducing the safe working 
load to which their structures might be subjected. Still, as experience accu- 
mulated it was observed that a higher factor of safety was required for a live 
load than for a dead one. 

In 1871 Wohler published the results of a number of experiments on bars 
of iron and steel subjected to live loads. In these experiments the stresses 
were put on and removed from the specimens without impact, but it was, 
nevertheless, found that the breaking stress of the materials was in every 
case much below the statical breaking load. Thus, a bar of Krupp 1 s axle 
steel having a tenacity of 49 tons per square inch broke with a stress of 28.6 
tons per square inch, when the load was completely removed and replaced 
without impact 170,000 times. These experiments were made on a large 



STRESS AND STRAIN". 239 

number of different brands of iron and steel, and the results were concor- 
dant in showing that a bar would break with an alternating stress of only, 
say, one third the statical breaking strength of the material, if the repetitions 
of stress were sufficiently numerous. At the same time, however, it ap- 
peared from the general trend of the experiments that a bar would stand an 
indefinite number of alternations of stress, provided the stress was kept 
below the limit. 

Prof. Bauschinger defines the elastic limit as the point at which stress 
ceases to be sensibly proportional to strain, the latter being measured with 

a mirror apparatus reading to ^-^th of a millimetre, or about in. 

This limit is always below the yield-point, and may on occasion be zero. On 
loading a bar above the yield-point, this point rises with the stress, and the 
rise continues for weeks, months, and possibly for years if the bar is left at 
rest under its load. On the other hand, when a bar is loaded beyond its true 
elastic limit, but below its yield-point, this limit rises, but reaches a maxi- 
mum as the yield-point, is approached, and then falls rapidly, reaching even 
to zero. On leaving the bar at rest under a stress exceeding that of its 
primitive breaking-down point the elastic limit begins to rise again, and 
may, if left a sufficient time, rise to a point much exceeding its previous 
value. 

This property of the elastic limit of changing with the history of a bar has 
done more to discredit it than auything else, nevertheless it now seems as if 
it, owing to this very property, were once more to take its former place in 
the estimation of engineers, and this time with fixity of tenure. It had long 
been known that the limit of elasticity might be raised, as we have said, to 
almost any point within the breaking load of a bar. Thus, in some experi- 
ments by Professor Styffe, the elastic limit of a puddled-steel bar was raised 
16,000 lbs. by subjecting the bar to a load exceeding its primitive elastic 
limit. 

A bar has two limits of elasticity, one for tension and one for compression. 
Bauschinger loaded a number of bars in tension until stress ceased to be 
sensibly proportional to strain. The load was then removed and the bar 
tested in compression until the elastic limit in this direction had been ex- 
ceeded. This process raises the elastic limit in compression, as would be 
found on testing the bar in compression a second time. In place of this, 
however, it was now again tested in tension, when it was found that the 
artificial raising of the limit in compression had lowered that in tension be- 
low its previous value. By repeating the process of alternately testing in 
. tension and compression, the two limits took up points at equal distances 
from the line of no load, both in tension and compression. These limits 
Bauschinger calls natural elastic limits of the bar, which for wrought iron 
correspond to a stress of about 8^ tons per square inch, but this is practically 
the limiting load to which a bar of the same material can be strained alter- 
nately in tension and compression, without breaking when the loading is 
repeated sufficiently often, as determined by Wohler's method. 

As received from the rolls the elastic limit of the bar in tension is above 
the natural elastic limit of the bar as defined by Bauschinger, having been 
artificially raised by the deformations to which it has been subjected in the 
process of manufacture. Hence, when subjected to alternating stresses, 
the limit in tension is immediately lowered, while that in compression is 
raised until they both correspond to equal loads. Hence, in Wohler's ex- 
periments, in which the bars broke at loads nominally below the elastic 
limits of the material, there is every reason for concluding that the loads 
were really greater than true elastic limits of the material. This is con- 
firmed by tests on the connecting-rods of engines, which of course work 
under alternating stresses of equal intensity. Careful experiments on old 
rods show that the elastic limit in compression is the same as that in ten- 
sion, and that both are far below the tension elastic limit of the material as 
received from the rolls. 

The common opinion that straining a metal beyond its elastic limit injures 
it appears to be untrue. It is not the mere straining of a metal beyond one 
elastic limit that injures it, but the straining, many times repeated, be3 7 ond 
its two elastic limits. Sir Benjamin Baker has shown that in bending a shell 
plate for a boiler the metal is of necessity strained beyond its elastic limit, 
so that stresses of as much as ? tons to 15 tons per square inch may obtain 
in it as it comes from the rolls, and unless the plate is annealed, these 
stresses will still exist after it has been built into the boiler. In such a case, 
however, when exposed to the additional stress due to the pressure inside 



240 STRENGTH OF MATERIALS. 

the boiler, the overstrained portions of the plate will relieve themselves by 
stretching and taking a permanent set, so that probably after a year's work- 
ing very little difference could be detected in the stresses in a plate built in- 
to the boiler as it came from the bending rolls, and in one which had been 
annealed, before riveting into place, and the first, in spite of its having been 
strained beyond its elastic limits, and not subsequently annealed, would be 
as strong as the other. 

Resistance of Metals to Repeated Shocks. 

More than twelve years were spent by Wohler at the instance of the Prus- 
sian Government in experimenting upon the resistance of iron and steel to 
repeated stresses. The results of his experiments are expressed in what is 
known as Wohler's law, which is given in the following words in Dubois's 
translation of Weyrauch: 

" Rupture may be caused not only by a steady load which exceeds the 
carrying strength, but also by repeated applications of stresses, none of 
which are equal to the carrying strength. The differences of these stresses 
are measures of the disturbance of continuity, in so far as by their increase 
the minimum stress which is still necessary for rupture diminishes." 

A practical illustration of the meaning of the first portion of this law may 
I e given thus: If 50,000 pounds once applied will just break a bar of iron or 
steel, a stress very much less than 50,000 pounds will break it if repeated 
sufficiently often. 

This is fully confirmed by the experiments of Fairbairn and Spangenberg, 
as well as those of Wohler; and, as is remarked by Weyrauch, it may be 
considered as along-known result of common experience. It pai-tially ac- 
counts for what Mr. Holley has called the " intrinsically ridiculous factor of 
safety of six." 

Another "long-known result of experience " is the fact that rupture may 
be caused by a succession of shocks or impacts, none of which alone would 
be sufficient to cause it. Iron axles, the piston-rods of steam hammers, and 
other pieces of metal subject to continuously repeated shocks, invariably 
break after a certain length of service. They have a " life " which is lim- 
ited. 

Several years ago Fairbairn wrote: " We know that in some cases wrought 
iron subjected to continuous vibration assumes a crystalline structure, and 
that the cohesive powers are much deteriorated, but we are ignorant of the 
causes of this change." We are still ignorant, not only of the causes of this 
change, but of the conditions under which it takes place. Who knows 
whether wrought iron subjected to very slight continuous vibration will en- 
dure forever? or whether to insure final rupture each of the continuous small 
shocks must amount at least to a certain percentage of single heavy shock 
(both measured in foot pounds), which would cause rupture with one applica- 
tion ? Wohler found in testing iron by repeated stresses (not impacts) that 
in one case 400,000 applications of a stress of 500 centners to the square inch 
caused rupture, while a similar bar remained sound after 48,000,000 applica- 
tions of a stress of 300 centners to the square inch (1 centner = 110.2 lbs.). 

Who knows whether or not a similar law holds true in regard to repeated 
shocks ? Suppose that a bar of iron would break under a single impact of 
1000 foot-pounds, how many times would it be likely to bear the repetition 
of 100 foot-pounds, or would it be safe to allow it to remain for fifty years 
subjected to a continual succession of blows of even 10 foot-pounds each ? 

Mr. William Metcalf published in the Metallurgical Rev ieiv, Dec. 1877, the 
results of some tests of the life of steel of different percentages of carbon 
under impact. Some small steel pitmans were made, the specifications for 
which required that the unloaded machine should run 4}4 hours at the rate 
of 1200 revolutions per minute before breaking. 

The steel was all of uniform quality, except as to carbon. Here are the 
results; The 

.30 C. ran 1 h. 21 m. Heated and bent before breaking. 

.49 0. " lh. 28 m., " " " 

.43 C. "4 h, 57 m. Broke without heating. 

.65 C. " 3 h. 50 m. Broke at weld where imperfect. 

.80C. " 5h. 40 m. 

.84 C. " 18 h. 

.87 C. Broke in weld near the end. 

.96 C. Ran 4.55 m., and the machine broke down. 

Some other experiments by Mr. Metcalf confirmed his conclusion, viz., 



STRESS AND STRAIK. 241 

that high-carbon steel was belter adapted to resist repeated shocks and vi- 
brations than low-carbon steel. 

These results, however, would scarcely be sufficient to induce any en- 
gineer to use .84 carbon steel in a car-axle or a bridge-rod. Further experi- 
ments are needed to confirm or overthrow them. 

(See description of proposed apparatus for such an investigation in the 
author's paper iu Trans. A. L M. E., vol. viii , p. 76, from which the above 
extract is taken.) 

Stresses Produced by Suddenly Applied Forces and 
Shocks. 

(Mansfield Merriman, R. R. <& Eng. Jour., Dec. 1889.) 
Let P be the weight which is dropped from a height h upon the end of a 
bar, and let y be the maximum elongation which is produced. The work 
performed by the falling weight, then, is 

W='P(h + y), 
and this must equal the internal work of the resisting molecular stresses. 
The stress in the bar, which is at first 0, increases up to a certain limit Q, 
which is greater than P; and if the elastic limit be not exceeded the elonga- 
tion increases uniformly with the stress, so that the internal work is equal 
to the mean stress 1/2Q multiplied by the total elongation y, or 

W=1/2Qij. 
Whence, neglecting the work that may be dissipated in heat, 

l/2Qy = Ph + Py. 
If e be the elongation due to the static load P, within the elastic limit 
y = — e ; whence 



?=p(iV i+2 ")' 



(1) 

which gives the momentary maximum stress. Substituting this value of Q, 
there results 



/ = e (l-f|/l+2^), 



(2) 

which is the value of the momentary maximum elongation. 

A shock results when the force P, before its action on the bar, is moving 
with velocity, as is the case when a weight P falls from a height h. The 
above formulas show that this height h may be small if e is a small quan- 
tity, and yet very great stresses and deformations be produced. For in- 
stance, let h = 4e, then Q = 4P and y = 4e ; also let h = 12e, then Q = 6P 
and y = 6 e. Or take a wrought-iron bar 1 in. square and 5 ft. long: under a 
steady load of 5000 lbs. this will be compressed about 0.0012 in., supposing 
that no lateral flexure occurs; but if a weight of 5000 lbs. drops upon its end 
from the small height of 0.0048 in. there will be produced the stress of 20,000 
lbs. 

A suddenly applied force is one which acts with the uniform intensity P 
upon the end of the bar, but which has no velocity before acting upon it. 
This corresponds to the case of h — in the above formulas, and gives Q = 
2P and y = 26 for the maximum stress and maximum deformation. Prob- 
ably the action of a rapidly-moving train upon a bridge produces stresses 
of this character. 

Increasing the Tensile Strength of Iron Bars by Twist- 
ing them.— Ernest L. Ransome of San Francisco has obtained an English 
Patent, No. 16221 of 1888, for an " improvement in strengthening and testing 
wrought metal and steel rods or bars, consisting in twisting the same in a 
cold state. . . . Any defect in the lamination of the metal which would 
otherwise be concealed is revealed by twisting, and imperfections are shown 
at once. The treatment may be applied to bolts, suspension-rods or bars 
subjected to tensile strength of any description." 

Results of tests of this process were reported by Lieutenant F. P. Gilmore, 
U. S. N.,in a paper read before the Technical Society of the Pacific Coast, 
published in the Transactions of the Society for the month of December, 
1888. 

Tests were also made in 1889 in the University of California. The exper- 
iments include trials with thirty-nine bars, twenty-nine of which were va- 



242 



STREHGTH OF MATERIALS. 



riously twisted, from three-eighths of one turn to six turns per foot. The 
test-pieces were cut from one and the same bar, and accurately measured 
and numbered. From each lot two pieces without twist were tested for ten- 
sile strength and ductility. One group of each set was twisted until the 
pieces broke, as a guide for the amount of twist to be given those to be 
tested for tensile strain. 

The following is the result of one set of Lieut. Gilmore's tests, on iron 
bars 8 in. long, .719 in. diameter. 



No. of 


Conditions. 


Twists 


Twists 


Tensile 


Tensile 


Gain per 


Bars. 


Turns. 


per ft. 


Strength. 


per sq. in. 


cent. 


2 


Not twisted. 








22,000 


54,180 




2 


Twisted cold. 


y 2 


% 


23,900 


59,020 


9 


2 


" " 


i 


m 


25,800 


63,500 


17 


2 


" " 


2 


3 


26,300 


64,750 


19 


1 




"* 


m 


26,400 


65,000 


20 



TENSILE STRENGTH. 

The following data are usually obtained in testing by tension in a testing- 
machine a sample of a material of construction : 

The load and the amount of extension at the elastic limit. 

The maximum load applied before rupture. 

The elongation of the piece, measured between gauge-marks placed a 
stated distance apart before the test; and the reduction of area at the 
point of fracture. 

The load at the elastic limit and the maximum load are recorded in pounds 
per square inch of the original area. The elongation is recorded as a per- 
centage of the stated length between the gauge-marks, and the reduction 
area as a percentage of the original area. The coefficient of elasticity is cal- 
culated from the ratio the extension within the elastic limit per inch of 
length bears to the load per square inch producing that extension. 

On account of the difficulty of making accurate measurements of the frac- 
tured area of a test-piece, and of the fact that elongation is more valuable 
than reduction of area as a measure of ductility and of resilience or work 
of resistance before rupture, modern experimenters are abandoning the 
custom of reporting reduction of area. The " strength per square inch of 
fractured section " formerly frequently used in reporting tests is now almost 
entirely abandoned. The data now calculated from the results of a tensile 
test for commercial purposes are: 1. Tensile strength in pounds per square 
incli of original area. 2. Elongation per cent of a stated length between 
gauge-marks, usually 8 inches. 3. Elastic limit in pounds per square inch 
of original area. 

The short or grooved test specimen gives with most metals, especially 
with wrought iron and steel, an apparent tensile strength much higher 
than the real strength. This form of test-piece is now almost entirely aban- 
doned. 

The following results of the tests of six specimens from the same 1*4" steel 

bar illustrate the apparent elevation of elastic limit and the changes in 

other properties due to change in length of stems which were turned down 

, in each specimen to .798" diameter. (Jas. E. Howard, Eng. Congress 1893, 

Section G.) 



Description of Stem. 



Elastic Limit, 
Lbs. per Sq. In. 



Tensile Strength, Contraction of 
Lbs. per Sq. In. Area, per cent. 



1.00" long 

.50 " 

.25 " 

Semicircular groove, 

A" radius 

Semicircular groove. 

Ys," radius 

V-sbaped groove 



64,900 
65,320 
68,000 



86,000, about 
90,000, about 



94,400 

97,800 
102,420 



134,960 
117,000 



Indeterminate. 



TENSILE STRENGTH. 243 

Tests plate made by the author in 1879 of straight and grooved test-pieces 
of boiler-plate steel cut from the same gave the following results : 

5 straight pieces, 56,605 to 59,012 lbs. T. S. Aver. 57,566 lbs. . , 
4 grooved " 64,341 to 67,400 " " " 65,452" 

Excess of the short or grooved specimen, 21 per cent, or 12,114 lbs. 

Measurement of Elongation.— In order to be able to compare 
records of elongation, it is necessary not only to have a uniform length of 
section between gauge-marks (say 8 inches), but to adopt a uniform method 
of measuring the elongation to compensate for the difference between the 
apparent elongation when the piece breaks near one of the gauge-marks, 
and when it breaks midway between them. The following method is rec- 
ommended (Trans. A. S. M. E., vol. xi., p. 622): 

Mark on the specimen divisions of 1/2 inch each. After fracture measure 
from the point of fracture the length of 8 of the marked spaces on each 
fractured portion (or 7 -f- on one side and 8 + on the other if the fracture is 
not at one of the marks). The sum of these measurements, less 8 inches, is 
the elongation of 8 inches of the original length. If the fracture is so 
near one end of the specimen that 7 + spaces are not left on the shorter 
portion, then take the measurement of as many spaces (with the fractional 
part next to the fracture) as are left, and for the spaces lacking add the 
measurement of as many corresponding spaces of the longer portion as are 
necessary to make the 7-f- spaces. 

Shapes of Specimens for Tensile Tests.— The shapes shown 
in Fig. 74 were recommended by the author in 1882 when he was connected 

f< 16-to-20" — »} 

b ~ ^3 No. 1. Square or flat bar, as 

1 m im rolled. 



m 



— 16-to-20- 



-=^ _ _ - -==^§11 \%0 No - 2 - Round bar, as rolled. 

H ; 16 "' t0 20 ~ • T^^l m No. 3. Standard shape for 

1 ' I I H flats or squares. Fillets 14 

1 r u g" u" a m inch radius. 

f* - 16-'to-20'' >| 

=gg^g^=g = - " I \j|| ^°- 4- Standard shape for 

1 ^ jj ^ ^ ^^ ^^) rounds. Fillets ]4 in - radius. 

U .jj-to-i-2- * No. 5. Government shape for 

i- ^^ > gj m marine boiler-plates of iron. 

I r - ^f|| || Not recommended for other 

~Z\~M~ tests, as results are generally 

I l \*~ in error. 

Fig. 75. 
with the Pittsburgh Testing Laboratory. They are now in most general 
use, the earlier forms, with 5 inches or less in length between shoulders, 
being almost entirely abandoned. 

Precautions Required in making Tensile Tests.— The 
testing-machine itself should be tested, to determine whether its weighing 
apparatus is accurate, and whether it is so made and adjusted that in the 
test of a properly made specimen the line of strain of the testing-machine 
is absolutely in line with the axis of the specimen. 

The specimen should be so shaped that it will not give an incorrect record 
of strength. 

It should be of uniform minimum section for not less than five inches of 
its length. 

Regard must be had to the time occupied in making tests of certain mate- 
rials. Wrought iron and soft steel can be made to show a higher than their 
actual apparent strength by keeping them under strain for a great length 
of time. 

Tn testing soft alloys, copper, tin, zinc, and the like, which flow under con- 
stant strain their highest apparent strength is obtained by testing them 
rapidly. In recording tests of such materials the length of time occupied in 
the test should be stated. 



244: STRENGTH OF MATERIALS. 

For very accurate measurements of elongation, corresponding to incre- 
ments of load during the tests, the electric contact micrometer, described 
in Trans. A. S. M. E., vol. vi., p. 479, will be found convenient. When read- 
ings of elongation are then taken during the test, a strain diagram may be 
plotted from the reading, which is useful in comparing the qualities of dif- 
ferent specimens. Such strain diagrams are made automatically by the new 
Olsen testing-machine, described in Jour. Frank. Inst. 1891. 

The coefficient of elasticity should be deduced from measurement ob- 
served between fixed increments of load per unit section, say between 2000 
and 12,000 pounds per square inch or between 1000 and 11,000 pounds instead 
of between and 10,000 pounds. 

COMPRESSIVE STRENGTH. 

What is meant by the term "compressive strength " has not yet been 
settled by the authorities, and there exists more confusion in regard to this 
term than in regard to any other used by writers on strength of materials. 
The reason of this may be easily explained. The effect of a compressive 
stress upon a material varies with the nature of the material, and with the 
shape and size of the specimen tested. While the effect of a tensile stress is 
to produce rupture or separation of particles in the direction of the line of 
strain, the effect of a compressive stress on a piece of material may be either 
to cause it to fly into splinters, to separate into two or more wedge-shaped 
pieces and fly apart, to bulge, buckle, or bend, or to flatten out and utterly re- 
sist rupture or separation of particles. A piece of speculum metal under 
compressive stress will exhibit no change of appearance until rupture takes 
place, and then it will fly to pieces as suddenly as if blown apart by gun- 
powder. A piece of cast iron or of stone will generally split into wedge- 
shaped fragments. A piece of wrought iron will buckle or bend. A piece of 
wood or zinc may bulge, but its action will depend upon its shape and size. 
A piece of lead will flatten out and resist compression till the last degree; 
that is, the more it is compressed the greater becomes its resistance. 

Air and other gaseous bodies are compressible to any extent as long as 
they retain the gaseous condition. Water not confined in a vessel is com- 
pressed by its own weight to the thickness of a mere film, while when con- 
fined in a vessel it is almost incompressible. 

It is probable, although it has not been determined experimentally, that 
solid bodies when confined are at least as incompressible as water. When 
they are not confined, the effect of a compressive stress is not only to 
shorten them, but also to increase their lateral dimensions or bulge them. 
Lateral strains are therefore induced by compressive stresses. 

The weight per square inch of original section required to produce any 
given amount or percentage of shortening of any material is not a constant 
quantity, but varies with both the length and the sectional area, with the 
shape of this sectional area, and with the relation of the area to the length. 
The " compressive strength" of a material, if this term be supposed to mean 
the weight in pounds per square inch necessary to cause rupture, may vary 
with every size and shape of specimen experimented upon. Still more diffi- 
cult would it be to state what is the " compressive strength " of a material 
which does not rupture at all, but flattens out. Suppose we are testing a 
cylinder of a soft metal like lead, two inches in length and one inch in diam- 
eter, a certain weight will shorten it one per cent, another weight ten per 
cent, another fifty per cent, but no weight that we can place upon it will 
rupture it, for it will flatten out to a thin sheet. What, then, is its compres- 
sive strength ? Again, a similar cylinder of soft wrought iron would prob- 
ably compress a few per cent, bulging evenly all around ; it would then com- 
mence to bend, but at first the bend would be imperceptible to the eye and 
too small to be measured. Soon this bend would be great enough' to be 
noticed, and finally the piece might be bent nearly double, or otherwise dis- 
torted. What is the " compressive strength " of this piece of iron ? Is it 
the weight per square inch which compresses the piece one per cent or five 
per cent, that which causes the first bending (impossible to be discovered;, 
or that which causes a perceptible bend ? 

As showing the confusion concerning the definitions of compressive 
strength, the following statements from different authorities on the strength 
of wrought iron are of interest. 

Wood's Resistance of Materials states, " comparatively few experiments 
have been made to determine how much wrought iron will sustain at the 
point of crushing. Hodgkinson gives 65,000, Rondulet 70,800, Weisbach 7'2,000 



COMPKESSIVE STRENGTH. 245 

Rankine 30,000 to 40,000. It is generally assumed that wrought iron will resist 
about two thirds as much crushing as to tension, but the experiments fail 
to give a very definite ratio." 

Mr. Whipple, in his treatise on bridge-building, states that a bar of good 
wrought iron will sustain a tensile strain of about 60,000 pounds per square 
inch, and a compressive strain, in pieces of a length not exceeding twice the 
least diameter, of about 90,000 pounds. 

The following values, said to be deduced from the experiments of Major 
Wade, Hodgkinson, and Capt. Meigs, are given by Haswell : 

American wrought iron 127,720 lbs. 

" " " (mean) 85,500 " 

English " } 65 ' 200 " 

Stoney states that the strength of short pillars of any given material, all 
having the same diameter, does not vary much, provided the length of the 
piece is not less than one and does not exceed four or five diameters, and 
that the weight which will just crush a short prism whose base equals one 
square inch, and whose height is not less than 1 to 1J4 and does not exceed 
4 or 5 diameters, is called the crushing strength of the material. It would 
be well if experimenters would all agree upon some such definition of the 
term " crushing strength," and insist that all experiments which are made 
for the purpose of testing the relative values of different materials in com- 
pression be made on specimens of exactly the same shape and size. An 
arbitrary size and shape should be assumed and agreed upon for this pur- 
pose. The size mentioned by Stoney is definite as regards area of section, 
viz., one square inch, but is indefinite as regards length, viz., from one to 
five diameters. In some metals a specimen five diameters long would bend, 
and give a much lower apparent strength than a specimen having a length of 
one diameter. The words " will just crush " are also indefinite for ductile 
materials, in which the resistance increases without limit if the piece tested 
does not bend. In such cases the weight which causes a certain percentage 
of compression, as five, ten, or fifty per cent, should be assumed as the 
crushing strength. 

For future experiments on crushing strength three things are desirable : 
First, an arbitrary standard shape and size of test specimen for comparison 
of all materials. Secondly, a standard limit of compression for ductile 
materials, which shall be considered equivalent to fracture in brittle mate- 
rials. Thirdly, an accurate knowledge of the relation of the crushing 
strength of a specimen of standard shape and size to the crushing strength 
of specimens of all other shapes and sizes. The latter can onlj' be 
secured by a very extensive and accurate series of experiments upon all 
kinds of materials, and on specimens of a great number of different shapes 
and sizes. 

The author proposes, as a standard shape and size, for a compressive test 
specimen for all metals, a cylinder one inch in length, and one half square 
inch in sectional area, or 0.798 inch diameter; and for the limit of compres- 
sion equivalent to fracture, ten per cent of the original length. The term 
"compressive strength," or "compressive strength of standard specimen," 
would then mean the weight per square inch required to fracture by Com- 
pressive stress a cylinder one inch long and 0.798 inch diameter,* or to 
reduce its length to 0.9 inch if fracture does not take place before that reduc- 
tion in length is reached. If such a standard, or any standard size whatever, 
had been used by the earlier authorities on the strength of materials, we 
never would have had such discrepancies in their statements in regard to 
the compressive strength of wrought iron as those given above. 

The reasons why this particular size is recommended are : that the sectional 
area, one-half square inch, is as large as can be taken in the ordinary test- 
ing-machines of 100,000 pounds capacity, to include all the ordinary metals 
of construction, cast and wrought iron, and the softer steels; and that the 
length, one inch, is convenient for calculation of percentage of compression. 
If the length were made two inches, many materials would bend in testing, 
and give incorrect results. Even in cast iron Hodgkinson found as the mean 
of several experiments on various grades, tested in specimens % inch in 
height, a compressive strength per square inch of 94,730 pounds, while the 
mean of the same number of specimens of the same irons tested in pieces \y% 
inches in height was only F8,800 pounds. The best size and shape of standard 
specimen should, however, be settled upon only after consultation and 
agreement among several authorities. 



246 



STKENGTH OF MATERIALS. 



The Committee on Standard Tests of the American Society of Mechanical 
Engineers say (vol. xi., p. 624) : 

'"Although compression tests nave heretofore been made on diminutive 
sample pieces, it is highly desirable that tests be also made on long pieces 
from 10 to 20 diameters in length, corresponding more nearly with actual 
practice, in order that elastic strain and change of shape may be determined 
by using proper measuring apparatus. 

The elastic limit, modulus or coefficient of elasticity, maximum and ulti- 
mate resistances, should be determined, as well as the increase of section at 
various points, viz., at bearing surfaces and at crippling point. 

The use of long compression-test pieces is recommended, because the in- 
vestigation of short cubes or cylinders has led to no direct application of 
the constants obtained by their use in computation of actual structures, 
which have always been and are now designed according to empirical for- 
mulae obtained from a few tests of long columns." 

COLUMNS, PILL.ARS, OR STRUTS. 

Hodgkinson's Formula for Columns. 

P = crushing weight in pounds; d = exterior diameter in inches; d x — in- 
terior diameter in inches; L = length in feet. 



Kind of Column. 



Both ends rounded, the Both ends flat, the 

length of the column length of the column 

exceeding 15 times exceeding 30 times 

its diameter. its diameter. 



Solid cylindrical col- j 

umns of cast iron ) 

Hollow cylindrical col- ) 

umns of cast iron j 

Solid cylindrical col- ) 

umns of wrought iron, f 
Solid square pillar of ) 

Dantzic oak (dry) ) 

Solid square pillar of ) 

red deal (dry) [ 



P = 33,5 



,^ 3-7 



W3-76_d i3 -76 



#3.65 



,/3-55 _ rl 3-55 

P = 99,320- ~* 



P = 299,600^- 

d 4 
P = 24,540^ 



P = 17,510 



L* 



The above formulae apply only in cases in which the length is so great that 
the column breaks by bending and not by simple crushing. If the column 
be shorter than that given in the table, and more than four or five times its 
diameter, the strength is found by the following formula : 



W : 



PCX 
P f MG'iT 



in which P= the value given by the preceding formulae, K= the transverse 
section of the column in square inches, C = the ultimate compressive resis- 
tance of the material, and W = the crushing strength of the column. 

Hodgkinsou's experiments were made upon comparatively short columns, 
the greatest length of cast-iron columns being W% inches, of wrought iron 
90% inches. 

The following are some of his conclusions: 

1. In all long pillars of the same dimensions, when the force is applied in 
the direction of the axis, the strength of one which has flat ends is about 
three times as great as one with rounded ends. 

2. The strength of a pillar with one end rounded and the other flat is an 
arithmetical mean between the two given in the pieced ing case of the same 
dimensions. 

3. The strength of a pillar having both ends firmly fixed is the same as 
one of half the length with both ends rounded. 

4. The strength of a pillar is not increased more than one seventh by en- 
larging it at the middle. 



MOMiM OF IKEilTiA Atfi) HADIUS OF GYRATION. Ml 

Gordon's formulae deduced from Hodgkinson's experiments are more 
generally used than Hodgkinson's own. They are: 

Columns with both ends fixed or fiat, P = — - — - ; 

Columns with one end flat, the other end round, P = - ; 

l + 1.8a^ 

Columns with both ends round, or hinged, P = — - 2 ; 

S = area of cross-section in inches; 
P — ultimate resistance of column, in pounds; 
/ = crushing strength of the material in lbs. per square inch; 
, ,. - ..'.-.'., „ Moment of inertia 

r = least radius of gyration, in inches,?' 2 = — — 5 : ; 

area of section 
I = length of column in inches; 
a — a coefficient depending upon the material; 
/ and a are usually taken as constants; they are really empirical variables, 
dependent upon the dimensions and character of the column as well as upon 
the material. (Burr.) 
For solid wrought-iion columns, values commonly taken are: / = 36,000 to 

40 ' 000;a = 3^ tO 40^00- 

For solid cast-iron columns, / = 80,000, a = ^-rr^. 

fin oon 
For hollow cast-iron columns, fixed ends, p = ^, I = length and 

1+800- 

d = diameter in the same unit, and p = strength in lbs. per square inch. 
Sir Benjamin Baker gives, 

For mild steel, / = 67,000 lbs., a = — * 
. For strong steel, / = 114,000 lbs., a = j^ Q 

Mr. Burr considers these only loose approximations for the ultimate resis- 
tances. 

MOMENT OF INERTIA AND RADIUS OF GYRATION. 

The moment of inertia of a section is the sum of the products of 
each elementary area of the section into the square of its distance from an 
assumed axis of rotation, as the neutral axis. 

The radius of gyration of the section equals the square root of the 
quotient of the moment of inertia divided by the area of the section. If 
E = radius of gyration, 1= moment of inertia and A = area, 

*=;# l=- 

The moments of inertia of various sections are as follows; 

d = diameter, or outside diameter; d t = inside diameter; 6 = breadth; 
h = depth;*!, h u inside breadth and diameter; - 

Solid rectangle I = l/126/i 3 ; Hollow rectangle I = 1/12(67* 3 - Mi 3 ) ; 

Solid square 1= 1/126 4 ; Hollow square ..J= 1/12(6* - 6, 4 ); 

Solid cylinder I = l/647rd 4 ; Hollow cylinder I = \/Mir(d* - d^). 

Moments of Inertia and Radius of Gyration for Various 
Sections, and their Use in tbe Formulas for Strength of 
Girders and Columns.-The strength of sections to resist strains, 
either as girders or as columns, depends not only on the area but also on the 
form of the section, and the property of the section which forms the basis 
of the constants used in the formulas for strength of girders and columns 
to express the effect of the form, is its moment of inertia about its neutral 
axis. Thus the moment of resistance of any section to transverse bending 



248 STKEKGTS OF MATERIALS. 

is its moment of inertia divided by the distance from the neutral axis to 
the fibres farthest removed from that axis; or 

,„. ,' . , Moment of inertia ,, I 

Moment of resistance = 7- - — • —. M = -. 

Distance ot extreme fibre from axis y 

Moment of Inertia of Compound Shapes. (Peneoyd Iron 
Works.)— The moment of inertia of any section about any axis is equal to the 
I about a parallel axis passing through its centre of gravity -f- (the area of 
the section X the square of the distance between the axes). 

By this rule, the moments of inertia or radii of gyration of any single sec- 
tions being known, corresponding values may be obtained for any combina- 
tion of these sections. 

Radius of Gyration of Compound Shapes.— In the case of a 
pair of any shape without a web the value of B can always be found with- 
out considering the moment of inertia. 

The radius of gyration for any section around an axis parallel to another 
axis passing through its centre of gravity is found as follows: 

Let r = radius of gyration around axis through centre of gravity; R = 
radius of gyration around another axis parallel to above; d = distance be- 
tween axes: 



R = \/d? + r*. 
When r is small, R may be taken as equal to d without material error. 
Graphical Method for Finding Radius of Gyration.— Ben j. 

F. La Hue, Eng. News, Feb. 2, 1893, gives a short graphical method for 
finding the radius of gyration of hollow, cylindrical, and rectangular col- 
umns, as follows: 

For cylindrical columns: 

Lay off to a scale of 4 (or 40) a right-angled triangle, in which the base 
equals the outer diameter, and the altitude equals the inner diameter of the 
column, or vice versa. The hypothenuse, measured to a scale of unity (or 
10), will be the radius of gyration sought. 

This depends upon the formula 



'W 



Mom. of Inertia ^D 2 + d 2 



Area 4 

in which A = area and D = diameter of outer circle, a = area and d = dia- 
meter of inner circle, and G = radius of gyration. ^D' 1 -f- d 2 is the expres- 
sion for the hypothenuse of a right-angled triangle, in which D and d are the 
base and altitude. 

The sectional area of a hollow round column is .7854(D 2 — d 2 ). By con- 
structing a right-angled triangle in which D equal s the hypothenuse and d 
equals the altitude, the base will equal |/D 2 - d\ Calling the value of this 
expression for the base B, the area will equal .7854.B 2 . 

Value of G for square columns: 

Lay off as before, but using a scale of 10, a right-angled triangle of which 
the base equals D or the side of the outer square, and the altitude equals d, 
the side of the inner square. With a scale of 3 measure the hypothenuse, 
which will be, approximately, the radius of gyration. 

This process for square columns gives an excess of slightly more than 4%. 
By deducting 4% from the result, a close approximation will be obtained. 

A very close result is also obtained by measuring the hypothenuse with 
the same scale by which the base and altitude were laid off, and multiplying 
by the decimal 0.29; more exactly, the decimal is 0.28867. 

The formula is 



■/ 



Mom. of inertia 1 



,_ \/D* + d\ = 0-28867 y D -x + d a 



This may also be applied to any rectangular column by using the lesser 
diameters of an unsupported column, and the greater diameters if the col- 
umn is supported in the direction of its least dimensions. 

ELEMENTS OF TJSUAX SECTIONS. 

Moments refer to horizontal axis through centre of gravity. This table is 
intended for convenient application where extreme accuracy is not impor- 
tant. Some of the terms are only approximate; those marked * are correct. 
Values for radius of gyration in flanged beams apply to standard minimum 
sections only. A = area of section; b = breadth; h = depth; D = diameter. 



ELEMENTS OF USUAL SECTIONS. 



249 



Shape of Section. 






Solid Rect- 
angle. 



Hollow Rect- 
angle. 



Moment 
of Inertia. 



12 



b h»-bjh^ 



12 



AD* * 
16 



Moment 

of 

Resistance, 



bhs-bji^ 



6/i 



AD* 

8 



Square of 

Least 
Radius of 
Gyration. 



h* -f hi- 
12 



2)2 4 

16 



Least 
Radius of 
Gyration. 



h±h i 
4.89 



f-Hri 



Hollow Circle. 
A, area of 
large section : 
a, area of 
small section 



D* + d* " 



5.64 



F=6 



Solid Triangle. 



6/i3 
36 



The least of 

of the two 

h* b* 

r 8 ° r 24 



The least of 
the two : 
h b 

— or — 
.24 4.9 



Even Angle. 



Ah* 
10.2 



Ah 
7.2 



62 
25 



9 



Uneven Angle 



Ah* 
9.5 



Ah 
6.5 



lZQl* + &2) 



2.6(/t 4- 6) 



JE3 



it* 



19 



9.5 



h* 
22.5 



4.74 




^2 
11.1 



Ah 

8 



^2 

6.66 



Ah 






Ah* 
7.34 



Ah 
3.67 



ta 



62 

22.5 



4.74 



6J 

21 



6 

4.58 



62 
12.5 



6 
3.54 



62 

36.5 



Distance of base from centre of gravity, solid triangle, ^; even angle, ~ • 
uneven angle, ^-i.; even tee, ^-5; deck beam, — -; all other shapes given in 

o.O Q,tJ 4-Q 

the table, --or -. 



250 



STRENGTH OF MATERIALS. 



Solid Cast-iron Columns. 

Hurst giv.'S the following table, based on Hodgkinson's formula (tons of 
2240 lbs.). 

The figures are the safe load or ^ of the breaking weight in tons, for solid 
columns, ends flat and fixed. 



■38. 


Length of Column in Feet. 


£"5 

a a 


6. 


8. 


10. 


12. 


14. 


16. 


18. 


20. 


25. 


Wz 


.82 


.50 


.34 


.25 


.19 


.15 


.13 


.11 


.07 


1% 


1.43 


.87 


.60 


.44 


.34 


.27 


.22 


.18 


.13 


,_£__ 


2.34 


- 1.41 


.97 


.71 


.55 


.44 


.36 


.30 


.20 


2J4 


3.52 


2.16 


1.48 


1.08 


.83 


.67 


.54 


.46 


.31 


2p 2 


5.15 


3.16 


2.16 


1.58 


1.22 


.97 


.80 


.66 


.56 


2'M 


7.26 


4.45 


3.05 


2.23 


1.72 


1.37 


1.12 


.94 


.64 


3 


9.93 


6.09 


4.17 


3.06 


2.35 


1.87 


1.53 


1.28 


.88 


3^ 


17.29 


10.60 


7.26 


5.32 


4.10 


3.26 


2.67 


2.23 


1.53 


4 


27.96 


17.15 


11.73 


8.61 


6.62 


5.28 


4.32 


3.61 


2.47 


#£ 


42.73 


26.20 


17.93 


13.15 


10.12 


8.07 


6.60 


5.52 


3.78 


5 2 


62.44 


38.29 


26.20 


19.22 


14.79 


11.79 


9.65 


8.06 


5.52 


5^ 


88.00 


53.97 


36.93 


27.09 


20.84 


16.61 


13.60 


31.37 


7.78 


6 


120.4 


73.82 


50.51 


37.05 


28.51 


22.72 


18.60 


15.55 


10.64 


6^ 


160.6 


98.47 


67.38 


49.43 


38.03 


30.31 


24.81 


20.74 


14.19 




209.7 


128.6 


87.98 


64.53 


49.66 


39.57 


32.33 


27.08 


18.53 


^ 


268.8 


164.8 


112.8 


82.73 


63.66 


50.73 


41.53 


34.72 


23.76 


8 


339.1 


207.9 


142.3 


104.4 


80.31 


64.00 


52.39 


43.80 


29.97 


- 8J^ 


421.8 


258.6 


177.0 


129.8 


99.90 


79.61 


65.16 


54.48 


37.28 


9 


518.2 


317.7 


217.4 


159.5 


122.7 


97.80 


80.05 


66.92 


45.80 


9^ 


629.5 


386.0 


264.2 


193.8 


149.1 


118.8 


97.25 


81.70 


55.64 


10 


757.2 


464.3 


317.7 


233.1 


179.3 


142.9 


117.0 


97.79 


66.92 


1<% 


902.6 


553.5 


378.7 


277.8 


213.8 


170.3 


139.4 


116.6 


79.77 


11 


1067.1 


654.4 


447.8 


328.5 


252.7 


201.4 


164.9 


137.8 


94.31 


-UM 


1252.3 


767.9 


525.5 


385.4 


296.6 


236.4 


193.5 


161.7 


110.7 


12 


1459.6 


895.1 


612.5 


449.3 


345.7 


275.5 


225.5 


188.5 


129.0 



The correction for short columns should be applied where the length is 
less than 30 diameters. 

SC 



Strength in tons of short columns j = 



106' + %C ' 



49 



S being the strength for long columns given in the above table, and C 
times the sectional area Of the metal in inches. 

Hollow Columns.— The strength nearly equals the difference be- 
tween that of two solid columns the diameters, of which are equal to the 
external and internal diameters of the hollow one. 

Ultimate Strength of Hollow, Cylindrical Wrought and 
Cast-iron Columns, when fixed at the ends. 



(Pottsville Iron and Steel Co.) 
Computed by Gordon's formula, p - 



C 



p = Ultimate strength in lbs. per square inch; 

J 40,000 lbs. for wrought-iron; I 
"!0,0001bs. for cast-iron; 



/ = 



C = 1/3000 for wrought-iron, and 1/800 for cast-iron. 



COLUMNS, PILLARS, OR STRUTS. 

80000 



251 



For cast-iron, p = - 

^ 1 



For wrought-iron, p - 



S00\h) 
40000 



^3000\hf 
HOLLOW CYLINDRICAL COLUMNS. 



Ratio 


Maximum Load per sq. in. 


Safe Load per square inch. 


of Length to 










Diameter. 










1 
h 


Cast Iron. 


Wrought Iron. 


Cast Iron, 
Factor of 6. 


Wrought Iron, 
Factor of 4. 


8 


74075 


39164 


12346 


9791 


10 


71110 


38710 


11851 


9677 


12 


67796 


38168 


11299 


9542 


14 


64256 


37546 


10709 


9386 


16 


60606 


36854 


10101 


9213 


18 


56938 


36100 


9489 


9025 


20 


53332 


35294 


8889 


8823 


22 


49845 


34442 


8307 


8610 


24 


46510 


33556 


7751 


8389 


26 


43360 


32642 


7226 


8161 


28 


40404 


31712 


6734 


7928 


30 


37646 


30768 


6274 


7692 


32 


35088 


29820 


5848 


7455 


34 


32718 


28874 


5453 


7218 


36 


30584 


27932 


5097 


6983 


38 


28520 


27002 


4753 


6750 


40 


26666 


26086 


4444 


6522 


42 


24962 


25188 


4160 


6297 


44 


23396 


24310 


3899 


6077 


46 


21946 


23454 


3658 


5863 


48 


20618 


22620 


3436 


5655 


50 


19392 


21818 


3262 


5454 


52 


18282 


21036 


3047 


5259 


54 


17222 


20284 


2870 


5071 


56 


16260 


19556 


2710 


4889 


58 


15368 


18856 


2561 


4714 


60 


14544 


18180 


2424 


4545 



Ultimate Strength of Wrought-iron Columns. 

p = ultimate strength per square inch; 
I = length of column in inches; 
r — least radius of gyration in inches. 

. . . 40000 

For square end-bearings, r> — 



1 + 



000 \r J 



For one pin and one square bearing, p = 



For two pin-bearings, 



40000 
40000 



1 -f 



KXX>\r/ 



30000' 
40000 



1 + 



-1-/1Y 2 

20000\rj 



For safe working load on these columns use a factor of 4 when used in 
buildings, or when subjected to dead load only; but when used in bridges 
the factor should be 5. 



252 STRENGTH OF MATERIALS. 

WROUGHT-IRON COLUMNS. 





Ultimate Strength 


in lbs. 




Safe Strength in lbs. per 


1 


per square inch. 


I 
r 


square inch— Factor of 5. 


r 


Square 
Ends. 


Pin and 

Square 

End. 


Pin 

Ends. 


Square 
Ends. 


Pin and 
Square 
End. 


Pin 
Ends. 


10 


39944 


39866 


39800 


10 


7989 


7973 


7960 


15 


39776 


39702 


39554 


15 


7955 


7940 


7911 


20 


39604 


39472 


39214 


20 


7921 


7894 


7843 


25 


39384 


39182 


38788 


25 


7877 


7836 


7758 


30 


39118 


38834 


38278 


30 


7821 


7767 


7656 


35 


38810 


38430 


37690 


35 


7762 


7686 


7538 


40 


38460 


37974 


37036 


40 


7692 


7595 


7407 


45 


38072 


37470 


36322 


45 


7614 


7494 


7264 


50 


37646 


36928 


35525 


50 


7529 


7386 


7105 


55 


37186 


36336 


34744 


55 


7437 


7267 


6949 


60 


36697 


35714 


33898 


60 


7339 


7143 


6780 


65 


36182 


34478 


33024 


65 


7236 


6896 


6605 


TO 


35634 


34384 


32128 


70 


7127 


6877 


6426 


75 


35076 


33682 


31218 


75 


7015 


6736 


6244 


80 


34482 


32966 


30288 


80 


6896 


6593 


6058 


85 


33883 


32236 


29384 


85 


6777 


6447 


5877 


90 


33264 


31496 


28470 


90 


6653 


6299 


5694 


95 


32636 


30750 


27562 


95 


6527 


6150 


5512 


100 


32000 


30000 


26666 


100 


6400 


6000 


5333 


105 


31357 


29250 


25786 


105 


6271 


5850 


5157 



maximum Permissible Stresses in columns used in buildings. 
(Building Ordinances of City of Chicago, 1893.) 
Maximum permissible loads : 
For cast-iron round columns : 



8 = 



10000a 



1+1 



I = length of column in inches; 
d = diameter of column in inches; 
a = area of column in square inches. 



For cast-iron rectangular columns; 

_ 10000a I and a as before ; 

" — 72 * d = least horizontal dimension of column. 



For riveted or other forms of wrought-iron columns: 
_ 12000a I = and a as before; 

^ - p • r = least radius of gyration in inches. 

1Jr 36000r 2 
For riveted or other steel columns, if less than 60r in length: 
60? 



8 = 17,000 - 



I and r as before. 



If more than 60r in length: 

S = 13,500a. a as before. 
For wooden posts: 



S = - 



1 + 1 



a = area of post in square inches; 

d = least side of rectangular post in inches ; 

I = length of post in inches; 

\ 600 for white or Norway pine ; 
c= < 800 for oak; 

( 900 for long-leaf yellow pine. 



HOLLOW CYLINDRICAL CAST IROK COLUMNS. 253 

SAFF LOAD OF HOLLOW CYLINDRICAL, CAST-IRON 
COLUMNS. (New Jersey Steel Iron Co.) 

(One fifth the breaking weight.) 

The following tables give the safe load in tons of 2,000 lbs., for columns 
having capitals and bases accurately turned to a true plane, and having a 
perfectly fair bearing on these surfaces. In the case of columns having 
turned ends, but set only with the degree of care usual in ordinary building, 
only one half of these loads should be taken; and for columns not turned at 
all, or having rounded ends, one third of these amounts should be taken for 
the safe load. Columns having one end accurately turned to a true plane, 
and the other rounded, may be loaded to two thirds the amount given in the 
tables. 

Safe Load, in Tons of 2000 lbs. for Cast-iron Columns 
with Turned Capitals and Rases. 





Outside 




Outside 




Outside 




Outside 




Diameter, 




Diameter, 


• 


Diameter, 




Diameter, 


* 


3 inches. 
Thickness in 


a 


3 inches. 


z 


4 inches. 




4 inches. 


£ 


Thickness in 


Thickness in 


Thickness in 


"Sx 


inches. 


ix 
PI 

a 

17 


inches. 


bx 

= 

S3 


inches. 


;x 

= 

17 


inches. 


1. 


H 


M i 


H 

3.0 


H 

3.6 


1 
3.9 


Hi 

24.9 


32.9 


1 

38.3 


41.7 


Y2 

7.0 


H 
8.9 


1 
10.1 


m 


7 


13 8 


15.917.2 


10.7 


8 


10 9 


13.0 14.0 


IS 


2.8 


3.3 


3.5 


8 


21.7 


2S.4 


33.0 


35.8 


is 


6.4 


8.1 


9.1 


9.7 


9 


8.9 


10.7 


11.4 


19 


2.5 


3.0 


3.2 


9 


19.0 


24.8 


28. 7 


31.0 


19 


5.8 


7.4 


8,3 


8.8 


10 


7 5 


8.9 


9.fi 


20 


2.3 


2.7 


2.9 


10 


17.4 


22.0 


24.9 


26.3 


20 


5.3 


6 8 


7.6 


8.1 


11 


6 4 


7.6 


8.1 


81 


2.1 


2.5 


2.7 


11 


14.8 


18.7 


21.1 


22.4 


21 


4.9 


6.2 


7 


7 5 


12 


5.4 


6.6 


7.0 


22 


1.9 


2.3 


2.5 


12 


12.7 


16.2 


18.2 


19.3 


22 


4.6 


5 8 


6.5 


6.9 


13 


4 R 


5.7 


6.1 


23 


1.8 


2.1 


2.3 


13 


11.1 


14.1 


15.9 


16.8 


23 


4.2 


5.3 


6.0 


6 4 


14 


4.2 


5.0 


5.4 


24 


1.7 


2.0 


2.1 


14 


9.8 


12.4 


14.0 


14.9 


24 


3.9 


5 


5 6 


5 9 


IS 


3 7 


4.5 


4.8 


25 


1.6 


1.9 


2.0 


15 


8.7 


11.1 


12.5 


13.2 


25 


3.7 


4.6 


5.2 


5 5 


16 


3.4 


4.0 


4.3 










16 


7.8 


9.9 


11.3 


11.8 











a 


Outside Diameter, 
5 inches. 


Outside Diameter, 
6 inches. 


Outside Diameter, 
7 inches. 


t, 


Thickness in inches. 


Thickness in inches. 


Thickness in inches. 


J 


39.5 


n 

53.8 


1 


m 


M 


1 


m 


W* 


Ya 


i 


150.7 


m 


7 


65.0 


73.3 


77.3 


95.5 


110.3 


122.1 


102.4 


128.7 


169.4 


8 


35.1 


47.6 


57.3 


64.4 


69.7 


85.7 


98.7 


108.8 


93 6 


117.0 


136 9 


153.5 


9 


31.3 


42.3 


50.7 


56.8 


62.8 


77.1 


88.5 


97.3 


85.6 


106.7 


124.6 


139 3 


10 


38.0 


37.7 


45.1 


50.4 


56.9 


69.6 


79.6 


87.4 


78.4 


97.5 


113.5 


1?fi 6 


11 


35 3 


33.8 


40 3 


44.9 


51.6 


63.0 


71.9 


78.7 


71.8 


89.2 


103.6 


115.3 


12 


33.7 


30.5 


36.2 


40.3 


46.9 


57.2 


65.2 


71.2 


66.0 


81.7 


94.8 


105.3 


13 


31.0 


27.6 


32.2 


35.2 


42.9 


52.1 


59.3 


64.6 


60.7 


75.1 


87 


96.5 


14 


18.5 


24.3 


28.3 


31.0 


39.3 


47.6 


54.1 


58.9 


56.0 


69.2 


80.0 


88.6 


15 


16.5 


21.6 


25.2 


27.6 


36.8 


43.9 


49.0 


52.6 


51.8 


63.9 


73 8 


81 6 


16 


14.8 


19.4 


22.6 


24.7 


33.0 


39.4 


44.0 


47.2 


48.1 


59.2 


68.2 


75.4 


17 


13.3 


17.5 


20.4 


22.3 


29.8 


35.5 


39.7 


42.5 


44.6 


54.9 


63.2 


69 8 


18 


12.1 


15.9 


18.5 


20.2 


27.0 


32.2 


36.0 


38.6 


42.0 


50.9 


57.8 


63.0 


19 


11.0 


14.5 


16.9 


18.4 


24.6 


29.4 


32.8 


35.2 


38.3 


46.4 


52 7 


57.4 


.20 


10.1 


13.3 


15.4 


16.9 


22.6 


26.9 


30 1 


32.3 


35.1 


42.5 


48.3 


53,6 


21 


9.3 


12.2 


14.2 


15.5 


20.8 


24.8 


27 7 


29.7 


32.3 


39.1 


44 5 


48 4 


22 


8.6 


11.3 


13.1 


14.4 


19.2 


22.9 


25.6 


27.4 


29.8 


36.2 


41 1 


44.7 


23 


8.0 


10.5 


12.2 


13.3 


17.8 


21.2 


23.7 


25.4 


27.7 


33.5 


38.1 


41.5 


24 


7.4 


9.7 


11.3 


12.4 


16.6 


19.7 


22.1 


23.7 


25.7 


31.2 


35.4 


38,6 


25 


6.9 


9.1 


10.6 


11.5 


15.4 


18.4 


20.6 


22.1 


24.0 


29.1 


33.1 


36.0 



254 



STRENGTH OE MATERIALS. 



Safe Load, in Tons of 2000 lbs. for Cast-iron Columns 
witli Turned Capitals and Bases. 





Outside Diameter, 


Outside Diameter. 


Outside Diameter, 





8 inches. 


9 inches. 






10 inches. 




Thickness in inches. 


Thickness in inches. 


Thickness 


in inches. 


J 


H 

128 8 


1 


Wa 


m 


H 


1 


m 


M 


M 


1 


m 


m 


7 


162.fi 


193 


219.5 


154.8 


197.7 


236.6 


271.4 


181.6 


233.4 


280.9 


324 2 


8 


118 7 


150.1 


177.7 


201.6 


144.7 


184.5 


220.2 


252.0 


171.1 


219.5 


263.8 


303 9 


9 


109.8 


138.5 


163.6 


185.2 


135.0 


171.8 


204.7 


233.9 


160.9 


206.2 


247.3 


284 5 


10 


101. B 


127,8 


150.7 


170.2 


126.0 


160.0 


190.3 


217.0 


151.2 


193.4 


231.6 


266.0 


11 


94 


118.0 


139.0 


156.7 


117.5 


149.0 


177.0 


201.4 


142.0 


181.4 


216.9 


248.7 


12 


87 


109.2 


128. 2 


144.3 


109.6 


138.8 


164.5 


187.0 


133.4 


170.1 


203.1 


232.6 


13 


80 7 


101 1 


118.5 


133.2 


102.4 


129.4 


153.2 


173.9 


125.3 


159.6 


190.3 


217.7 


14 


75 


93 8 


109.8 




95.7 


120.8 


142.8 


161.9 


117.8 


149.8 


178.4 


203 . S 


IS 


69 8 


87 1 


101.9 


114.2 


89.5 


112.9 


133.3 


150.9 


110.8 


140.7 


167.5 


191.1 


Hi 


65.0 


81 1 


94.7 


106.1 


83.9 


105.7 


124.6 


140.9 


104.3 


132.4 


157.3 


179.3 


17 


60.7 


75.7 


88.3 


98.7 


78.7 


99.0 


116.7 


131.8 


98.3 


124.6 


148.0 


168.5 


18 


56.8 


70.7 


82.4 


92.1 
86.1 


73.9 
69.6 


92.9 

87.4 


109.4 
102.7 


123.5 
115.9 


92.7 

87.5 


117.4 


139.3 


158.5 










110.8 


131.3 




20 


51.1 


62.7 


72.1 


79.5 


65.5 


82.3 


96.7 


108.9 


82.7 


104.6 


124.0 


140.8 


21 


47 


57; 7 


66\4 


73.2 


61.8 


75.5 


91.0 


102.6 


78.3 


99.0 


117.2 


133.0 




43.5 


53.3 


61.3 


67.6 


58.4 


73.2 


85.9 


96.7 


74.2 


93.7 


110.9 


125.8 


:>H 


40.3 


49.4 


56.8 


62.7 


55.9 


69.3 


80.4 


89.5 


70.4 


88.9 


105.1 


119.1 




37.5 


46.0 


52.9 


58.3 


52.0 


64.4 


74.8 


83.3 


66.9 


84.3 


99.7 


112.9 




35.0 


42.9 49.3 


54.4 


48.5 


60.1 


69.8 


77.7 


64.9 


81.0 


94.2 


106.3 





Outside Diameter, 


Outside Diameter, 


Outside Diameter. 


a 


11 inches. 


12 inches. 




13 inches. 




Thickness in inches. 


Thickness in inches. 


Thickness in inches. 


c 
J 


1 
269.4 


M 


377.6 


.2 


1 


1M 


m 


2 


1 


m 


IK 


2 


7 


325,9 


469.5 


305.3 


370.8 


431.7 


540.9 


341.5 


414.4 


485.7 


612.7 


H 


255 1 


30S 1 


356.8 


442.2 


290.9 


352.8 


410.2 


512.8 


327.0 


396.3 


464.1 


583.9 


9 


241.2 


290 8 


336,3 


415.6 


276.6 


335.0 


389.1 


485.0 


312.4 


378.4 


442.5 


555.5 


10 


227.8 


274.2 


316.7 


390.3 


262.7 


317.7 


368.6 


458.3 


298.0 


360.6 


421.3 


527.8 


11 


214.9 


258.4 


298.1 


366.3 


249.2 


301.0 


348.8 


432.9 


284.0 


343.4 


400.6 


501.1 


12 


202.7 


243.5 


280.5 


343.9 


236.3 


285.1 


330.0 


408.6 


270.5 


326.7 


380.8 


475 3 


13 


191.2 


229.4 


264.0 


322.8 


223.9 


270.0 


312.2 


385.7 


257.5 


310.8 


361.8 


450.7 


14 


180.5 


216.2 


248.5 


303.3 


212.3 


255.6 


295.3 


364.1 


245.0 


295.5 


343.7 


427.4 


15 


170.3 


203.9 


234.1 


285.1 


201.2 


242.1 


279.4 


343.9 


233.2 


281.1 


326.5 


405.4 


16 


160.9 


192.4 


220.7 


268.3 


190.8 


229.4 


264.5 


325.0 


222.0 


267.3 


310.3 


384.6 


17 


152 1 


181.7 


208.2 


252.7 


181.1 


217.5 


250.6 


307.4 


211.3 


254.4 


295.0 


365.1 


18 


143.9 


171.7 


196.7 


238.3 


171.9 


206.3 


237.5 


290.9 


201.3 


242.1 


280.5 


316.7 


19 


136.2 


162.5 


185.9 


225.0 


163.3 


195.8 


225.3 


275.6 


191.8 


230.6 


267.0 


329.5 


20 


129.1 


153 9 


176.0 


212.6 


155.2 


186.0 


213.9 


261.3 


182.8 


219.7 


254.2 


313.3 


21 


122.4 


145.9 


166.7 


201.2 


147.7 


176.9 


203.2 


247.9 


174.4 


209.5 


242.2 


298.2 


22 


116,3 


138.4 


158.1 


190.6 


140.6 


168.3 


193.3 


235.5 


166.5 


199.9 


230.9 


284.0 


23 


110.5 


131.5 


150.1 


180.7 


134.0 


160.3 


184.0 


224.0 


159.0 


190.8 


220.4 


270.7 


24 


105 2 


125 1 


142.7 


171.6 


127.8 


152.8 


175.3 


213.2 


152.0 


182.3 


210.4 


258.3 


25 


100.2 


119.1 


135 7 


163.1 


122.0 


145.8 


167.1 


203.1 


145.4 


174.3 


201.0 


246.6 



ECCENTRIC LOADING OF COLUMNS. 



255 







Safe Load of Cast-iron Columns- (Continued). 






Outside Diameter, 


Outside Diameter, 


Outside Diameter, 


c 


14 inches. 




15 inches. 


16 inches. 


tc 


Thickness in inches. 


Thickness in inches. 


Thickness in inches. 


c 


1 


m 


M 


2 

684.6 


1 


1M 


1^ 


2 


1 


m 


648.0 


2 


7 


877.7 


461.1 


539.9 


413.7 


506.1 


594.0 


756.7 


449.8 


551.1 


828.6 


8 


363.1 


442.8 


518.0 


655.9 


399.3 


487.9 


572.2 


727.7 


435.3 


532.8 


626.3 


799.8 


9 


848.5 


424.4 


-196.3 


627.0 


384.4 


469.5 


550.1 


698.4 


420.5 


514.4 


604.1 


770.4 


10 


888.8 


406.3 


474.6 


598.5 


369.7 


451.0 


528.2 


669.3 


405.6 


496.0 


581.8 


740.9 


11 


819.4 


388.5 


453.4 


570.7 


355.1 


433.0 


506.3 


640.9 


390.6 


477.4 


559.8 


711.7 


12 


305.4 


371.1 


432.6 


543.6 


340.6 


415.0 


485.0 


612.8 


376.0 


459.3 


538.0 


683.4 


IS 


291.8 


354.3 


412.7 


517.7 


326.6 


397.6 


464.5 


585.9 


361.6 


441.2 


516.7 


655,1 


14 


278.8 


338.2 


393.6 


493.0 


313.0 


380.7 


444.4 


559.7 


347.6 


423.8 


495.9 


628 


15 


266 2 


322.7 


375.3 


469.4 


299.9 


364.5 


425.2 


534.9 


333.9 


406.9 


475.9 


601 8 


16 


254.3 


308.0 


357.9 


446.9 


287.2 


348.9 


406.7 


510.9 


320.7 


390.6 


456.6 


576.6 


17 


242.9 


294.0 


341 .4 


425.7 


275.1 


334.0 


389.1 


488.1 


308.0 


374.9 


438.0 


552.5 


18 


232.0 


280.6 


325.6 


405.5 


263.6 


319.7 


372.2 


466.5 


295.8 


359.9 


420.1 


529.4 


19 


221.7 


268.0 


310.8 


386.5 


252.5 


306.2 


356.2 


445.9 


284.1 


345.4 


403.0 


507.3 


20 


212.0 


256.1 


296.7 


368.6 


242.0 


293.3 


341.0 


426.3 


272.9 


331.6 


386.8 


486.3 


21 


202.7 


244.7 


283.5 


351.8 


232.0 


281.0 


326.5 


407.8 


262.1 


318.4 


371.2 


466 2 




194.0 


284.0 


270.9 


335.9 


222.5 


269.3 


312.8 


390.3 


251.9 


305.9 


356.4 


447 2 




185.7 


224.0 


259.1 


320.9 


213.4 


258.3 


299.8 


373.7 


242.2 


293.9 


342.3 


429.1 


24 


177.9 


214.4 


248.0 


306.8 


204.9 


247.8 


287.5 


358.1 


232.9 


282.5 


328.8 


411,9 


25 


170.5 


205.4 


237.5 


294.1 


196.7 


237.8 


275.9 


343.2 


224.0 


271.6 


316.1 


395.6 



ECCENTRIC LOADING OF COLUMNS. 

In a given rectangular cross-section, such as a masonry joint under press- 
ure, the stress will be distributed uniformly over the section only when the 
resultant passes through the centre of the section ; any deviation from such 
a central position will bring a maximum unit pressure to one edge and a 
minimum to the other; when the distance of the resultant from one edge is 
one third of the entire width of the joint, the pressure at the nearer edge is 
twice the mean pressure, while that at the farther edge is zero, and that 
when the resultant approaches still nearer to the edge the pressure at the 
farther edge becomes less than zero; in fact becomes a tension, if the 
material (mortar, etc., there is capable of resisting tension. Or, if, as usual 
in masonry joints, the material is practically incapable of resisting tension, 
the pressure at the nearer edge, when the resultant approaches it nearer 
than one third of the width, increases very rapidly and dangerously, becom- 
ing- theoretically infinite when the resultant reaches the edge. 

With a given position of the resultant relatively to one edge of the joint or 
section, a similar redistribution of the pressures throughout the section may 
be brought about by simply adding to t or diminishing the width of the 
section. 

Let P = the total pressure on any section of a bar of uniform thickness. 
= the width of that section = the area of the section, when thickness 



: 1. 



: the mean unit pressure on the section. 



M — the maximum unit pressure on the section. 
m — the minimum unit pressure on the section. 

d = the eccentricity of the resultant — its distance from the centre of 
the section. 



Thenilf = p (l4~ ) and m = p (l - ^). 



When d is greater than l/6w, the resultant in that case being less than 
one third of the width from one edge, p becomes negative. (J. C. Traut- 
wine, Jr., Engineering News, Nov, 23, 1893.) 



256 



STRENGTH OF MATERIALS. 
BUILT COLUMNS. 



From experiments by T. D. Lovett, discussed by Burr, the values of /and 
a in several cases are determined, giving empirical forms of Gordon's for- 
mula as follows: p = pounds crushing strength per square inch of section, 
I = length of column in inches, r — radius of gyration in inches. 





Keystone 
Columns. 



39,500 



(1) 



! + ? 



Flat Ends. 



Square 
Columns. 

39,000 . 



Phoenix 
Columns. 



American Bridge 
Co. Columns. 



F <8) 



1 + 



i<*> 



35,000 r* 



Flat Ends, Swelled. 



i+s 



! + f 



2 (7) 



1 + 



17,000 r 2 ' 22,700 r 

Pin Ends, Swelled. 



Round Ends. 



r<8> 



i + 



12,500 1 



(10) 



l + o 



36,000 



(ID 



1 + 



11,500 ? 



With great variations of stress a factor of safety of as high as 6 or 8 may 
be used, or it may be as low as 3 or 4, if the condition of stress is uniform or 
essentially so. 

Burr gives the following general principles which govern the resistance of 
built columns : 

The material should be disposed as far as possible from the neutral axis 
of the cross-section, thereby increasing r; 

There should be no initial' internal stress; 

The individual portions of the column should be mutually supporting; 

The individual portions of the column should be so firmly secured to each 
other that no relative motion can take place, in order that the column may 
fail as a whole, thus maintaining the original value of r. 

Stoney says: "When the length of a rectangular wrought-iron tubular 
column does not exceed 30 times its least breadth, it fails by the bulging or 
buckling of a short portion of the plates, not by the flexure of the pillar as a 
whole." 

In Trans. A. S. C. E., Oct. 1880, are given the following formulae for the 
ultimate resistance of wrought-iron columns designed by C. Shajer Smith : 



BUILT COLUMNS. 257 



Flat Ends. 



Square Phoenix American Bridge Common 

Column. Column. Co. Column. Column. 



(15) -1 w (18) 



1 i J_ J? 1 i J_ l l ,,JLi! X4.JL.* 

"^5820 d 2 ^~4500 d 2 '^3750 d 2 "^OO d 2 
One Pin End. 

_^ (18) _i2|^ (16> _i^ <19) -Sf^ , 

1- *~3000 d a 1 + 2250 d 2 1 + 2250 d 2 1 + 1500 d 2 
Two Pin Ends. 

37,500 n - { 36,600 (1?) 36,500 ^ 36,500 



l-4-_ i_l__L_± l + _i_ ± 1J_J_ ± 

M900 d 2 M500 d 2 ^1750 d 2 ^1200 d 2 

The " common " column consists of two channels, opposite, with flanges 
outward, with a plate on one side and a lattice on the other. 

The formula for " square " columns may be used without much error for 
the common-chord section composed of two channel-bars and plates, with 
the axis of the pin passing through the centre of gravity of the cross- 
section. (Burr). 

Compression members composed of two channels connected by zigzag 
bracing may be treated by formulae 4 and 5, using / = 36,000 instead of 
39,000. 

Experiments on full-sized Phoenix columns in 1873 showed a close agree- 
ment of the results with formulas 6-8. Experiments on full-sized Phoenix 
columns on the Watertovvn testing-machine in 1881 showed considerable dis- 
crepancies when the value of I h- r became comparatively small. The fol- 
lowing modified form of Gordon's formula gave tolerable results through 
the whole range of experiments : 

40,000 (l-f y) 

Phoenix columns, flat end, p = --^ — (24) 



1 + 50,000 r 2 



Plotting results of three series of experiments on Phoenix columns, a 
more simple formula than Gordon's is reached as follows : 

Phoenix columns, flat ends, p = 39,640 - 

p = 64,700 - 4600 \ 1 when I -h r is less than 30. 

Dimensions of Phoenix Columns, 

(Phoenix Iron Co.) 

The dimensions are subject to slight variations, which are unavoidable in 
rolling iron shapes. 

The weights of columns given are those of the 4, 6, or 8 segments of 
which they are composed. The rivet-heads add from 2 to 5 per cent to the 
weights given. Rivets are spaced 3, 4, or 6 inches apart from centre to 
centre, and somewhat more closely at the ends than towards the centre of 
the column. 

Q columns have 8 segments, .£7 columns 6 segments, O, B 2 , B l , and A have 
4 segments. Least radius of gyration - D X -3636, 



258 






Phoenix Columns. 








One Segment. 


Diameters in inches. 


One Column. 
















Safe 
Load in 
net tons 


a 


CO 










<K 


•i| . 


w 


S-g 




1 


to 


M 


cS cS © 


for 




tab 


6 
a 

T3 


"3 
o 






.Sfft 
$3 




16-feet 
Lengths. 


3-16 


9% 


r 


4 


6 1-16 


3.8 


12.6 


1.45 


17.72 


5-16 


12 


A j 


4% 


6 3-16 


4.8 


16.0 


1.50 


22.65 


14% 


3% 1 


434 

4% 


6 5-16 


5.8 


19.3 


1.55 


27.66 


% 


17 


6 7-16 


6.8 


22.6 


1.59 


32.58 


u 


16 


r 


5 5-16 


8 1-16 


6.4 


21.3 


1.92 


32.00 


5-16 


19% 


l 


5 7-16 


8% 


7.8 


26.0 


1.96 


39.15 


s i, 6 


23 

26% 


m 1 


5 9-16 
5 11-16 


8% 


9.2 
10.6 


30.6 
35.3 


2.02 
2.07 


46.45 
53.72 


% 


30 


5 13-16 


8 7-16 


12.0 


40.0 


2.11 


61.08 


9-16 


33% 




5 15-16 


8% 


13.4 


44.6 


2.16 


68.48 


% 


37 


i 


6 1-16 


8% 


14.8 


49.3 


2.20 


70.88 


M 


i8y 2 


r 


6 7-16 


9% 


7.4 


24.6 


2.34 


45.72 


5-16 


22y 2 




6 9-16 


9M 


9.0 


30.0 


2.39 


55.77 


% 


w& 


r5 2 I 

■5il 


6 11-16 


9 5-16 


10.6 


35.3 


2.43 


65.82 


7-16 


my 2 


6 13-16 


9% 


12.2 


40.6 


2.48 


75.95 


% 


3*y 2 


6 15-16 


9% 


13.8 


46.0 


2.52 


86.08 


9-16 


ssy 2 


1 


7 1-16 


9% 


15.4 


51.3 


2.57 


96.30 


% 


42% 


I 


7 3-16 


9 11-16 


17.0 


56.6 


2.61 


106.49 


M 


25% 




7 11-16 


11 9-16 


10.2 


34. 


2.80 


64.41 


5-16 


31 




7 13-16 


11% 


12.4 


41.3 


2.85 


78.45 


% 


36 




7 15-16 


11 11-16 


14.4 


48.0 


2.90 


91.28 


7-16 


41 




8 1-16 


11% 


16.4 


54.6 


2.94 


104.09 


1-16 


46 




8 3-16 


11 13-16 


. 18.4 


61.3 


2.98 


116.94 


51 




8 5-16 


11% 


20 4 


68. 


3.03 


129.87 


% 


56 




8 7-16 


12 


22.4 


74.6 


3.08 


142.83 


11-16 


62 


8 9-16 


12 1-16 


24.8 


82.6 


3.12 


158.34 


M 


68 


8 11-16 


12 3-16 


27.2 


90.6 


3.16 


173.86 


13-16 


73 




8 13-16 


12 5-16 


29.2 


97.3 


3.21 


186.93 


% 


78 




8 15-16 


12 7-16 


31.2 


104. 


3.26 


200.02 


1 


89 




9 3-16 


12 9-16 


35.6 


118.6 


3.34 


228.72 


IS 


99 




9 7-16 


12% 


39.6 


132. 


3.43 


255.02 


109 




9 11-16 


12 15-16 


43.6 


145.3 


3.52 


281.41 


34 


28 




11% 


15 7-16 


16.8 


56. 


4.18 


109.88 


5-16 


32^ 




11% 


15 9-16 


19.5 


65. 


4.23 


127.64 


% 


37 




11% 


15 11-16 


22.2 


74. 


4.28 


145.48 


7-16 


42 




11% 


15 13-16 


25.2 


84. 


4.32 


165.21 


% 


47 




12 


15% 


28.2 


94. 


4.36 


184.98 


9-16 


52 




12% 


16 


31.2 


104. 


4.40 


205.33 


% 


57 


E 


12M 

12% 


16 1-16 


34.2 


114. 


4.45 


224.64 


11-16 


62 


11 ' 


16 3-16 


37.2 


124. 


4.50 


244.53 


% 


68 




12% 


16 5-16 


40.8 


136. 


4.55 


268.37 


13-16 


73 




12% 


16 7-16 


43.8 


146. 


4.60 


288.30 


% 


78 




12% 


16% 


46.8 


156. 


4.64 


308.16 


1 


88 




13 


16% 


52.8 


176. 


4.73 


348.15 


in 


98 




13J4 


17 


58.8 


196. 


4.82 


388.15 


108 




13% 


17 3-16 


64.8 


216. 


4.91 


428.26 


5-16 


31 


f 


15 


19% 


24.8 


82.6 


5.45 


164.87 


% 
7-16 


36 


\ 


15% 


1934 


28.8 


96. 


5.50 


191.54 


41 


G J 


15J4 


19% 


32.8 


109.3 


5.55 


218.25 


M 


46 


14% 1 


ii 


19 7-16 


36.8 


122.6 


5.59 


244.95 


9-16 


51 


1 


19% 


40.8 


136. 


5.63 


271.69 


% 


56 


1 


15% 


19% 


44.8 


149.3 


5.68 


298.45 



FORMULAE FOR IROtf AKD STEEL STRUTS. 



259 



One Segment. 


Diameters in 


nches. 


One Column. 




















Safe 
Load in 




w* 








& 


£ 


s a 




S-6 




a3 




c &■ 
5 a" 


^49 




net tons 
for 


® 03 




<D 


"a 


is 

5 s 






£££ 


16-feet 








3 
O 






3 Us 


Lengths. 


11-16 


61 


r 


\W A 


19% 


48.8 


162.6 


5.72 


325.21 


13-16 


66 


I 


15% 


19% 


52.8 


176. 


5.77 


352.02 


71 


1 


16 


20 


56.8 


189.3 


5.82 


378.85 


%. 


76 


G J 

14% 1 


16% 


20% 
203^ 


60.8 


202.6 


5.87 


405.70 


1 


86 


16% 


68.8 


229.3 


5.95 


464.38 


1M 


96 


1 


16% 


20% 


76.8 


256. 


6.04 


513.17 


m 

i% 


106 




16% 


2034 


84.8 


282.6 


6.14 


567.06 


116 


I 


17% 


21 


92.8 


309.3 


6.23 


620.98 



Working Formulae for Wr ought-Iron and Steel Struts 
of various Forms.— Burr gives the followiug practical formulae, which 
he believes to possess advantages over Gordon's: 



p = Ultimate 

Strength, 

lbs. per sq. in. 

of Section. 



Kind of Strut. 
Flat and fixed end iron angles and tees 44000-140 — 

Hinged-end iron angles and tees 46000—175 — 

r 

Flat-end iron channels and I beams. .. .40000— 110 — 



Pj = Working 
Strength = 
1/5 Ultimate, 
lbs. per sq. 

in. of Section. 



Flat-end mild-steel angles 52000-180 

Flat-end high-steel angles 76000-290 

Pin-end solid wrought iron columns 32000- 80 

32000 



(1) 
(3) 

(5) 
(7) 
(9) 



8800-28 



(2) 

(4) 
(6) 



(10) 



n 

■ leu) 



277 5 I 



Equations (1) to (4) are to be used only between — = 40 and — = 200 



(5) and (6) ■ 
(7) to (10) ' 
(11) and (12) ' 



= 40 " ' 
.20 " ' 

: 6 and - 



= 200 
= 200 
= 200 

- = 65 



Steel columns, properly made, of steel ranging in specimens from 65,000 to 
73,000 lbs. per square inch should give a resistance 25 to 33 per cent in ex- 
cess of that of wrought-iron columns with the same value of I ■+- r, provided 
that ratio does not exceed 140. 

The unsupported width of a plate in a compression member should not 
exceed 30 times its thickness. 

In built columns the transverse distance between centre lines of rivets 
securing plates to angles or channels, etc., should not exceed 35 times the 
plate thickness. If this width is exceeded, longitudinal buckling of the 



260 



STRENGTH OF MATERIALS. 



plate takes place, and the column ceases to fail as a whole, but yields in 
detail. 

The same tests show that the thickness of the leg of an angle to which 
latticing is riveted should not be less than 1/9 of the length of that leg or 
side if the column is purely and wholly a compression member. The above 
limit may be passed somewhat in stiff ties and compression members de- 
signed to cany transverse loads. 

The panel points of latticing should not be separated by a greater distance 
than 60 times the thickness of the angle-leg to which the latticing is riveted, 
if the column is wholly a compression member. 

The rivet pitch should never exceed 16 times the thickness of the thinnest 
metal pierced by the rivet, and if the plates are very thick it should never 
nearly equal that value. 

Merriman's Rational Formula for Columns (Eng. News, 
July 19, 1894). 

- ,-Jg.il • ' (0 



1 "T" n^E r 2 



(2) 



B = unit-load on the column = total load P -*- area of cross-section A ; 
C — maximum compressive unit-stress on the concave side of the column ; 
I = lengtb of the column; r = least radius of gyration of the cross-section; 
E = coefficient of elasticity of the material; n = 1 for both ends round; 
n = 4/9 for one end round and one fixed; n = % for both ends fixed. This 
formula is for use with strains within the elastic limit only: it does not 
hold good when the strain C exceeds the elastic limit. 

Prof. Merrimau takes the mean value of E for timber = 1,500,000, for cast 
iron = 15,000,000, for wrought-iron = 25,000,000, and for steel = 30,000,000, 
and 7T 2 = 10 as a close enough approximation. With these values he com- 
putes the following tables from formula (1): 



I.- 


-Wrought-iron Columns with 


Round Ends 


. 


Unit- 
load. 


Maximum Compressive Unit-stress C. 


p 

^rOTB. 
A 


-H" 


f=*> 


i-ao 

r 


7 = 80 


- = 100 
r 


- = 120 
r 


1 = 140 


- = 160 


5,000 
6,000 
7,000 
8,000 
9,000 
10,000 
11.000 


5,040 
6,055 
7,080 
8,100 
9,130 
10,160 
11.200 
12,240 
13,280 


5,170 
6,240 
7,330 
8,430 
9,550 
10,680 
11,750 
13,000 
14,180 


5,390 
6,560 
7,780 
9,040 
10,340 
11,680 
13,070 
14,500 
15,990 


5,730 
7,090 
8,530 
10,060 
11,690 
13,440 
15,310 
17,320 
19,480 


6,250 
7,890 
9,720 
11,660 
14,060 
16,670 
19,640 
23,080 


6,980 
9,090 
11,610 
14,640 
18,380 
23,090 


8,220 
11,330 
15,510 
21,460 


10.250 
15,560 
24,720 


i2;ooo 








13,000 



















STRENGTH OF WROUGHT IRON AND STEEL COLUMNS. 261 
II.— Wrought-iron Columns with Fixed Ends. 



Unit- 
load. 




Maximum Compressive Unit-stress 


C. 




— or B. 
A 


H° 


£=*> 


1 = 60 

V 


7=M 


- = 100 


- = 120 


-=140 
r 


- = 160 
r 


6,000 


6,010 


6,060 


6,130 


6,240 


6,380 


6,570 


6,800 


7,090 


7,000 


7,020 


7,080 


7,180 


7,330 


7,530 


7,780 


8,110 


8,530 


8,000 


8,025 


8,100 


8,240 


8,430 


8,700 


9,040 


9,490 


10,060 


9,000 


9,030 


9,130 


9,300 


9,550 


9,890 


10,340 


10,930 


11,690 


10,000 


10,040 


10,160 


10,370 


10,710 


11,110 


11,680 


12,440 


13,440 


11,000 


11,050 


11,200 


11,450 


11,830 


12,360 


13,070 


14,020 


15,310 


12,000 


12,060 


12,240 


12,540 


13,000 


13,640 


14,510 


15,690 


17,320 


13,000 


13,070 


13,280 


13.640 


14,210 


14,940 


15,990 


17,440 


19,480 


14,000 


14,080 


14,320 


14,740 


15,380 


16,280 


17,530 


19,290 


21,820 





III.- 


-Steel Columns with Round Ends. 




Unit- 
load. 


Maximum Compressive Unit-stress C. 


p 

- A or B. 

A 


7 = 2 ° 


| = 40 


7 = 60 


1 = 80 


-=100 

V 


1 = 120 


- = 140 

r , 


— = 160 


6,000 

7,000 
8,000 
9,000 
10,000 
11,000 
12,000 
13,000 
14,000 


6,050 
7,070 
8,090 
9,110 
10,130 
11,160 
12.200 
13,330 
14,250 


6,200 
7,270 
8,380 
9,450 
10,560 
11,690 
12,820 
13,970 
15,130 


6,470 
7,650 
8,770 
10,090 
11,360 
12.670 
14,020 
15,400 
16,830 


6,880 
8,230 
9,650 
11,140 
12,710 
14 370 


7,500 
9,130 
10,870 
12,850 
15,000 
17,370 


8,430 
10,540 
12,990 
15.850 
19,230 
23,300 
28,300 


9,870 
12,900 
16,760 
20,930 
28,850 


12,300 
17,400 
24,590 


16,130 


20.000 






18,000 | 22,940 
19,960 1 26.250 























IV.— Steel Columns with Fixed Ends. 



Unit- 
load. 




Maximum Compressive Unit-stress C. 




p 
~otB. 

A 


r 


1 = 40 
r 


1 = 60 
r 


- = 80 
r 


- = 100 


— =120 
r 


-=140 
r 


— = 160 


7,000 


7.020 


7,070 


7,150 


7,270 


7,430 


7,650 


7,900 


8,230 


8,000 


8.020 


8,090 


8,200 


8,380 


8,570 


8,770 


9,200 


9,650 


9,000 


9,030 


9,110 


9,250 


9,450 


9,730 


10,090 


10,550 


11,140 


10,000 


10,030 


10,130 


10,310 


10,560 


10,910 


11,360 


11,810 


12,710 


11,000 


11,040 


11,160 


11,380 


11,690 


12,110 


12,670 


13,410 


14,370 


12,000 


12.050 


12,200 


12,450 


12,820 


13.330 


14,020 


14,930 


16,130 


13,000 


13,060 


13,230 


13,530 


13,970 


14,580 


15,400 


16,500 


17.990 


14,000 


14,070 


14,250 


14,610 


15,130 


15,850 


16,830 


18,150 


19,960 


15,000 


15,080 


15,310 


15,710 


16,310 


17,140 


18,290 


19,870 


22,060 



The design of the cross-section of a column to carry a given load with 
maximum unit-stress C may be made by assuming dimensions, and then 



262 STRENGTH OF MATEEIALS. 

computing C by formula (1). If the agreement between the specified and 
computed values is not sufficiently close, new dimensions must be chosen, 
and the computation be repeated. By the use of the above tables the work 
will be shortened. 

The formula (1) may be put in another form which in some cases will ab- 
breviate the numerical work. For B substitute its value P-f- A, and for 
Ar"* write I, the least moment of inertia of the cross-section; then 

P nPl% 

x -^ = % < 3 > 

in which I and r* are to be determined. 

For example, let it be required to find the size of a square oak column 
with fixed ends when loaded with 24.000 lbs. and 16 ft, long, so that the 
maximum compressive stress C shall be 1000 lbs. per square inch. Here 
/= 24,000, C = 1000, n = y 4 , tt 2 = 10, E = 1,500,000, I = 16 x 12, and (3) be- 
comes 

I - 24r 2 = 14.75. 

Now let x be the side of the square; then 

I=- and ,* = -, 

so that the equation reduces to x* - 24a: 2 = 177, from which a 2 is found to be 
29.92 sq. in., and the side x = 5.47 in. Thus the unit-load B is about 802 
lbs. per square inch. 

WORKING STRAINS ALLOWED IN BRIDGE 
MEMBERS. 

Theodore Cooper gives the following in his Bridge Specifications : 
Compression members shall be so proportioned that the maximum load 
shall in no case cause a greater strain than that determined by the follow- 
ing formula : 

p 8000 „ 

■r = — for square-end compression members ; 

1 + 40,000r 2 

P = ~ for compression members with one pin and one square end ; 

^ 30,000r 2 

P= „ — for compression members with pin-bearings; 

1-j 

~ 20,000r 2 

(These values may be increased in bridges over 150 ft. span. See Cooper's 
Specifications.) 

P = the allowed compression per square inch of cross-section; 
I = the length of compression member, in inches; 
r = the least radius of gyration of the section in inches. 

No compression member, however, shall have a length exceeding 45 times 
its least width. 

The Phoenix Bridge Company give the following : 

The greatest working stresses in wrought-iron compression members of 
spans 150 feet in length and under shall be the following: 

Flat Ends. Pin Ends. 

Phoenix column P = = — P = — 

T 50,000r 2 T 30,000r a 

T *.*■ i i t> 800 ° « 7800 
Latticed or common column P = — — - P = — — 

H i-j - 

T 40,000r 2 ^ 30,000r 2 

Angle-iron struts P = 9000 - 30- P = 9000 - 34 - 



WORKING STRAINS ALLOWED IN BRIDGE MEMBERS. 263 

Upper chords shall be proportioned by the flat-end formula. 

A mean between flat-end and pin-end results shall be used for one pin end 
and one flat end. 

Lateral and trausverse struts shall be designed by taking working stresses 
equal to one and four tenths those given by the preceding formulae. 

Working Stresses allowed in Bridge Tension Members. 

(Theodore Cooper's Specifications.) 

All parts of the structure shall be so proportioned that the maximum * 
loads shall in no case cause a greater tension than the following (except in 
spans exceeding 150 feet) : 

Pounds per 
sq. in. 

Ou lateral bracing 15,000 

On solid rolled beams, used as cross floor-beams and stringers. 9,000 

On bottom chords and main diagonals (forged eye-bars) 10,000 

On bottom chords and main diagonals (plates or shapes), net 

section 8,000 

On counter rods and long verticals (forged eye-bars) 8,000 

On counter and long verticals (plates or shapes), net section.. 6,500 

On bottom flange of riveted cross-girders, net section 8,000 

On bottom flange of riveted longitudinal plate girders over 

20 ft. long, net section 8,000 

On bottom flange of riveted longitudinal plate girders under 

20 ft. long, net section 7,000 

On floor-beam hangers, and other similar members liable to 

sudden loading (bar iron with forged ends) 6,000 

On floor beam hangers, and other similar members liable to 

sudden loading (plates or shapes), net section 5,000 

Members subject to alternate strains of tension and compression shall be 
proportioned to resist each kind of strain. Both of the strains shall, how- 
ever, be considered as increased by an amount equal to 8/10 of the least of 
the two strains, for determining the sectional area by the above allowed 
strains. 

The Phoenix Bridge Company specify : The greatest working stresses in 
all wrought-iron tensile members of railway spans 150 feet in length and 
under, shall be as follows: 

Pounds per 
sq. in. 

In, counter web members 8,000 

In long verticals 8,000 

In main-web and lower-chord members (eye-bars) 10,000 

In suspension loops 7,000 

In suspension plates (net section) 7,000 

In tension members of lateral and transverse bracing 15,000 

In counter rods and long verticals of lattice girders (net sec- 
tion) 7,000 

In lower chords and main tension members of lattice girders 

(net section) 8,000 

In bottom flange of plate girders (net section) 8,000 

In bottom flange of rolled beams 8,000 

In angle-iron lateral ties (net section) 12,000 

In spans over 150 feet in length, the greatest working tensile stresses per 
square inch of wrought iron, lower-chord and end main-web eye-bars, shall 
be 

total stress\ 



V- max. total stress / 

whenever this quantity exceeds 10,000. 

Working Stresses for Steel. 

The greatest allowed working stresses for steel tension members, for 
spans of 200 feet in length and less, shall be as follows ; 



264 STRENGTH OF MATERIALS. 

Pounds per 
sq. in. 

In counter web members 10,500 

In long verticals 10,000 

In all main-web and lower-chord eye-bars 13,200 

In plate hangers (neb section) 9,000 

In tension members of lateral and transverse bracing 19,000 

In steel-angle lateral ties (net section) 15,000 

For spans over 200 feet in length the greatest allowed working stresses 
per square inch, in lower-chord and end main-web eye-bars, shall be taken at 



/ mm^otalst^X 
V max. total stress / 



max. total stress s 

whenever this quantity exceeds 13,200. 

The greatest allowable stress in the main-web eye-bars nearest the centre 
of such spans shall be taken at 13,200 pounds per square inch ; and those 
for the intermediate eye-bars shall be found by direct interpolation between 
the preceding values. 

The greatest allowable working stresses in steel plate and lattice girders 
and rolled beams shall be taken as follows : 

Pounds per 
sq. in. 

Upper flange of plate girders (gross section) 10,000 

Lower flange of plate girders (net section) 10,000 

In counters and long verticals of lattice girders (net section) . . 9,000 
In lower chords and main diagonals of lattice girders (net 

section) 10,000 

In bottom flanges of rolled beams 10,000 

In top flanges of rolled beams 10,000 

RESISTANCE OF HOLLOW CYLINDERS TO 
COLLAPSE. 

Fairbairn's empirical formula {Phil. Trans. 1858) is 

p- 9,675,600^, . . . . . .(1) 

where p = pressure in lbs. per square inch, t = thickness of cylinder, d = 
diameter, and I — length, all in inches ; or, 

p = 806,600 tj-z-i if L is in feet (2) 

He recommends the simpler formula 

p = 9,675,600^ ( 3 ) 

as sufficiently accurate for practical purposes, for tubes of considerable 
diameter and length. 

The diameters of Fairbairn's experimental tubes were 4", 6", 8", 10", and 
12", and their lengths, between the cast-iron ends, ranged between 19 inches 
and 60 inches. 

His formula (3) has been generally accepted as the basis of rules for 
ascertaining the strength of boiler flues. In some cases, however, limits are 
fixed to its application by a supplementary formula. 

Lloyd's Register contains the following formula for the strength of circular 
boiler-flues, viz., 

"^ « 

The English Board of Trade prescribes the following formula for circular 
flues, when the longitudinal joints are welded, or made with riveted butt- 
straps, viz., 

P - M' 000 *" (5) 

r -(L + Dd- • ; (5) 

For lap-joints and for inferior workmanship the numerical factor may be 
reduced as low as 60,000, 



RESISTANCE OF fiOLLOW CYLINDERS TO COLLAPSE. 265 

The rules of Lloyd's Register, as well as those of the Board of Trade, pre- 
scribe further, that in uo case the value of P must exceed the amount given 
by the following equation, viz.* 

P=*£ <6> 

In formulae (4), (5), (6) P is the highest working pressure in pounds per 
square inch, t and d are the thickness and diameter in inches, L is the 
length of the flue in feet measured between the strengthening rings, in case 
it is fitted with such. Formula (4) is the same as formula (3), with a factor 
of safety of 9. In formula (5) the length L is increased by 1 ; the influence 
which this addition has on the value of P is, of course, greater for short 
tubes than for long ones. 

Nystrom has deduced from Fairbairn's experiments the following formula 
for the collapsing strength of flues : 

4Tt* 

P= ^£' <T) 

where p. t, and d have the same meaning as in formula (1), L is the length in 
feet, and T is the tensile strength of the metal in pounds per square inch. 

If we assign to T the value 50,000, and express the length of the flue in 
inches, equation (7) assumes the following form, viz., 

p = 692,800 -¥— (8) 

dVT 

Nystrom considers a factor of safety of 4 sufficient in applying his formula. 
(See "A New Treatise on Steam Engineering," by J. W. Nystrom, p. 106.) 

Formula (1), (4), and (8) have the common defect that they make the 
collapsing pressure decrease indefinitely witli increase of length, and vice 
versa. M. Love has deduced from Fairbairn's experiments an equation of 
a different form, which, reduced to English measures, is as follows, viz., 

p = 5,358,150 ^ + 41,906^+ 1323 j, (9) 

where the notation is the same as in formula (1). 

D. K. Clark, in his " Manual of Rules," etc., p. 696, gives the dimensions of 
six flues, selected from the reports of the Manchester Steam-Users Associa- 
tion, 1862-69, which collapsed while in actual use in boilers. These flues 
varied from 24 to 60 inches in diameter, and from 3-16 to % inch in thickness. 
They consisted of rings of plates riveted together, with one or two longitud- 
inal seams, but all of them unfortified by intermediate flanges or strength- 
ening rings. At the collapsing pressures the flues experienced compressions 
ranging from 1.53 to 2.17 tons, or a mean compression of 1.82 tons per square 
inch of section. From these data Clark deduced the following formula 
"for the average resisting force of common boiler-flues," viz., 

50 ' 000 -500), (iO, 

where p is the collapsing pressure in pounds per square inch, and d and t 
are the diameter and thickness expressed in inches. 

C. R. Roelker, in Van Nosti-and^s Magazine, March, 1881, discussing the 
above and other formnlae, shows that experimental data are as yet insuffi- 
cient to determine the value of any of the formulas. He says that Nystrom's 
formula, (8), gives a closer agreement of the calculated with the actual col- 
lapsing pressures in experiments on flues of every description than any of 
the other formulae. 

Collapsing Pressure of Plain Iron Tubes or Flues. 

(Clark, S. E., vol. i. p. 643.) 
The resistance to collapse of plain-riveted flues is directly as the square of 
the thickness of the plate, and inversely as the square of the diameter. The 
support of the two ends of the flue does not practically extend over a length 
of tube greater than twice or three times the diameter. The collapsing 
pressure of long tubes is therefore practically independent of the length. 



> = t* (J 



266 STRENGTH OF MATERIALS. 

Instances of collapsed flues of Cornish and Lancashire boilers collated by 
Clark, showed that the resistance to collapse of flues of %-inch plates, 18 to 
43 feet long, and 30 to 50 inches diameter, varied as the J, 75 power of the 
diameter. Thus, 

for diameters of 30 35 40 45 50 inches, 

the collapsing pressures were 76 58 45 37 30 lbs. per sq. in; 

for 7-16-inch plates the collapsing 

pressures were 60 49 42 " " " 

For collapsing pressures of plain iron flue-tubes of Cornish and Lanca- 
shire steam-boilers, Clark gives: 

_ 200,000^ 

P = collapsing pressure, in pounds per square inch; 
t = thickness of the plates of the furnace tube, in inches. 
d = internal diameter of the furnace tube, in inches. 

For short lengths the longitudinal tensile resistance may be effective in 
augmenting the resistance to collapse. Flues efficiently fortified by flange- 
joints or hoops at intervals of 3 feet may be enabled to resist from 50 lbs. 
to 60 lbs. or 70 lbs. pressure per square inch more than plain tubes, accord- 
ing to the thickness of the plates. 

Strength of Small Tubes.— The collapsing resistance of solid- 
drawn tubes of small diameter, and from .134 inch to .109 inch in thickness, 
has been tested experimentally by Messrs. J. Russell & Sons. The results 
for wrought-iron tubes varied from 14.33 to 20.07 tons per square-inch sec- 
tion of the metal, averaging 18.20 tons, as against 17.57 to 24.28 tons, averag- 
ing 22.40 tons, for the bursting pressure. 

(For strength of Segmental Crowns of Furnaces and Cylinders see Clark, 
S. E., vol. i, pp.- 649-651 and pp. 627, 628.) 

Formula for Corrugated Furnaces (Eng^g, July 24, 1891, p. 
102).— As the result of a series of experiments on the resistance to collapse 
of Fox's corrugated furnaces, the Board of Trade and Lloyd's Registry 
altered their formulae for these furnaces in 1891 as follows: 

Board of Trade formula is altered from 

i a ,5ooxr =w , fto i4.oooxr = trp 

T = thickness in inches; 

D = mean diameter of furnace; 

WP = working pressure in pounds per square inch. 

Lloyd's formula is altered from 

1000 x(T*) 1884 X(r») p 

D D ~~ 

T = thickness in sixteenths of an inch; 

D = greatest diameter of furnace; 

WP = working pressure in pounds per square inch. 

TRANSVERSE STRENGTH. 

In transverse tests the strength of bars of rectangular section is found to 
vary directly as the breadth of the specimen tested, as the square of its 
depth, and inversely as its length. The deflection under any load varies as 
the cube of the length, and inversely as the breadth and as the cube of the 
depth. Represented algebraically, if S = the strength and D the deflection, 
I the length, b the breadth, and d the depth, 

a "• bd * a r, Z 3 

S varies as ~r and D varies as —ts> 
I bd 3 

For the purpose of reducing the strength of pieces of various sizes to 
a common standard, the term modulus of rupture (represented by R) is 
used. Its value is obtained by experiment on a bar of rectangular section 



TRANSVERSE STRENGTH. 267 

supported at the ends and loaded in the middle and substituting numerical 
values in the following formula : 

3 PI 
E - «6d5' 

in which P — the breaking load in pounds, I = the length in inches, 6 the 
breadth, and d the depth. 

The modulus of rupture is sometimes defined as the strain at the instant 
of rupture upon a unit of the section which is most remote from the neutral 
axis on the side which first ruptures. This definition, however, is based 
upon a theory which is yet in dispute among authorities, and it is better to 
define it as a numerical value, or experimental constant, found by the ap- 
plication of the formula above given. 

1 rom the above formula, making I 12 inches, and b and d each 1 inch, it 
follows that the modulus of rupture is 18 times the load required to break a 
bar one inch square, supported at two points one foot apart, the load being 
applied in the middle. 

„ ™ . . „ , , t , span in feet X load at middle in lbs. 

Coefficient of transverse strength = _— — ^r-. — : — : — — : — - 

breadth in inches x (depth m inches) 2 . 

Fundamental Formulae for Flexure of Beams (Merriman). 

Resisting shear = vertical shear; 

Resisting moment = bending moment; 

Sum of tensile stresses = sum of compressive stresses; 

Resisting shear = algebraic sum of all the vertical components of the in- 
ternal stresses at any section of the beam. 

Tf A be the area of the section and Ss the shearing unit stress, then resist- 
ing shear = ASs; and if the vertical shear = V, then V = ASs.' 

The vertical shear is the algebraic sum of all the external vertical forces 
on one side of the section considered. It is equal to the reaction of one sup- 
port, considered as a force acting upward, minus the sum of all the vertical 
downward forces acting between the support and the section. 

The resisting moment = algebraic sum of all the moments of the inter- 
nal horizontal stresses at any section with reference to a point in that sec- 
tion, = ' — , in which S — the horizontal unit stress, tensile or compressive 

as the case may be, upon the fibre most remote from the neutral axis, c = 
the shortest distance from that fibre to said axis, and I = the moment of 
inertia of the cross-section with reference to that axis. 

The bending moment M is the algebraic sum of the moment of the ex- 
ternal forces on one side of the section with reference to a point in that sec- 
tion = moment of the reaction of one support minus sum of moments of 
loads between the support and the section considered. 



The bending moment is a compound quantity = product of a force by the 
distance of its point of application from the section considered, the distance 
being measured on a line drawn from the section perpendicular to the 
direction of the action of the force. 

Concerning the above formula, Prof. Merriman, Eng. News, July 21, 1894, 
says: The formula just quoted is true when the unit-stress S on the part of 
the beam farthest from the neutral axis is within the elastic limit of the 
material. It is not true when this limit is exceeded, because then the neutral 
axis does not pass through the centre of gravity of the cross-section, and 
because also the different longitudinal stresses are not proportional to their 
distances from that axis, these two requirements being involved in the de- 
duction of the formula. But in all cases of design the permissible unit- 
stresses should not exceed the elastic limit, and hence the formula applies 
rationally, without regarding the ultimate strength of the material or any 
of the circumstances regarding rupture. Indeed so great reliance is placed 
upon this formula that the practice of testing beams by rupture has been 
almost entirely abandoned, and the allowable unit-stresses are mainly de- 
rived from tensile and compressive tests. 



268 



STRENGTH OF MATERIALS. 















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APPROXIMATE SAFE LOADS IN LBS. ON STEEL BEAMS. 269 



Formulae for Transverse Strength of Beams.— Referring to 
table on pieeeding page, 

P = load at middle; 

W= total load, distributed uniformly; 
I = length, o = breadth, d = depth, in inches; 

E = modulus of elasticity; 

R ■= modulus of rupture, or stress per square inch of extreme fibre; 

/ = moment of inertia; 

c = distance between neutral axis and extreme fibre. 

For breaking load of circular section, replace bd 2 by 0.59<2 3 . 

For good wrought iron the value of R is about 80,000, for steel about 120,000, 
the percentage of carbon apparently having no influence. (Thurston, lion 
and Steel, p. 491). 

For cast iron the value of R varies greatly according to quality. Thurston 
found 45,740 and 67,980 in No. 2 and No. 4 cast iron, respectively. 

For beams fixed at both ends and loaded in the middle, Barlow, by experi- 
ment, found the maximum moment of stress = 1/6PI instead of %Pl, the 
result given by theory. Prof. Wood (Resist. Matls. p. 155) says of this case: 
The phenomena are of too complex a character to admit of a thorough and 
exact analysis, and it is probably safer to accept the results of Mr. Barlow 
in practice than to depend upon theoretical results. 

APPROXIMATE GREATEST SAFE LOADS IN LBS. ON 

STEEL. BEAMS. (Pencoyd Iron Works.) 

Based on fibre strains of 16,800 lbs. for steel. (For iron the loads should be 
one sixth less, corresponding to a fibre strain of 14,000 lbs. per square inch). 
L = length in feet between supports; a = interior area in square 

A — sectional area of beam in square inches; 

inches; d = interior depth in inches. 

D = depth of beam in inches w = working load in net tons. 



Shape of 
Section. 


Greatest Safe Load in Pounds 


Deflection 


in Inches. 


Load in 

Middle. 


Load 
Distributed. 


Load in " 

Middle. 


Load 
Distributed. 


Solid Rect- 
angle. 


940.4 D 
L 


1880.4D 
L 


wL 3 
32AD* 


ivL 3 
52 AD* 


HollowRect- 


940( AD -ad) 
L 


1880(AD-ad) 
L 


wU 


wL 3 


angle. 


32(AD 2 -ad' 2 ) 


52(AD*-ad*) 


Solid Cylin 
der. 


700AD 
L 


1400 AD 
L 


ivL 3 
24AD* 


wL 3 

38AD'* 


Hollow 


700( AD- ad) 
L 


UOO(AD-ad) 
L 


tvL 3 


wL 3 




24(AD2-ad*) 


38(AD*-ad*) 


Even-legged 
Angle or 
Tee. 


930AD 


moAD 

L 


tvL 3 
32AD* 


wL 3 


L 


52AD* 


Channel or 
Z bar. 


1600AD 
L 


3200AD 
L 


wL 3 
53AD 2 


ivL 3 

85AD 2 


Deck Beam. 


1450 AD 
L 


2900,4 D 
L 


wL 3 
50AD* 


ivL 3 
80AD* 


I Beam. 


1780AD 
L 


mQAD 
L 


wL 3 

58 A D 2 


wL 3 
93,4 D 2 


I 


II 


III 


IV 


V 



270 



STRENGTH OF MATERIALS. 



The above formulae for the strength and stiffness of rolled beams of va- 
rious sections are intended for convenient application in cases where 
strict accuracy is not required. 

The rules for rectangular and circular sections are correct, while those for 
the flanged sections are approximate, and limited in their application to the 
standard shapes as given in the Pencoyd tables. When the section of any 
beam is increased above the standard minimum dimensions, the flanges re- 
maining unaltered, and the web alone being thickened, the tendency will be 
for the load as found by the rules to be in excess of the actual; but within 
the limits that it is possible to vary any section in the rolling, the rules 
will apply without any serious inaccuracy. 

The calculated safe loads will be approximately one half of loads that 
would injure the elasticity of the materials. 

The rules for deflection apply to any load below the elastic limit, or less 
than double the greatest safe load by the rules. 

If the beams are long without lateral support, reduce the loads for the 
ratios of width to span as follows : 



Length of Beam. 
20 times flange width. 
30 " 

40 " « « 

50 " 
60 ■" 



Proportion of Calculated Load 
forming Greatest Safe Load. . 
Whole calculated load. 
9-10 

8-10 " 

7-10 
6-10 
5-10 



These rules apply to beams supported at each end. For beams supported 
otherwise, alter the coefficients of the table as described below, referring to 
the respective columns indicated by number. 

Changes of Coefficients for Special Forms of Beams. 



Kind of Beam. 



Fixed at one end, loaded 
at the other. 



One fourth of the coeffi- 
cient, col. II. 



Coefficient for Safe 
Load. 



One sixteenth of the co- 
efficient of col. IV. 



Coefficient for Deflec 
tion. 



Fixed at one end, load 
evenly distributed. 



One fourth of the coeffi- 
cient of col. III. 



Five forty-eighths of the 
coefficient of col. V. 



Both ends rigidly fixed, 
or a continuous beam, 
with a load in middle. 



Twice the coefficient of 
col. II. 



Four times the coeffi- 
cient of col. IV. 



Both ends rigidly fixed, 
or a continuous beam, 
with load evenly dis- 
tributed. 



One and one-half times 
the coefficient of col. 
III. 



Five times the coefficient 
of col. V. 



ELASTIC RESILIENCE. 

In a rectangular beam tested by transverse stress, supported at the ends 
and loaded in the middle, 

2 Rbd* 
r ~ 3 I 5 

_ 1 PV 
A _ 4 Eod 3 ' 
in which, if P is the load in pounds at the elastic limit, R = the modulus of 
transverse strength, or the strain on the extreme fibre, at the elastic limit, 
E = modulus of elasticity, A = deflection, L b, and d = length, breadth, and 
depth in inches. Substituting for P in (2) its value in (1), we have 

A-- — 
6 Ed • 



BEAMS OF UNIFORM STRENGTH THROUGHOUT LENGTH. 271 



ELEVATION. 



The elastic resilience = half the product of the load and deflection = J^jPA, 
and the elastic resilience per cubic inch 

_ 1 PA 
~ 2 Ibd' 

Substituting the values of P and A, this reduces to elastic resilience per 

1 P 2 
cubic inch = ^j-j™. which is independent of the dimensions; and therefore 

the elastic resilience per cubic inch for transverse strain may be used as a 
modulus expressing one valuable qualit}' of a material. 
Similarly for tension: 

Let P = tensile stress in pounds per square inch at the elastic limit; 
e = elongation per unit of length at the elastic limit; 
E = modulus of elasticity = P -f- e; whence e = P 4- E. 

1 P 2 
Then elastic resilience per cubic inch = J^Pe = 5-^'. 

BEAMS OF UNIFORM STRENGTH THROUGHOUT 
THEIR LENGTH. 

The section is supposed in all cases to be rectangular throughout. The 
beams shown in plan are of uniform depth throughout. Those shown in 
elevation are of uniform breadth throughout. 

B = breadth of beam. D = depth of beam. 

Fixed at one end, loaded at the other; 
curve parabola, vertex at loaded end ; PP 2 
proportional to distance from loaded end. 
The beam may be reversed, so that the up- 
per edge is parabolic, or both edges may be 
parabolic. 

Fixed at one end, loaded at the other; 
triangle, apex at loaded end; PP 2 propor- 
tional to the distance from the loaded end. 

Fixed at one end; load distributed; tri- 
angle, apex at unsupported end; BD' 1 pro- 
portional to square of distance from unsup- 
ported end. 

Fixed at one end; load distributed ; curves 
two parabolas, vertices touching each other 
at unsupported end; PP 2 proportional to 
distance from unsupported end. 

Supported at both ends; load at any one 
point; two parabolas, vertices at the points 
of support, bases at point loaded ; BD 2 pro- 
portional to distance from nearest point of 
support. The upper edge or both edges 
may also be parabolic. 

Supported at both ends; load at any one 
point; two triangles, apices at points of sup- 
port, bases at point loaded; BD 2 propor- 
tional to distance from the nearest point of 
support. 

Supported at both ends; load distributed; 
curves two parabolas, vertices at the middle 
of the beam; bases centre line of beam; PP 2 
proportional to product of distances from 
points of support. 

Supported at both ends; load distributed; 
curve semi-ellipse; BD 2 proportional to the 
product of the distances from the points of 
support. 





PASE S? 

ELEVATION. |9| 




£72 STRENGTH OF MATERIALS. 

PROPERTIES OF ROLLED STRUCTURAL SHAPES. 

Explanation of Tables of the Properties of Carnegie 
I Beams, Channels, and Z Bars. 

The tables of I beams are calculated for the minimum weight to which 
each pattern can be rolled. The tables of channels are calculated for the 
minimum and maximum weights of the various shapes, while the properties 
of Z bars are given for thicknesses differing by 1/16 inch. 

Columns 11 and 13, in the tables fori beams and channels, give coefficients 
by the help of which the safe uniformly-distributed load may readily be de- 
termined. To do this, divide the coefficient given by the span or distance 
between suppoi-ts in feet. If the weight of the section is intermediate be- 
tween the minimum and maximum weights given, add to the coefficient 
for the minimum weight the value given in columns 12 or 14 (for one pound 
increase of weight), multiplied by the number of pounds the section is 
heavier than the minimum. 

If a section is to be selected (as will usually be the case) intended to 
carry a certain load, for a length of span already determined on, ascertain 
the coefficient which this load and span will require, and refer to the table 
for a section having a coefficient of this value. The coefficient is obtained 
by multiplying the load, in pounds uniformly distributed, by the span 
length in feet. 

In case the load is not tmiformly distributed, but is concentrated at the 
middle of the span, multiply the load by 2 and then consider it as uni- 
formly distributed. The deflection will be 8/10 of the deflection for the 
latter' load. 

For other cases of loading obtain the bending moment in foot-pounds; this 
multiplied by 8 will give the coefficient required. 

If the loads are quiescent, the coefficients for a fibre strain of 16,000 lbs. 
per square inch for steel and 12,000 lbs. for iron may be used ; but if 
moving loads are to be provided for, the coefficients for 12,500 and 10,000 
lbs., respectively, should be taken. Inasmuch as the effects of impact may 
be very considerable (the strains produced in an unyielding, inelastic 
material by a load suddenly applied being double those produced by the 
same load in a quiescent state), it will sometimes be advisable to use still 
smaller fibre strains than those given in the tables. In such cases the co- 
efficients can readily be determined by proportion. Thus, for a fiber strain 
of 8000 lbs. per square inch the coefficient will equal the coefficient for 
10,000 lbs. fibre strain, from the table, multiplied by 8/10. 

The moments of resistance given in column 9 are used to determine the 
fibre strain per square inch in a beam, or other shape, subjected to bending 
or transverse strains, by dividing the same into the bending moment 
expressed in inch-pounds. 

For Carnegie Z bars, complete tables of moments of inertia, moments of 
resistance, radii of gyration, and values of the coefficients (C) are given for 
thicknesses varying by 1/16 inch. These coefficients may be applied, as 
explained above, for cases where the Z bars are subjected to transverse 
loading, as, for example, in the case of roof-purlins. 

For more complete and detailed information concerning structural shapes, 
consult the pocket-books and circulars issued by the manufacturers. 



PROPERTIES OF ROLLED STRUCTURAL SHAPES. 2?3 



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2?7 



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278 



STRENGTH OF MATERIALS. 



TRENTON IRON BEAMS AND CHANNELS. 

(New Jersey Steel and Iron Co.) 





*■• »j 




,JD 






u . 


„ 


.0 






-1 


a . 


"Irf 


JD 03 




« a 


§0 

. 




|& 




II 




o &> 




o 

a 


fl 


53 




.5 > 

2^M 


d 






cc^ 


a c§^ 


a 




ei_i a 






£4 


°.S 


v a 


<» S ac 




Erf 


a 


© a 


I 
S 

m 




a"" 
a a 
o— ' 


O 


1 


5 fl 









I Beams. 




Channels. 


20 


272 


m 


11-16 


1,320,000 


15 


190 


m 


M 


625,000 


20 


200 


6 


H 


990,000 


15 


120 


4 


H 


401,000 


K% 


200 


5% 


.6 


748,000 


mi 


140 


4 


11-16 


381,000 


15 3-16 


150 


5 


^ 


551,000 


1214 
mi 


70 


3 


.33 


200.100 


15^ 


125 


5 


.42 


460,000 


60 


2% 


% 


134,750 


12 5-16 


170 


1 


.6 


511,000 


10 


48 


2y 2 


5-16 


102,000 


12^ 


125 


.47 


377,000 


9 


70 


m 


7-16 


146,000 


12 


120 


5j| 


.39 


375,000 


9 


50 


2^ 


.33 


104,000 


12 


96 


.32 


306,000 


8 


45 


2^ 


.26 


88,950 


10^ 


135 


5 


.47 


360,000 


8 


33 


2.2 


.20 


65,800 


10^ 


105 


4« 


% 


286.000 


7 


36 


2^ 


H 


62,000 


« 


90 


4^ 


5-16 


250,000 


7 


25y 2 


2 


.20 


39,500 


9 


125 


4V 2 


.57 


268,000 


6 


45 


2^ 


.40 


58,300 


9 


85 


iy 2 


% 


199.000 


6 


33 


2M 


.28 


45,700 


9 


70 


4 


.3 


167,000 


6 


22^ 


1% 


.18 


33,680 


8 


80 


4J^ 


% 


168,000 


5 


19 


1% 


.20 


22,800 


8 


65 


4 


.3 


135,000 


4 


16H 


1M 


.20 


15,700 


7 


55 


SH 


.3 


101,000 
172,000 


3 


15 


ifc 


.20 


10,500 


6 


120 


5J4 












6 


90 


5 


% 


132,000 




De 


ck Be* 


ims. 




6 
6 


50 


3^ 
3 


3 


76,800 












40 


M 


62,600 












5 


40 


3 


5-16 


49,100 


8 


65 


4^2 


% 


91,800 


5 


30 
37 


m 

3 


M 

5-16 


38,700 
36,800 


7 


55 


4J^ 


5-16 


63,500 


4 












4 


30 


2% 


J4 


30,100 












4 


18 1 


2 


3-16 1 


18,000 













Trenton Beams and Channels. 

To find which beam, supported at both ends, will be required to support 
with safety a given uniformly distributed load: 

Multiply the load in pounds by the span in feet, and take the beam whose 
" Coefficient for Strength " is nearest to and exceeds the number so found. 
The weight of the beam itself should be included in the load. 

The deflection in inches, for such distributed load, will be found by divid- 
ing the square of the span taken in feet, by 70 times the depth of the beam, 
taken in inches, for iron beams, and by 52.5 times the depth for steel. 

Example.— Which beam will be required to support a uniformly distrib- 
uted load of 12 tons (= 24,000 lbs.) on a span of 15 feet ? 

24,000 X 15 = 360,000, which is less than the coefficient of the 1214-inch 125- 
lb. iron beam. The weight of the beam itself would be 625 lbs., which, ad- 
ded to the load and multiplied by the span, would still give a product less 
than the coefficient; thus, 

24,625 X 15 = 369,375. 

The deflection will be 

15 X 15 _ ■ , 
-■ „ , = 0.26 inch. 

70 X Vl\i 

The safe distributed load for each beam can be found by dividing the co- 
efficient by the span in feet, and subtracting the weight of the beam. 



'TkEXTOtf AKGLfi-BAftS. 



279 



When the load is concentrated entirely at the centre of the span, one half 
of this amount must be taken. 

The beams must be secured against yielding sideways, or the safe loads 
will be much less. 

For beams used with plastered ceilings, the deflection allowed should not 
exceed 1/80 inch per foot of span, to avoid cracking of the plaster. 

TRENTON ANGLE-BARS. 



Size of 


Approximate Weight, in pounds per yard, for 


Coeff. for 
Transverse 
Strength. 


Bar. 


each thickness in inches. 




7/16 


V> 


9/16 


% 


11/16 


% 


13/16 


% 


Thinnest Bar. 


6 x6 




ft*. 5 


64.3 


71.1 


77.8 


84.4 


91.0 


97.3 


36,900 lbs. 


4^x4^2 


fe° 


42\5 

716 


47 5 


52.3 

9/16 


57.2 


61.9 

11/16 






18,000 " 


% 






4 x4 


28.6 


33 1 


37.5 


41.8 


46.1 


50.5 


54.4 




12,184 " 


3^x3^ 


24.8 


28.7 
5/16 


32.5 

% 


36.2 

7/16 


39.8 

Yo, 


43.4 

9/16 






9,200 " 


% 


11/16 




3 x3 


14.4 


17.7 


21.1 


24 4 


27.5 


30.6 


33.6 


36.5 


4,611 " 




5/16 


% 


13/32 


7/16 


15/32 


% 


17/32 


9/1& 




2Mx2% 


16.2 


19.2 


20.7 


22.2 


23.6 


25.0 


26.3 


27.7 


4,710 " 




M 


5/16 


11/32 


% 


13/32 


7/16 


15/32 


M, 




2^x2^ 


11.9 


14.7 


16.0 


17.3 


18.6 


20.0 


21.2 


22.5 


3,156 " 




M 


9/32 


5/16 


11/32 


% 


13/32 


7/16 






2J4x2M 


10.6 


11.9 


13.1 


14.3 


15.5 


16.8 


17.8 




2,530 " 




7/32 


% 


9/32 


5/16 


11/33 


H 








2 x2 


8.3 


9.4 


10.4 


11.5 


12.6 


13 6 






1,752 " 




3/16 


7/32 


Va 


9/32 


5/16 


11/ ,2 


% 






m*m 


6.21 


7.18 


8.13 


9 05 


9.96 


10 8 


11.7 




1,150 " 


1%*1}4 


5.27 


6.09 


6.88 


7.64 


8.40 


9.13 






832 " 




% 


5/32 


3/16 


7/32 


l A 










1 xl 


2.97 
2.34 


3.66 

2.88 


4 34 
3.40 


4.99 
3.91 


5.63 
4 38 








393 " 








246 " 


%x % 


2.03 
1.72 


3.48 
2.09 


2.93 
2.46 












186 '-' 












133 " 















Uneven Legs. 



5 x3J, 
4^x3 

4 x3 

3^x3 
3^x2^ 

3^x1^ 
3 x2^ 
3 x2 







7/16 
41.8 

35.3 
30.9 


40.0 
35.0 


9/16 
53.1 

44.7 
39 


% 
58.6 

49.2 
43.0 


11/16 
64.0 

53.7 
46. S 


69.4 

58.1 
50.6 




% 

30.5 
26.7 


14.4 


5/16 
20.9 


24.8 


7/16 

28.7 


32.5 


9/16 
36.2 


% 
39.8 


11/16 
43.4 


19.3 
17.7 


23.0 
21.1 


26.5 
24.4 


30.0 
27.5 


33.4 
30.6 


36.7 
33.6 


40 
36.5 


13.1 


H 

11.9 

5/16 
16.2 
7/32 
10.4 


il/32 
17.7 

% 
11.9 


19.2 

9/32 
13.3 


13/3*2 
20.7 
5/16 
14.6 




7/16 
20.0 


9/16 
27.7 

22.5 



{ 30,680, 
) 14,750, 
j 18,353, 
I 9,651, 
j 14,580, 
1 7,020, 
j 9,850, 
1 5,871, 



j 5,515, m 



1,148, 
4,490, 



2Y2" 



230 



STRENGTH 0£ MATERIALS. 



TRENTON TEE BARS. 



Designation of 
Bar. 



Table. Leg. 
4" x4" 
3^"x3^" 

2^"x2^"" 

2" x 2" 



v 



2H" 



. 2" 



<2y 2 " 

<2" 
«3" 



Approximate Weight, in pounds per 
yard, for each thickness in inches. 



5-16" 
5-16" 

7-1" 
5-32" 



7 lbs. 
I 4 ' 



5-16" 14.6 lbs. 



5.6 



Y 2 " 37.5 lbs. 

\4t" 27.5 ." 
%" 17.3 " 



5-16" 11.5 
" ' ' 5.5 ' 



3-16' 
W 35.0 
%" 17.3 



Coefficient 

for Transverse 

Strength. 



Thinnest Bar. 
15,800 ibs. 
10,550 
6,680 
3,850 
3,087 
1.970 
1,033 

596 

268 
6,344 
2,540 
6,404 
6,173 
1,355 

604 

457 

421 



SIZE OF BEAMS, AND THEIR DISTANCE APART, 

Suitable for Floors having Loads per square 
foot from 100 lbs. to 300 lbs. 

(New Jersey Steel and Iron Co.) 





Load 


per 
ft. 


Load 


per 
ft. 


Load 


r 


Load per 


Load per 
sq. ft. 




sq. 


sq. 


sq. 


sq. ft. 
250 lbs. 


.ja- 


100 lbs. 


150 lbs. 


200 lbs. 


300 lbs. 


il 


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CO 


P 




in. lb. 


feet 


in. lb. 


feet 


in. lb. 


feet 


in. lb. 


feet 


in. lb. 


feet 


H 


4 30 


4.6 


4 30 


3.1 


5 30 


3.0 


6 40 3.9 


6 40 


3 2 


5 30 


5,9 


5 30 


4.0 


6 40 


4.8 


6 501 4.7 
6 50J 3.0 


5 50 


3 9 


i»i 


5 30 


3.8 


6 40 


4.1 


6 40 


3.0 


7 55 


3 3 


5 40 


4.8 


6 50 


5.0 


6 50 


3.7 


7 551 4.0 


8 65 


4 4 


iij 


6 40 


4.2 


6 50 


3.4 


7 55 


3.4 


8 65| 3.6 


8 65 


a 


6 50 


5.2 


7 55 


4.6 


8 65 


4.5 


9 70 4.5 


9 70 


3.8 


14{ 


7 55 


5.0 


7 55 


3.3 


8 65 


3.3 


9 70 3.3 


9 85 


3 3 


8 65 


6.7 


8 65 


4.5 


9 70 


4.1 


10^ 90, 5.0 


10^ 90 


4 2 


16 \ 


8 65 


5.0 8 65 


3.3 


9 85 


3.7 


10*6 90, 3.8 


1<% 105 


3.6 


9 70 


6.3 


9 70 


4.2 


10V£ 90 


4.7 


10^ 105, 4.3 


I214 125 


4,8 


H 


9 70 


4.9 


9 85 


3.9 


10}^ 105 


4.2 


10^ 105; 3.4 
12*4 125! 4.5 


10^ 135 


3 6 


9 85 


5.9 


10J6 90 


4.9 


12 96 


4.6 


12J4 125 


3 7 


20 -j 


10J^ 90 


6.0 


10^ 105 
12J4 125 


4.5 


10^ 105 


3.4 


1234 125 3.6 


I214 125 


3 






6.0 


12i4 125 


4.5 


12J4 170 4.9 


15 150 


4.4 


22 -j 


10fc5 90 


4.9 


12 96 


4.0 


1214 125 


3.7 


12J4 1251 3.0 


1234 170 


8.3 


lOfc 105 


5.6 


lSi/4 125 


4.9 


15 125 


4.5 


15 125! 3.6 


15 150 


3.6 


u\ 


12 96 


5.0 


I2J4 125 


4.1 


12J4 125 


3.0 


1214 170, 3.3 


[5 150 


3.0 


1234 125 


6.1 


15 125 


5.0 


15 150 


4.5 


15 150, 3.6 


[5 200 


4.1 


26--J 


12J4 125 


5.1 


15 425 


4.3 


15 150 


3.8 


15 150, 3.0 


i5 200 


3.5 






15 150 
15 150 


5.1 
4.3 


15 200 
15 200 


5.2 
4.4 


15 200 4.2 
15 200! 3.5 


20 200 
20 200 


4.7 


28] 


15 125 


5 5 


3 9 






15 200 
15 150 


5.9 
3.7 


20 200 
15 200 


6.0 

3.8 


20 200! 4.8 
20 200 4.1 


20 272 
20 200 


5 3 


30 1 


15 150 


5.6 


3.4 






15 .200 


5.1 


20 200 


5.2 20 2T2 5.5 


20 272 


4 6 






1 


i 







TORSIONAL STRENGTH. 281 

FLOORING MATERIAL. 

For fire-proof flooring, the space between the floor-beams may be spanned 
with brick arches, or with hollow brick made especially for the purpose, the 
latter being much lighter than ordinary brick. 

Arches 4 inches deep of solid brick weigh about 70 lbs. per square foot, 
including the concrete levelling material, and substantial floors are thus 
made up to 6 feet span of arch, or much greater span if the skew backs at 
the springing of the arch are made deeper, the rise of the arch being prefer- 
ably not less than 1/10 of the span. Hollow brick for floors are usually in 
depth about % of the span, and. are used up to, and even exceeding, spans 
of 10 feet. The weight of the latter material will vary from 20 lbs. per 
square foot for 3-foot spans up to 60 lbs. per square foot for spans of 10 feet. 
Full particulars of this construction are given by the manufacturers. For 
supporting brick floors the beams should be securely tied with rods to resist 
the lateral pressure. 

In the following cases the loads, in addition to the weight of the floor 
itself, may be assumed as: 

For street bridges for general public traffic 80 lbs. per sq. ft. 

For floors of dwellings 40 lbs. " " 

For churches, theatres, and ball-rooms 80 lbs. " " 

For hay-lofts 80 lbs. " " 

For storage of grain 100 lbs. " " 

For warehouses and general merchandise 250 lbs. " " 

For factories . 200 to 400 lbs. " " 

For snow thirty inches deep . ... 16 lbs. " " 

For maximum pressure of wind 50 lbs. " " 

For brick walls 112 lbs. per cu. ft. 

For masonry walls 116-144 lbs. " " 

Roofs, allowing thirty pounds per square foot for wind and snow: 
For corrugated iron laid directly on the purlins. . . 37 lbs. per sq. ft. 

For corrugated iron laid on boards 40 lbs. " " 

For slate nailed to laths 43 lbs. " " 

For slate nailed on boards 46 lbs. " " 

If plastered below the rafters, the weight will be about ten pounds per 
square foot additional. 

TIE-RODS FOR REAMS SUPPORTING BRICK 
ARCHES. 

The horizontal thrust of brick arches is as follows: 

1 5TFS 2 

-^—^ — = pressure in pounds, per lineal foot of arch: 

W — load in pounds, per square foot; 
8 — span of arch in feet; 
B = rise in inches. 

Place the tie-rods as low through the webs of the beams as possible and 
spaced so that the pressure of arches as obtained above will not produce a 
greater stress than 15,000 lbs. per square inch of the least section of the bolt. 

TORSIONAL STRENGTH. 

Let a horizontal shaft of diameter — d be fixed at one end, and at the 
other or free end, at a distance = I from the fixed end, let there be fixed a 
horizontal lever arm with a weight — P acting at a distance = a from the 
axis of the shaft so as to twist it; then Pa = moment of the applied force. 

Resisting moment = twisting moment = — , in which S = unit shearing 

resistance, J = polar moment of inertia of the section with respect to the 
axis, and c = distance of the most remote fibre from the axis, in a cross- 
section. For a circle with diameter d, 



^--55-; c = %d; 



32 ' 



s; <* = i/- 



282 STRENGTH OF MATERIALS. 

For hollow shafts of external diameter d and internal diameter d lv 



d = 



3 / 5 -l-Pa 



For a square whose side = d, 

J=±; c = 4VU; f =Pa =1 ^!=0. 28M » S . 
For a rectangle whose sides are 6 and d, 

j=^ + ™. c = y 2 Vi* + d>; *U J b = < M ' + '»9g . 

12 12 c 6 |/62 + d 2 

The above formulae are based on the supposition that the shearing resist- 
ance at any point of the cross-section is proportional to its distance from the 
axis; but this is true only within the elastic limit. In materials capable of 
flow, while the particles near the axis are strained within the elastic limit 
those at some distance within the circumference may be strained nearly to 
the Ultimate resistance, so that the total resistance is something greater 
than that calculated by the formulae. (See Thurston. " Matls. of Eng.," Part 
II. p. 527.) Saint Venant finds for square shafts Pa = 0.281d 3 S (Rankine, 
"Mach. and Millwork," p. 504). For working strength, however, the for- 
mulae may be used, with S taken at the safe working unit resistance. 

The ultimate torsional shearing resistance <S is about the same as the di- 
rect shearing resistance, and may be taken at 20,000 to 25,000 lbs. per square 
inch for cast iron, 45,000 lbs. for wrought iron, and 50,000 to 150,000 lbs. for 
steel, according to its carbon and temper. Large factors -of safety should 
be taken, especially when the direction of stress is reversed, as in reversing 
engines, and when the torsional stress is combined with other .stresses, as is 
usual in shafting. (See "Shafting. 1 ') 

Elastic Resistance to Torsion.— Let I = length of bar being 
twisted, d — diameter, P = force applied at the extremity of a lever arm 
of length — a, Pa = twisting moment, G = torsional modulus of elasticity, 
= angle through which the free end of the shaft is twisted, measured in 
arc of radius = 1 . 

For a cylindrical shaft 

nOGd* ■ S2Pal „ 32Pal 32 

Pa = OI „ ; 6 = —3177 ; G = -5-31-; — = 10.186. 

If a = angle of torsion in degrees, 

an 1800 180 X S2Pal 583.6PaZ 



The value of G is given by different authorities as from \& to 2/5 of E, the 
modulus of elasticity for tension. 

COMBINED STRESSES. 
(From Merriman's "Strength of Materials.") 

Combined Tension and Flexure.— Let A = the area of a bar 
subjected to both tension and flexure, P = tensile stress applied at the ends, 
P -+- A = unit tensile stress, S = unit stress at the fibre on the tensile side most 
remote from the neutral axis, due to flexure alone, then maximum tensile 
unit stress = (P-i-A)-\- S. A beam to resist combined tension and flexure 
should be designed so that (P-^- A) + S shall not exceed the proper allow- 
able working unit stress. 

Combined Compression and Flexure.— If P-^-.4 = unit stress 
due to compression alone, and S = unit compressive stress at fibre most 
remote from neutral axis, due to flexure alone, then maximum compressive 
unit stress 

Combined Tension {or Compression) and Shear,— If ap- 



STRENGTH OF FLAT PLATES. 283 

plied tension (or compression) unit stress = p, applied shearing unit stress 
= v, then from the combined action of the two forces 

Max. S = ± Vv" 1 + J4P 2 » Maximum shearing unit stress; 

Max. t = y»p + Vv* + MP 2 > Maximum tensile (or compressive) unit stress. 

Combined Flexure and Torsion.— If S — greatest unit stress 
due to flexure alone, and <Ss = greatest torsional shearing unit stress due to 
torsion alone, then for the combined stresses 

Max. tension or compression unit stress t = %S + VSs* -f- 246^; 

Max. shear s= ± \ZSa* -f Y^S*. 

Formula for diameter of a round shaft subjected to transverse load while 
transmitting a given horse-power (see also Shafts of Engines): 

jl 16M , 16 /ilf 2 , 402,500,000i?2 

d3 = ^ + Tf^- j - * ' 

where M = maximum bending moment of the transverse forces in pound- 
inches, H = horse-power transmitted, n = No. of revs, per minute, and t = 
the safe allowable tensile or compressive working strength of the material. 
Combined Compression and Torsion.— For a vertical round 
shaft carrying a load and also transmitting a given horse-power, the result- 
ant maximum compressive unit stress 



4P /if 2 16P 2 



in which P is the load. From this the diameter d may be found when t and 
the other data are given. 

Stress due to Temperature.— Let I = length of a bar, A = its sec- 
tional area, c — coefficient of linear expansion for one degree, t = rise or 
fall in temperature in degrees, E = modulus of elasticity, A the change of 
length due to the rise or fall t; if the bar is free to expand or contract, A — 
ctl. 

If the bar is held so as to prevent its expansion or contraction the stress 
produced by the change of temperature = S = ActE. The following are 
average values of the coefficients of linear expansion for a change in temper- 
ature of one degree Fahrenheit: 

For brick and stone. . . .a = 0.0000050, 

For cast iron a = 0.0000062, 

For wrought iron a = 0.0000067, 

For steel a = 0.0000065. 

The stress due to temperature should be added to or subtracted from the 
stress caused by other external forces according as it acts to increase or to 
relieve the existing stress. 

What stress will be caused in a steel bar 1 inch square in area by a change 
of temperature of 100° F. ? S = ActE = I X .0000065 X 100 X 30,000,000 = 
19,500 lbs. Suppose the bar is under tension of 19,500 lbs. between rigid abut- 
ments before the change in temperature takes place, a cooling of 100° F. 
will double the tension, and a heating of 100° will reduce the tension to zero. 

STRENGTH OF FLAT PL.ATES. 

For a circular plate supported at the edge, uniformly loaded, according to 
Grashof, 

6ft* 
'' 5r** 



. 5r 2 , /5r 2 p 

For a circular plate fixed at the edge, uniformly loaded, 






in which /denotes the working stress; r, the radius in inches; t, the thick 
ness in inches; and p, the pressure in pounds per square inch. 



284 STRENGTH OF MATERIALS. 

For mathematical discussion, see Lanza, "Applied Mechanics," p. 900, etc. 
Lanza gives the following table, using a factor of safety of 8, with tensile 
strength of cast iron 20,000, of wrought iron 40,000, and of steel 80,000 : 

Supported. Fixed. 

Cast iron t = .0182570)- Vp t = .0163300r Vp 

Wrought iron t = .0117850r \/p t = .0105410r \/p 

Steel. t= .0091287r \fp t = .0081649r \/p 

For a circulate plate supported at the edge, and loaded with a concen- 
trated load P applied at a circumference the radius of which is r : 



I, r . \ P P 



for — = 10 20 30 40 50; 
c = 4.07 5.00 5 4 53 5.92 6.22; 

The above formulae are deduced from theoretical considerations, and give 
thicknesses much greater than are generally used in steam-engine cylinder- 
heads. (See empirical formulae under Dimensions of Parts of Engines.) The 
theoretical formulae seem to be based on incorrect or incomplete hypoth- 
eses, but they err in the direction of safety. 

The Strength of Unstayed Flat Surfaces.— Robert Wilson 
(Eng'g, Sept. 24, 1877) draws attention to the apparent discrepancy between 
the results of theoretical investigations and of actual experiments on the 
strength of unstayed flat surfaces of boiler-plate, such as the unstayed flat 
crowns of domes and of vertical boilers. 

Rankine's " Civil Engineering " gives the following rules for the strength 
of a circular plate supported all round the edge, prefaced by the remark 
that " the formula is founded on a theory which is only approximately true, 
but which nevertheless may be considered to involve no error of pi'actical 
importance:" 

,, Wb P6 3 : 

Here 

M = greatest bending moment ; 

P6 2 7T 

W — total load uniformly distributed == —r-\ 

b = diameter of plate in inches ; 
P = bursting pressure in pounds per square inch. 
Calling t the thickness in inches, for a plate supported round the edges, 

1 P6 2 

M = -i42,000M 2 ; .-. -rr = 7000< 2 . 

D 4Ht 

For a plate fixed round the edges, 

2 P&2 *2 x 6 3 000 
- — - = 7000< 2 : whence P = —? — > 

3 24 r 2 

where r = radius of the plate. 
Dr. Grashof gives a formula from which we have the following rule: 

< 2 X 72,000 

This formula of Grashof's has been adopted by Professor Unwin in his 
"Elements of Machine Design." These formulae by Rankine and Grashof 
may be regarded as being practically the same. 

On trying to make the rules given by these authorities agree with the 
results of his experience of the strength of unstayed 'flat ends of cylindrical 
boilers and domes that had given way after long use, Mr. Wilson was led to 
believe that the above rules give the breaking strength much lower than it 



STRENGTH OF FLAT PLATES. 285 

actually is. He describes a number of experiments made by Mr. Nichols of 
Kirkstall, which gave results varying widely from each other, as the method 
of supporting the edges of the plate was varied, and also varying widely 
from the calculated bursting pressures, the actual results being in all cases 
very much the higher. 
Some conclusions drawn from these results are: 

1. Although the bursting pressure has been found to be so high, boiler- 
makers must be warned against attaching any importance to this, since the 
plates deflected almost as soon as any pressure was put upon them and 
sprang back again on the pressure being taken off. This springing of the 
plate in the course of time inevitably results in grooving or channelling, 
which, especially when aided by the action of the corrosive acids in the 
water or steam, will in time reduce the thickness of the plate, and bring 
about the destruction of an unstayed surface at a very low pressure. 

2. Since flat plates commence to deflect at very low pressures, they should 
never be used without stays; but it is better to dish the plates when they are 
not stayed by flues, tubes, etc. 

3. Against the commonly accepted opinion that the limit of elasticity 
should never be reached in testing a boiler or other structure, these experi- 
ments show that an exception should be made in the case of an unstayed 
flat end-plate of a boiler, which will be safer when it has assumed a perma- 
nent set that will prevent its becoming grooved by the continual variation 
of pressure in working. The hydraulic pressure in this case simply does 
what should have been done before the plate was fixed, that is, dishes it. 

4. These experiments appear to show that the mode of attaching by flange 
or by an inside or outside angle-iron exerts an important influence on the 
manner in which the plate is strained by the pressure. 

When the plate is secured to an angle-iron, the stretching under pressure is, 
to a certain extent, concentrated at the line of rivet-holes, and the plate par- 
takes rather of a beam supported than fixed round the edge. Instead of the 
strength increasing as the square of the thickness, when the plate is attached 
by an an^le-iron, it is probable that the strength does not increase even 
directly as the thickness, since the plate gives way simply by stretching at 
the rivet-holes, and the thicker the plate, the less uniformly is the strain 
borne by the different layers of which the plate may be considered to be 
made up. When the plate is flanged, the flange becomes compressed by the 
pressure against the body of the plate, and near the rim, as shown by the 
contrary flexure, the inside of the plate is stretched more than the outside, 
and it may be by a kind of shearing action that the plate gives way along 
the line where the crushing and stretching meet. 

5. These tests appear to show that the rules deduced from the theoretical 
investigations of Lame, Rankine, and Grashof are not confirmed by experi- 
ment, and are therefore not trustworthy. 

Unbraced Wrought-iron Heads of Boilers, etc. {The Loco- 
motive, Feb. 1890). — Few experiments have been made on the strength of 
flat heads, and our knowledge of them comes largely from theory. Experi- 
ments have been made on small plates 1-16 of an inch thick, yet the data so 
obtained cannot be considered satisfactory when we consider the far thicker 
heads that are used in practice, although the results agreed well with Ran- 
kine's formula. Mr. Nichols has made experiments on larger heads, and 
from them he has deduced the following rule: " To find the proper thick- 
ness for a flat unstayed head, multiply the area of the head by the pressure 
per square inch that it is to bear safely, and multiply this by the desired 
factor of safety (say 8); then divide the product by ten times the tensile 
strength of the material used for the head." His rule for finding the burst- 
ing pressure when the dimensions of the head are given is: " Multiply the 
thickness of the end-plate in inches by ten times the tensile strength of the 
material used, and divide the product by the area of the head in inches." 

In Mr. Nichols's experiments the average tensile strength of the iron used 
for the heads was 44,800 pounds. The results he obtained are given below, 
with the calculated pressure, by his rule, for comparison. 

1. An unstayed flat boiler-head is 34^ inches in diameter and 9-16 inch 
thick. What is its bursting pressure? The area of a circle 34J^ inches in 
diameter is 935 square inches; then 9-1(5 X 44,800 x 10 = 252,000, and 252,000 -*- 
935 = 270 pounds, the calculated bursting pressure. The head actually burst 
at 280 pounds. 

2. Head 34^ inches in diameter and % inch thick. The area = 935 
square inches; then, % x 44,800 x 10 = 168.000, and 168,000 -h 935 = 180 pounds, 
calculated bursting pressure. This head actually burst at 200 pounds. 



286 BTEEKGTH 0£ MATERIALS. 

3. Head 26J4 inches in diameter, and % inch thick. The area 541 square 
inches. Then, % x 41,800 x 10 = 168,000, and 168,000 -h 541 = 311 pounds. 
This head burst at 370 pounds. 

4. Head 28J^ inches in diameter and % inch thick. The area = 638 
square inches; then, % x 44,800 X 10 = 168,000, and 168,000 -f- 638 = 263 
pounds. The actual bursting pressure was 300 pounds. 

In the third experiment, the amount the plate bulged under different 
pressures was as follows : 

At pounds per sq. in.... 10 20 40 80 120 140 170 200 
Plate bulged 1/32 1/16 Y 8 U % Yn % % 

The pressure was now reduced to zero, " and the end sprang back 3-16 
inch, leaving it with a permanent set of 9-16 inch. The pressure of 200 lbs. 
was again applied on 36 separate occasions during an interval of five days, 
the bulging and permanent sec being noted on each occasion, but without 
any appreciable difference from that noted above. 

The experiments described were confined to plates not widely different in 
their dimensions, so that Mr. Nichols's rule cannot be relied upon for heads 
that depart much from the proportions given in the examples. 

Thickness of Flat Cast-iron Plates to resist Bursting 
Pressures. — In Church's Life of Ericsson is found the following letter: 

'* My dear Sir: The proper thickness of a square cast-iron plate will be ob- 
tained by following: Multiply the side in feet (or decimals of a foot) by J4 
of the pressure in pounds and divide by 850 times the side in inches; the 
quotient is the square of the thickness in inches. 

" Example.— A plate 5 feet or 60 inches square, with a pressure of 30 lbs. 
per square inch. 

" Thickness - 5 X %* jf?* 3 ° - 2.64. a/2M = 1.62 inches. 
85(1 X w ' 

" For a circular plate, multiply 11-14 of the diameter in feet by J4 of the 
pressure on the plate in pounds. Divide by 850 times 11-14 of the diameter 
in inches. [Extract the square root.] 

" Example.— Plate 5 feet diameter, pressure 30 lbs. per square inch. 

" Area 2827 X — = 8 -^^ = 21,202; diam. 60 x — = 47.1; 5 x — = 3.92. 
4 4 14 14 

3.92 X 21,202 = 83,811 
8.50 X 4.71 = 41,035 = 
" A great mathematician would cover half a dozen sheets with figures to 
solve this problem." 

Strength of Stayed Surfaces.— A flat plate of thickness t is sup- 
ported uniformly by stays whose distance from centre to centre is a, uniform 
load p lbs. per square inch. Each stay supports pa* lbs. The greatest 
stress on the plate is 

. 2a2 /TT . 
f= 9 JiP- (Unwin). 

SPHERICAL SHELLS AND DOMED BOILER-HEADS. 

To find the Thickness of a Spherical Shell to resist a 
given Pressure.— Let d = diameter in inches, and p the internal press- 
ure per square inch. The total pressure which tends to produce rupture 
around the great circle will be \fcrd' l p. Let S = safe tensile stress per 
square inch, and t the thickness of metal in inches; then the resistance to the 
pressure will be irdtS. Since the resistance must be equal to the pressure. 

Y^d^p = ndtS. Whence t = ||-. 

The same rule is used for finding the thickness of a hemispherical head 
to a cylinder, as of a cylindrical boiler. 

Thickness of a Domed Head of a boiler.— If S = safe tensile 
stress per square inch, d = diameter of the shell in inches, and t = thickness 
of the shell, t = pd -s- 2S ; but the thickness of a hemispherical head of the 
same diameter is t = pd-r-4S. Hence if we make the radius of curvature 
of a domed head equal to the diameter of the boiler, we shall have t = 

— = — , or the thickness of such a domed head will be equal to the thick- 

4S 2S 

ness of the shell. 



THICK CYLINDERS UNDER TENSION. 



287 



Stresses in Steel Plating due to Water-pressure, as in 

plating of vessels and bulkheads (Engineering, May 22, 1891, page 629). 

Mr. J. A. Yates has made calculations of the stresses to which steel plates 
are subjected by external water-pressure, and arrives at the following con- 
clusions : 

Assume 2a inches to be the distance between the frames or other rigid 
supports, and let d represent the depth in feet, below the surface of the 
water, of the plate under consideration, t = thickness of plate in inches, 
D the deflection from a straight line under pressure in inches, and P — stress 
per square inch of section. 



For outer bottom and ballast- tank plating, a = 4 



, D should not be 



greater than .05 — , and — not greater than 2 to 3 tons ; while for bulkheads, 

etc., a = 2352-, D should not be greater than .1—, and — not greater than 

7 tons. To illustrate the application of these formulae the following cases 
have been taken : 



For Outer Bottom, etc. 


For Bulkheads, etc. 


Thick- 


Depth 


Spacing of 


Thick- 


Depth of 
Water. 


Maximum Spac- 


ness of 


below 


Frames should 


ness of 


ing of Rigid 


Plating. 


Water. 


not exceed 


Plating 


Stiffeners. 




ft. 


in. 


in. 


ft. 


ft. in. 


H 


20 


About 21 


\ 


20 


9 10 




10 


" 42 


20 


7 4 


§1 


18 


" 18 


10 


14 8 


% 


9 


" 36 


y* 


20 


4 10 


8 


10 


" 20 


H 


10 


9 8 


5 


" 40 


% 


10 


4 10 



It would appear that the course which should be followed in stiffening 
bulkheads is to fit substantially rigid stiffening frames at comparatively 
wide intervals, and only work such light angles between as are necessary 
for making a fair job of the bulkhead, 

THICK HOLLOW CYLINDERS UNDER TENSION. 

Burr, " Elasticity and Resistance of Materials," p. 36, gives 
. t — thickness; r = interior radius ; 

( A + A 5 1 I ft = maximum allowable hoop tension at the 
t = r < \T~-z). — * \ ' interior of the cylinder; 

{ p p = intensity of interior pressure. 

Merriman gives 

s = unit stress at inner edge of the annulus; 
r = interior radius ; t = thickness ; 
I = length. 



(1) 






The total interior pressure which tends to rupture the cylinder is 2rl — p. 
If p be the unit pressure, then p = ^rqTf f rom wmcn one of tn e quantities 
s, p, r, or t can be found when the other three are given. 
P)t . t _ rp 
P' 



t = - 



288 STRENGTH OF MATERIALS. 

In eq. (1), if t be neglected in comparison with r, it reduces to 2slt, which 
is the same as the formula for thin cylinders. If t = r, it becomes sit, or 
only half the resistance of the thin cylinder. 

The formulas given by Burr and by Merriman are quite different, as will 
be seen by the following example : Let maximum unit stress at the inner 
edge of the annulus = 8000 lbs. per square inch, radius of cylinder = 4 inches, 
interior pressure = 4000 lbs. per square inch. Required the thickness. 

BrBurr, t^K^^)* - . \ = ^ - .,- I.H. inches. 

~ ™ . 4 X 4000 , . . 

By Mernman, t = mQ _ mQ = 4 inches. 

Limit to Useful Thickness of Hollow Cylinders (Eng'g, 
Jan. 4, 1884).— Professor Barlow lays down the law of the resisting powers 
of thick cylinders as follows : 

" In a homogeneous cylinder, if the metal is incompressible, the tension 
on every concentric layer, caused by an internal pressure, varies inversely 
as the square of its distance from the centre. 1 ' 

Suppose a twelve-inch gun to have walls 15 inches thick. 

Pressure on exterior _ 6*_ _ _ 
Pressure on interior ~ 21 2 ~~ 

So that if the stress on the interior is 12J4 tons per square inch, the stress 
on the exterior is only 1 ton. 

Let s = the stress on the inner layer, and s, that at a distance x from the 
axis ; r = internal radius, R = external radius. 

r 2 

5, : s : : r 2 : a; 2 , or S* = s — . 
1 x* 

The whole stress on a section 1 inch long, extending from the interior to 
R — r 
the exterior surface, is <S= sr X — „ — . 

In a 12-inch gun, let s = 40 tons, r = 6 in., R = 21 in. 

s = 40 X 6 X ^- 6 = 172 tons. 

Suppose now we go on adding metal to the gun outside: then R will be- 
come so large compared with r, that R — r will approach the value R, so 

R — r 
that the fraction — — becomes nearly unity. 

Hence for an infinitely thick cylinder the useful strength could never 
exceed Sr (in this case 240 tons). 
Barlow's formula agrees with the one given by Merriman. 
Another statement of the gun problem is as follows : Using the formula 

st 

r -j- t 

s = 40 tons, t = 15 in., r = 6 in., p = — ^ — = 28f tons per sq. in., 28f X 

radius = 172 tons, the pressure to be resisted by a section 1 inch long of the 
thickness of the gun on one side. Suppose thickness were doubled, making 

t = 30 in.: p — — ^ — = 33^j tons, or an increase of only 16 per cent. 

For short cast-iron cylinders, such as are used in hydraulic presses, it is 
doubtful if the above formulas hold true, since the strength of the cylindri- 
cal portion is reinforced by the end. In that case the bursting strength 
would be higher than that calculated by the formula. A rule used in 
practice for such presses is to make the thickness = 1/10 of the inner cir- 
cumference, for pressures of 3000 to 4000 lbs. per square inch. The latter 
pressure would bring a stress upon the inner layer of 10,350 lbs. per square 
inch, as calculated by the formula; which would necessitate the use of the 
best charcoal-iron to make the press reasonably safe. 



HOLDING-POWER OF NAILS, SPIKES, AND SCREWS. 289 

THIN CYL1NBERS IJNBER TENSION. 

Let p = safe working pressure in lbs. per sq. in. ; 
d = diameter in inches; 

T = tensile strength of the material, lbs. per sq. in.; 
t = thickness in inches; 
/ = factor of safety ; 
c = ratio of strength of riveted joint to strength of solid plate. 

If T = 50000, / = 5, and c = 0.7; then 

- 14000t . _ d P 

P ~ d ' ~ 14000" 

The above represents the strength resisting rupture along a longitudinal 

seam. For resistance to rupture in a circumferential seam, due to pressure 

, * ... , »7rd 2 Ttndc 

on the ends of the cylinder, we have = — - — ; 

4Ttc 
whence p = —r—. 
"J 
Or the strength to resist rupture around a circumference is twice as great 
as that to resist rupture longitudinally ; hence boilers are commonly single- 
riveted in the circumferential seams and double-riveted in the longitudinal 
seams. 

HOLLOW COPPER BALLS. 
Hollow copper balls are used as floats in boilers or tanks, to control feed 
and discharge valves, and regulate the water-level. 

They are spun up in halves from sheet copper, and a rib is formed on one 
half. Into this rib the other half fits, and the two are then soldered or 
brazed together. In order to facilitate the brazing, a hole is left on one side 
of the ball, to allow air to pass freely in or out; and this hole is made use of 
afterwards to secure the float to its stem. The original thickness of the 
metal may be anything up to about 1-16 of an inch, if the spinning is done 
on a hand lathe, though thicker metal may be used when special machinery 
is provided for forming it. In the process of spinning, the metal is thinned 
down in places by stretching; but the thinnest place is neither at the equator 
of the ball (i.e., along the rib) nor at the poles. The thinnest points lie along 
two circles, passing around the ball parallel to the rib, one on eacU side of it, 
from a third to a half of the way to the poles. Along these lines the thick- 
ness may be 10, 15, or 20 per cent less than elsewhere, the reduction depend 
ing somewhat on the skill of the workman. 

The Locomotive for October, 1891, gives two empirical rules for determin- 
ing the thickness of a copper ball which is to work under an external 
pressure, as follows: 

_ diameter in inches X pressure in pounds per sq in. 
1. Thickness _ — ^i^m " 

2 Thickness = diameter x ^/pressure _ 
1240 
These rules give the same result for a pressure of 166 lbs. only. Example: 
Required the thickness of a 5-inch copper ball to sustain 

Pressures of 50 100 150 166 200 250 lbs. per sq. in. 

Answer by first rule. .. .0156 .0312 .0469 .0519 .0625 .0781 inch. 
Answer by second rule .0285 .0403 .0494 .0518 .0570 .0637 " 

HOLBING-POWER OF NAILS, SPIKES, ANB 
SCREWS. 
(A. W. Wright, Western Society of Engineers, 1881.) 
Spikes.— Spikes driven into dry cedar (cut 18 months): 

Size of spikes 5 X H in. sq. 6 X M 6 X \i 5 X% 

Length driven in 4J4 in. 5 in. 5 in. 4J4 in. 

Pounds resistance to drawing. Av'ge, lbs. 857 821 1691 1202 

„ ■ .. n ^ . . jMax. " 1159 923 2129 1556 

From 6 to 9 tests each -j Mhl _ « Tg § 76 6 1120 687 



290 STRENGTH OF MATERIALS. 

A. M. Wellington found the force required to draw spikes 9/16 X 9/16 in., 
driven 4J4 inches into seasoned oak, to be 4281 lbs.; same spikes, etc., in un- 
seasoned oak, 6523 lbs. 

" Professor W. R. Johnson found that a plain spike % inch square 
driven 3% inches into seasoned Jersey yellow pine or unseasoned chestnut 
required about 2000 lbs. force to extract it; from seasoned white oak about 
4000 and from well-seasoned locust 6000 lbs." 

Experiments in Germany, by Funk, give from 2465 to 3940 lbs. (mean of 
many experiments about 3000 lbs.) as the force necessary to extract a plain 
5^-inch square iron spike 6 inches long, wedge-pointed for one inch and 
driven 4^ inches into white or yellow pine. When driven 5 inches the force 
required was about 1/10 part greater. Similar spikes 9/16 inches square, 7 
inches long, driven 6 inches deep, required from 3700 to 6745 lbs. to extract 
them from pine; the mean of the results being 4873 lbs. In all cases about 
twice as much force was required to extract them from oak. The spikes 
were all driven across the grain of the wood. When driven with the grain, 
spikes or nails do not hold with more than half as much force. 

Boards of oak or pine nailed together by from 4 to 16 tenpenny common cut 
nails and then pulled apart in a direction lengthwise of the boards, and 
across the nails, tending to break the latter in. two by a shearing action, 
averaged about 300 to 400 lbs. per nail to separate them, as the result of 
many trials. 

Resistance of Drift-bolts in Timber.— Tests made by Rust and 
Coolidge, in 1878. 

Pounds. 

hole 26,400 

" 16,800 

" 14,600 

" 13,200 

" 18,720 

" 19,200 

" 15,600 

" 14,400 



1st Test. 


1 in. square iron drove 


2d " 


1 in. round " " 


3d " 


1 in. square " " 


4th " 


1 in. round " " 


5th " 


1 in. round " " 


6th " 


1 in. square " " 


7th " 


1 in. square " " 


8th " 


1 in. round " " 



30 in. in white pine, 15/16-in 

34 " " " " 13/16-in. 

18 " " " " 15/16-in. 

22 " " " " 13/16-in. 

34 " "Norw'y pine, 13/16-in. 

30 " " " " 15/16-in. 

18 " " " " 15/16-in 

22 " " " " 13/16-in 



Note.— In test No. 6 drift-bolts were not driven properly, holes not being 
in line, and a piece of timber split out in driving. 
Force required to draw Screws out of Norway Pine. 

}4" diam. drive screw 4 in. in wood. Power required, average 2424 lbs. 

" " 4 threads per in. 5 in. in wood. " " " 2743 " 

" " D'blethr'd,3perin.,4 in. in " " " " 2730 " 

" " Lag-screw, 7 per in., 1% " " " " " 1465 " 

6 " " 2' 7 i " " "' " " 2026 " 

Y% inch R.R. spike 5 " " " " " 2191 " 

Force required to draw Wood Screws out of Dry Wood. 

—Tests made by Mr. Bevan. The screws were about two inches in length, 
.22 diameter at the exterior of the threads, .15 diameter at the bottom, the 
depth of the worm or thread being .035 and the number of threads in one 
inch equal 12. They were passed through pieces of wood half an inch in 
thickness and drawn out by the weights stated: Beech, 460 lbs.: ash, 790 
lbs.: oak. 760 lbs.: mahogany, 770 lbs.; elm, 665 lbs.; sycamore, 830 lbs. 

Tests of Ijag-screws in Various Woods were made by A. J. 
Cox, University of Iowa, 1891: 

g gr w . 1st #& £& t no- 

Seasoned white oak. % in. ^in. 4}4 in. 8037 3 

" 9/16" 7/16" 3 " 6480 1 

" M " %" Q& " 8780 2 

Yellow-pine stick % u y% " 4 " 3800 2 

White cedar, unseasoned % " J^" 4 " 3405 2 

In figuring area for lag-screws, the surface of a cylinder whose diameter is 
equal to that of the screw was taken. The length of the screw part in each 
case was 4 inches.— Engineering Netcs, 1891. 

Cut versus Wire Nails.— Experiments were made at the Watertown 
Arsenal in 1893 on the comparative direct tensile adhesion, in pine and 
spruce, of cut and wire nails. The results are stated by Prof, W, H. Burr 
as follows: 



HOLDING-POWER OF NAILS, SPIKES, AND SCREWS. 291 



There were 58 series of tests, ten pairs of nails (a cut and a wire nail in each) 
being: used, making a total of 1160 nails drawn. The tests were made in 
spruce wood in most instances, but some extra ones were made in white 
pine, with "box nails. 1 ' The nails were of all sizes, varying from 1% inches to 
6 inches in length. In every case the cut nails showed the superior holding 
strength by a large percentage. In spruce, in nine different sizes of nails, 
both standard and* light weight, the ratio of tenacity of cut to wire nail 
was about 3 to 2, or, as he terms it, "a superiority of 47.45$ of the former." 
With the " finishing 1 ' nails the ratio was roughly 3.5 to 2; superiority 72$. 
"With box nails (134 to 4 inches long) the ratio was roughly 3 to 2; superiority 
51$. The mean superiority in spruce wood was 61$. In white pine, cut nails, 
driven with taper along the grain, showed a superiority of 100$, and with 
taper across the grain of 135$. Also when the nails were driven in the end 
of the stick, i.e., along the grain, the superiority of cut nails was 100$, or the 
ratio of cut to wire was 2 to 1. The total of the results showed the ratio of 
tenacity to be about 3.2 to 2 for the harder wood, and about 2 to 1 for the 
softer, and for the whole taken together the ratio was 3.5 to 2. We are 
led to conclude that under these circumstances the cut nail is superior to 
the wire nail in direct tensile holding-power by 72.74$. 

Nail-holding Power of Various Woods. 
(Watertown Experiments.) 

Holding-power per square inch of 



Kind of Wood. 



Surface in Wood, lbs. 



Wire Nail. Cut Nail. Mean. 
I f 450 I 

455 



White pine . 



Yellow pine. 



10" 
50" 
60" 



20' 



651 



340 
695 
755 
596 
604 
1340 
1292 
1018 
664 
702 
1179 
1221 



Nail-holding Power of Various Woods. 

(F. W. Clay's Experiments. Eng'g News, Jan. 11,1 
Wood. 



-Tenacity of 6d nails- 



White pine'.. 
Yellow pine . 

Basswood 

White oak... 
Hemlock 



D lain. Barbed. 


Blued. 


Mean. 


106 94 


135 


Ill 


190 130 


270 


196 


78 132 


219 


143 


226 300 


555 


360 


141 201 • 


319 


220 


e the resistance of a 1- 


n. round 



rod in a 15/16-inch hole perpendicular to the grain, as 6000 lbs. per lin. ft. in 
pine and 15,600 lbs. in oak. Experiments made at the East River Bridge 
gave resistances of 12,000 and 15,000 lbs. per lin. ft. for a 1-in. round rod in 
holes 15/16-in. and 14/16-in. diameter, respectively, in Georgia pine. 
Holding-power of Bolts in White Pine. 
(Eng'g News, September 26, 1891.) 

Round. Square. 

Lbs. Lbs. 

Average of all plain 1-in. bolts 8224 8200 

Average of all plain bolts, % to 1% in 7805 8110 

Average of all bolts 8383 8598 

Round drift-bolts should be driven in holes 13/16 of their diameter, and 
square drift-bolts in holes whose diameter is 14/16 of the side of the square. 



292 



STRENGTH OF MATERIALS. 



STRENGTH OF WROUGHT IRON BOLTS. 

(Computed by A. F. Nagle.) 





,0 0) 


CO 


Q _© 




Stress upon Bolt upon Basis of 




o a> 
£ a 


. o 


w C 


w a 


5* . 

ax 


CD 

£ a 


© be 






£-2^3 




fi"" 1 


&-~ 


_Q.« 


jQ" H 




XoSO 


43 "3 
AM 


£J3 






© CO 


8 ST 


o o 1 

1" 


o c? 
o m 


© & 
© 








lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


y* 


13 


.38 


.12 


350 


460 


580 


810 


1160 


5800 


9-16 


12 


.44 


.15 


450 


600 


750 


1050 


1500 


7500 


% 


11 


.49 


.19 


560 


750 


930 


1310 


1870 


9000 


u 


10 


.60 


.28 


750 


1130 


1410 


1980 


2830 


14000 


Vs 


9 


.71 


.39 


1180 


1570 


1970 


2760 


3940 


19000 


l 


8 


.81 


.52 


1550 


2070 


2600 


3630 


5180 


25000 


m 


7 


.91 


.65 


1950 


2600 


3250 


4560 


6510 


30000 


m 


7 


1.04 


.84 


2520 


3360 


4200 


5900 


8410 


39000 


Ws 


6 


1.12 


1.00 


3000 


4000 


5000 


7000 


10000 


46000 


w% 


6 


1.25 


1.23 


3680 


4910 


6140 


8600 


12280 


56000 


w% 


5^ 


1.35 


1.44 


4300 


5740 


7180 


10000 


14360 


65000 


m 

Ws 


5 


1.45 


1.65 


4950 


6600 


8250 


11560 


16510 


74000 


5 


1.57 


1.95 


5840 


7800 


9800 


13640 


19500 


85000 


2 


4^ 


1.66 


2.18 


6540 


8720 


10900 


15260 


21800 


95000 


m 


4^ 


1.92 


2.88 


8650 


11530 


14400 


20180 


28800 


125000 


n 


4 


2.12 


3.55 


10640 


14200 


17730 


24830 


35500 


150000 


4 


2.37 


4.43 


13290 


17720 


22150 


31000 


44300 


186000 


3 


&A 


2.57 


5.20 


15580 


20770 


26000 


36360 


52000 


213000 


&A 


3^ 


3.04 


7.25 


21760 


29000 


36260 


50760 


72500 


290000 


4 


3 


3.50 


9.62 


28860 


38500 


48100 


67350 


96200 


385000 



When it is known what load is to be put upon a bolt, and the judgment of 
the engineer has determined what stress is safe to put upon the iron, look 
down in the proper column of said stress until the required load is found. 
The area at the bottom of the thread will give the equivalent area of a flat 
bar to that of the bolt. 

Effect of Initial Strain in Bolts.— Suppose that bolts are used 
to connect two parts of a machine and that they are screwed up tightly be- 
fore the effective load comes on the connected parts. Let P 3 = the initial 
tension on a bolt due to screwing up, and P 2 = the load afterwards added. 
The greatest load may vary but little from Pj or P 2 , according as the 
former or the latter is greater, or it may approach the value P x -f- P^ de- 
pending upon the relative rigidity of the bolts and of the parts connected. 
Where rigid flanges are bolted together, metal to metal, it is probable that 
the extension of the bolts with any additional tension relieves the initial 
tension, and that the total tension is Pj or P 2 , but in cases where elastic 
packing, as india rubber, is interposed, the extension of the bolts may very 
little affect the initial tension, and the total strain may be nearly P x + P 2 . 
Since the latter assumption is more unfavorable to the resistance of the 
bolt, this, contingency should usually be provided for. (See Unwin, "Ele- 
ments of Machine Design " for demonstration.) 

STAND-PIPES AND THEIR DESIGN. 

(Freeman C. Coffin, New England Water Works Assoc, Eng. News, March 
16, 1893.) See also papers by A. H. Howland, Eng. Club of Phil. 1887; B. F. 
Stephens, Amer. Water Works Assoc, Eng. News, Oct. 6 and 13, 1888; W. 
Kiersted, Rensselaer Soc of Civil Eng., Eng'g Record, April 25 and May 2, 
1891, and W. D. Pence, Eng. Neivs, April and May, 1894. 

The question of diameter is almost entirely independent of that of height. 
The efficient capacity must be measured by the length from the high-water 
line to a point below which it is undesirable to draw the water on account of 
loss of pressure for fire-supply, whether that point is the actual bottom of 
the stand-pipe or above it. This allowable fluctuation ought not to exceed 
50 ft., in most cases. This makes the diameter dependent upon two condi- 



STAND-PIPES AND THEIR DESIGN. 293 

tions, the first of which is the amount of the consumption during the ordi- 
nary interval between the stopping and starling of the pumps. This should 
never draw the water below a point that will give a good fire stream and 
leave a margin for still further draught for fires. The second condition is 
the maximum number of fire streams and their size which it is considered 
necessary to provide for, and the maximum length of time which they are 
liable to have to run before the pumps can be relied upon to reinforce 
them. 

Another reason for making the diameter large is to provide for stability 
against wind-pressure when empty. 

The following table gives the height of stand-pipes beyond which they are 
not safe against wind pressures of 40 and 50 lbs. per square foot. The area 
of surface taken is the height multiplied by one half the diameter. 

Heights of Stand-pipe that will Resist Wind-pressure 
l>y its Weight alone, when Empty. 

Diameter, Wind, 40 lbs. Wind, 50 lbs. 

feet. per sq. ft. per sq. ft. 

20 45 35 

25 ... 70 55 

30 150 80 

35 160 

To have the above degree of stability the stand-pipes must be designed 
with the outside angle-iron at the bottom connection. 

Any form of anchorage that depends upon connections with the sid3 
plates near the bottom is unsafe. By suitable guys the wind-pressure is re- 
sisted by tension in the guys, and the stand-pipe is relieved from wind 
strains that tend to overthrow it. The guys should be attached to a band 
of angle or other shaped iron that completely encircles the tank, and rests 
upon some sort of bracket or projection, and not be riveted to the tank. 
They should be anchored at a distance from the base equal to the height of 
the point at which they are attached, if possible. 

The best plan is to build the stand-pipe of such diameter that it will resist 
the wind by its own stability. 

Thickness of the Side Plates. 

The pressure on the sides is outward, and due alone to the weight of the 
water, or pressure per square inch, and increases in direct ratio to the 
height, and also to the diameter. The strain upon a section 1 inch in height 
at any point is the total strain at that point divided bj r two — for each side is 
supposed to bear the strain equally. The total pressure at any point is 
equal to the diameter in inches, multiplied by the pressure per square inch, 
due to the height at that point. It may be expressed as follows: 

H = height in feet, and/ = factor of safety; 

d — diameter in inches; 

p — pressure in lbs. per square inch; 
.434 = p for 1 ft. in height; 

s = tensile strength of material per square inch; 

T — thickness of plate. 

Then the total strain on each side per vertical inch 

_ AMHd _pd_ .AMHdf _ pdf_ 

~ 2 ' ~ 2 ' ~ 2s ~ 2s ' 

Mr. Coffin takes/ = 5. not counting reduction of strength of joint, equiv- 
alent to an actual factor of safety of 3 if the strength of the riveted joint is 
taken as 60 per cent of that of the plate. 

The amount of the wind strain per square inch of metal at any joint can 
be found by the following formula, in which 

H = height of stand-pipe in feet above joint; 

T = thickness of plate in inches; 

p — wind-pressure per square foot: 
W — wind-pressure per foot in height above joint; 
W = Dp where D is the diameter in feet; 
m — average leverage or movement about neutral axis 

or central points in the circumference; or, 
m = sine of 45°, or .707 times the radius in feet. 



294 



STRENGTH OF MATERIALS. 



Then the strain per square inch of plate 



(Hrv)~ 



circ. in ft. X mT 



Mr. Coffin gives a number of diagrams useful in the design of stand-pipes, 
together with a number of instances of failures, with discussion of their 
probable causes. 

Mr. Kiersted's paper contains the following : Among the most prominent 
strains a stand-pipe has to bear are: that due to the static pressure of the 
water, that due to the overturning effect of the wind on an empty stand- 
pipe, and that due to the collapsing effect, on the upper rings, of violent 
wind storms. 

For the thickness of metal to withstand safely the static pressure of 
water, let 

t — thickness of the plate iron in inches; 
H = height of stand-pipe in feet; 
D = diameter of stand-pipe in feet. 

Then, assuming: a tensile strength of 48.000 lbs. per square inch, a factor 
of safety of 4, and efficiency of double-riveted lap-joint equalling 0.6 of the 
strength of the solid plate, 

(-.MJoaurxi* *=4S«; 

d.bD 
which will give safe heights for thicknesses up to % to % of an inch. The 
same formula may also apply for greater heights and thicknesses within 
practical limits, if the joint efficiency be increased by triple riveting. 

The conditions for the severest overturning wind strains exist when the 
stand-pipe is empty. 

Formula for Rand-pressure of 50 pounds per square foot, when 

d = diameter of stand-pipe in inches; 
x = any unknown' height of stand-pipe; 
x = VWndt = 15.85 VdT. 

The following table is calculated by these formulae. The stand-pipe is 
intended to be self-sustaining; that is, without guys or stiffeners. 



Heights of Stand-pipes for Various Diameters and 
Thicknesses of Plates. 


Thickness of 


Diameters in Feet. 


Plate in Frac- 
tions of an Inch. 


5 


6 


7 


8 


9 


10 


12 


14 


15 


16 


18 


20 


25 


3-16 

7 32 


50 
55 
CO 
70 
75 
SO 
85 


55 


60 


65 


55 

05 
75 
90 
100 
110 
115 
125 
130 


50 
00 
70 
85 
100 
115 
120 
130 
135 
145 
150 


35 

50 
55 
70 
85 
100 
115 
130 
145 
155 
1G5 


40 
50 

60 
75 
85 
100 
110 
120 
135 
145 
100 


' 40 
45 
55 
70 

80 
90 
100 
115 
125 
135 
150 
160 


"40 

50 
65 
75 
85 
95 
105 
120 
130 
140 
150 
160 


"35 
45 
55 
05 
75 
85 
95 
105 
115 
125 
135 
145 
155 


35 

40 
50 
00 
70 
80 
85 
95 
105 
110 
120 
130 
140 




4-16 


65 

75 

SO 
90 
95 


70 

SO 
90 
95 

100 


75 
85 
95 
100 
110 
115 


25 


5 16 


35 


6 16 


40 


7-16 

S--16 


45 
55 


9 16 


60 


10-16 








65 


11-16 










75 


12-16 

13 16 












80 

qo 


14 16 














95 


15 16 


















105 


16-16 




















no 



Heights to nearest 5 feet. Rings are to build 5 feet vertically. 

Failures of Stand-pipes have been numerous in recent years. A 
list showing 23 important failures inside of nine years is given in a paper by 
Prof. W. D. Pence, Eng'g News, April 5, 12, 19 and 26, May 3, 10 and 24, and 
June 7. 1894. His discussion of the probable causes of the failures is most 
valuable. 



WROUGHT-IRON AND STEEL WATER-PIPES. 



295 



Kenneth Allen, Engineers Club of Philadelphia, 1886, gives the following 
rules for thickness of plates for stand pipes. 

Assume: Wrought iron plate T. S. 48,000 pounds in direction of fibre, and 
T. S. 4 r »,000 pounds across the fibre. Strength of single riveted joint .4 that 
of the plate, and of double riveted joint, .7 that of the plate ; wind pressure 
= 50 pounds per square foot ; safety factor = 3. 

Let /(. = total height in feet ; r — outer radius in feet ; r' = inner radius 
in feet ; p — pressure per square inch ; t = thickness in inches ; d — outer 
diameter in feet. 

Then for pipe filled and longitudinal seams double riveted 



t -. 



pr x 12 
48,000 X.7XH 



hcl . 
: 4301' 



and for pipe empty and lateral seams, single riveted, we have by equating 
I moments : 



3 X 2f (|)» = 144 



X 6000 (r 4 - ?•"») 



, whence r 4 — r' 4 = 



27144" 



Table showing required Thickness of Bottom Plate. 









Diameter. 






Height in 














Feet. 
















5 feet, 


10 feet. 


15 feet. 


20 feet. 


25 feet. 


30 feet. 


50 


t 7-64* 


H * 


11-64* 


15-64 


19-64 


23-64 


60 


+11-64* 


9-64* 


7-32 


9-32 


23-64 


27-64 


70 


+ 7-32 


11-64* 


9-32 


21-64 


13-32 


31-64 


80 


+19-64 


3-16 


% 


15-32 


9-16 


90 


t % 


7-32 


5-16 


27-64 


17-32 


% 


100 


+29-64 


+15-64 


23-64 


15-32 


37-64 


45-64 


•125 




+23-64 


7-16 


37-64 


47-64 


Vs 


150 




+33-64 


17-32 


45-64 


% 


1 3-64 


175 




+11-16 


39-64 


13-16 


1 1-32 


1 7-32 


200 




+29-32 


45-64 


15-16 


1 11-64 


1 25-64 



* The minimum thickness should = 3-16". 

N.B.— Dimensions marked + determined by wind-pressure. 

Water Tower at Xonkers, N. Y.— This tower, with a pipe 122 feet 
high and 20 feet diameter, is described in Engineering Neivs, May 18, 1892. 

The thickness of the lower rings is 11-16 of an inch, based on a tensile 
strength of 60,000 lbs. per square inch of metal, allowing 65$ for the strength 
of riveted joints, using a factor of safety of 3}^ and adding a constant of 
% inch. The plates diminish in thickness by 1-16 inch to the last four 
plates at the top, which are J4 i'icli thick. 

The contract for steel requires an elastic limit of at least 33,000 lbs. per 
square inch ; an ultimate tensile strength of from 56,000 to 66,000 lbs. per 
square inch ; an elongation in 8 inches of at least 20%, and a reduction of 
area of at least 45%. The inspection of the work was made by the Pittsburgh 
Testing Laboratory. According to their report the actual conditions de- 
veloped were as follows : Elastic limit from 34,020 to 39,420 ; the tensile 
strength from 58,330 to 65,390 ; the elongation in 8 inches from 22J4 to 32% ; 
reduction in area from 52.72 to 71.32% ; 17 plates out of 141 were rejected in 
the inspection. 

WBOUGHT'IRON AND STEEL, WATER-PIPES. 
Riveted Steel Water-pipes {Engineering News, Oct, 11, 1890, and 
Aug. 1, 1891.)— The use of riveted wrought-iron pipe has been common in 
the' Pacific States for many years, the largest being a 44-inch conduit in 
connection with the works of' the Spring Valley Water Co., which supplies 
San Francisco. The use of wrought iron and steel pipe has been neces- 
sary in the West, owing to the extremely high pressures to be withstood 
and the difficulties of transportation. As an example : In connection with 



296 STRENGTH OF MATERIALS. 

the water supply of Virginia City and Gold Hill, Nev., there was laid in 
1872 an llj^-inch riveted wrought-iron pipe, a part of which is under a head 
of 1720 feet". 

Irs the East, the most important example of the use of riveted steel water 
pipe is that of the East Jersey Water Co., which supplies the city of Newark. 
The contract provided for a maximum high service supply of 25,000,000 gal- 
lons daily. In this case 21 miles of 48-inch pipe was laid, some of it under 340 
feet head. The plates from which the pipe is made are about 13 feet long 
by 7 feet wide, open-hearth steel. Four plates are used to make one section 
of pipe about 27 feet long. The pipe is riveted longitudinally with a double 
row, and at the end joints with a single row of rivets of varying diameter, 
corresponding to the thickness of the steel plates. Before being rolled into 
the trench, two of the 27-feet lengths are riveted together, thus diminishing 
still further the number of joints to be made in the trench and the extra 
excavation to give room for jointing. All changes in the grade of the pipe- 
line are made by 10° curves and all changes in line by 2^£, 5, 7J^ and 10° 
curves. To lay on curved lines a standard bevel was used, and the different 
curves are secured by varying the number of beveled joints used on a 
certain length of pipe. 

The thickness of the plates varies with the pressure, but only three thick- 
nesses are used, J4, 5-16, and % inches, the pipe made of these thicknesses 
having a weight of 160, 185, and 225 lbs. per foot, respectively. At the works 
all the pipe was tested to pressure \y% times that to which it is to be sub- 
jected when in place. 

Mamiesmaim Tubes for High Pressures.— At the Mannes- 
maiin Works at Komotau, Hungary, more than 600 tons or 25 miles of 3-iuch 
and 4-inch tubes averaging J4 inch in thickness have been successfully 
tested to a pressure of 2000 lbs. per square inch. These tubes were intended 
for a high-pressure water-main in a Chilian nitrate district. 

This great tensile strength is probably due to the fact that, in addition to 
being: much more worked than most metal, the fibres of the metal run 
spirally, as has been proved by microscopic examination. While cast-iron 
tubes will hardly stand more than 200 lbs. per square inch, and welded tubes 
are not safe above 1000 lbs. per square inch, the Mannesmann tube easily 
withstands 2000 lbs. per square inch. The length up to which they can 
be readily made is shown by the fact that a coil of 3-inch tube 70 feet long 
was made recently. 

For description of the process of making Mannesmann tubes see Trans. 
A. I. M. E , vol. xix., 384. 

STRENGTH OF VARIOUS M ATERIALS. EXTRACTS 
FROM KIRKALDY'S TESTS. 

The recent publication, in a book by W. G. Kirkaldy, of the results of many 
thousand tests made during a quarter of a century by his father, David Kir- 
kaldy, has made an important contribution to our knowledge concerning 
the range of variation in strength of numerous materials. A condensed 
abstract of these results was published in'the American Machinist, May 11 
and 18, 1893, from which the following still further condensed extracts are 
taken: 

The figures for tensile and compressive strength, or, as Kirkalds* calls 
them, pulling and thrusting stress, are given in pounds per square inch of 
original section, and for bending strength in pounds of actual stress or 
pounds per BD 2 (breadth x square of depth) for length of 36 inches between 
supports. The contraction of area is given as a percentage of the original 
area, and the extension as a percentage in a length of 10 inches, except when 
otherwise stated. The abbreviations T. S., E. L., Contr., and Ext. are used 
for the sake of brevity, to represent tensile strength, elastic limit, and per- 
centages of contraction of area, and elongation, respectively. 

Cast Iron.— 44 tests: T. S. 15.468 to 28,740 pounds; 17 of these were un- 
sound, the strength ranging from 15,468 to 24,357 pounds. Average of all, 
23,805 pounds. 

Thrusting stress, specimens 2 inches long, 1.34 to 1.5 in. diameter: 43 tests, 
all sound, 94,352 to 131,912; one, unsound, 93,759; average of all, 113,825. 

Bending stress, bars about 1 in. wide by 2 in deep, cast on edge. Ulti- 
mate stress 2876 to 3854; stress per BD 2 — 725 to 892; average, 820. Average 
modulus of rupture, R, = stress per BD 2 X length, = 29,520. Ultimate de- 
flection, .29 to .40 in.; average .34 inch. 

Other tests of cast iron, 460 tests, 16 lots from various sources, gave re-. 



EXTRACTS FROM KIRK ALB Y*S TESTS. 29? 

suits with total range as follows: Pulling stress, 12,688 to 33,616 pounds; 

thrusting stress, 66,363 to 175,950 pounds; bending stress, per i>'D 2 , 505 to 

1128 pounds; modulus of rupture, R, 18,180 to 40,608. Ultimate deflection, 

.21 to .45 inch. 

The specimen which was the highest in thrusting stress was also the high- 
est in bending, and showed the greatest deflection, but its tensile strength 

was only 26,502. 
The specimen with the highest tensile strength had a thrusting stress of 

143,939, and a bending strength, per 7?D 2 , of 979 pounds with 0.41 deflection. 

The specimen lowest in T. S. was also lowest in thrusting and bending, but 

gave .38 deflection. The specimen which gave .21 deflection had T. S., 19,188: 

thrusting. 101.281; and bending, 561. 
Iron Castings.— 69 tests; tensile strength, 10,416 to 31,652; thrusting 

stress, ultimate per square inch, 53,502 to 132,031. 
Channel Irons.— Tests of 18 pieces cut from channel irons. T. S. 

40,693 to 53,141 pounds per square inch; contr. of area from 3.9 to 32.5 %. 

Ext. in 10 in. from 2.1 to 22.5 %. The fractures ranged all the way from 100 % 

fibrous to 100^ crystalline. The highest T. S., 53,141, with 8.1 % contr. and 
5.3 % ext., was 100 $ crystalline; the lowest T. S.,- 40,693, with 3.9 contr. and 
2.1 % ext., was 75 % crystalline. All the fibrous irons showed from 12.2 to 
R.5* ext., 17.3 to 32.5 contr., and T. S. from 43,426 to 49,615. The fibrous 
irons are therefore of medium tensile strength and high ductility. The 
crystalline irons are of variable T. S., highest to lowest, and low ductility. 

Lowmoor Iron Bars.— Three rolled bars 2% inches diameter; ten- 
sile tests: elastic, 23,200 to 24,200; ultimate, 50,875 to 51,905; contraction, 44.4 
to 42.5; extension, 29.2 to 24.3. Three hammered bars, 414 inches diameter, 
elastic 25,100 to 24,200; ultimate, 46,S10 to 49,223; contraction, 20.7 to 46.5-; 
extension, 10.8 to 31.6. Fractures of all, 100 per cent fibrous. In the ham- 
mered bars the lowest T. S. was accompanied by lowest ductility. 

Iron Bars, Various.— Of a lot of 80 bars of various sizes, some rolled 
and some hammered (the above Lowmoor bars included) the lowest T. S. 
(except one) 40,808 pounds per square inch, was shown by the Swedish 
"hoop L " bar 314 inches diameter, rolled. Its elastic limit was 19,150 
pounds; contraction 68.7 % and extension 37.7$ in 10 inches. It was also 
the most ductile of all the bars tested, and was 100 % fibrous. The highest 
T. S., 60,780 pounds, with elastic limit, 29,400; contr., 36.6; and ext., 24.3 %, 
was shown by a " Farnley " 2-inch bar, rolled. It was also 100 % fibrous. 
The lowest ductility 2M% contr., and 4.1 % ext., was shown by a 3%-inch 
hammered bar, without brand. It also had the lowest T. S., 40,278 pounds, 
but rather high elastic limit, 25,700 pounds. Its fracture was 95 % crystal- 
line. Thus of the two bars showing the lowest T. S., one was the most duc- 
tile and the other the least ductile in the whole series of 80 bars. 

Generally, high ductility is accompanied by low tensile strength, as in the 
Swedish bars, but the Farnley bars showed a combination of high ductility 
and high tensile strength. 

Locomotive Forgings, Iron.— 17 tests: average, E. L., 30,420; T. S., 
50.521; contr., 36.5: ext. in 10 inches, 23.8. 

Broken Anchor Forgings, Iron.— 4 tests: average, E. L., 23,825; 
T. S , 40,083; contr., 3.0; ext. in 10 inches, 3.8. 

■ Kirkaldy places these two irons in contrast to show the difference between 
good and bad work. The broken anchor material, he says, is of a most 
treacherous character, and a disgrace to any manufacturer. 

Iron Plate Girder. —Tensile tests of pieces cut from a riveted iron 
girder after twenty years 1 service in a railway bridge. Top plate, average 
of 3 tests, E. L., 26,600; T. S., 40,806; contr. 16 1; ext. in 10 inches, 7.8. 
Bottom plate, average of 3 tests, E. L., 31,200; T. S., 44,288; contr., 13.3; ext. 
in 10 inches, 6.3. Web-plate, average of 3 tests, E. L., 28,000; T. S., 45,902; 
contr., 15 9; ext. in 10 inches, 8.9. Fractures all fibrous. The results of 30 
lists from different parts of the girder prove- that the iron has undergone 
«o change during twenty years of use. 

Steel Plates.— Six plates 100 inches long, 2 inches wide, thickness vari- 
ous, .36 to .97 inch T. S., 55,485 to 60,805; E. L., 29,600 to 33,200; contr., 52.9 
to 59.5; ext.. 17.05 to 18.57. 

Steel Bridge Links.— 40 links from Hammersmith Bridge, 1886. 



298 



STREHGTH -OF MATERIALS. 





02 

Eh" 


£ 

H 


u 

o 
o 


1 

a 


Fracture. 






c 




67.294 
60,753 
75,936 
64,044 
63,745 
65,980 
63,980 


38,294 
36,030 
44,166 
32,441 
38,118 
36,792 
39,017 


u.bf. 

30.1 
31.2 
34.7 
52.8 
40.8 
6.0 


14.11$ 
15.51 
12.42 
13.43 
15.46 
17.78 
6.62 


15 
30 
100 
35 







70$ 


Highest T. S. and E. L 

Lowest E. L 

Greatest Contraction 

Greatest Extension 

Least Contr. and Ext 


85 
70 

65 
100 



The ratio of elastic to ultimate strength ranged from 50.6 to 65.2 per cent; 
average, 56.9 per cent. 

Extension in lengths of 100 inches. At 10,000 lbs. per sq. in., .018 to .024; 
mean, .0^0 inch; at 20.000 lbs. per sq. in. .049 to .063; mean, .055 inch; at 
30,000 lbs. per sq. in., .083 to .100; mean, .090; set at 30,000 pounds per sq. in., 
to .002; mean, 0. 

The mean extension between 10,000 to 30,000 lbs. per sq. in. increased regu- 
larly at the rate of .007 inch for each 2000 lbs. per sq. in. increment of strain. 
This corresponds to a modulus of elasticity of 28,571, 4J9. The least increase 
of extension for an increase of load of 20,000 lbs. per sq. in., .065 inch, cor- 
responds to a modulus of elasticity of 30,769,231, and the greatest, .076 inch, 
to a modulus of 26,315,789. 

Steel Rails.— Bending tests, 5 feet between supports, 11 tests of flange 
rails 72 pounds per yard, 4.63 inches high. 

Elastic stress. Ultimate stress. Deflection at 50,000 Ultimate 



Pounds. 
34,200 
32,000 



Pounds. 
60,960 
56,740 



Pounds. 
3.24 ins. 
3.76 " 
3.53 " 



Hardest. 
Softest . . 
Mean .... 

All uncracked at 8 inches deflection. 

Pulling tests of pieces cut from same rails. Mean results. 

Elastic Ultimate Contraction of 

Stress. Pounds. area of frac- 

per sq. in. per sq. in. ture. 

Top of rails 44,200 83,110 19.9* 

Botton of rails 40,900 77,820 30.9$ 



Deflection. 
3 ins. 



Extension 

in 10 ins. 

13.5* 

22.8* 



Steel Tires.— Tensile tests of specimens cut from steel tires. 



Krupp Steel.— 262 Tests. 



Highest. . 

Mean 

Lowest . . 



E. L. 

69,250 
52,869 
41,700 



T. S. 
119,079 
104,112 

90,523 



Contr. 
31.9 
29.5 
45.5 



Vickers, Sons & Co.— 70 Tests. 



Highest. . 

Mean 

Lowest.. 



E. L. 

58,600 
51,066 
43,700 



T. S. 

120,789 
101,264 



Contr. 

11.8 



14.-7 



Ext. in 
5 inches. 

18.1 

19.7 

23.7 



Ext. in 

5 inches. 

8.4 

12.4 

16.0 



Note the correspondence between Krupp's and Vickers , steels as to ten- 
sile strength and elastic limit, and their great difference in contraction and 
elongation. The fractures of the Krupp steel averaged 22 per cent silky, 
78 per cent granular; of the Vicker steel, 7 per cent silky, 93 per cent granu- 
lar. 



EXTRACTS FROM KIRKALDY^S TESTS. 



299 



Steel Axles. — Tensile tests of specimens cut froni steel axles- 
Patent Shaft and Axle Tree Co. — 157 Tests. 

Ext. in 
E. L. T. S. Contr. 5 inches. 

Highest 49,800 99,009 81.1 16.0 

Mean... 36,267 72,099 33.0 23.6 

Lowest 31,800 61,382 34.8 25.3 

Vickers, Sons & Co.— 125 Tests. 

Ext. in 
E. L. T. S. Contr. 5 inches. 

Highest 42,600 83,701 18.9 13.2 

Mean 37,618 70,572 41.6 27.5 

Lowest 30,250 56,388 49.0 37.2 

The average fracture of Patent Shaft and Axle Tree Co. steel was 33 per 
cent silky, 67 per cent granular. 

The average fracture of Vickers' steel was 88 per cent silky, 12 per cent 
granular. 
Tensile tests of specimens cut from locomotive crank axles. 
Vickers'.— 82 Tests, 1879. 

Ext. in 
E. L. T. S. Contr. 5 inches. 

Highest 26,700 68,057 28.3 18.4 

Mean 24,146 57,922 32.9 24.0 

Lowest 21,700 50,195 52.7 36.2 

Vickers'.— 78 Tests, 1884. 

Ext. in 
E. L. T. S. Contr. 5 inches. 

Highest 27,600 64,873 27.0 20.8 

Mean 23,573 56,207 32.7 25.9 

Lowest 17,600 47,695 35.0 27.2 

Fried. Krupp.— 43 Tests, 1889. 

Ext. in 
E. L. T. S. Contr. 5 inches. 

Highest 31,650 66,868 48.6 35.6 

Mean 29,491 61,774' 47.7 32.3 

Lowest 21,950 55,172 55.3 35.6 

Steel Propeller Shafts.— Tensile tests of pieces cut from two shafts, 
mean of four tests each. Hollow shaft, Whitworth. T. S.. 61,290; E. L., 
30,575; contr., 52.8; ext. in 10 inches, 28 6. Solid Shaft, Vickers', T. S., 
46,870; E. L. 20,425; contr., 44.4; ext. in 10 inches, 30.7. 

Thrusting tests, Whitworth, ultimate, 56,201; elastic, 29,300; set at 30,000 
lbs., 0.18 per cent; set at 40,000 lbs., 2.04 per cent; set at 50,000 lbs., 3.82 per 
cent. 

Thrusting tests, Vickers', ultimate, 44,602; elastic, 22,250; set at 30,000 lbs., 
2.29 per cent; set at 40,000 lbs., 4.69 per cent. 

Shearing strength of the Whitworth shaft, mean of four tests, was 40,654 
lbs. per square inch, or 66.3 per cent of the pulling- stress. Specific gravity 
of the Whitworth steel. 7.867: of the Vickers', 7.856. 

Spring Steel.— Untempered, 6 tests, average, E. L., 67,916; T. S., 

115,668; contr., 37.8; ext. in 10 inches, 16.6. Spring steel untempered. 15 

tests, average, E. L., 38,785; T. S., 69,496; contr., 19.1 ; ext. in 10 inches, 29 8. 

'These two lots were shipped for the same purpose, viz., railway carnage 

leaf springs. 

Steel Castings.— 44 tests, E. L., 31,816 to 35,567; T. S., 54,928 to 63,840; 
contr., 1.67 to i5.8; ext., 1.45 to 15.1. Note the great variation in ductility. 
The steel of the highest strength was also the most ductile. 

Riveted Joints, Pulling Tests of Riveted Steel Plates, 

Triple Riveted I^ap Joints, Machine Riveted, 

Holes J>rilled. 

Plates, width and thickn ss, inches : 

13.50 X .25 13.00 X .51 11.75 X .78 12.25 X 1.01 14.00 X .77 
Plates, gross sectional area square inches : 

3.375 6.63 9.165 12.372 10.780 

Stress, total, pounds : 

199,320 332,640 423,180 528,000 455,210 



300 STRENGTH OF MATERIALS. 

Stress per square inch of gross area, joint : 

59,058 50,172 46,173 42,696 42,227 

Stress per square inch of plates, solid : 

70,765 65,300 64,050 62,280 68,045 

Ratio of strength of joint to solid plate : 

83.46 76.83 72.09 68.55 62.06 

Ratio net area of plate to gross : 

73.4 65.5 62.7 , 64.7 72.9 

Where fractured : 

plate at plate at plate at plate at rivets 

holes. holes. holes. holes. sheared. 

Rivets, diameter, area and number : 

.45, .159, 24 .64, .321, 21 .95, .708, 12 1.08, .918, 12 .95, .708, 12 
Rivets, total area : 

3.816 6.741 8.496 10.992 8.496 

Strength of Welds.— Tensile tests to determine ratio of strength of 
weld to solid bar. 

Iron Tie Bars.— 28 Tests. 

Strength of solid bars varied from 43,201 to 57,065 lbs. 

Strenth of welded bars varied from 17,816 to 44,586 lbs. 

Ratio of weld to solid varied from 37.0to79.1$ 

Iron Plates.— 7 Tests. 

Strength of solid plate from 44,851 to 47,481 lbs. 

Strength of welded plate from 26,442 to 38,931 lbs. 

Ratio of weld to solid 57.7 to 83.9$ 

Chain Links.— 216 Tests. 

Strength of solid bar from 49,122 to 57,875 lbs. 

Strength of welded bar from 39,575 to 48,824 lbs. 

Ratio of weld to solid 72.1 to 95.4$ 

Iron Bars.— Hand and Electric Machine Welded. 

32 tests, solid iron, average 52,444 

17 " electri- welded, average 46,836 ratio 89.1$ 

19 " hand " " 46,899 " 89.3$ 

Steel Bars and Plates.— 14 Tests. 

Strength of solid 54,226 to 64,580 

Strength of weld 28,553 to 46,019 

Ratio weld to solid 52.6 to 82.1$ 

The ratio of weld to solid in all the tests ranging from 37.0 to 95.4 is proof 
of the great variation of workmanship in welding. 

Cast Copper.— 4 tests, average, E. L., 5900; T. S., 24,781; ccntr., 24.5; 
ext., 21.8. 

Copper Plates.— As rolled, 22 tests, .26 to .75 in. thick; E. L.,9766 to 
18,650; T. S., 30,99.; to 34,281 ; contr., 31.1 to 57.6; ext., 39.9 to 52.2. The va- 
riation in elastic limit is due to difference in the heat at which the plates 
were finished. Annealing reduces the T. S. only about 1000 pounds, but the 
E. L. from 3000 to 7000 pounds. 

Another series, .38 to .52 thick; 148 tests, T. S., 29,099 to 31,924; contr., 28.7 
to 56.7; ext. in 10 inches, 28.1 to 41.8. Note the uniformity in tensile 
strpngth. 

Drawn Copper.— 74 tests (0.88 to 1.08 inch diameter); T. S., 31,634 to 
40,557; contr., 37.5 to 64.1; ext. in 10 inches, 5.8 to 48.2. 

Bronze from a Propeller Blade.— Means of two tests each from 
centre and edge. Central portion (sp. gr. 8.320). E. L., 7550; T. S., 26,312; 
contr., 25.4; ext. in 10 inches, 32.8. Edge portion (sp. gr. 8550). E. L., 8950; 
T. S., 35,960; contr., 37.8; ext. in 10 inches, 47.9. 

Cast German Silver.— 10 tests: E. L., 13,400 to 29,100; T. S., 23,714 to 
46,540; contr., 3.2 to 21.5; ext. in 10 inches, 0.6 to 10.2. 

Thin Sheet Metal.— Tensile Strength. 

German silver, 2 lots 75.816 to 87,129 

Bronze, 4 lots 73,380 to 92,086 

Brass. 2 lots , 44,398 to 58,188 

Copper, 9 lots.... 30,470 to 48,450 

Iron, 13 lots, lengthway 44,331 to 59,484 

Iron. 13 lots, crossway 39,838 to 57,350 

Steel, 6 lots 49,253 to 78,251 

Steel, 6 lots, crossway 55,948 to 80,799 



EXTRACTS FROM KIRKALD1 S TESTS. 



301 



"Wire.— Tensile Strength. 

German silver, 5 lots 81 ,735 to 92,224 

Bronze, 1 lot . . 78,049 

Brass, as drawn, 4 lots 81,114 to 98,578 

Copper, as drawn, 3 lots 37,607 to 46,494 

Copper annealed, 3 lots 34,936 to 45,210 

Copper (another lot), 4 lots 35,052 to 62,190 

Copper (extension 36.4 to 0.6$). 

Iron,81ots 59,246 to 97,908 

Iron (extension 15.1 to 0.7$). 

Steel, Slots 103,272 to 318,823 

The Steel of 318,823 T. S. was .047 inch diam., and had an extension of only 
0.3 per cent; that of 103,272 T. S, was .107 inch diam. and had an extension 
of 2.2 per cent. One lot of .044 inch diam. had 267,114 T. S., and 5.2 per cent 
extension. 

Wire Ropes. 

Selected Tests Showing Range of "Variation. 





of 
o 




Strands. 


*+« 








4> gj 


A B 






u O 




®5 








Description. 


a! 

5 


II 


Kb 


o p 


-2.2 
2 © 


Hemp Core. 


e3 60 • 

a §£ 

5oq 


Galvanized 


7.70 


53.00 


6 


19 


.1563 


Main 


339,780 


Ungalvanized 


7.00 


53.10 


7 


19 


.1495 


Main and Strands 


314,860 


Ungalvanized 


6.38 


42.50 


7 


19 


.1347 


Wire Core 


295,920 


Galvanized.. 


7.10 


37.57 


« 


30 


.1004 


Main and Strands 


272,750 


Ungalvanized 


6.18 


40.46 


7 


19 


.1302 


Wire Core 


268,470 


Ungalvanized 


6.19 


40.33 


7 


19 


.1316 


Wire Core 


221,820 


Galvanized.. 


4.92 


20.86 


6 


30 


.0728 


Main and Strands 


190,890 


Galvanized 


5.36 


18.94 


fi 


12 


.1104 


Main and Strands 


136,550 


Galvanized 


4.82 


21.50 


6 


7 


.1693 


Main 


129,710 


Ungalvanized 


3.65 


12.21 


6 


19 


.0755 


Main 


110,180 


Ungalvanized 


3.50 


12.65 


7 


7 


.122 


Wire Core 


101,440 


Ungalvanized 


3 . 82 


14.12 


fi 


7 


.135 


Main 


98,670 


Galvanized 


4.11 


11.35 


6 


12 


.080 


Main and Strands 


75,110 


Galvanized 


3.31 


7.27 


6 


12 


.068 


Main and Strands 


55,095 


Ungalvanized 


3.02 


8.62 


6 


7 


.105 


Main 


49,555 


Ungalvanized 

Galvanized 


2.68 


6.26 


H 


« 


.0963 


Main and Strands 


41,205 


2.87 


5.43 


6 


12 


.0560 


Main and Strands 


38,555 


Galvanized 


2.46 


3.85 


(J 


12 


.0472 


Main and Strands 


28,075 


Ungalvanized...,. 


1.75 


2.80 


6 


7 


.0619 


Main 


24,552 


Galvanized 


2.04 


2.72 


6 


12 


.0378 


Main and Strands 


20,415 


Galvanized 


1.76 


1.85 


6 


12 


.0305 


Main 


14,634 



Hemp Ropes, Untarred.— 15 tests of ropes from 1.53 to 6.90 inches 
circumference, weighing 0.42 to 7.77 pounds per fathom, showed an ultim- 
ate strength of from 1670 to 33,808 pounds, the strength per fathom weight 
varying; from 2872 to 5534 pounds. 

Hemp Ropes, Tarred. —15 tests of ropes from 1.44 to 7.12 inches 
circumference, weighing from 0.38 to 10.39 pounds per fathom, showed an 
ultimate strength of from 1046 to 31.549 pounds, the strength per fathom 
weight varying from 1767 to 5149 pounds. 

Cotton Ropes.— 5 ropes, 2.48 to 6.51 inches circumference, 1.08 to 8.17 
pounds per fathom. Strength 3089 to 23,258 pounds, or 2474 to 3346 pounds 
per fathom weight. 

Manila Ropes.— 35 tests: 1.19 to 8.90 inches circumference, 0.20 to 
11.40 pounds per fathom. Strength 1280 to 65,550 pounds, or 3003 to 7394 
pounds per fathom weight. 



302 



STRENGTH OF MATERIALS. 



Belting. 

No. of Tensile strength 

lots. per square inch. 

11 Leather, single, ordinary tanned 3248 to 4824 

4 Leather, single, Helvetia 5631 to 5944 

7 Leather, double, ordinary tanned 2160 to 3572 

8 Leather, double Helvetia 4078 to 5412 

6 Cotton, solid woven 5648 to 8869 

14 Cotton, folded, stitched 4570 to 7750 

1 Flax, solid, woven 9946 

1 Flax, folded, stitched 6389 

6 Hair, solid, woven 3852 to 5159 

2 Rubber, solid, woven 4271 to 4343 

Canvas.— 35 lots: Strength, lengthwise, 113 to 408 pounds per inch; 

crossways, 191 to 468 pounds per inch. 

The grades are numbered 1 to 6, but the weights are not given. The 
strengths vary considerably, even in the same number. 

Marbles.— Crushing strength of various marbles. 38 tests, 8 kinds. 
Specimens were 6-inch cubes, or columns 4 to 6 inches diameter, and 6 and 

12 inches high. Range 7542 to 13,720 pounds per square inch. 
Granite.— Crushing strength, 17 tests; square columns 4X4 and 6x4, 

4 to 24 inches high, 3 kinds. Crushing strength ranges 10,026 to 13,271 
pounds per square inch. (Very uniform.) 

Stones.— (Probably sandstone, local names only given.) 11 kinds, 42 
tests, 6x6, columns 12, 18 and 24 inches high. Crushing strength ranges 
from 2105 to 12,122. The strength of the column 24 inches long is generally 
from 10 to 20 per cent less than that of the 6-inch cube. 

Stones.— (Probably sandstone) tested for London & Northwestern Rail- 
way. 16 lots, 3 to 6 tests in a lot. Mean results of each lot ranged from 
3785 to 11.956 pounds. The variation is chiefly due to the stones being from 
different lots. The different specimens in each lot gave results which gen- 
erally agreed within 30 per cent. 

Bricks.— Crushing strength, 8 lots; 6 tests in each lot; mean results 
ranged from 1835 to 9209 pounds per square inch. The maximum variation 
in the specimens of one lot was over 100 per cent of the lowest. In the most 
uniform lot the variation was less than 20 per cent. 

Wood..— Transverse and Thrusting Tests. 



Sizes abt. in 
square. 



Span 
inches. 



Ultimate 

Stress. 


8 = 
LW 




4BD* 


45,856 


1096 


to 


to 


80,520 


1403 


37,948 


657 


to 


to 


54,152 


790 


32,856 


1505 


to 


to 


39,084 


1779 


23,624 


1190 


to 


to 


26,952 


1372 



Pitch pine 

Dantzic fir 

English oak 

American white 
oak 



uy 2 to 12^ 

12 to 13 
4^X 12 
4Y 2 X 12 



120 
120 



5438 
2478 



3423 
2473 



Demerara greenheart, 9 tests (thrusting) 8169 to 10,785 

Oregon pine, 2 tests 5888 and 7284 

Honduras mahogany, 1 test 6769 

Tobasco mahogany, 1 test 5978 

Norway spruce, 2 tests 5259 and 5494 

American yellow pine, 2 tests 3875 and 3993 

English ash, 1 test 3025 

Portland Cement.— (Austrian.) Cross-sections of specimens 2 x 2^ 
inches for pulling testsonly; cubes, 3x3 inches for thrusting tests; weight, 



MISCELLANEOUS TESTS OF MATERIALS. 



303 



98.8 pounds per imperial bushel; residue, 0.7 per cent with sieve 2500 meshes 
per square inch: 88.8 per cent by volume of water required for mixing; time 
of setting, 7' days; 10 tests to each lot. The mean results in lbs. per sq. in. 
were as follows: 

Cement Cement 1 Cement, ' 1 Cement, 1 Cement, 

alone, alone, 2 Sand, 3 Sand, 4 Sand, 

Age. Pulling. Thrusting. Thrusting. Thrusting. Thrusting. 

10 days 376 2910 893 407 228 

20 days 420 3342 1023 494 275 

30 days 451 3724 1172 594 338 

Portland Cement.- Various samples pulling tests, 2 x 2% inches 
cross-section, all aged 10 days, 180 tests; ranges 87 to 643 pounds per square 
inch. 

TENSILE STRENGTH OF WIRE. 
(From J. Bucknall Smith's Treatise on Wire.) 

Tons per sq. Pounds per 
in. sectional sq. in. sec- 
area, tional area. 

Black or annealed iron wire 25 56,000 

Bright hard drawn 35 78,400 

Bessemer, steel wire 40 89,600 

Mild Siemens-Martin steel wire.... 60 134,000 

High carbon ditto (or " impi'oved ") 80 179,200 

Crucible cast-steel ,l improved " wire 100 224,000 

"Improved" cast-steel "plough" 120 268,800 

Special qualities of tempered and improved cast- 
steel wire may attain 150 to 170 336,000 to 380,800 

MISCELLANEOUS TESTS OF MATERIALS. 
Reports of Work of the Watertown Testing-machine in 

1883. 
TESTS OF RIVETED JOINTS, IRON AND STEEL PLATES. 





6 




.. 








•^ 6 a3 


•B 2 


a 




a 
H 


> 

2> V 

"£.2 

1 

fi 


©o . 

S'S-g 

3 
P-l 


0J z. 


> 
6 




Tensile Streng 

Joint in Net St 

tion of Plate p 

square inch, 

pounds. 


w .2 o 


'5 
1-5 « 

a © 

•Sfi 
o 
& 
H 


* 


% 


11-16 


u 


10% 


6 


m 


39,300 


47,180 


47.0 X 


* 


% 


11-16 


n 


wy 2 


6 


i% 


41,000 


47,180 


49.0 t 


* 


Vk 


% 


13-16 


10 


5 


2 


35,650 


44,615 


45.6 % 


* 


M 


U 


13-16 


10 


5 


2 


35,150 


44,615 


44.9 % 


* 


Ys 


11-16 


H 


10 


5 


2 


46,360 


47,180 


59.9 § 


* 




11-16 


H 


10 


5 


2 


46.875 


47,180 


60.5 § 


* 


M& 


H 


13-16 


10 


5 


2 


46,400 


44,615 


59.4 § 


* 


* 


% 


13-16 


10 


5 


2 


46,140 


44,615 


59.2 § 


* 


l 


1 1-16 


10^ 


4 


m 


44,260 


44,635 


57.2 § 


* 


% 


l 


1 1-16 


ioy 2 


4 


2% 


42,350 


44,635 


54.9 § 


* 


% 


*H 


1 3-16 


11.9 


4 


2.9 


42,310 


46.590 


52.1 § 


* 


% 


1*4 


1 3-16 


11.9 


4 


2.9 


41,920 


46,590 


51.7 § 


* 


Ys 


M 


13-16 


10^ 


6 


m 


61,270 


53.330 


59.5 X 


t 


% 


M 


13-16 


10^ 


6 


m 


60,830 


53,330 


59.1 X 


t 


y* 


15-16 


1 


10 


5 




47,530 


57,215 


40.2 X 


t 


y* 


15-16 


1 


10 


5 


2 


49,840 


57,215 


42.3 % 


t 


% 


11-16 


S 


10 


5 


2 


62,770 


53,330 


71.7 § 


t 


Vs 


11-16 


10 


5 


2 


61,210 


53,330 


69.8 § 


t 


y* 


15-16 


1 


10 


5 


2 


68,920 


57,215 


57.1 § 


t 


14 


15-16 


1 


10 


5 


2 


66,710 


57,215 


55.0 § 


t 


% 


1 


1 1-16 


9V£ 


4 


2% 


62,180 


52,445 


63.4 § 


t 


% 


1 


1 1-16 


Wz 


4 


2% 


62,590 


52,445 


63.8 § 


t 


u 


m 


1 3-16 


10 


4 


2y 2 


54,650 


51,545 


54 § 


+ 


% 


m 


1 3-16 


10 


4 


2y 2 


54,200 


51,545 


53.4 § 



X Lap-joint. 



5 Butt-joint. 



304 



STRENGTH OP MATERIALS. 



The efficiency of the joints is found by dividing the maximum tensile 
stress on the gross sectional area of plate by the tensile strength of the 
material. 

COMPRESSION TESTS OF 3 X 3 INCH WROUGHT-IRON BARS. 





Tested with Two Pin Ends, Pins 
V/z inch in Diameter. 


Tested with One 
Flat and One Pin 


Length, inches. 


Ultimate Com- 
pressive Strength 
pounds per square 
inch. 


Tested with Two 
Flat Ends, Ulti- 
mate Compressive 
Strength, pounds 
per square inch. 


End, Ultimate 

Compressive 

Strength, pounds 

per square inch. 




$ 28,260 
(31,990 
J 26,310 
| 26,640 
j 24.030 
j 25,380 
j 20,660 
1 20,200 
j 16,520 
1 17,840 
I 13,010 
\ 15,700 
























90 


j 26,780 
1 25,580 
J 23,010 
1 22,450 


j 25,120 


120... 


| 25,190 
j 22,450 
1 21,870 




150 



















Tested with two pin- j 
ends. Length o f bars -< 
120 inches. 



I 2^ 



Diameter 

of Pins. 

inch 

inches. 



Ult. Comp. Str., 
per sq. in., lbs. 

16,250 

17,740 

21,400 

22,210 



TENSILE TEST OF SIX STEEL EYE-BARS. 

COMPARED WITH SMALL TEST INGOTS. 

The steel was made by the Cambria Iron Company, and the eye-bar heads 
made by Keystone Bridge Company by upsetting and hammering. All the 
bars were made from one ingot. Two test pieces, %-inch round, rolled from 
a test-ingot, gave elastic limit 48,040 and 42,210 pounds; tensile strength, 
73,150 and 69,470 pounds, and elongation in 8 inches, 22.4 and 25.6 per cent, 
respectively. The ingot from which the eye-bars were made was 14 inches 
square, rolled to billet, 7x6 inches. The eye-bars were rolled to Q% X 1 inch. 
Cnemical tests gave carbon .27 to .30; manganese, .64 to .73; phosphorus, 
.074 to .098. 

Elongation 
per cent, in 
Gauged Length. 
15.8) 
6.96 
8.6 
12.3 
12.0 
16.4 
13.9 

The average tensile strength of the %-inch test pieces was 71,310 lbs., that 
of the eye-bars 67,230 lbs., a decrease of 5.7$. The average elastic limit of 
the test pieces was 45,150 lbs., that of the eye-bars 36,402 lbs., a decrease of 
19.4$. The elastic limit of the test pieces was 63.3^ of the ultimate strength, 
that of the eye-bars 54.2% of the ultimate strength. 



Gauged 


Elastic 


Tensile 


Length, 


limit, lbs. 


strength per 


inches. 


per sq. in. 


sq. in., lbs. 


160 


37,480 


67,800 


160 


36,650 


64,000 


160 




71,560 


200 


37,600 


68,720 


200 


35,810 


65,850 


200 


33,230 


64,410 


200 


37,640 


68,290 



MISCELLANEOUS TESTS OE MATERIALS. 



305 



COMPRESSION OF WROUGHT-IRON COLUMNS, LATTICED BOX 
AND SOLID WEB. 

ALL TESTED WITH PIN ENDS. 



Columns made of 



6 inch channel, solid web.. 



8-inch channels, with 5-16-in. continuous 
plates 

5-16-inch continuous plates and angles. 
Width of plates, 12 in., 1 in. and 7.35 in. 

7-16-inch continuous plates and angles. 
Plates 12 in. wide . 

8-inch channels, latticed 



8-inch channels, latticed, swelled sides. 



10 " " " 

10 " " " 

10-inch channels, latticed, swelled sides. 

* 10-inch channels, latticed one side; con- 
tinuous plate one side 

1 10 inch channels, latticed one side; con- 
tinuous plate one side 





e3 


^ 


<D 


2-3 






< a 


S S-a 


-fl 


$? 


M§ 


a 


O g3 

o a 1 
w 




10.0 


9.831 


432 


15.0 


9.977 


592 


20.0 


9.762 


755 


20.0 


16.281 


1,290 


26.8 


16.141 


1,645 


26.8 


19.417 


1,940 


26.8 


16.168 


1,765 


26.8 


20.954 


2,242 


13.3 


7.628 


679 


20.0 


7.621 


924 


26.8 


7.673 


1,255 


13.4 


7.624 


684 


20.0 


7.517 


921 


26.8 


7.702 


1,280 


16.8 


11.944 


1,470 


25.0 


12.175 


1,926 


16.7 


12.366 


1,549 


25.0 


11.932 


1,962 


25.0 


17.622 


1,848 


25.0 


17.721 


1,827 



30,220 
21,050 
16,220 
22,540 
17,570 

25,290 

28,020 

25,770 
33,910 
34,120 
29.870 
33,530 
33,390 
30,770 
33,740 
32,440 
31,130 
32,740 

26,190 

17,270 



* Pins in centre of gravity of channel bars and continuous plate, 1.63 
inches from centre line of channel bars. 
+ Pins placed in centre of gravity of channel bars. 

EFFECT OF COLD-DRAWING ON STEEL. 

Three tensile bars and two compression bars, cut from the same bar of 
hot-rolled steel, from the Norway Steel and Iron Company: 



Elonga- 
tion, 
per cent. 



Tensile 
strength per 
sq. in., lbs. 

1. Piece of the oi'iginal hot-rolled bar, length 

66 inches, diameter 2.03 inches. Gauged 

length 30 inches 55,400 23.9 

2. Diameter reduced in compressing dies (one 

pass), .094 inch. Gauged length 20 inches. 70,420 2.7 

3. Diameter reduced in compression dies (one 

pass), .222 inch. Gauged length 20 inches. 81,890 0.075 

Compress. Amount Corn- 
Stress, lbs. of Com- press, 
per sq. in. press., in. set, in. 

4. Compression test of cold-drawn bar (same 

as No. 3). Length 4 inches, diameter 

1.808inches 75,000 .0562 .0395 

5. Do., same as No. 4 75,000 .0578 .0400 

Pieces 4 and 5 both had diameters increased in the middle to 1.821 inches, 
and at the ends to 1.813 inches. 



306 



STRENGTH 0$ MATERIALS. 



TESTS OF AMERICAN WOODS. (See also page 309.) 
In all cases a large number of tests were made of each wood. Minimum 
and maximum results only are given. All of the test specimens had a sec- 
tional area of 1.575 x 1.575 inches. The transverse test specimens were 39.37 
inches between supports, and the compressive test specimens were 12.60 

inches long. Modulus of rupture calculated from formula R = --,— n:; P = ! 

2 btV- 
load in pounds at the middle, I = length in inches, b = breadth, d = depth: 

Compression 

Parallel to 
Grain, pounds I 
per square inch. 



Name of Wood. 



Cucumber tree (Magnolia acuminata).. 

Yellow poplar white wood (Lirioden- 
dron tulipifera) 

White wood, Basswood (Tilia Ameri- 
cana) 

Sugar-maple, Rock-maple (Acer sac- 
char inum 

Red maple (Acer rubrum) 

Locust (Robinia pseudacacia) 

Wild cherry (Primus serotina) 

Sweet gum (Liquidambar styraciflua) . . 

Dogwood (Cornusflorida) 

Sour gum, Pepperidge (Nyssa sylvatica), 

Persimmon (Diospyros Virginiana). ... 

White ash (Fraxunis Americana) 

Sassafras (Sassafras officinale) 

Slippery elm ( Ulmus fulva) . . 

White elm (Ulmus Americana) 

Sycamore; Buttonwood (Platanus occi- 
dentalis) 

Butternut; white waluut (Juglans ci- 
nerea) 

Black walnut (Juglans nigra) 

Shellbark hickory (Carya alba) ... 

Pignut (Carya porcina) 

White oak (Quercus alba) 

Red oak ( Quercus rubra) . 

Black oak (Quercus tinctorin) 

Chestnut (Castanea vulgaris) . 

Beech (Fag us ferruginea) 

Canoe-birch, paper-birch (Betula papy- 
racea) 

Cottonwood (Populus mouilifera) 

White cedar (Thuja occidentalis) . 

Red cedar (Juniper us Virginiana). 

Cypress (Saxodium Distichum) 

White pine (Pinus strobus) 

Spruce pine ( Pinus glabra) 

Long-leaved pine, Southern pine (Pinus 
pal ustris) 

White spruce (Picea alba), 

Hemlock (Tsuga Canadensis) 

Red fir, yellow fir (Pseudotsuga Doug- 
lasii) 

Tamarack ( Larix Americana) 



Transverse Tests. 


Modulus of 


Rupture. 


Min. 


Max. 


7,440 


12,050 


6,560 


11,756 


6,720 


11,530 


9,680 


20,130 


8,610 


13,450 


12,200 


21,730 


8,310 


16,800 


7,470 


11,130 


10,190 


14.560 


9,830 


14,300 


18.500 


10,290 


5,950 


15,800 


5,180 


10,150 


10,220 


13,952 


8,250 


15,070 


6,720 


11,360 


4,700 


11,740 


8,400 


16,320 


14,870 


20,710 


11,560 


19,430 


7,010 


18,360 


9,760 


18,370 


7,900 


18,420 


5,950 


12,870 


13,850 


18,840 


11,710 


17,610 


8,390 


13,430 


6,310 


9,530 


5,640 


15,100 


9,530 


10,030 


5,610 


11,530 


3,780 


10,980 


9,220 


21,060 


9,900 


11,650 


7,590 


14,680 


8,220 


17,920 


10,080 


16,770 



Min. 



4,560 

4,150 

3,810 

7,460 
6,010 
8,330 
5,830 
5,630 
6,250 
6,240 
6,650 
4,520 
4,050 



4,960 

5,480 
6,940 
7,650 
7,460 
5,810 
4,960 
4,540 
3,680 
5,770 

5,770 
3,790 
2,660 
4,400 
5,060 
3,750 
2,580 

4,010 
4,150 
4,500 

4,880 
6,810 



SHEARING STRENGTH OF IRON AND STEEL. 

H. V. Loss in American Engineer and Railroad Journal, March and April, 
1893, describes an extensive series of experiments on the shearing of iron 
and steel bars in shearing machines. Some of his results are : 



307 



Depth of penetration at point of maximum resistance for soft steel bars 
is independent. of the width, but varies with the thickness. If d = depth of 
penetration and t = thickness, d — M for a flat knife, d — .25 t for a 4° bevel 
knife, and d = .16 \/i^for an 8° bevel knife. The ultimate pressure per inch 
of width in flat steel bars is approximately 50,000 lbs. X t. The energy con- 
sumed in foot pounds per inch width of steel bars is. approximately: 1" 
thick, 1300 ft.-lbs.; \y 2 ", 2500; \%", 3700; 1%", 4500; the energy increasing 
at a slower rate than the thickness. Iron angles require more energy 
than steel angles of the same size; steel breaks while iron has to be 
cut off. For hot-rolled steel the resistance per square inch for rectan- 
gular sections varies from 4400 lbs. to 20,500 lbs., depending partly upon its 
hardness and partly upon the size of its cross-area, which latter element 
indirectly but greatly indicates the temperature, as the smaller dimensions 
require a considerably louger time to reduce them down to size, which time 
again means loss of heat. 

It is not probable that the resistance in practice can be brought very 
much below the lowest figures here given — viz., 4400 lbs. per square inch- 
as a decrease of 1000 lbs. will henceforth mean a considerable increase in 
cross-section and temperature. 

HOIiDING-POWER OF BOILER-TUBES EXPANDED 
INTO TUBE-SHEETS. 

Experiments by Chief Engineer W. H. Shock, U. S. N., on brass tubes, 2% 
inches diameter, expanded into plates %-inch thick, gave results ranging 
from 5850 to 46.000 lbs. Out of 48 tests 5 gave figures under 10,000 lbs., 12 
between 10,000 and 20,000 lbs., 18 between 20,000 and 30,000 lbs., 10 between 
30,000 and 40,000 lbs., and 3 over 40,000 lbs. 

Experiments by Yarrow & Co. , on steel tubes, 2 to 2J4 inches diameter, 
gave results similarly varying, ranging from 7900 to 41,715 lbs., the majority 
ranging from 20X00 to 30,000 lbs. In 15 experiments on 4 and 5 inch tubes 
the strain ranged from 20,720 to 68,040 lbs. Beading the tube does not neces- 
sarily give increased resistance, as some of the lower figures were obtained 
with beaded tubes. (See paper on Rules Governing the Construction of 
Steam Boilers, Trans. Engineering Congress, Section G, Chicago, 1893.) 

CHAINS. 
Weight per Foot, Proof Test and Breaking; Weigiit. 

(Pennsylvania Railroad Specifications.) 



Nominal 


Description. 


Specifications. 


Diameter 
of Wire, 
inches. 


Weight per 
foot, lbs. 


Proof Test, 
lbs. 


Breaking 

Weight, 
lbs. 


5/32 




0.20 
0.35 
0.70 
1.10 
1.50 
1.50 
1.90 
1.90 • 
2.50 
2.50 
4.00 
4.00 
5.50 
5.50 
7.40 
9.50 
12.00 
15.00 
21.00 






3/16 


Fire-door chain 

Crossing-gate chain 

Sprocket-wheel chain 






H 

5/16 

3 

7/16 
7/16 

y* 

% 
% 

M 
% 

Vs 


1500 

3000 

3500 

4000 

5000 

5500 

7000 

7500 

11,000 

11,000 

16,000 

16,000 

22,000 

30,000 

40,000 

50,000 

70,000 


3000 
5500 
7000 




7500 


Drop-bottom branch chain. 

Crane-chain 

Drop-bottom main chain 


9500 
10,000 
12,500 
13,000 
20,000 
20 000 








29,000 
29 000 






40,000 
55,000 
66,000 
82,000 
116,000 


m 
i^ 


mi it "■ 


U It 





Elongation of all sizes, 10 per cent. All chain must stand the prescribed 
proof test without deformation. 



308 



STRENGTH OF MATERIALS. 



British Admiralty Proving Tests of Chain Cables.— Stud- 
links. Minimum size in inches and 16ths. Proving test in tons of 2240 lbs. 
Min. Size: i| \% \% \% \% 1 life 1 T % 1 T % 1t% 1& Irs Its- 
Test, tons: 8£g 10& 11£§ 13£§ 15*{j 18 20 5 6 n 22J§ 25^ 28& 31 34 37 2 %. 
Min. Size: l 8 l 9 I 10 l 11 l 12 l 13 l 14 l 15 2 2* 2 2 2 3 . 
Test, tons: 40^ 43|| 47'i§ 51& 55^ 59 2 2 ff 63& 67^ 72 76^ 81 & 91,%. 

Wrought-iron Chain Cables.— The strength of a chain link is 
less than twice that of a straight bar of a sectional area equal to that of one 
side of the link. A weld exists at one end and a bend at the other, each re- 
quiring at least one heat, which produces a decrease in the strength. The 
report of the committee of the U. S. Testing Board, on tests of wrought-iron 
and chain cables contains the following conclusions. That beyond doubt, 
when made of American bar iron, with cast-iron studs, the studded link is 
inferior in strength to the unstudded one. 

"That when proper care is exercised in the selection of material, a varia- 
tion of 5 to 17 per cent of the strongest may be expected in the resistance 
of cables. Without this care, the variation may rise to 25 per cent. 

" That with proper material and construction the ultimate resistance of 
the chain may be expected to vary from 155 to 170 per cent of that of the 
bar used in making the links, and show an average of about 163 per cent. 

" That the proof test of a chain cable should be about 50 per cent of the 
ultimate resistance of the weakest link.' 1 

The decrease of the resistance of the studded below the unstudded cable 
is probably due to the fact that in the former the sides of the link do not 
remain parallel to each other up to failure, as they do in the latter. The re- 
sult is an increase of stress in the studded link over the unstudded in the 
proportion of unity, to the secant of half the inclination of the sides of the 
former to each other. 

From a great number of tests of bars and unfinished cables, the commit- 
tee considered that the average ultimate resistance, and proof tests of chain 
cables made of the bars, whose diameters are given, should be such as are 
shown in the accompanying table. 

ULTIMATE RESISTANCE AND PROOF TESTS OF CHAIN CABLES. 



Diam. 
of 
Bar. 


Average resist. 
= 163^ of Bar. 


Proof Test. 


Diam. 
of 
Bar. 


Average resist. 
= 163$ of Bar. 


Proof Test. 


Inches. 


Pounds. 


Pounds. 


Inches. 


Pounds. 


Pounds. 


1 1/16 


71,172 


33,840 


1 9/16 


162,283 


77,159 


1 1/16 


79,544 


37,820 


m 


174,475 


82,956 


W8 


88,445 


42,053 


1 11/16 


187,075 


88,947 


1 3/16 


97,731 


46,468 


1% 


200,074 


95,128 


M 


107,440 


51,084 


1 13/16 


213,475 


101,499 


1 5/16 


117,577 


55,903 


Ws 


227,271 


108,058 


m 


128,129 


60,920 


1 15/16 


241,463 


114,806 


1 7/16 


139,103 


66,138 


2 


256,040 


121,737 


1H 


150,485 


71,550 









STRENGTH OF GLASS. 

(Faii-bairn's " Useful Information for Engineers," Second Series.) 

• Best Common Extra White 

Flint Glass. Green Glass. Crown Glass. 

Mean specific gravity 

Mean tensile strength, lbs. per sq. in., bars.. 

do. thin plates. 

Mean crush 'g strength, lbs. p. sq. in., cyl'drs. 

do. cubes. 

The bars in tensile tests were about y% inch diameter. The crushing tests 

were made on cylinders about % inch diameter and from 1 to 2 inches high, 

and on cubes approximately 1 inch on a side. The mean transverse strength 

of glass, as calculated by Fairbairn from a mean tensile strength of 2560 

lbs. and a mean compressive strength of 30,150 lbs. per sq. in., is, for a bar 

supported at the ends and loaded in the middle, 



3.078 


2.528 


2.450 


2,413 


2,896 


2,546 


4,200 


4,800 


6,000 


27,582 


39.876 


31,003 


13,130 


20,206 


21,867 



w = 3140- 



I 



STRENGTH OF TIMBER. 



309 



in which to = breaking weight in lbs., b = breadth, d = depth, and I = length, 
in inches. Actual tests will probably show wide variations in both direc- 
tions from the mean calculated strength. 

STRENGTH OF COPPER AT HIGH TEMPERATURES. 

The British Admiralty conducted some experiments at Portsmouth Dock- 
yard in 1877, on the effect of increase of temperature on the tensile strength 
of copper and various bronzes. The copper experimented upon was in rods 
.72-in. diameter, having a tensile strength of about 25 tons per square inch. 

The following table shows some of the results: 



Temperature 
Fahr. 


Tensile Strength 
in lbs. per sq. in. 


Temperature 
Fahr. 


Tensile Strength 
in lbs. per sq. in. 


Atmospheric. 
100° 
200° 
300° 


23,115 
23,366 
22,110 
21,607 


Atmospheric. 
400° 
500° 


21,105 
19,597 



Up to a temperature of 400° F. the loss of strength was only about 10 per 
cent, and at 500° F. the loss was 16 per cent. The temperature of steam at 
200 lbs. pressure is 382° F., so that according to these experiments the loss 
of strength at this point would not be a serious matter. Above a tempera- 
ture of 500° the strength is seriously affected. 

STRENGTH OF TIMBER. 

Strength of IiOng-leaf Pine (Yellow Pine, Pinus Palustris) from 
Alaban a (Bulletin No. 8, Forestry Div., Dept. of Agriculture, 1893. Tests 
by Prof. J. B. Johnson.) 

The following is a condensed table of the range of results of mechanical 
tests of over 2000 specimens, from 26 trees from four different sites in 
Alabama ; reduced to 15 per cent moisture : 



Specific gravity 

Transversestrength,- — — 

do do. atelast. limit 
Mod. of elast., thous. lbs. 
Relative elast. resilience, 

inch-pounds per cub. in. 
Crushing endwise, str. per 

sq. in. -lbs 

Crushing across grain, 

strength per sq. in., lbs. 
Tensile strength per sq. in. 
Shearing strength (with 

grain), mean per sq. ' 



Butt Logs. 



Middle Logs. 



0.449 to 1 

4,762 to II 

4,930 to 
1,119 to 

0.23 to 4 

4,781 to 

675 to 

8,600 to 



0.575 to 
7,640 to 



13,110 
3,11" 



2,094 
31,890 



i to IS 



Top Logs. 



5,540 to 
1,136 to 



0.859 
17,128 
11,790 



5,030 to 9,3 



656 to 
0,330 to 



1,445 
29,500 



539 to 1,230 



2,553 to 

842 to 



1 09 to 

4,587 to 



584 to 
4,170 to 



0.907 

15,554 

11,950 
2,697 

4.65 

9,100 

1,766 

23,280 



484 to 1156 



Av'g of 

all Butt 

Logs. 



0.767 

12,614 

9,460 
1,926 



7,452 



Some of the deductions from the tests were as follows : 

1. With the exception of tensile strength a reduction of moisture is ac- 
companied by an increase in strength, stiffness, and toughness. 

2. Variation in strength goes generally hand-in-hand with specific gravity. 

3. In the first 20 or 30 feet in height the values remain constant ; then 
occurs a decrease of strength which amounts at 70 feet to 20 to 40 per cent 
of that of the butt-log. 

4. In shearing parallel with the grain and crushing across and parallel 
with the grain, practically no difference was found. 

5. Large beams appear 10 to 20 per cent weaker than small pieces. 

6. Compression tests endwise seem to furnish the best average statement 
of the value of wood, and if one test only can be made, this is the safest, as 
was also recognized by Bauschinger. 

7. Bled timber is in no respect inferior to unbled timber. 



310 



STRENGTH OF MATERIALS. 



The figures for crushing across the grain represent the load required to 
cause a compression of 15 per cent. The relative elastic resilience, in inch- 
pounds per cubic inch of the material, is obtained by measuring the area 
of the plotted-strain diagram of the transverse test from the origin to the 
point in the curve at which the rate of deflectiou is 50 per cent greater than 
the rate in the earlier part of the test where the diagram is a straight line. 
This point is arbitrarily chosen since there is no definite "elastic limit " in 
timber as there is in iron. The "strength at the elastic limit" is the 
strength taken at this same point. Timber is not perfectly elastic for any 
load if left on any great length of time. 

The long-leaf pine is found in all the Southern coast states from North 
Carolina to Texas. Prof. Johnson says it is probably the strongest timber 
in large sizes to be had in the United States. In small selected specimens, 
other species, as oak and hickory, may exceed it in strength and tough- 
ness. The other Southern yellow pines, viz., the Cuban, short-leaf and 
the loblolly pines are inferior to the long-leaf about in the ratios of their 
specific gravities ; the long-leaf being the heaviest of all the pines. It 
averages (kiln-dried) 48 pounds per cubic foot, the Cuban 47, the short-leaf 
40, and the loblolly 34 pounds. 

Strength, of Spruce Timber,— The modulus of rupture of spruce 
is given as follows by different authors : Hatfield, 9900 lbs. per square inch ; 
Rankine, 11,100 ; Laslett, 9045 ; Trautwine, 8100 ; Rodman, 6168. Traut- 
wine advises for use to deduct one-third in the case of knotty and poor 
timber. 

Prof. Lanza, in 25 tests of large spruce beams, found a modulus of 
rupture from 2995 to 5666 lbs.; the average being 4613 lbs. These were 
average beams, ordered from dealers of good repute. Two beams of 
selected stock, seasoned four years, gave 7562 and 8748 lbs. The modulus 
of elasticity ranged from 897,000 to 1,588,000, averaging 1,294,000. 

Time tests show much smaller values for both modulus of rupture and 
modulus of elasticity. A beam tested to 5800 lbs. in a screw machine was 
left over night, and the resistance was found next morning to have dropped 
to about 3000, and it broke at 3500. 

Prof. Lanza remarks that while it was necessary to use larger factors of 
safety, when the moduli of rupture w r ere determined from tests with smaller 
pieces, it will be sufficient for most timber constructions, except in factories, 
to use a factor of four. For breaking strains of beams, he states that it is 
better engineering to determine as the safe load of a timber beam the load 
that will not deflect it more than a certain fraction of its span, say about 
1/300 to 1/400 of its length. 

Properties of Timber. 

(N. J. Steel & Iron Co.'s Book.) 



Description. 



Ash 

Beech 

Cedar 

Cherry 

Chestnut , 

Elm 

Hemlock 

Hickory 

Locust 

Maple 

Oak, White.... 

Oak, Live 

Pine, White.... 
Pine, Yellow.., 

Spruce 

Walnut, Black 



43 to 55, 
43 to 53.4 
50 to 56.8 



Weight 
per 
cubic 

foot, in 
lbs. 



4 i 

49 
45 to 54.5 

70 

30 
28.8 to 33 

' ' ' 42 



Tensile 

Strength 

per sq. inch, 

in lbs. 



Crushing 

Strength per 

sq. inch, 

in lbs. 



11,000 to 17,20" 
11,500 to 18,000 
10.300 to 11,400 



10,500 
13,400 to 13,489 

8,700 
12,800 to 18,000 
20,500 to 24,800 
10,500 to 10,584 
10,253 to 19,500 



10,000 to 12,000 
12,600 to 19,200 
10,000 to 19,500 
9,286 to 16.000 



4,400 to 9,363 
5,800 to 9,363 
5,600 to 6,000 



5,350 to 5,600 
5,831 to 10,331 

5.700 

8,925 
5,113 to 11,700 

8,150 
4,684 to 9,509 

6,850 
5,000 to 6,650 
5,400 to 9,500 
5,050 to 7,850 

7,500 



Relative 
Strength 
for Cross 
Breaking. 

White 
Pine- 100. 



130 to 180 
100 to 144 

55 to 63 
130 

96 to 123 
96 

88 to 95 
150 to 210 
132 to 227 
122 to 220 
130 to 177 
155 to 189 
100 . 

98 to 170 

86 to 110 



Shearing- 
Strength 
with the 
Grain, 
lbs. per 
sq. inch 



458 to 700 



367 to 647 
752 to 966 

225 to 423 

286 to 415 
253 to 374 



STftEKGTH OF TIMBER. 



311 



The above table should be taken with caution. The range of variation in 
the species is apt to be much greater than the figures indicate. See Johnson's 
tests on long-leaf pine, and Lanza's on spruce, above. The weight of yellow- 
pine in the table is much less than that given by Johnson. (W. K.) 

Compressive Strengths of American Woods, when slowly 
and carefully seasoned.— Approximate averages, deduced from many exper- 
iments made with the U. S. Government testing-machine at Watertown, 
Mass., by Mr. S. P. Sharpless, for the Census of 1880. Seasoned woods resist 
crushing much better than green ones; in many cases, twice as well. Differ- 
ent specimens of the same wood vary greatly. The strengths may readily 
vary as much as one-third part more or less from the average. 



Ash, red and white 

Aspen 

Beech 

Birch 

Buckeye 

Butternut 

Buttonwood 

(sycamore) 

Cedar, red 

Cedar,white (arbor- 
vitas) 

Catalpa (Ind.bean) 

Cherry, wild 

Chestnut.. . 

Coffee-tree, Ky 

Cypress, bald 

Elm, Am. or white 

i4 red 

Hemlock 

Hickory 

Lignum-vitce 

Linden, American, 
Locust: 

black and yellow. 

honey 

Mahogany ... 

Maple: 
broad-leafed, Ore, 



End- 
wise,* 
bs. pei 
sq. in. 



4400 
7000 
8000 
4400 
5400 

6000 
6000 

4400 

5000 
8000 
5300 
5200 
6000 
6800 
7700 
5300 
8000 
10000 
5000 



7000 
9000 



5300 



Side- 
wise^ 
lbs. per 



3000 
1400 
1900 
2600 
1400 
1600 

2600 
1000 



1300 
2600 
1600 
2600 
1200 
2600 
2600 
1100 
4000 
13000 
900 

4400 
2600 
5300 

2600 



Maple : 

sugar and black.. 

white and red.... 
Oak : 

white, post (or 
iron), swamp 
white, red, and 
black 

scrub and basket. 

chestnut and live 

pin 

Pine : 

white 

red or Norway 

pitch and Jersey 
scrub 

Georgia 

Poplar 

Sassafras 

Spruce, black 

" white 

Sycamore (button- 
wood 

Walnut : 

black 

white (butternut). 
Willow 



End- 



bs. per 
sq. in. 



7000 
6000 
7500 
6500 

5400 
6300 

5000 
8500 
5000 
5000 
5700 
4500 

6000 

8000 
5400 
4400 



Side- 
wise^ 
lbs. per 
sq. in. 



.01 



A 



1900 4300 
1300 2900 



1600 4000 
1700 4200 
1600 4500 
1300 3000 



1200 
1400 

2000 
2600 
1100 
2100 
1300 
1200 



2600 
1600 
1400 



* Specimens 1.57 ins. square X 12.6 ins. long. 

t Specimens 1.57 ins. square X 6.3 ins. long. Pressure applied at mid-length 
by a punch covering one-fourth of the length. The first column gives the 
loads producing an indentation of .01 inch, the second those producing an 
indentation of .1 inch. (See also page 306). 

Expansion of Timber Due to tlie Absorption of Water. 

(De Volson Wood, A. S. M. E., vol. x.) 
Pieces 36 X 5 in., of pine, oak, and chestnut, were dried thoroughly, and 
then immersed in water for 37 days. 
The mean per cent of elongation and lateral expansion were: 

Pine. Oak. Chestnut. 

Elongation, per cent 0.065 0.0S5 0.165 

Lateral expansion, per cent. . .2.6 3.5 3.65 

Expansion ofWood by Heat.— Trautwine gives for the expansion 
of white pine for 1 degree Fahr. 1 part in 440,530, or for 180 degrees 1 part in 
2447, or about one-third of the expansion of iron. 



3ia 



STRENGTH OF MATERIALS. 



Shearing Strength of American Woods, adapted tor 
Pins or Treenails. 

J. C. Trautwine {Jour. Franklin Inst.). (Shearing across the grain.) 



per sq. m. 

Ash .. 6280 

Beech 5223 

Birch ; 5595 

Cedar (white) 1372 

" 1519 

Cedar (Central American) 3410 

Cherry 2945 

Chestnut 1536 

Dogwood 6510 

Ebony 7750 

Gum 5890 

Hemlock 2750 

Locust 7176 



per sq. in. 

Hickory 6045 

" 7285 

Maple 6355 

Oak 4425 

Oak (live) 8480 

Pine (white) 2480 

Pine (Northern yellow 4340 

Pine (Southern yellow) . 5735 

Pine (very resinous yellow) 5053 

Poplar 4418 

Spruce 3255 

Walnut (black) 4728 

Walnut (common) 2830 



the: strength of brick, stone, etc. 

A great advance has recently been made in the manufacture of brick, in 
the direction of increasing their strength. Chas. P. Chase, in Engineering 
Neios, says: " Taking the tests as given in standard engineering books eight 
or ten years ago, we find in Trautwine the strength of brick given as 500 to 
4200 lbs. per sq. in. Now, taking recent tests in experiments made at 
Watertown Arsenal, the strength ran from 5000 to 22,000 lbs. per sq. in. In 
the tests on Illinois paving brick, by Prof. I. O. Baker, we find an average 
strength in hard paving brick of over 5000 lbs. per square inch. The average 
crushing strength of ten varieties of paving-brick much used in the West, I 
find to be 7150 lbs. to the square inch." 

A recent test of brick made by the dry-clay process at Watertown Areenal, 
according to Paving, showed an average compressive strength of 3972 lbs. 
per sq. in. In one instance it reached 4973 lbs. per sq. in. A test was made 
at the same place on a "fancy pressed brick." The first crack developed 
at a pressure of 305,000 lbs., and the brick crushed at 364,300 lbs., or 11,130 
lbs. per sq. in. This indicates almost as great compressive strength as 
granite paving-blocks, which is from 12,000 to 20,000 lbs. per sq. in. 

The following notes on bricks are from Trautwine's Engineer's Pocket- 
book : 

Strength of Brick.— 40 to 300 tons per sq. ft., 622 to 4668 lbs. per sq. in. 
A soft brick will crush under 450 to 600 lbs. per sq. in., or 30 to 40 tons per 
square foot, but a first-rate machine-pressed brick will stand 200 to 400 tons 
per sq. ft. (3112 to 6224 lbs. per sq. in.). 

"Weight of Bricks. — Per cubic foot, best pressed brick, 150 lbs.; good 
pi-essed brick, 131 lbs.; common hard brick, 125 lbs.; good common brick, 
118 lbs. ; soft inferior brick, 100 lbs. 

Absorption of Water.— A brick will in a few minutes absorb J^ to 
% lb. of water, the last being 1/7 of the weight of a hand-moulded one, or y & 
of its bulk. 

Tests of Bricks, full size, on flat side. (Tests made at Water- 
town Arsenal in 1883.)— The bricks were tested between flat steel buttresses. 
Compressed surfaces (the largest surface) ground approximately flat. The 
bricks were all about 2 to 2.1 inches thick, 7.5 to 8.1 inches long, and 3.5 to 
3.76 inches wide. Crushing strength per square inch: One lot ranged from 
11,056 to 16,734 lbs.; a second, 12,995 to 22,351; a third, 10,390 to 12,709. Other 
tests gave results from 5960 to 10,250 lbs. per sq. in. 

Crushing Strength of Masonry Materials. (From Howe's 
" Retaining- Walls.") 

tons per sq. ft. tons per sq. ft. 

Brick, best pressed. . 40 to 300 Limestones and marbles. 250 to 1000 

Chalk 20to 30 Sandstone 150 to 550 

Granite 300 to 1200 Soapstone 400 to 800 

Strength of Granite. — The crushing strength of granite is commonly 
rated at 12,000 to 15,000 lbs. per sq. in. when tested in two-inch cubes, and 
only the hardest and toughest of the commonly used varieties reach a 
strength above 20,000 lbs. Samples of granite from a quarry on the Con- 



STRENGTH OF LIME AND CEMENT MORTAR. 



313 



necticut River, tested at the Watertown Arsenal, have shown a strength of 
35,965 Jbs. per sq. in. (Engineering News, Jan. 12, 1893). 

Strength, of Avondale, Pa., Limestone— (Engineering Neivs, 
Feb. 9, 1893).— Crushing strength of 2-in. cubes: light stone 12,112, gray stone 
18,040. lbs. per sq. in. , 

Transverse test of lintels, tool-dressed, 42 in. between knife-edge bear- 
ings, load with knife-edge brought upon the middle between bearings: 
Gray stone, section 6 in. wide X 10 in. high, broke under a load of 20,950 lbs. 

Modulus of rupture 2,200 " 

Light stone, section 8J4 in. wide X 10 in. high, broke under 14,720 " 

Modulus of rupture 1,170 " 

Absorption.— Gray stone 051 of \% 

Light stone 052 of 1% 

Transverse Strength of Flagging. 

(N. J. Steel & Iron Co.'s Book.) 
Experiments made by R. G. Hatfield and Others. 
b = width of the stone in inches; d — its thickness in inches; I — distance 
between bearings in inches. 

The breaking loads in tons of 2000 lbs., for a weight placed at the centre 
of the space, will be as follows : 



Bluestone flagging 744 

Quincy granite 624 

Little Falls freestone. 576 

Belleville, N. J., freestone 480 

Granite (another quarry) 432 

Connecticut freestone 312 

Thus a block of Quincy granite 80 inches wide and 6 inches thick, resting 
on beams 36 inches in the clear, would be broken by a load resting midway 

between the beams = — — — X .624 == 49.92 tons. 



Dorchester freestone .264 

Aubigny freestone 216 

Caen freestone 144 

Glass 1.000 

Slate 1 .2 to 2.7 



STRENGTH OF LIME AND CEMENT MORTAR. 

{Engineering, October 2, 1891.) 
Tests made at the University of Illinois on the effects of adding cement to 
lime mortar. In all the tests a good quality of ordinary fat lime was used, 
slaked for two days in an earthenware jar, adding two parts by weight of 
water to one of lime, the loss by evaporation being made up by fresh addi- 
tions of water. The cements used were a German Portland, Black Diamond 
(Louisville), and Rosendale. As regards fineness of grinding, 85 per cent of 
the Portland passed through a No. 100 sieve, as did 72 per cent of the Rosen- 
dale. A fairly sharp sand, thoroughly washed and dried, passing through a 
No. 18 sieve and caught on a No. 30, was used. The mortar in all cases con- 
sisted of two volumes of sand to one of lime paste. The following results 
were obtained on adding various percentages of cement to the mortar: 

Tensile Strength, pounds per square inch. 



Age } 


4 


7 


14 


21 


28 


50 


84 


Days. 


Days. 


Days. 


Days. 


Days. 


Days. 


Days. 




4 
5 


8 

83^ 


10 
9Kj 


13 
12 


18 
17 


21 
17 


26 


20 per cent Rosendale.. 


18 


20 " " Portland.... 


5 


8y 2 


14 


20 


25 


24 


26 


30 " " Rosendale.. 


7 


ii 


13 


18^ 


21* 


22^ 


23 


30 " " Portland.... 


8 


16 


18 


22 


25 


28 


27 


4D " " Rosendale.. 
40 " " Portland.. . 


10 


12 


16^ 


mi 


22y 2 


24 


36 


27 


39 


38 


43 


47 


59 


57 


60 " " Rosendale.. 


9 


13 


20 


16 


22 


22^3 


23 


60 " " Portland.... 


45 


58 


55 


68 


67 


102 


78 


80 " " Rosendale.. 


12 


18J6 


22^ 


27 


29 


31^ 


33 


80 " " Portland.... 


87 


91 


103 


124 


94 


210 


145 


100 " " Rosendale.. 


18 


23 


26 


31 


34 


46 


48 


100 " " Portland.... 


90 


120 


146 


152 


181 


205 


202 



314 STRENGTH OF MATERIALS. 

MODULI OF ELASTICITY OF VARIOUS MATERIALS. 

The modulus of elasticity determined from a tensile test of a bar of any 
material is the quotient obtained by dividing the tensile stress in pounds per 
square inch at any point of the test by the elongation per inch of length 
produced by that stress ; or if P = pounds of stress applied, K = the sec- 
tional area, I = length of the portion of the bar in which the measure- 
ment is made, and A = the elongation in that length, the modulus, of 

elasticity E — — -*- - = — . The modulus is generally measured within the 

elastic limit only, in materials that have a well-defined elastic limit, such as 
iron and steel, and when not otherwise stated the modulus is understood to 
be the modulus within the elastic limit. Within this limit, for such materials 
the modulus is practically constant for any given bar, the elongation being 
directly proportional to the stress. In other materials, such as cast iron, 
which have no well-defined elastic limit, the elongations from the beginning 
of a test increase in a greater ratio than the stresses, and the modulus is 
therefore at its maximum near the beginning of the test, and continually 
decreases. The moduli of elasticity of various materials have already been 
given above in treating of these materials, but the following table gives 
some additional values selected from different sources : 

Brass, cast 9,170,000 

wire 14,230,000 

Copper 15,000,000 to 18,000,000. 

Lead 1,000,000 

Tin, cast 4,600,000 

Iron, cast 12,000,000 to 27,000,000 (?) 

Iron, wrought 22,000,000 to 29,000,000 

Steel 26,000,000 to 32,000,000 

Marble 25,000,000 

Slate 14,500,000 

Glass 8,000,000 

Ash 1,600,000 

Beech 1,300,000 

Birch 1,250,000 to 1,500,000 

Fir 869,000 to 2,191,000 

Oak 974,000 to 2,283.000 

Teak 2,414,000 

Walnut 306,000 

Pine, long-leaf (butt-logs)... 1,119,000 to 3,117,000 A vge. 1,926,000 
The maximum figures given by many writers for iron and steel, viz., 
40,000,000 and 42,000,000, are undoubtedly erroneous. 

Prof. J. B. Johnson, in his report on Long-leaf Pine, 1893, says: "The 
modulus of elasticity is the most constant and reliable property of all 
engineering materials. The wide range of value of the modulus of elasticity 
of the various metals found in public records must be explained by erro- 
neous methods of testing.' 1 

In a tensile test of cast iron by the author (Van Noetrand's Science Series, 
No. 41, page 45), in which the ultimate strength was 23,285 lbs. per sq. in., 
the measurements of elongation were made; + o .0001 inch, and the modulus 
of elasticity was found to decrease from the beginning of the test, as 
follows: At 1000 lbs. per sq. in., 25,000,000 ; at 2000 lbs., 16,666,000 ; at 4000 
lbs.. 15,384,000 ; at 6000 lbs., 13,636,000; at 8000 lbs., 12,500,000 ; at 12,000 lbs., 
11,250,000; at 15,000 lbs., 10,000,000 ; at 20,000 lbs., 8.000.000; at 23,000 lbs., 
6,140,000. The modulus of elasticity of steel (within the elastic limit) is 
remarkably constant, notwithstanding great variations in chemical analysis, 
temper, etc. It rarely is found below 28,000,000 or above 31,000,000. It is 
generally taken at 30,000,000 in engineering calculations. 
FACTORS OF SAFETY. 
A factor of safety is the ratio in which the load that is just sufficient to 
overcome instantly the strength of a piece of material is greater than the 
greatest safe ordinary working load. (Rankine.) 

Rankine gives the following " examples of the values of those factors 
which occur in machines ": 

Dead T oad Live Load > Live Load » 

Dead Load. Greatest> Mean . 

Iron and steel 3 6 from 6 to 40 

Timber 4 to 5 8 to 10 

Masonry ,. 4 8 



FACTORS OF SAFETY. 315 

The great factor of safety, 40, is for shafts in millwork which transmit 
very variable efforts. 

Unwin gives the following "factors of safety which have been adopted in 
certain cases for different materials." They " include an allowance for 
ordinary contingencies." 

T)pad ' Live Load - > 

T oad In Temporary In Permanent In Structures 

• Structures. Structures, subj. to Shocks. 

Wrought iron and steel. 3 4 4 to 5 10 

Cast iron 3 4 5 10 

Timber 4 10 

Brickwork 6 .... 

Masonry 20 .... 20 to 30 

Unwin says says that " these numbers fairly represent practice based on 
experience in many actual cases, but they are not very trustworthy." 

Prof. Wood in his " Resistance of Materials " says : " In regard to the 
margin that should be left for safety, much depends upon the character of 
the loading. If the load is simply a dead weight, the margin may be com- 
paratively small; but if the structure is to be subjected to percussive forces 
or shocks, the margin should be comparatively large on account of the 
indeterminate effect produced by the force. In machines which are sub- 
jected to a constant jar while in use, it is very difficult to determine the 
proper margin which is consistent with economy and safety. Indeed, in 
such cases, economy as well as safety generally consists in making them 
excessively strong, as a single breakage may cost much more than the extra 
material necessary to fully insure safety." 

For discussion of the resistance of materials to repeated stresses and 
shocks, see pages 238 to 240. 

Instead of using factors of safety it is becoming customary in designing 
to fix a certain number of pounds per square inch as the maximum stress 
which will be allowed on a piece. Thus, in designing a boiler, instead of 
naming a factor of safety of 6 for the plates and 10 for the stay-bolts, the 
ultimate tensile strength of the steel being from 50,000 to 60,0001bs. persq. in., 
an allowable working stress of 10,000 lbs. per sq. in. on the plates and 6000 
lbs. per sq. in. on the stay-bolts may be specified instead. So also in 
Merriman's formula for columns (see page 260) the dimensions of a column 
are calculated after assuming a maximum allowable compressive stress per 
square inch on the concave side of the column. 

The factors for masonry under dead load as given by Rankine and by Unwin, 
viz., 4 and 20, show a remarkable difference, which may possibly be explained 
as follows : If the actual crushing strength of a pier of masonry is known 
from direct experiment, then a factor of safety of 4 is sufficient for a pier of 
the same size and quality under a steady load; but if the crushing strength 
is merely assumed from figures given by the authorities (such as the crush- 
ing strength of pressed brick, quoted above from Howe's Retaining Walls, 40 
to 300 tons per square foot, average 170 tons), then a factor of safety of 20 
may be none too great. In this case the factor of safety is really a " factor 
of ignorance." 

The selection of the proper factor of safety or the proper maximum unit 
stress for any given case is a matter to be largely determined by the judg- 
ment of the engineer and by experience. No definite rules can be given. 
The customary or advisable factors in many particular cases will be found 
where these cases are considered throughout this book. In general the 
following circumstances are to be taken into account in the selection of 
a factor : 

1. When the ultimate strength of the material is known within narrow 
limits, as in the case of structural steel when tests of samples have been 
made, when the load is entirely a steady one of a known amount, and there 
is no reason to fear the deterioration of the metal by corrosion, the lowest 
factor that should be adopted is 3. 

2. When the circumstances of 1 are modified by a portion of the load being 
variable, as in floors of warehouses, the factor should be not less than 4. 

3. When the whole load, or nearly the whole, is apt to be alternately put 
on and taken off, as in suspension rods of floors of bridges, the factor should 
be 5 or 6. 

4. When the stresses are reversed in direction from tension to compres- 
sion, as in some bridge diagonals and parts of machines, the factor should 
be not less than 6. 



316 STRENGTH OF MATERIALS. 

5. When the piece is subjected to repeated shocks, the factor should be 
not less than 10. 

6. When the piece is subject to deterioi'ation from corrosion the section 
should be sufficiently increased to allow for a definite amount of corrosion 
before the piece be so far weakened by it as to require removal. 

7. When the strength of the material, or the amount of the load, or both 
are uncertain, the factor should be increased by an allowance sufficient to 
cover the amount of the uncertainty. 

8. When the strains are of a complex character and of uncertain amount, 
such as those in the crank-shaf t of a reversing engine, a very high factor is 
necessary, possibly even as high as 40, the figure given by Rankine for shafts 
in millwork. 

THE MECHANICAL PROPERTIES OF CORK. 

Cork possesses qualities which distinguish it from all other solid or liquid 
bodies, namely, its power of altering its volume in a very marked degree in 
consequence of change of pressure. It consists, practically, of an aggrega- 
tion of minute air-vessels, having thin, water-tight, and very strong walls, 
and hence, if compressed, the resistance to compression rises in a manner 
more like the resistance of gases than the resistance of an elastic solid such 
as a spring. In a spring the pressure increases in proportion to the dis- 
tance to which the spring is compressed, but with gases the pressure in- 
creases in a much more rapid manner; that is, inversely as the volume 
which the gas is made to occupy. But from the permeability of cork to 
air, it is evident that, if subjected to pressure in one direction only, it will 
gradually part with its occluded air by effusion, that is, by its passage 
through the porous walls of the cells in which it is contained. The gaseous 
part of cork constitutes 53$ of its bulk. Its elasticity has not only a very 
considerable range, but it is very persistent. Thus in the better kind of corks 
used in bottling the corks expand the instant they escape from the bottles. 
This expansion may amount to an increase of volume of 75%, even after the 
corks have been kept in a state of compression in the bottles for ten years. 
If the cork be steeped in hot water, the volume continues to increase till 
it attains nearly three times that which it occupied in the neck of the bottle. 

When cork is subjected to pressure a certain amount of permanent defor- 
mation or "permanent set" takes place very quickly. This property is 
common to all solid elastic substances when strained beyond their elastic 
limits, but with cork the limits are comparatively low. Besides the perma- 
nent set, there is a certain amount of sluggish elasticity— that is, cork on; 
being released from pressure springs back a certain amount at once, but 
the complete recovery takes an appreciable time. 

Cork which had been compressed and released in water many thousand 
times had not changed its molecular structure in the least, and had contin- 
ued perfectly serviceable. Cork which has been kept under a pressure of 
three atmospheres for many weeks appears to have shrunk to from 80$ to 
85% of its original volume.— Van NostrancVs Eng'g Mag. 1886, xxxv. 307. 
TESTS OF VULCANIZED INDIA-RUBBER. 

Lieutenant L. Vladomiroff, a Russian naval officer, has recently carried 
out a series of tests at the St. Petersburg Technical Institute with a view to 
establishing rules for estimating the quality of vulcanized india-rubber. 
The following, in brief, are the conclusions arrived at, recourse being had 
to physical properties, since chemical analysis did not give any reliable re- 
sult: 1. India-rubber should not give the least sign of superficial cracking 
when bent to an angle of 180 degrees after five hours of exposure in a closed 
air-bath to a temperature of 125° C. The test-pieces should be 2.4 inches 
thick. 2. Rubber that does not contain more than half its weight of metal- 
lic oxides should stretch to five times its length without breaking. 3. Rub- 
ber free from all foreign matter, except the sulphur used in vulcanizing it, 
should stretch to at least seven times its length without rupture. 4. The 
extension measured immediately after rupture should not exceed \2% of the 
original length, with given dimensions. 5. Suppleness may be determined 
by measuring the percentage of ash formed in incineration. This may form 
the basis for deciding between different grades of rubber for certain pur- 
poses. 6. Vulcanized rubber should not harden under cold. These rules 
have been adopted for the Russian navy.— Iron Age, June 15, 1893. 

XTLOLITH, OR WOODSTONE 
is a material invented in 1883, but only lately introduced to the trade by 
Otto Serrig & Co., of Pottschappel, near Dresden. It is made of magnesia 



ALITMIHUM — ITS PROPERTIES AHD USES. 317 

cement, or calcined magnesite, mixed with sawdust and saturated with a 
solution of chloride of calcium. This pasty mass is spread out into sheets 
and submitted to a pressure of about, 1000 ibs. to the square inch, and then 
simply dried in the air. Specific gravity 1.553. The fractured surface shows 
a uniform close grain of a yellow color. It has a tensioual resistance when 
dry of 100 lbs. per square inch, and when wet about 66 lbs. When immersed 
in water for 12 hours it takes up 2.1$ of its weight, and 3.8$ when immersed 
216 hours. 

When treated for several days with hydrochloric acid it loses 2.3$ in 
weight, and shows no loss of weight under boiling in water, brine, soda-lye, 
and solution of sulphates of iron, of copper, and of ammonium. In hardness 
the material stands between feldspar and quartz, and as a non-conductor of 
heat it ranks between asbestos and cork. 

It stands fire well, and at a red heat it is rendered brittle and crumbles at 
the edges, but retains its general form and cohesion. This xylolith is sup- 
plied in sheets from *4 in - to iy Q in. thick, and up to one metre square. It 
is extensively used in Germany for floors in railway stations, hospitals, etc., 
and for decks of vessels. It can be sawed, bored, and shaped with ordinary 
woodworking tools. Putty in the joints and a good coat of paint make it 
entirely water-proof. It is sold in Germany for flooring at about 7 cents per 
square foot, and the cost of laying adds about 4 cents more.— Eng'g News, 
July 28, 1892, and July 27, 1893. 

ALUMINUM-ITS PROPERTIES AND USES. 
(By Alfred E. Hunt, Pres't of the Pittsburgh Reduction Co.) 

The specific gravity of pure aluminum in a cast state is 2.58 ; in rolled 
bars of large section it is 2 6 ; in very thin sheets subjected to high com- 
pression under chilled rolls, it is as much as 2.7. Taking the weight of a 
given bulk of cast aluminum as 1, wrought iron is 2.90 times heavier ; struc- 
tural steel, 2.95 times ; copper, 3.60 ; ordinary high brass, 3.45. Most wood 
suitable for use in structures has about one third the weight of aluminum, 
which weighs 0.092 lb. to the cubic inch. 

Pure aluminum is practically not acted upon by boiling water or steam. 
Carbonic oxide or hydrogen sulphide does not act upon it at any tempera- 
ture under 600° F. It is not acted upon by most organic secretions. 

Hydrochloric acid is the best solvent for aluminum, and strong solutions 
of caustic alkalies readily dissolve it. Ammonia has a slight solvent action, 
and concentrated sulphuric acid dissolves aluminum upon heating, with 
evolution of sulphurous acid gas. Dilute sulphuric acid acts but slowly on 
the metal, though the presence of any chlorides in the solution allow rapid 
decomposition. Nitric acid, either concentrated or dilute, has very little 
action upon the metal, and sulphur has no action unless the metal is at a red 
heat. Sea-water has very little effect on aluminum. Strips of the metal 
placed on the sides of a wooden ship corroded less than 1/1000 inch after six 
months' exposure to sea- water, corroding less than copper sheets similarly 



In malleability pure aluminum is only exceeded by gold and silver. In 
ductility it stands seventh in the series, being exceeded by gold, silver, 
platinum, iron, very soft steel, and copper. Sheets of aluminum have been 
rolled down to a thickness of 0.0005 inch, and beaten into leaf nearly as 
thin as gold leaf. The metal is most malleable at a temperature of between 
400° and 600° F., and at this temperature it can be drawn down between 
rolls with nearly as much draught upon it as with heated steel. It has also 
been drawn down into the very finest wire. By the Mannesmann process 
aluminum tubes have been made in Germany. 

Aluminum stands very high in the series as an electro-positive metal, and 
contact with other metals should be avoided, as it would establish a galvanic 
couple. 

The electrical conductivity of aluminum is only surpassed by pure copper, 
silver, and gold. With silver taken at 100 the electrical conductivity of 
aluminum is 54.20 ; that of gold on the same scale is 78; zinc is 29.90; iron is 
only 16, and platinum 10.60. Pure aluminum has no polarity, and the 
metal in the market is absolutely non-magnetic. 

Sound castings can be made of aluminum in either dry or " green " sand 
moulds, or in metal "chills.'" It must not be heated much beyond its 
melting-point, and must be poured with care, owing to the ready absorption 
of occluded gases and air. The shrinkage in cooling is 17/64 inch per foot, 
or a little more than ordinary brass. It should be melted in plumbago 
crucibles, and the metal becomes molten at a temperature of 1120° F. ac- 
cording to Professor Roberts-Austen, or at 1300° F. according to Richards. 



318 STRENGTH OP MATERIALS. 

The coefficient of linear expansion, as tested on %-ineh round aluminum 
rods, is 0.00002295 per degree centigrade between the freezing and boiling 
point of water. The mean specific heat of aluminum is higher than that of 
any other metal, excepting only magnesium and the alkali metals. From 
zero to the melting-point it is 0.2185; water being taken as 1, and the latent 
heat of fusion at 28.5 heat units. The coefficient of thermal conductivity of 
unannealed aluminum is 37.96; of annealed aluminum, 38.37. As a conductor 
of heat aluminum ranks fourth, being exceeded only by silver, copper, and 
gold. 

Aluminum, under tension, and section for section, is about as strong as 
cast iron. The tensile strength of aluminum is increased by cold rolling or 
cold forging, and there are alloys which add considerably to the tensile 
strength without increasing the specific gravity to over 3 or'3.25. 

The strength of commercial aluminum is given in the following table as 
the result of many tests : 

Elastic Limit Ultimate Strength Percentage 
per sq. in. in per sq. in. in of Reducfn 

Form. Tension, Tension, of Area in 

lbs. lbs. Tension. 

Castings 6,500 15,000 15 

Sheet 12,000 24,000 35 

Wire 16,000-30,000 30,000-65,000 60 

Bars 14,000 28,000 40 

The elastic limit per square inch under compression in cylinders, with 
length twice the diameter, is 3500. The ultimate strength per square inch 
under compression in cylinders of same form is 12,000. The modulus of 
elasticity of cast aluminum is about 11,000,000. It is rather an open metal in 
its texture, and for cylinders to stand pressure an increase in thickness must 
be given to allow for this porosity. Its maximum shearing stress in castings 
is about 12,000, and in forgings about 16,000, or about that of pure copper. 

Pure aluminum is too soft and lacking in tensile strength and rigidity for 
many purposes. Valuable alloys are now being made which seem to give 
great promise for the future. They are alloys containing from 2% to 7$ or 8$ 
of copper, manganese, iron, and nickel. As nickel is one of the principal 
constituents, these alloys have the trade name of " Nickel-aluminum." 

Plates and bars of this nickel alloy have a tensile strength of from 40,000 to 
50,000 pounds per square inch, an elastic limit of 55$ to 60$ of the ultimate ten- 
sile strength, an elongation of 20$ in 2 inches, and a reduction of area of 25$. 

This metal is especially capable of withstanding the punishment and 
distortion to which structural material is ordinarily subjected. Nickel- 
aluminum alloys have as much resilience and spring as the very hardest of 
hard-drawn brass. 

Their specific gravity is about 2.80 to 2.85, where pure aluminum has a 
specific gravity of 2.72. 

In castings, more of the hardening elements are necessary in order to give 
the maximum stiffness and rigidity, together with the strength and ductility 
of the metal; the favorite alloy material being zinc, iron, manganese, and 
copper. Tin added to the alloy reduces the shrinkage, and alloys of alumi- 
num and tin can he made which have less shrinkage than cast iron. 

The tensile strength of hardened aluminum-alloy castings is from 20,000 
to 25,000 pounds per square inch. 

Alloys of aluminum and copper form two series, both valuable. The 
first is aluminum-bronze, containing from 5$ to 11^>$ of aluminum; and the 
second is copper-hardened aluminum, containing from 2% to 15$ of copper. 
Aluminum-bronze is a very dense, fine-grained, aud strong alloy, having good 
ductility as compared with tensile strength. The 10$ bronze in forged bars 
will give 100,000 lbs. tensile strength per square inch, with 60,000 lbs. elastic 
limit per square inch, and 10$ elongation in 8 inches. The 5$ to 7J^$ bronze 
has a specific gravity of 8 to 8.30, as compared with 7.50 for the 10$ to \\y>% 
bronze, a tensile strength of 70,000 to 80,000 lbs., an elastic limit of 40,000 
lbs. per square inch, and an elongation of 30$ in 8 inches. 

Aluminum is used by steel manufacturers to prevent the retention of the 
occluded gases in the steel, and thereby produce a solid ingot. The propor- 
tions of the dose range from y% lb. to several pounds of aluminum per ton of 
steel. Aluminum is also used in giving extra fluidity to steel used in castings, 
making them sharper and sounder. Added to cast iron, aluminum causes 
the iron to be softer, free from shrinkage, and lessens the tendency to "chill.'" 

With the exception of lead and mercury, aluminum unites with all metals, 



ALLOYS. 



319 



though it unites with antimony with great difficulty. A small percentage 
of silver whitens and hardens the metal, and gives it added strength; and 
this alloy is especially applicable to the manufacture of fine instruments 
and apparatus. The following alloys have been found recently to be useful 
in the arts: Nickel-aluminum, composed of 20 parts nickel to 8 of aluminum; 
rosine, made of 40 parts nickel, 10 parts silver, 30 parts aluminum, and 20 
parts tin, for jewellers'' work; mettaline, made of 35 parts cobalt, 25 parts 
aluminum, 10 parts iron, and 30 parts copper. The aluminum-bourbounz 
metal, shown at the Paris Exposition of 1889, has a specific gravity of 2.9 to 
2.96, and can be cast in very solid shapes, as it has very little shrinkage. 
From analysis the following composition is deduced: Aluminum, 85. 74$; tin, 
12.94^; silicon, 1.32$; iron, none. 

The metal can be readily electrically welded, but soldering is still not sat- 
isfactory. The high heat conductivity of the aluminum withdraws the heat 
of the molten solder so rapidly that it " freezes " before it can flow suffi- 
ciently. A German solder said to give good results is made of 80# tin to 20# 
zinc, using a flux composed of 80 parts stearic acid, 10 parts chloride of 
zinc, and 10 parts of chloride of tin. Pure tin, fusing at 250° C, has also 
been used as a solder. The use of chloride of silver as a flux has been 
patented, and used with ordinary soft solder has given some success. A 
pure nickel soldering-bit should be used, as it does not discolor aluminum 
as copper bits do. 

ALLOYS. 

ALLOYS OF COPPER AND TIN. 

(Extract from Report of U. S. Test Board.*) 





Mean Com- 
position by 


tcfl 
c ~ 

Eg 

W u 

£& 

§.3 


o p, 




05 

CD 

§»« 

CD~ 3 
> S-p 

cos 

H 


5 o 

c. c * 
'+3?' a3 

"gffl.9 

Q 


d 

.Sep. 
jq <d . 

o 


Torsion 

Tests. 


4= 

s 


Analysis. 


3 £% 

e3 O c 




Cop- 
per. 


Tin. 


O § CD 

.2 'is 


1 

la 

9, 


100. 
100. 
97.89 
96.06 
94.11 
92.11 
90.27 
88.41 
87.15 
82.70 
80.95 
77.56 
76.63 
72.89 
69.84 
68.58 
67.87 
65.34 
56.70 
44.52 
34.22 
23.35 
15.08 
11.49 
8.57 
3.72 
0. 


"i!90* 
3.76 
5.43 

7.80 
9.58 
11.59 
12.73 
17.34 
18.84 
22.25 
23.24 
26.85 
29.88 
31.26 
32.10 
34.47 
43.17 
55.28 
65.80 
76.29 
84.62 
88.47 
91.39 
96.31 
100. 


27,800 
12,760 
24,580 
32,000 


14,000 
11,000 
10,000 
16,000 


6.47 
0.47 
13.33 
14.29 


29,848 
21,251 


bent. 
2.31 


42,000 
39,000 
34,000 
42,048 


143 
65 
150 
157 


153 
40 
317 


3 
4 


33,232 

38,659 
43,731 
49,400 
60,403 
34,531 
67,930 
56,715 
29,926 
32.210 
9,512 
12,076 
9,152 
9,477 
4,776 
2,126 
4,776 
5,384 
12,408 
9,063 
10,706 
5,305 
6,925 
3,740 


bent. 

4.00 
0.63 
0.49 
0.16 
0.19 
0.05 
0.06 
0.04 
0.05 
0.02 
0.02 
0.03 
0.04 
0.27 
0.86 
5.85 
bent. 


247 


5 
6 


28,540 
26,860 


19,000 
15,750 


5.53 
3.66 


42,000 
38,000 


160 
175 


126 

114 


8 
9 


29,430 


20,000 


3.33 


53,666 


182 


100 


10 
11 


32,980 




0.04 

0. 

0. 

0. 

0. 

0. 

0. 

0. 

0. 

0. 

0. 

0. 


78,000 


190 


16 


12 
13 


22,010 


22,010 


114,000 


122 


3.4 


14 

15 


5,585 


5,585 


147,000 


18 


1.5 


16 












17 

18 


2,201 
1,455 
3.010 
3,371 
6,775 


2,201 
1,455 
3,010 
3,371 
6,775 


84,700 


16 


1 


19 
20 
21 


35.800 
19,600 


23 
17 


1 

2 


W, 


6,500 
10,100 
9,800 
9,800 
6,400 


2-5 
23 
23 
23 

12 


25 


23 

24 
25 
26 


6,380 
6,450 
4,780 
3,505 


3,500 
3,500 
2,750 


4.10 
6.87 
12.32 
35.51 


62 
132 
220 
557 



* The tests of the alloys of copper and tin and of copper and zinc, the re- 
sults of which are published in the Report of the U. S. Board appointed to 
test Iron, Steel, and other Metals, Vols. I and II, 1879 and 1881, were made 
by the author under direction of Prof. R. H. Thurston, chairman of the 
Committee on Alloys. See preface to the report of the Committee, in Yol. J, 



320 ALLOYS. 

Nos. la and 2 were full of blow-holes. 

Tests Nos. 1 and la show the variation in cast copper due to varying con- 
ditions of casting. In the crushing tests Nos. 12 to 20, inclusive, crushed and 
broke under the strain, but all the others bulged and flattened out. In these 
cases the crushing strength is taken to be that which caused a decrease of 
10% in the length. The test-pieces were 2 in. long and % in. diameter. The 
torsional tests were made in Thurston's torsion-machine, on pieces % in. 
diameter and 1 in. long between heads. 

Specific Gravity ©f the Copper-tin Alloys.— The specific 
gravity of copper, as found in these tests, is 8.874 (tested in turnings from 
the ingot, and reduced to 39.1° F.). The alloy of maximum sp. gr. 8.956 
contained 62.42 copper, 37.48 tin, and all the alloys containing less than 37$ 
tin varied irregularly in sp. gr. between 8.65 and 8.93, the density depending 
not on the composition, but on the porosity of the casting. It is probable 
that the actual sp. gr. of all these alloys containing less than 37$ tin is about 
8.95, and any smaller figure indicates porosity in the specimen. 

From 37$ to 100$ tin, the sp. gr. decreases regularly from the maximum of 
8.956 to that of pure tin, 7.293. 

Note on the Strength of the Copper-tin Alloys. 

The bars containing from 2$ to 24$ tin, inclusive, have considerable 
strength, and all the rest are practically worthless for purposes in which 
strength is required. The dividing line between the strong and brittle alloys 
is precisely that at which the color changes from golden yellow to silver- 
white, viz., at a composition containing between 24$ and 30$ of tin. 

It appears that the tensile and compressive strengths of these alloys are 
in no way related to each other, that the torsional strength is closely pro- 
portional to the tensile strength, and that the transverse strength may de- 
pend in some degree upon the compressive strength, but it is much more 
nearly related to the tensile strength. The modulus of rupture, as obtained 
by the transverse tests, is, in general, a figure between those of tensile and 
compressive strengths per square inch, but there are a few exceptions in 
which it is larger than either. 

The strengths of the alloys at the copper end of the series increase rapidly 
with the addition of tin till about 4$ of tin is reached. The transverse 
strength continues regularly to increase to the maximum, till the alloy con- 
taining about 17J^$ of tin is reached, while the tensile and torsional 
strengths also increase, but irregularly, to the same point. This irregularity 
is probably due to porosity of the metal, and might possibly be removed by 
any means which would make the castings more compact. The maximum 
is reached at the alloy containing 82.70 copper, 17.34 tin, the transverse 
strength, however, being very much greater at this point than the tensile 
or torsional strength. From the point of maximum strength the figures 
drop rapidly to the alloys containing about 27.5$ of tin, and then more slowly 
to 37.5$, at which point the minimum (or nearly the minimum) strength, by 
all three methods of test, is reached. The alloys of minimum strength are 
found from 37.5$ tin to 52.5$ tin. The absolute minimum is probably about 
45$ of tin. 

From 52.5$ of tin to about 77.5$ tin there is a rather slow and irregular in- 
crease in strength. From 77.5$ tin to the end of the series, or all tin, the 
strengths slowly and somewhat irregularly decrease. 

The results of these tests do not seem to corroborate the theory given by 
some writers, that peculiar properties are possessed by the alloys which 
are compounded of simple multiples of their atomic weights or chemical 
equivalents, and that these properties are lost as the compositions vary 
more or less from this definite constitution. It does appear that a certain 
percentage composition gives a maximum strength and another certain 
percentage a minimum, but neither of these compositions is represented by 
simple multiples of the atomic weights. 

There appears to be a regular law of decrease from the maximum to the 
minimum strength which does not seem to have any relation to the atomic 
proportions, but only to the percentage compositions. 

Hardness.— The pieces containing less than 24$ of tin were turned in 
the lathe without difficulty, a gradually increasing hardness being noticed, 
the last named giving a very short chip, and requiring frequent sharpening 
of the tool. 

With the most brittle alloys it was found impossible to turn the test-pieces 
in the lathe to a smooth surface. No. 13 to No. 17 (26.85 to 34.47 tin) could 
not be cut with a tool at all. Chips would fly off in advance of the tool and 



ALLOYS OF COPPER AND ZINC. 



321 



beneath it, leaving a rough surface; or the tool would sometimes, apparently, 
crush off portions of the metal, grinding it to powder. Beyond 40$ tin the 
hardness decreased so that the bars could be easily turned. 

ALLOYS OF COPPER AND ZINC. (U. S. Test Board). 









Elastic 


•fes. . 


Trans- 


„ 




Torsional 




Mean Com- 




Limit 


- K 


?H^ 


Crush- 


Tests. 




position by 


Tensile 


$of 


O Jh 


verse 
Test 
Modu- 
lus of 
Rup- 
ture. 


fl£* . 




■ 


No. 


Analysis. 


Strength, 
lbs. per 
sq. in. 


Break- 
ing 
Load, 
lbs. per 
sq. in. 


be-- 


.2 s- .2 
C -a 


ing 
Str'gth 
per sq. 
in., lbs. 


° S S 


o§ 




Cop- 
per. 


Zinc. 




1 


97.83 
82.93 


1.88 
16.98 


27,240 
32,600 


"26!i" 


36 1 7 


'23,197 


- 




130 
155 


357 


2 


Bent 




329 


3 


81.91 


17.99 


32,670 


30.6 


31.4 


21,193 


" 




166 


345 


4 


77.39 


22.45 


35,630 


20.0 


35.5 


25,374 


" 




169 


311 


5 


76.65 


23.08 


30,520 


24.6 


35.8 


22,325 


" 


' 42,000 


165 


267 


6 


73.20 


26.47 


31,580 


23.7 


:^8.5 


25,894 


" 




168 


293 


7 


71.20 


28.54 


30,510 


29.5 


21). 2 


24,468 


" 




164 


269 


8 


69.74 


30.06 


28,120 


28.7 


20.7 


26,930 


" 




143 


202 


9 


66.27 


33.50 


37,800 


25.1 


37.7 


28,459 


' k 




176 


257 


10 


63.44 


36.36 


48,300 


32.8 


31.7 


43,216 


" 




202 


230 


11 


60.94 


38.65 


41,065 


40.1 


20.7 


38,968 


" 


' 75,000 


194 


202 


12 


58.49 


41.10 


50,450 


54.4 


10.1 


63,C04 


" 




227 


93 


13 


55.15 


44.44 


44,280 


44.0 


15.3 


42,463 


" 


' 78,000 


209 


109 


14 


54.86 


44.78 


46,400 


53.9 


8.0 


47,955 


" 




223 


72 


15 


49. 6G 


50.14 


30,990 


54.5 


5.0 


33,467 


1.26 


117,400 


172 


38 


16 


48.99 


50.82 


26,050 


100. 


0.8 


40,189 


0.61 




176 


16 


17 


47.56 


52.28 


24,150 


100. 


0.8 


48,471 


1.17 


121,000 


155 


13 


18 


43.36 


56.22 


9,170 


100. 




17,691 


0.10 




88 


2 


19 


41.30 


58.12 


3,727 


100. 




7,761 


0.04 




18 


2 


20 


32.94 


66.23 


1,774 


100. 




8,296 


0.04 




29 


1 


21 


29.20 


70.17 


6,414 


100. 




16,579 


0.04 




40 


2 


22 


20.81 


77.63 


9,000 


100. 


'o.k 


22,972 


0.13 


' 52,152 


65 


1 


23 


12.12 


86.67 


12,413 


100. 


4 


35,026 


0.31 




82 


3 


24 


4.35 


94.59 


18,065 


100. 


0.5 


26,162 


0.46 




81 


22 


25 


Cast Zinc. 


5.400 


75. 


0.7 


7.539 


0.12 


' 22,000 


37 


142 



Variation in Strength of Gun-bronze, and Means of 
Improving the Strength.— The figures obtained for alloys of from 
7.8$ to 12.7$ tin, viz., from 26,860 to 29,430 pounds, are much less than are 
usually given as the strength of gun-metal. Bronze guns are usually cast 
under the pressure of a head of metal, which tends to increase the strength 
and density. The strength of the upper part of a gun casting, or sinking 
head, is not greater than that of the small bars which have been tested in 
these experiments. The following is an extract from the report of Major 
Wade concerning the strength and density of gun-bronze (1850): — Extreme 
variation of six samples from different parts of the same gun (a 32-pounder 
howitzer): Specific gravity, 8.487 to 8.835; tenacity, 26,428 to 52,192. Extreme 
variation of all the samples tested: Specific gravity, 8.308 to 8.850; tenacity, 
23,108 to 54,531. Extreme variation of all the samples from the gun heads: 
Specific gravity, 8.308 to 8.756; tenacity, 23,529 to 35,484. 

Major Wade says: The general results on the quality of bronze as it is 
found in guns are mostly of a negative character. They expose defects in 
density and strength, develop the heterogeneous texture of the metal in dif- 
ferent parts of the same gun, and show the irregularity and uncertainty of 
quality which attend the casting of all guns, although made from similar 
materials, treated in like manner. 

Navy ordnance bronze containing 9 parts copper and 1 part tin, tested at 
Washington, D. C, in 1875-6, showed a variation in tensile strength from 
29,800 to 51,400 lbs. per square inch, in elongation frcm 3% to 58$, and in spe- 
cific gravity from 8.39 to 8.88. 

That a great improvement may be made in the density and tenacity of 
gun-bronze by compression has been shown by the experiments of Mr. S. B. 
Dean in Boston, Mass., in 1869, and bj^ those of General Uchatius in Austria 
in 1873. The former increased the density of the metal next the bore of the 
gun from 8,321 to 8,875, and the tenacity from 27,238 to 41,471 pounds per 



322 



square inch. The latter, by a similar process, obtained the following figures 

for tenacity: 

"*"■"- Pounds per sq. in. > 

Bronze with 10^ tin 72,053 

Bronze with 8^ tin 73,958 

Bronze with 6% tin 77,656 

AI.L.OYS OF COPPER, TIN, AND ZINC. 

(Report of U. S. Test Board, Vol. II, 1881.) 



No. 


Analysis, 
Original Mixture. 


Transverse 
Strength. 


Tensile 
Strength per 
square inch. 


Elongation 

per cent in 

5 inches. 


in 
Report. 








Modulus 


Deflec- 












Cu. 


Sn. 


Zn. 


of 
Rupture 


tion, 
ins. 


A. 


B. 


A. 


B. 


72 


90 


5 


5 


41,334 


2.63 


23,660 


30,740 


2.34 


9.68 


5 


88.14 


1.86 


10 


31,986 


3 67 


3-',000 


33,000 


17.6 


19.5 


70 


85 


5 ; 


10 


44,457 


2.85 


28,840 


28,560 


6.80 


5.28 


71 


85 


10 


5 


62,470 


2.56 


35,680 


36,000 


2.51 


2.25 


89 


85 


12:5 


2.5 


62,405 


2.83 


34,500 


32,800 


1.29 


2.79 


88 


82.5 


12.5 


5 


69,960 


1.61 


36,000 


34,000 


.86 


.92 


77 


82.5 


15 


2.5 


69,045 


1.09 


33,600 


33,800 




.68 


67 


80 


5 


15 


42,618 


3.88 


37,560 


32,300 


ii!6" 


3.59 


68 


80 


10 


10 


67,117 


2.45 


32.830 


31, "950 


1.57 


1.67 


69 


80 


15 


5 


54,476 


.44 


32,350 


30,760 


.55 


.44 


86 


77.5 


10 


12.5 


63,849 


1.19 


35,500 


36,000 


1.00 


1.00 


87 


77.5 


12.5 


10 


61.705 


.71 


36,000 


32,500 


.72 


.59 


63 


75 


5 


20 


55,355 


2.91 


33,140 


34,960 


2.50 


3.19 


85 


75 


7.5 


17.5 


62,607 


1.39 


33,700 


39,300 


1.56 


1.33 


64 


75 


10 


15 


58,345 


.73 


35,320 


34,000 


1.13 


1.25 


65 


75 


15 


10 


51.109 


.31 


35,440 


28,000 


.59 


.54 


66 


75 


20 


5 


40; 235 


.21 


23,140 


27,660 


.43 




83 


72.5 


7.5 


20 


51,839 


2.86 


32,700 


34,800 


3.73 


"3! 78" 


84 


72.5 


10 


17.5 


53,230 


.74 


30,000 


30,000 


.48 


.49 


59 


70 


5 


25 


57,349 


1.37 


38,000 


32,940 


2.06 


.99 


82 


70 


7.5 


22.5 


48,836 


.36 


38,000 


32,400 


.84 


.40 


60 


70 


10 


20 


36,520 


.18 


33,140 


26,300 


.31 




6t 


70 


15 


15 


37,924 


.20 


33,440 


27.800 


.25 




62 


70 


20 


10 


15,126 


.08 


17,000 


12,900 


.03 




81 


67.5 


2.5 


30 


58,343 


2.91 


34,720 


45.850 


7.27 


"3I69" 


74 


67.5 


5 


27.5 


55,976 


.49 


34,000 


34,460 


1.06 


.43 


75 


67.5 


7.5 


25 


46,875 


.32 


29,500 


30,000 


.36 


.26 


80 


65 


2.5 


32.5 


56,949 


2.36 


41,350 


38.300 


3.26 


3.02 


55 


65 


5 


30 


51 ,369 


.56 


37,140 


36,000 


1.21 


.61 


56 


65 


10 


25 


27,075 


.14 


25.720 


22.500 


.15 


.19 


57 


65 
65 
62.5 


15 
20 
2.5 


20 
15 
35 


13,591 
11,932 
69,255 


.07 

.05 

2.34 


6,820 
3,765 
44,400 


7,231 
2,665 
45,000 






58 






79 


"2". 15;"! 


*2!i9" 


78 


60 


2.5 


37.5 


69,508 


1.46 


57,400 


52,900 


4.87 


3.02 


52 


60 


5 


35 


46,076 


.28 


41,160 


38,330 


.39 


.40 


53 


60 


10 


30 


24,699 


.13 


21,780 


21,240 


.15 




54 


60 


15 


25 


18,248 


.09 


18,020 


12,400 






12 


58.22 


2.30 


39.48 


95,623 


1.99 


66,500 


67,600 


"k'.\h" 


"3.15* 


3 


58.75 


8.75 


32.5 


35,752 


.18 


Broke 


before t 


est ; ver 


y brittle 


4 


57.5 
55 


21.25 
0.5 


21.25 
44.5 


2,752 
72,308 


.02 
3.05 


725 
68,900 


1,300 
68,900 






73 


"9.43"' 


"2. 88*" 


50 


55 


5 


40 


38,174 


.22 


27,400 


30,500 


.46 


.43 


51 


55 


10 


35 


28,258 


.14 


25,460 


18,500 


.29 


.10 


49 


50 


5 


45 


20,814 


.11 


23,000 


31,300 


.66 


.45 



The transverse tests were made in bars 1 in. square, 22 in. between sup- 
ports. The tensile tests were made on bars 0.798 in. diam. turned from the 
two halves of the transverse-test bar, one half being marked A and the 
other B, 



ALLOYS OF COPPER, TIN, AND ZINC. 323 

Ancient Bronzes.— The usual composition of ancient bronze was the 
same as that of modern gun-metal— 90 copper, 10 tin; but the proportion of 
tin varies from 5$ to 15$, and in some cases lead has been found. Some an- 
cient Egyptian tools contained 88 copper, 12 tin. 

Strength of the Copper-zinc Alloys.— The alloys containing less 
than 15$ of zinc by original mixture were generally defective. The bars 
were full of blow-holes, and the metal showed signs of oxidation. To insure 
good castings it appears that copper-zinc alloys should contain more than 
15$ of zinc. 

From No. 2 to No. 8 inclusive, 16.98 to 80.06$ zinc the bars show a remark- 
able similarity in all their properties. They have all nearly the same 
strength and ductility, the latter decreasing slightly as zinc increases, and 
are nearly alike in color and appearance. Between Nos. 8 and 10, 30.06 and 
36.36$ zinc, the strength by all methods of test rapidly increases. Between 
No. 10 and No. 15, 36.36 and 50.14^ zinc, there is another group, distinguished 
by high strength and diminished ductility. The alloy of maximum tensile, 
transverse and torsional strength contains about 41$ of zinc. 

The alloys containing less than 55$ of zinc are all yellow metals. Beyond 
55$ the color changes to white, and the alloy becomes weak and brittle. Be- 
tween 70$ and pure zinc the color is bluish gray, the brittleness decreases 
and the strength increases, but not to such a degree as to make them useful 
for constructive purposes. 

Difference hetween Composition hy Mixture and hy 
Analysis.— There is in every case a smaller percentage of zinc in the 
average analysis than in the original mixture, and a larger percentage of 
copper. The loss of zinc is variable, but in general averages from 1 to 2$. 

Liquation or Separation of the Metals.— In several of the 
bars a considerable amount of liquation took place, analysis showing a 
difference in composition of the two ends of the bar. In such cases the 
change in composition was gradual from one end of the bar to the other, 
the upper end in general containing the higher percentage of copper. A 
notable instance was bar No. 13, in the above table, turnings from the upper 
end containing 40.36^ of zinc, and from the lower end 48.52$. 

Specific Gravity. — The specific gravity follows a definite law. varying 
with the composition, and decreasing with the addition of zinc. From the 
plotted curve of specific gravities the following mean values are taken: 

Per cent zinc 10 20 30 40 50 60 70 80 90 100. 

Specific gravity 8.80 8.72 8.60 8.40 8.36 8.20 8.00 7.72 7.40 7.20 7.14. 

Graphic Representation of the L<aw of Variation of 
Strength of Copper-Tin-Zinc Alloys.— In an equilateral triangle 
the sum of the perpendicular distances from any point within it to the three 
sides is equal to the altitude. Such a triangle can therefore be used to 
show graphically the percentage composition of any compound of three 
parts, such as a triple alloy. Let one side represent copper, a second 
tin, and the third zinc, the vertex opposite each of these sides repre- 
senting 100 of each element respectively. On points in a triangle of wood 
representing different alloys tested, wires were erected of lengths propor- 
tional to the tensile strengths, and the triangle then built up with plaster to 
the height of the wires. The surface thus formed has a characteristic 
topography representing the variations of strength with variations of 
composition. The cut shows the surface thus made. The vertical section 
to the left represents the law of tensile strength of the copper-tin allojs, 
the one to the right that of tin-zinc alloys, and the one at the rear that of 
the copper-zinc alloys. The high point represents the strongest possible 
alloys of the three metals. Its composition is copper 55, zinc 43, tin 2, and its 
strength about 70,000 lbs. The high ridge from this point to the point of 
maximum height of the section on the left is the line of the strongest alloys, 
represented by the formula zinc + (3 X tin) = 55. 

All alloys lying to the rear of the ridge, containing more copper and less 
tin or zinc are alloys of greater ductility than those on the line of maximum 
strength, and are the valuable commercial alloys; those in front on the decliv- 
ity toward the central valley are brittle, and those in the valley are both brit- 
tle and weak. Passing from the valley toward the section at the right the 
alloys lose their brittleness and become soft, the maximum softness being 
at tin = 100, but they remain weak, as is shown by the low elevation of the 
surface. This model was planned and constructed by Prof. Thurston in 
1877. (See Trans. A. S. C. E. 1881, Report of the U. S. Board appointed to 



324 



Washington, 1881, and Thurston's Materials 



test Iron, Steel, etc., vol. 
of Engineering, vol. iii.) 

The best alloy obtained in Thurston's research for the U. S. Testing Board 
has the composition, Copper 55, Tin 0.5, Zinc 44.5. The tensile strength in a 
cast bar was 68,900 lbs. per sq. in., two specimens giving the same result; the 
elongation was 47 to 51 per cent in 5 inches. Thurston's formula for copper- 
tin- zinc alloys of maximum strength (Trans. A. S. C. E., 1881) is z-\- U = 55, 




Fig. 77. 

in which z is the percentage of zinc and t that of tin. Alloys proportioned 
according to this formula should have a strength of about 40,000 lbs. 
per sq. in. + 5002. The formula fails with alloys containing less than 1 per 
cent of tin. 

The following would be the percentage composition of a number of alloys 
made according to this formula, and their corresponding tensile strength in 
castings : 

Tensile 
Tin. Zinc. Copper, <*«■«£ 



34 



47 


66,000 


49 


64,500 


51 


63,000 


53 


61,500 


55 


60,000 


57 


58,500 


59 


57,000 



Tensile 



Zinc. 


Copper. 


strengtn 
lbs. per 
sq. in. 


31 


61 


55,500 


28 


63 


54,000 


25 


65 


52,500 


19 


69 


49,500 


13 


73 


46,500 


7 


77 


43,500 


1 


81 


40,500 



These alloys, while possessing 1 maximum tensile strength, would in general 
be too hard for easy working by machine tools. Another series made on 
the formula z -f- 4 t — 50 would have greater ductility, together with con- 
siderable strength, as follows, the strength being calculated as before, 
tensile sjtrengh in lbs. per sq. in. = 40,000 -f 5002. 



ALLOYS OP COPPER, TIN, AND ZINC. 



325 



Tin. Zinc. Copper. 



34 
30 
26 



59 

62 



Tensile 

Strength, 

lbs. per 

sq. in. 

63,000 
61.000 
59,000 
57.000 
55,000 
53,000 





Tensile 


inc. Copper. 


Strength, 
lbs. per 




sq. in. 


22 71 


51,000 


18 74 


49,000 


14 77 


47,000 


10 80 


45,000 


6 83 


43,000 


2 86 


41,000 



Composition of Alloys in Everyday Use in Brass 
Foundries. {American Machinist.) 



Cop- 
per. 


Zinc. 


lbs. 

87 


lbs. 

5 


16 
16 


..„.. 


64 
32 


8 
1 


20 


1 


16 
60 


"40 


92 




90 




16 
50 


3 

50 



Admiralty metal 

Bell metal 

Brass (yellow) 

Bush metal 

Gun metal 



Steam metal.. 



Hard gun metal. . 
Muntz metal 



Phosphor bronze. 



I phos. tin 



Brazing metal... 
" solder. . 



For parts of engines on board 

naval vessels. 
Bells for ships and factories. 
For plumbers, ship and house 

brass work. 
For bearing bushesf or shafting. 
For pumps and other hydraulic 

purposes. 
Castings subjected to steam 

pressure. 
For heavy bearings. 
Metal from which bolts and nuts 

are forged, valve spindles, etc. 
For valves, pumps and general 

work. 
For cog and worm wheels, 

bushes, axle hearings, slide 

valves, etc. 
Flanges for copper pipes. 
Solder for the above f 



Gurley's Bronze.— 16 parts copper, 1 tin, 1 zinc, Y% lead, used by 
W. & L. E. Gurley of Troy for the framework of their engineer's transits. 
Tensile strength 41,114 lbs. per sq. in., elongation 27$ in 1 inch, sp. gr. 8.696. 
(W. J. Keep, Trans. A. I. M. E. 1890.) 

Useful Alloys of Copper, Tin, and Zine. 

(Selected from numerous sources.) 

U. S. Navy Dept. journal boxes ) 

and guide-gibs ) 

Tobin bronze 

Naval brass 

Composition, U. S. Navy 

Brass bearings (J. Rose) 

Gun metal 



Tough brass for engines 

Bronze for rod-boxes (Laf ond) 

" " pieces sub ject to shock . . 

Red brass parts 

" " per cent 

Bronze for pump casings (Lafond)... 

" " eccentric straps. " 

" " shrill whistles 

" " low-toned whistles 



opper. 


Tin. 


Zinc. 


6 


1 


J4 parts. 


82.8 


13.8 


3.4 per cent. 


58 22 


2.30 


39.48 " " 


62 


1 


37 " " 


88 


10 


2 " " 


64 


8 


1 parts. 


87.7 


11.0 


1.3 per cent. 


92.5 


5 


2.5 " " 


91 


7 


2 " " 


87.75 


9.75 


2.5 " " 


85 


5 


10 " " 


83 


2 


15 " " 


13 


2 


2 parts. 


76.5 


11.8 


11.7 percent. 


82 


16 


2 slightly malleable. 


83 


15 


1.50 0.50 lead. 


20 


1 


1 1 


87 


4.4 


4.3 4.3 " 


88 


10 


2 


84 


14 


2 


80 


18 


2.0 antimony. 


81 


17 


.... 2.0 



326 



ALLOYS. 



Copper. Tin. Zinc. 

Art bronze, dull red fracture 97 2 1 

Gold bronze 89.5 2.1 5.6 2.8 lead. 

Bearing metal — 89 8 3 

I' 89 2y 2 sy 2 

" 86 14 

" 85^ im 2 

" 80 18 2 

" 79 18 2)4 i/o lead. 

74 914 ^A 7 lead. 

English brass of a.d. 1504 04 3 29}^ 3^ lead. 

Copper-Nickel Alloys, German Silver. 

Copper. Nickel. Tin. Zinc. 

German silver. 51.6 25.8 22.6 

" 50.2 14.8 3.1 31.9 

" 51.1 13.8 3.2 31.9 

" 52 to 55 18 to 25 20 to 30 

Nickel " 75 to 66 25 to 33 

A refined copper-nickel alloy containing 50$ copper and 49$ nickel, with 
very small amounts of iron, silicon and carbon, is produced direct from 
Bessemer matte in the Sudbury (Canada) Nickel Works. German silver 
manufacturers purchase a ready-made alloy, which melts at a low heat and 
requires simple addition of zinc, instead of buying the nickel and copper 
separately. This alloy, "50-50" as it is called, is almost indistinguishable 
from pure nickel. Its cost is less than nickel, its melting jDoint much lower, 
it can be cast solid in any form desired, and furnishes a casting which works 
easily in the lathe or planer, yielding a silvery white surface unchanged by- 
air or moisture. For bullet casings now used in various British and conti- 
nental rifles, a special alloy of 80$ copper and 20$ nickel is made. 

Special Alloys. {Engineer, March 24, 1893.) 
Japanese Alloys for art work ; 



Copper. Silver. Gold. 



Lead. 



Zinc. 



Iron. 



Shaku-do. . 
Shibu-ichi. 



94.50 
67.31 



1.55 
32.07 



3.73 

traces. 



Gilbert's Alloy for cera-perduta process, for casting in plaster-of-paris • 
Copper 91.4 Tin 5.7 Lead 2.9 Very fusible. 

COPPER-ZINC-IRON AliliOYS. 

(F. L. Garrison, Jour. Frank. Inst., June and July, 1891.) 
Delta Uletal. — This alloy, which was formerly known as sterro-metal, 
is composed of about 60 copper, from 34 to 44 zinc, 2 to 4 iron, and 1 to 2 tin. 
The peculiarity of all these alloys is the content of iron, which appears to 
have the property of increasing their strength to an unusual degree. Tn 
making delta metal the iron is previously alloyed with zinc in known and 
definite proportions. When ordinary wrought-iron is introduced into 
molten zinc, the latter readily dissolves or absorbs the former, and will take 
it up to the extent of about 5$ or more. By adding the zinc-iron alloy thus 
obtained to the requisite amount of copper, it is possible to introduce any 
definite quantity of iron up to 5% into the copper alloy. Garrison gives the 
following as the range of composition of copper-zinc-iron, and copper-zinc- 
tin-iron alloys : 

I. II. 

Percent. Percent. 

Iron 0.1 to 5 Iron 0.1 to 5 

Copper 50 to 65 Tin 0.1 to 10 

Zinc 49.9 to 30 Zinc 1.8 to 45 

Copper 98 to 40 

The advantages claimed for delta metal are great strength and toughness. 
It produces sound castings of close grain. It can be rolled and forged hot 
and can stand a certain amount of drawing and hammering when cold. It 
takes a high polish, and when exposed to the atmosphere tarnishes less than 
brass. 



PHOSPHOR-BRONZE AND OTHER SPECIAL BRONZES. 327 

When cast in sand delta metal has a tensile strength of about 45,000 pounds 
per square inch, and about 10$ elongation ; when rolled, tensile strength of 
60,000 to 75,000 pounds per square inch, elongation from 9$ to 17$ on bars 1.128 
inch in diameter and 1 inch area. 

Wallace gives the ultimate tensile strength 33,600 to 51,520 pounds per 
square inch, with from 10$ to 20$ elongation. 

Delta metal can be forged, stamped and rolled hot. It must be forged at 
a dark cherry-red heat, and care taken to avoid striking when at a black 
heat. 

According to Lloyd's Proving House tests, made at Cardiff, December 20, 
1887, a half-inch delta metal-rolled bar gave a tensile strength of 88,400 
pounds per square inch, with an elongation of 30$ in three inches. 

Tohin Bronze.— This alloy is practically a sterro or delta metal with 
the addition of a small amount of lead, which tends to render copper softer 
and more ductile. 

The following analyses of Tobin bronze were made by Dr. Chas. B. Dudley: 

Pig Metal, Test Bar (Rolled), 
per cent. per cent. 

Copper 59.00 61.20 

Zinc 38.40 37.14 

Tin 2.16 0.90 

Iron 0.11 0.18 

Lead 0.31 0.35 

Dr. Dudley writes, " We tested the test bars and found 78,500 tensile 
strength with 15$ elongation in two inches, and 40V£$ in eight inches. This 
high teusile strength can only be obtained when the metal is manipulated. 
Such high results could hardly be expected with cast metal." 

The original Tobin bronze in 1875, as described by Thurston, Trans. 
A. S. C. E 1881, had, composition of copper 58.22, tin 2.30, zinc 39.48. As 
cast it had a tenacity of 66,000 lbs. per sq. in., and as rolled 79,000 lbs. ; cold 
rolled it gave 104,000 lbs. 

A circular of Ansonia Brass & Copper Co. gives the following : — The tensile 
strength of six Tobin bronze one-inch round rolled rods, turned down to a 
diameter of % of an inch, tested by Fairbanks, averaged 79,600 lbs. per sq. 
in., and the elastic limit obtained on three specimens averaged 54,257 lbs. per 
sq. in. 

At a cherry-red heat Tobin bronze can be forged and stamped as readily 
as steel. Bolts and nuts can be forged from it, either by hand or by ma- 
chinery, with a marked degree of economy. Its great tensile strength, and 
resistance to the corrosive action of sea-water, render it a most suitable 
metal for condenser plates, steam-launch shafting, ship sheathing and 
fastenings, nails, hull plates for steam yachts, torpedo and life boats, and 
ship deck fittings. 

The Navy Department has specified its use for certain purposes in the 
machinery of the new cruisers. Its specific gravity is 8.071. The weight of 
a cubic inch is .291 lb. 

PHOSPHOR-BRONZE AND OTHER SPECIAL 
BRONZES. 

Phosphor-bronze.— In the year 1868, Montefiore & Kunzel of Liege, 
Belgium, found by adding small proportions of phosphorus or "phosphoret 
of tin or copper" to copper that the oxides of that metal, nearly always 
present as an impurity, more or less, were deoxidized and the copper much 
improved in strength and ductility, the grain of the fracture became finer, 
the color brighter, and a greater fluidity was attained. 

Three samples of phosphor-bronze tested by Kirkaldy gave : 

Elastic limit, lbs. per sq. in 23.800 24,700 16,100 

Tensile strength, lbs. per sq. in. .. . 52,625 46,100 44.448 
Elongation, per cent 8.40 1.50 33.40 

The strength of phosphor-bronze varies like that of ordinal bronze 
according to the percentages of copper, tin, zinc, lead, etc., in the alloy. 

Deoxidized Bronze.— This alloy resembles phosphor bronze some- 
Avhat in composition and also delta metal, in containing zinc and iron. The 
following analysis gives its average composition: 



Copper 82 

Tin 12.40 

Zinc 3.23 

Lead 2.14 



Iron 0.10 

Silver 0.07 

Phosphorus 0.005 



328 



Comparison of Copper, Silicon-bronze, and Phosphor- 
bronze Wires, 

(Engineering, Nov. 23, 1883.) 



Description of Wire. 


Tensile Strength per 
square inch in 

Tons. Lbs. 


Relative 
Conductivity. 




17.78 
18.27 
48.25 
45.71 


39,827 
41,696 
108,080 
102,390 


100 per cent. 

96 

34 


Silicon bronze (telegraph) 


Phosphor Bronze (telephone) ... 


26 



ALUMINUM ALIiOYS. 
(Aluminum Bronze. Cowles Electric Smelting and Al. Co.'s circular.) 

The standard A No. 2 grade of aluminum bronze, containing 10$ of alumi- 
num and 90$ of copper, has many remarkable characteristics which dis- 
tinguish it from all other metals. 

The tenacity of castings of A No. 2 grade metal varies between 75,000 
and 90,000 lbs. to the square inch, with from 4$ to 14$ elongation. 

Increasing the proportion of aluminum in bronze beyond 110 produces a 
brittle alloy; therefore nothing higher than the A No. 1, which contains 11$, 
is made. 4 

The B, C, D, and E grades, containing 7}4%, 5$, %V&%, and 1J4$ of aluminum, 
respectively, decrease in tenacity in the order named, that of the former 
being about 65,000 pounds, while the latter is 25,000 pounds. While there is 
also a proportionate decrease in transverse and torsional strengths, elastic 
limit, and resistance to compression as the percentage of aluminum is low- 
ered and that of copper raised, the ductility on the other hand increases in 
the same proportion. The specific gravity of the A No. 1 grade is 7.56. 

Bell Bros., Newcastle, gave the specific gravity of the aluminum bronzes 
as below: 

3$ aluminum 8.691 

4$ " 8.621 

5$ " 8.369 

10$ " 7.689 

Casting;.— The melting point of aluminum bronze varies slightly with 
the amount of aluminum contained, the higher grades melting at a some- 
what lower temperature than the lower grades. The A No. 1 grades melt 
at about 1700° F., a little higher than ordinary bronze or brass. 

Aluminum bronze shrinks more than ordinary brass. As the metal solidi- 
fies rapidly it is necessary to pour it quickly and to make the feeders amply 
large, so that there will be no "freezing "in them before the casting is 
properly fed. Baked-sand moulds are preferable to green sand, except for 
small castings, and when fine skin colors are desired in the castings. (See 
paper by Thos. D. West, Trans. A. S. M. E. 1886, vol. viii.) 

All grades of aluminum bronze can be rolled, swedged, spun, or drawn 
cold except A 1 and A 2. They can all be worked at a bright red heat. 

In rolling, swedging, or spinning cold, it should be annealed very often, and 
at a brighter reel heat than is used for annealing brass. 

Brazing.— Aluminum bronze will braze as well as any other metal, 
using one quarter brass solder (zinc 500, copper 500 (and three quarters 
borax, or, better, three quarters cryolite. 

Soldering.— To solder aluminum bronze with ordinary soft (pewter) 
solder: Cleanse well the parts to be joined free from grease and dirt. Then 
place the parts to be soldered in a strong solution of sulphate of copper and 
place in the bath a rod of soft iron touching the parts to be joined. After 
a while a coppery-like surface will be seen on the metal. Remove from 
bath, rinse quite clean, and brighten the surfaces. These surfaces can then 
be tinned by using a fluid consisting of zinc dissolved in hydrochloric acid, in 
the ordinary way, with common soft solder. 

Mierzinski recommends ordinary hard solder, and says that Hulot uses 
an alloy of the usual half-and-half lead-tin solder, with 12.5$, 25j£ or 50$ of 
zinc amalgam. 



ALUMINUM BRONZE. 



329 



Tests of Aluminum Bronzes. 

(By John H. J. Dagger, in a paper read before the British Association, 1889.) 



Per cent 

of 

Aluminum. 



Tensile Strength. 



Tons per 
square inch. 



Pounds per 
square inch. 



Elonga- 
tion, 
per cent. 



Specific 
Gravity. 



89,600 to H in, son 
73,9-20 " 89, GOO 
5(5,000 " 67,200 
33,600 " 40,320 
29,120 
24,640 



33.600 
29,120 



40 
50 
55 



7.23 
7.69 
8.00 
8.37 



5-5V* 

^ 

m 

Both physical and chemical tests made of samples cut from various sec - 
tions of 2]4%, 5$, 7}&%, or 10$ aluminized copper castings tend to prove that 
the aluminum unites itself with each particle of copper with uniform pro- 
portion in each case, so that we have a product that is free from liquation 
and highly homogeneous. (R. C. Cole, Iron Age, Jan. 16, 1890.) 

Aluminum-Brass (E. H. Cowles, Trans. A. I. M. E., vol. xviii.)— 
Cowles aluminum-brass is made by fusing together equal weights of A 1 
aluminum-bronze, copper, and zinc. The copper and bronze are first thor- 
oughly melted and mixed, and the zinc is finally added. The material is left 
in the furnace until small test-bars are taken from it and broken. When 
these bars show a tensile strength of 80,000 pounds or over, with 2 or 3 per 
cent ductility, the metal is ready to be poured. Tests of this brass, on small 
bars, have at times shown as high as 100,000 pounds tensile strength. 

The screw of the United States gunboat Petrel is cast from this brass, 
mixed with a trifle less zinc in order to increase its ductility. 
Tests of Aluminum-Brass. 
(Cowles E. S. & Al. Co.) 



Specimen (Castings.) 


Diameter 

of Piece, 

Inch. 


Area, 
sq. in. 


Tensile 

Strength, 

lbs. per 

sq. in. 


Elastic 
Limit, 
lbs. per 
sq. in. 


Elonga- 
tion, 
per ct. 


Remarks. 


15$ A grade Bronze. 1 
17$.Zinc V 


.465 
.465 
.460 


.1698 
.1698 
.1661 


41,225 

78,327 
72,246 


17,668 


4iy 2 


% &c 


68$ Copper ) 

1 part A Bronze ) 

1 part Zinc V 

1 part Copper ) 

1 part A Bronze j 

1 part Zinc > 

1 part Copper ) 


.2 2. % ,a 



The first brass on the above list is an extremely tough metal with low 
elastic limit, made purposely so as to " upset " easily. The other, which is 
called Aluminum-brass No 2, is very hard. 

We have not in this country or in England any official standard by which 
to judge of the physical characteristics of cast metals. There are two con- 
ditions that are absolutely necessary to be known before we can make a 
fair comparison of different materials: namely, whether the casting was 
made in dry or green sand or in a chill, and whether it was attached to a 
larger casting or cast by itself. It has also been found that chill-castings 
give higher results than sand-castings, and that bars cast by themselves 
purposely for testing almost invariably run higher than test-bars attached 
to castings. It is also a fact that bars cut out from castings are generally 
weaker than bars cast alone, (E. H. Cowles.) 

Caution as to Reported Strength of Alloys.— The same 
variation in strength which has been fouud in tests of gun-metal (copper 
and tin) noted above, must be expected in tests of aluminum bronze and in 
fact of all alloys. They are exceedingly subject to variation in density and 
in grain, caused by differences in method of molding and casting, tempera- 
ture of pouring, size and shape of casting, depth of " sinking head," etc. 



330 



Alloys. 



Aluminum Hardened by Addition of Copper Rolled 
Sheets .04 Inch Thick. (The Engineer, Jau 2, 1891.) 











Tensile Strength 


Al. 


Cu. 


Sp. Gr. 


Sp. Gr. 


in pounds per 


Per cent. 


Per cent. 


Calculated. 


Determined. 


square inch. 


100 






2.67 


26,535 


98 


2 


2.78 


2.71 


43,563 


96 


4 


2.90 


2.77 


44,130 


94 


6 


3.02 


2.82 


54,773 


92 


8 


3.14 


2.85 


50,374 



Tests of Aluminum Alloys. 

(Engineer Harris, U. S. N., Trans. A. I. M. E., vol. xviii.) 



Composition. 


Tensile 

Strength, 

per sq. in. 

lbs. 


Elastic 
Limit, 

lbs. per 
sq.in. 


Elonga- 
tion, 
per ct. 


Reduc- 
tion of 
Area, 
per ct. 


Cop- 
per. 


Alumi- 
num. 


Silicon. 


Zinc. 


Iron. 


91.50* 

88.50 
91.50 
90.00 


6.50* 

9.33 

6.50 

9.00 

3.33 

3.33 

6.50 

6.50 

9.33 

6.50 


1.75* 

1.66 

1.75 

1.00 

0.33 

0.33 

1.75 

0.50 

1.66 

0.50 




0.25* 
0.50 
0.25 


60,700 
66,000 
67,600 
72,830 
82,200 
70,400 
59,100 
53,000 
69,930 
46,530 


18,000 
27,000 
24,000 
33,000 
60,000 
55,000 
19,000 
19,000 
33,000 
17,000 


23.2 
3.8 

13. 
2.40 
2.33 
0.4 

15.1 
6.2 
1.33 
7.8 


30.7 
7.8 

21.62 
5.78 


63.00 
63.00 
91.50 
93.00 


33.33* 
33.33 


"6!25"' 


9.88 
4.33 
23.59 
15.5 


88.50 
92.00 




0.50 


3.30 
19.19 











For comparison with the above 6 tests of " Navy Yard Bronze," Cu 88, 
Sn 10, Zn 2, are given in which the T. S. ranges from 18,000 to 24,590, E. L. 
from 10,000 to 13,000, El. 2.5 to 5.8*. Red. 4.7 to 10.89. 

Alloys of Aluminum, Silicon and Iron. 

M. and E. Bernard have succeeded in obtaining through electrolysis, by 
treating directly and without previous purification, the aluminum earths 
(red and white bauxites) the following : 

Alloys such asferro-aluminum, ferro-silicon-aluminum and silicon-alumi- 
num, where the proportion of silicon may exceed 10% which are employed 
in the metallurgy of iron for refining steel and cast-iron. 

Also silicon-aluminum, where the proportion of silicon does not exceed 
10%, which may be employed in mechanical constructions in a rolled or 
hammered condition, in place of steel, on account of their great resistance, 
especially where the lightness of the piece in construction constitutes one 
of the main conditions of success. 

The following analyses are given: 

1. Alloys applied to the metallurgy of iron, the refining of steel and cast 
iron: 

Types. Aluminum. Iron. Silicon. Manganese. 

No. 1 70* 25* 5* 0* 

No.2 ... 70 20 10 

No. 3 70 15 15 

No.4 70 10 20 

No. 5 70 10 10 10 

No. 6 70 trace 20 10 

2. Mechanical alloys: 

Types. Aluminum. Silicon. Iron. 

No. 1 92* 6.75* 1.25* 

No.2 90 9.25 0.75 

No.3 90 10.00 trace. 

Up to this time it has been thought that silicon was rather injurious when 
alloyed with aluminum. From numerous experiences it has been demon- 
strated that it gives to aluminum some remarkable properties of resistance; 
the best results were with alloys where the proportion of iron was very low, 
and the proportion of silicon in the neighborhood of 10*. Above that pro- 



ALLOYS OF MANGANESE AND COPPER. 331 

portion the alloy becomes ciTstalline and can no longer be employed. The 
density of the alloys of silicon is approximately the same as that of alumi- 
num. — La Metallurgie, 1892. 

Tungsten and Aluminum.— Mr. Leinhardt Mannesmann says that 
the 'addition of a little tungsten to pure aluminum or its alloys communi- 
cates a remarkable resistance to the action of cold and hot water, salt water 
and other re-agents. Wh-n the proportion of tungsten is sufficient the 
alloys offer great resistance to tensile strains. 

Aluminum and Tin.— M. Bourbouze has compounded an alloy of 
aluminum and tin, by fusing together 100 parts of the former with 10 parts 
of tbe latter. This alloy is paler than aluminum, and has a specific gravity 
of 2.85. The alloy is not as easily attacked by several reagents as alumi- 
num is, and it can also be worked more readily. Another advantage is that 
it can be soldered as easily as bronze, without further preliminary prepara- 
tions. 

Aluminum-Antimony Alloys.— Dr. C. R. Alder Wright describes 
some aluminum-antimony alloys in a communication read before the Society 
of Chemical Industry. The results of his researches do not disclose the 
existence of a commercially useful alloy of these two metals, and have 
greater scientific than practical interest. A remarkable point is that the 
alloy with the chemical composition Al Sb has a higher melting point than 
either aluminum or antimony alone, and that when aluminum is added to 
pure antimony the melting-point goes up from that of antimony (450° C.) 
to a certain temperature rather above that of silver (1000° C). 

ALLOYS OF MANGANESE AND COPPER, 

Various Manganese Alloys.— E. H. Cowles, in Trans. A. I. M. E., 
vol. xviii, p. 495, states that as the result of numerous experiments on 
mixtures of the several metals, copper, zinc, tin. lead, aluminum, iron, and 
manganese, and the metalloid silicon, and experiments upon the same in 
ascertaining tensile strength, ductility, color, etc., the most important 
determinations appear to be about as follows : 

1. That pure metallic manganese exerts a bleaching effect upon copper 
more radical in its action even than nickel. In other words, it was found 
that ~18]4% of manganese present in copper produces as white a color in the 
resulting alloy as 25$ of nickel would do, this being the amount of each 
required to remove the last trace of red. 

2. That upwards of 20$ or 25% of manganese may be added to copper with- 
out reducing its ductility, although doubling its tensile streugth and chang- 
ing its color. 

3. That manganese, copper, and zinc when melted together and poured 
into moulds behave very much like the most " yeasty " German silver, 
producing an ingot which is a mass of blow-holes, and which swells up 
above the mould before cooling. 

4. That the alloy of manganese and copper by itself is very easily 
oxidized. 

5. That the addition of 1.25$ of aluminum to a manganese-copper alloy 
converts it from one of the most refractory of metals in the casting process 
into a metal of superior casting qualities, and the noncorrodibility of which 
is in many instances greater than that of either German or nickel silver. 

A "silver-bronze '" alloy especially designed for rods, sheets, and wire 
has the following composition : Manganese, 18; aluminum, 1.20; silicon, 0.5 ; 
zinc, 13; and copper, 67.5$. It has a tensile strength of about 57,000 pounds 
on small bars, and 20% elongation. It has been rolled into thin plate and 
drawn into wire .008 inch in diameter. A test of the electrical conductivity 
of this wire (of size No. 32) shows its resistance to be 41.44 times that of pure 
copper. This is far lower conductivity than that of German silver. 

Manganese Bronze. (F. L. Garrison, Jour. F. I., 1891.)— This alloy 
has been used extensively for casting propeller-blades. Tests of some made 
by B. H. Cramp & Co., of Philadelphia, gave an average elastic limit of 
30,000 pounds per square inch, tensile strength of about 60,000 pounds per 
square inch, with an elongation of 8% to 10$ in sand castings. When rolled, 
the elastic limit is about 80,000 pounds per square inch, tensile strength 
95,000 to 106,000 pounds per square inch, with an elongation of 12$ to 15$. 

Compression tests made at United States Navy Department from the 
metal in the pouring-gate of propeller-hub of U. S. S. Maine gave in two tests 
a crushing stress of 126,450 and 135,750 lbs. per sq. in. The specimens w r ere 
1 inch high by 0.7 X 0.7 inch in cross-section = 0.49 square inch. Both speci- 



332 ALLOYS. 

mens gave way by shearing, on a plane making an angle of nearly 45° with 
the direction of stress. 

A test on a specimen 1 X 1 X 1 inch was made from a piece of the same 
pouring-gate. Under stress of 150,000 pounds it was flattened to 0.72 inch 
high by about 1J4 X 134 inches, hut without rupture or any sign of distress. 

One of the great objections to the use of manganese bronze, or in fact 
any alloy except iron or steel, for the propellers of iron ships is on account 
of the galvanic action set up between the propeller and the stern-posts. 
This difficulty has in great measure been overcome by putting strips of 
rolled zinc around the propeller apertures in the stern. frames. 

The following analysis of Parsons' manganese bronze No. 2 was made 
from a chip from the propeller of Mr. W. K. Vanderbilt's yacht Alva. 

Copper i 88.644 

Zinc 1 .570 

Tin 8.700 

Iron 0.720 

Lead : 0.295 

Phosphorus trace 

99.923 

It will be observed there is no manganese present and the amount of zinc 
is very small. 

E. H. Cowles, Trans. A. I. M. E., vol. xviii, says : Manganese bronze, so 
called, is in reality a manganese brass, for zinc instead of tin is the chief 
element added to the copper. Mr. P. M. Parsons, the proprietor of this 
brand of metal, has claimed for it a tensile strength of from 24 to 28 tons on 
small bars when cast in sand. Mr. W. C. Wallace states that brass-founders 
of high repute in England will not admit that manganese bronze has more 
than from 12 to 17 tons tensile strength. Mr. Horace See found tensile 
strength of 45,000 pounds, and from 6$ to 12J^$ elongation. 

GERMAN-SILVER AND OTHER NICKEL ALLOYS. 

Copper. Nickel. Zinc. 

Chinese packfong 40.4 31.6 6.5 parts, 

tutenag 8 3 6.5 " 

German silver , 2 1 1 " 

" " (cheaper) 8 2 3.5 " 

" " (closely resembles sil). 8 .3 3.5 " 

For analyses of some German-silvers see page 326. 

German Silver.— The composition of German silver is a very uncertain 
thing and depends largely on the honesty of the manufacturer and the 
price the purchaser is willing to pay. It is composed of copper, zinc, and 
nickel in varying propor lions. The best varieties contain from 18$ to 25$ of 
nickel and from 20$ to 30$ of zinc, the remainder being copper. The more 
expensive nickel silver contains from 25$ to 33$ of nickel and from 75$ to 66$ 
of copper. The nickel is used as a whitening element; it also strengthens 
the alloy and renders it harder and more non-corrodible than the brass 
made without it, of copper and zinc. Of all troublesome alloys to handle in 
the foundry or rolling-mill, German silver is the worst. It is unmanageable 
and refractory at every step in its transition from the crude elements into 
rods, sheets, or wire. (E. H. Cowles, Trans. A. I. M. E., vol. xviii. p. 494.) 

ALLOYS OF BISMUTH. 

By adding a small amount of bismuth to lead that metal may be hard- 
ened and toughened. An alloy consisting of three parts of lead and two of 
bismuth has ten times the hardness and twenty times the tenacity of lead. 
The alloys of bismuth with both tin and lead are extremely fusible, and 
take fine impressions of casts and moulds. An alloy of one part bismuth, 
two parts tin, and one part lead is used by pewter-workers as a soft solder, 
and by soap-makers for moulds. An alloy of five parts bismuth, two parts 
tin, and three parts lead melts at 199° F , and is somewhat used for ster- 
eotyping, and for metallic writing-pencils. Thorpe gives the following 
proportions for the better-known fusible metals: 



BEARING-METAL ALLOYS. 



333 



Name of Alloy. 


Bismuth. 


Lead. 


Tin. 


Cad- 
mium 


Mer- 
cury. 


Melting- 
point. 




50 
50 
50 
50 
50 
50 
50 


31.25 

28.10 
25.00 
25.00 
25.00 
26.90 
20.55 


18.75 
24.10 
25.00 
25.00 
12.50 
12.78 
21.10 






202° F. 




203° " 








201° " 


D'Arcet's with mercury. 
Wood's 


12 ".50 
10.40 
14.03 


250.6 


113° " 
149° " 
149° " 


Guthrie's l ' Entectic ' ' . . . 


" Very low. 1 ' 



The action of heat upon some of these alloys is remarkable. Thus, Lipo- 
witz's alloy, which solidifies at 149° Fah., contracts very rapidly at first, as 
it cools from this point. As the cooling goes on the contraction becomes 
slower and slower, until the temperature falls to 101.3° Fah. From this 
point the alloy expands as it cools, until the temperature falls to about 77° 
Fah., after which it again contracts, so that at 32° F. a bar of the alloy has 
the same length as at 115° F. 

Alloys of bismuth have been used for making fusible plugs for boilers, but 
it is found that they are altered by the continued action of heat, so that one 
cannot rely upon them to melt at the proper temperature. Pure Banca tin 
is used by the U. S. Government for fusible plugs. 

FUSIBLE ALLOYS. (From various sources.) 

Sir Isaac Newton's, bismuth 5, lead 3, tin 2, melts at 212° F. 

Rose's, bismuth 2, lead 1, tin 1, melts at 200 " 

Wood's, cadmium 1, bismuth 4, lead 2. tin 1, melts at 165 " 

Guthrie's, cadmium 13.29, bismuth 47.38, lead 19.36, tin 19.97, melts at. 160 " 

Lead 3, tin 5, bismuth 8, melts at , . 208 " 

Lead 1, tin 3, bismuth 5, melts at 212 " 

Lead 1, tin 4, bismuth 5, melts at 240 " 

Tin 1, bismuth 1, melts at , 286 " 

Lead 2, tin 3, melts at 334 " 

Tin 2, bismuth 1, melts at 336 " 

Lead 1, tin 2, melts at 360 " 

Tin 8, bismuth 1, melts at 392 " 

Lead 2, tin 1, melts at 475 " 

Lead 1, tin 1, melts at 466 " 

Lead 1, tin 3, melts at 334 " 

Tin 3, bismuth 1, melts at 392 " 

Lead 1, bismuth 1, melts at 257 " 

Lead 1, Tin 1, bismuth 4, melts at 201 " 

Lead 5, tin 3, bismuth 8," melts at 202 " 

Tin 3, bismuth 5, melts at , 202 " 

BEARING-METAL ALLOYS. 

(G B. Dudley, Jour. F. I., Feb. and March, 1892.) 
Alloys are used as bearings in place of wrought iron, cast iron, or steel, 
partly because wear and friction are believed to be more rapid when two 
metals of the same kind work together, partly because the soft metals are 
more easily worked and got into proper shape, and partly because it is de- 
sirable to use a soft metal which will take the wear rather than a hard 
metal, which will wear the journal more rapidly. 

A good bearing-metal must have five characteristics: (1) It must be strong 
enough to carry the load without distortion. Pressures on car-journals are 
frequently as high as 350 to 400 lbs. per square inch. 

(2) A good bearing-metal should not heat readily. The old copper-tin 
bearing, made of seven parts copper to one part tin, is more apt to heat 
than some other allovs. In general, research seems to show that the harder 
the bearing-metal, the more likely it is to heat. 

(3) Good bearing-metal should work well in the foundry. Oxidatiou while 
melting causes spongy castings. It can be prevented by a liberal use of 
powdered charcoal while melting. The addition of \% to 2% of zinc or a 
small amount of phosphorus greatly aids in the production of sound cast- 
ings. This is a principal element of value in phosphor- brgnz§, 



334 



(4) Good bearing-metals should show small friction. It is true that friction 
is almost wholly a question of the lubricant used; but the metal of the bear- 
ing has certainly some influence. 

(5) Other things being equal, the best bearing-metal is that which wears 
slowest. 

The principal constituents of bearing-metal alloys are copper, tin, lead, 
zinc, antimony, iron, and aluminum. The following table gives the constitu- 
ents of most of the prominent bearing-metals as analyzed at the Pennsyl- 
vania Railroad laboratory at Altoona. 

Analyses of Bearing-metal Alloys. 



Cop- 
per. 



Tin. Lead. Zinc 



Camelia metal 

Anti-friction metal 

White metal 

Car-brass lining 

Salgee anti-friction 

Graphite bearing-metal 

Antimonial lead 

Carbon bronze 

Cornish bronze 

Delta metal 

♦Magnolia metal 

American anti-friction metal.. 

Tobin bronze 

Graney bronze 

Damascus bronze 

Manganese bronze 

Ajax metal 

Anti-friction metal 

Harrington bronze 

Car-box metal 

Hard lead 

Phosphor-bronze 

Ex. B. metal 



70.21 
1.61 



trace 
9 91 
14.38 



75.47 
77.83 
92.39 
trace 



59.00 
75.80 
76.41 
90.52 
81.24 



2.K 
9.20 

10.60 
9.58 

10.98 



87.92 
84 

1.15 
67.73 
80.69 
14.5' 
12.40 

5.10 
83.55 
78.44 

0.31 
15.06 
12.52 



12.08 
15.10 



16.73 
18.83 



0.55 
trace 



? (1) 



trace 

0.98 
38.40 



16.45 
19.60 



(2) 

trace(3) 

0.07 
trace(4) 

0.65 

0.11 



0.97 



8.00 



7.2^ 
88.32 

84.33 
94.40 
9.61 
15.00 



.(5) 

• (6) 



42j 

trace 



• (7) 

• (8) 



Other constituents: 

(1) No graphite. (5) No manganese. 

(2) Possible trace of carbon. (6) Phosphorus or arsenic, 0.37. 

(3) Trace of phosphorus. (7) Phosphorus, 0.94. 

(4) Possible trace of bismuth. (8) Phosphorus, 0.20. 

* Dr. H. C. Torrey says this analysis is erroneous and that Magnolia 
metal always contains tin. 

As an example of the influence of minute changes in an alloy, the Har- 
rington bronze, which consists of a minute proportion of iron in a copper- 
zinc alloy, showed after rolling a tensile strength of 75,000 lbs. and 20$ elon- 
gation in 2 inches. 

In experimenting on this subject on the Pennsylvania Railroad, a certain 
number of the bearings were made of a standard bearing-metal, and the 
same number were made of the metal to be tested. These bearings were 
placed on opposite ends of the same axle, one side of the car having the 
standard bearings, the other the experimental. Before going into service 
the bearings were carefully weighed, and after a. sufficient time they were 
again weighed. 

The standard bearing-metal used is the " S bearing-metal " of the Phos- 
phor-bronze Smelting Co. It contains about 79.70$ copper, 9.50$ lead. 10$ 
tin, and 0.80$ phosphorus. A large number of experiments have shown that 
the loss of weight of a bearing of this metal is 1 lb. to each 18,000 to 25,000 
miles travelled. Besides the measurement of wear, observations were made 
on the frequency of " hot boxes " with the different metals. 

The results of the tests for wear, so far as given, are condensed into the 
following table: 



BEARIKG-METAL ALLOYS. 335 

Composition. Rate 

Metal. . -*• — -^ of 

Copper. Tin. Lead. Phos. Arsenic. Wear. 

Standard 79.70 10.00 9.50 0.80 — 100 

Copper-tin 87.50 12.50 .... .... 148 

Copper-tin, second experiment, same metal 153 

Copper-tin, third experiment, same metal. .... .147 

Arsenic-bronze 89.20 10.00 ... .... 0.80 142 

Arsenic-bronze 79.20 10.00 7.00 .... 0.80 115 

Arsenic-bronze 79.70 10.00 9.50 .... 0.80 101 

"K"bronze 77.00 10.50 12.50 .... .... 92 

" K " bronze, second experiment, same metal : 92.7 

Alloy"B" 77.00 8.00 15.00 .... .... 86.5 

The old copper-tin alloy of 7 to 1 lias repeatedly proved its inferiority to the 
phosphor-bronze metal. Many more of the copper-tin bearings heated 
than of the phosphor-bronze. The showing of these tests was so satisfac- 
tory that phosphor-bronze was adopted as the standard bearing-metal of 
the Pennsylvania R.R , and was used for a long time. 

The experiments, however, were continued. It was found that arsenic 
practically takes the place of phosphorus in a copper-tin alloy, and three 
tests were made with arsenic- bronzes as noted above. As the proportion 
to lead is increased to correspond with the standard, the durability increases 
as well. In view of these results the "K " bronze was tried, in which neither 
phosphorus nor arsenic were used, and in which the lead was increased 
above the proportion in the standard phosphor-bronze. The result was that 
the metal wore 7.30% slower than the phosphor-bronze. No trouble from 
heating was experienced with the " K " bronze more than with the standard. 
Dr. Dudley continues: 

At about this time we began to find evidences that wear of bearing-metal 
alloys varied in accordance with the following law: " That alloy which has 
the greatest power of distortion without rupture (resilience), will best resist 
wear." It was now attempted to design an alloy in accordance with this 
law, taking first the proportions of copper and tin, 9V£ parts copper to 1 of 
tin was settled on by experiment as the standard, although some evidence 
since that time tends to show that 12 or possibly 15 parts copper to 1 of tin 
might have been better. The influence of lead on this copper-tin alloy seems 
to be much the same as a still further diminution of tin. However, the 
tendency of the metal to yield under pressure increases as the amount of 
tin is diminished, and the amount of the lead increased, so a limit is set to 
the use of lead. A certain amount of tin is also necessary to keep the lead 
alloyed with the copper. 

Bearings were c ist of the metal noted in the table as alloy " B," and it 
wore 13.5% slower than the standard phosphor-bronze. This metal is now 
the standard bearing-metal of the Pennsylvania Railroad, being slightly 
changed in composition to allow the use of phosphor-bronze scrap. The 
formula adopted is: Copper, 105 lbs.; phosphor-bronze, 60 lbs. ; tin, 9% lbs.; 
lead, 25J4 lbs. By using ordinary care in the foundry, keeping the metal 
well covered with charcoal during: the melting, no trouble is found in casting 
good bearings with this metal. The copper and the phosphor-bronze can be 
put in the pot before putting it in the melting-hole. The tin and lead should 
be added after the pot is taken from the fire. 

It is not known whether the use of a little zinc, or possibly some other 
combination, might not give still better results. For the present, however, 
this alloy is considered to fulfil the various conditions required for good 
bearing-metal better than any other alloy. The phosphor-bronze had an 
ultimate tensile strength of 30,000 lbs., with 6% elongation, whereas the alloy 
" B " had 24,000 lbs. tensile strength and 11% elongation. 

(For other bearing-metals, see Alloys containing antimony, on next page. 



S36 



ALLOYS CONTAINING ANTIMONY. 

Various Analyses of Babbitt Metal and other Alloys containing 
Antimony. 





Tin. 

50 

=89.3 
96 

=88.9 
85.7 
81.9 
81.0 
70.5 
22 
45.5 
89.3 
85 


Copper 


Antimony. 


Zinc. 


Lead. 


Bismuth. 




1 

1.8 

4 

3.7 

1.0 

"2"" 

4 
10 
1.5 
1.8 
5 


5 parts 
8.9perct. 
8 parts 
7.4 per ct. 

10.1 

16.2 

16. 

25.5 

62. 

13. 
7.1 

10. 
















Harder Babbitt | 
for bearings* j 










2.9 
1.9 
1. 












»i 






« 






" 


6. 






"Babbitt" 


40.0 




Plate pewter. . 
White metal. . . 




1 ft 


Bearings 


on Ger. locomotives. 



* It is mixed as follows: Twelve parts of copper are first melted and then 
36 parts of tin are added; 24 parts of antimony are put in, and then 36 parts 
of tin, the temperature being lowered as soon as the copper is melted in 
order not to oxidize the tin and antimony, the surface of the bath being 
protected from contact with the air. The alloy thus made is subsequently 
remelted in the proportion of 50 parts of alloy to 100 tin. (Joshua Rose.) 

White-metal Alloys.— The following alloys are used as lining metals 
by the Eastern Railroad of France (1890): 



Number. 

1 

2 

3 


Lead. 

65 



70 


Antimony. 
25 

11.12 
20 
8 


Tin. 


83.33 
10 
12 


Copper. 

5.55 



4 


80 






No. 1 is used for lining cross-head slides, rod-brasses and axle-bearings: 
No. 2 for lining axle-bearings and connecting-rod brasses of heavy engines; 
No. 3 for lining eccentric straps and for bronze slide-valves; and No. 4 for 
metallic rod-packing. 

Some o£ the best-known white-metal alloys are the following (Circular 
of Hoveler & Dieckhaus, London, 1893): 

Tin. Antimony. 



1. Parsons' 

2. Richards' 

3. Babbitt's 

4. Fentons' 

5. French Navy.. 

6. German Navy . . 



70 
55 
16 

85 



1 

15 
18 





7^ 



Lead. 

2 
10^ 



Copper. Zinc. 
2 27 

4% 



79 

87^ 



" There are engineers who object to white metal containing lead or zinc. 
This is, however, a prejudice quite unfounded, inasmuch as lead and zinc 
often have properties of great use in white alloys." 

It is a further fact that an "easy liquid" alloy must not contain more 
than 18$ of antimony, which is an invaluable ingredient of white metal for 
improving its hardness; but in no case must it exceed that margin, as this 
would reduce the plasticity of the compound and make it brittle. 

Hardest alloy of tin and lead: 6 tin, 4 lead. Hardest of all tin alloys (?): 74 
tin, 18 antimony, 8 copper. 

Alloy for thin open-work, ornamental castings: Lead 2, antimony 1. 
White metal for patterns: Lead 10, bismuth 6, antimony 2, common brass 8, 
tin 10. 

Type-metal is made of various proportions of lead and antimony, from 
17$ to 20% antimony according to the hardness desired. 

Babbitt Metals. (C. R. Tompkins, Mechanical Netvs, Jan. 1891.) 
The practice of lining journal-boxes with a metal that is sufficiently fusi- 
ble to be melted in a common ladle is not always so much for the purpose 
of securing anti-friction properties as for the convenience and cheapness of 
forming a perfect bearing in line with the shaft without the necessity of 



ALLOYS CONTAINING ANTIMONY. 337 

boring them. Boxes that are bored, no matter how accurate, require great 
care in fitting and attaching them to the frame or other parts of a machine. 

It is not good practice, however, to use the shaft for the purpose of cast- 
ing the bearings, especially if the shaft be steel, for the reason that the hot 
metal is apt to spring it; the better plan is to use a mandrel of the same 
size or a trifle larger for this purpose. For slow-running journals, where 
the load is moderate, almost any metal that may be conveniently melted 
and will run free will answer the purpose. For wearing properties, with a 
moderate speed, there is probably nothing superior to pure zinc, but when 
not combined with some other metal it shrinks so much in cooling that it 
cannot be held firmly in the recess, and soon works loose; and it lacks those 
anti-friction properties which are necessary in order to stand high speed. 

For line-shafting, and all work where the speed is not over 300 or 400 r. p. 
m., an alloy of 8 parts zinc and 2 parts block-tin will not only wear longer 
than any composition of this class, but will successfully resist the force of 
a heavy load. The tin counteracts the shrinkage, so that the metal, if not 
overheated, will firmly adhere to the box until it is worn out. But this 
mixture does not possess sufficient anti-friction properties to warrant its use 
in fast-running journals. 

Among all the soft metals in use there are none that possess greater anti- 
. friction properties than pure lead; but lead alone is impracticable, for it is so 
soft that it cannot be retained in the recess. But when by any process lead 
can be sufficiently hardened to be retained in the boxes without materially 
injuring its anti-friction properties, there is no metal that will wear longer 
in light fast-running journals. With most of the best and most popular 
anti-friction metals in use and sold under the name of the Babbitt metal, 
the basis is lead. 

Lead and antimony have the property of combining with each other in 
all proportions without impairing the anti-friction properties of either. The 
antimony hardens the lead, and when mixed in the proportion of 80 parts 
lead by weight with 20 parts antimony, no other known composition of 
metals possesses greater anti-friction or wearing properties, or will stand a 
higher speed without heat or abrasion. It runs free in its melted state, has 
no shrinkage, and is better adapted to light high-speeded machinery than 
any other known metal. Care, however, should be manifested in using it, 
and it should never be heated beyond a temperature that will scorch a dry 
pine stick. 

Many different compositions are sold under the name of Babbitt metal. 
Some are good, but more are worthless; while but very little genuine Babbitt 
metal is sold that is made strictly according to the original formula. Most 
of the metals sold under that name are the refuse of type-foundries and 
other smelting-works, melted and cast into fancy ingots with special brands, 
and sold under the name of Babbitt metal. 

It is difficult at the present time to determine the exact formulas used by 
the original Babbitt, the inventor of the recessed box, as a number of differ, 
ent formulas are given for that composition. Tin, copper, and antimony 
were the ingredients, and from the best sources of information the original 
proportions were as follows : 

Another writer gives: 

50partstin = 89. 3# 83. 3# 

2parts copper = 3.6% 8.3% 

4 parts antimony = t.\% 8.3% 

The copper was first melted, and the antimony added first and then about 
ten or fifteen pounds of tin, the whole kept at a dull-red heat and constantly 
stirred until the metals were thoroughly incorporated, after which the 
balance of the tin was added, and after being thoroughly stirred again it 
was then cast into ingots. When the copper is thoroughly melted, and 
before the antimony is added, a handful of powdered charcoal should be 
thrown into the crucible to form a flux, in order to exclude the air and pre- 
vent the antimony from vaporizing; otherwise much of it will escape in the 
form of a vapor and consequently be wasted. This metal, when carefully 
prepared, is probably one of the best metals in use for lining boxes that are 
subjected to a heavy weight and wear; but for light fast-running journals 
the copper renders it more susceptible to friction, and it is more liable to 
heat than the metal composed of lead and antimony in the proportions just 
given. 



338 



STRENGTH OF MATERIALS. 



SOLDERS. 

Common solders, equal parts tin and lead ; fine solder, 2 tin to 1 lead ; cheap 
solder, 2 lead, 1 tin. 
Fusing-point of tin- lead alloys: 



Tin 1 to lead 25 . . 
" 1 " " 10.. 



1 



. .334° F. 
..340 
. 356 



.558° F. Tin 1^ to lead 

..541 " 2 ' 

..511 " 3 

3 482 " 4 

2 441 " 5 

" 1 " " 1 370 " 6 

Common pewter contains 4 lead to 1 tin. 

Gold solder: 14 parts gold, 6 silver, 4 copper. Gold solder for 14-carat 
gold: 25 parts gold, 25 silver, 12^ brass, 1 zinc. 

Silver solder: Yellow brass 70 parts, zinc 7, tin 11^. Another: Silver 145 
parts, brass (3 copper, 1 zinc) 73, zinc 4. 
German-silver solder: Copper 38, zinc 54, nickel 8. 
Novel's solders for aluminum: 

Tin 100 parts, lead 5; melts at 536° to 572° F. 

" 100 " zinc 5; " 536 to 612 

"1000 " copper 10 to 15; " 662 to 842 
"1000 " nickel 10 to 15; " 662 to 842 
Novel's solder for aluminum bronze: Tin 900 parts, copper 100, bismuth 2 
to 3. It is claimed that this solder is also suitable for joining aluminum to 
copper, brass, zinc, iron, or nickel. 

ROPES AND CABLES. 

STRENGTH OF ROPES. 

(A S. Newell & Co., Birkenhead. Klein's Translation of Weisbach, vol. iii, 

part 1, sec. 2.) 



\ 



Hemp. 


Iron. 


Steel. 
















Tensile 




Weight 




Weight 




Weight 


Strength. 


Girth. 


per 
Fathom. 


Girth. 


per 
Fathom. 


Girth. 


per 

Fathom. 




Inches. 


Pounds. 


Inches. 


Pounds. 


Inches. 


Pounds. 


Gross tons. 


2% 


2 


1 


1 






2 






m 


M 


1 


1 


3 


m 


4 


m 


2 






4 






m 


2^ 


m 


w% 


5 


m 


5 


3 






6 






2 


Wz 


\% 


2 


7 


5^ 


7 


m 


4 


m 


2^ 


8 
9 


6 


9 


2y* 


5 


■m 


3 


10 
11 


6^2 


10 


m 


6 


2 


m 


12 






m 


§y% 


2^ 


4 


13 


7 


12 


7 


2J4 


4^ 


14 






3 


Wa 






15 


7^ 


14 


3^ 

m 
m 


8 


2% 


5 


16 
17 


8 


16 


9 


m 


5^ 


18 






m 


10 


2% 


6 


20 


m 


18 


3% 

m 


11 
12 


2% 


6^ 


22 
24 


w* 


22 


m 


13 


3J4 


8 


26 


10 


26 


4 


14 






28 


11 


30 


m 


15 


m, 


9 


30 






4^ 


16 

18 


m 


10 


32 
36 


12 


34 


m 


20 


m 


12 


40 



STRENGTH OF ROPES. 
Flat Ropes. 



339 



Hemp. 


Iron. 


Steel. 






Weight 




Weight. 




Weight 


Tensile 
Strength. 


Girth. 


per 
Fathom. 


Girth. 


per 
Fathom . 


Girth. 


per 
Fathom . 




Inches. 


Pounds. 


Inches. 


Pounds. 


Inches. 


Pounds. 


Gross tons. 


4 xl^ 


20 


2y 4 xy% 


11 






20 


5 x \y A 
5V£ x 1% 


24 


y&*% 


13 






23 


26 


m*% 


15 






27 


m * w* 


28 


3 x% 


16 


2 x^ 


10 


28 


6 x 1^ 


30 


m* 5 /8 


18 


2yi*y 2 


11 


32 


7 xl% 


36 


3y 2 *y 8 


20 


2y^y 2 


12 


36 


m*m 


40 


3^x11/16 


22 


2y 2 xy 2 


13 


40 


sy 2 x 2y 4 


45 


4 x 11/16 


25 


2%x% 


15 


45 


9 x 2i/ 2 


50 


4^x34 


28 


3 x% 


16 


50 


9^x2% 


55 


4^x?4 


32 


3^x3^ 


18 


56 


10 x 2^ 


60 


4%xH 


34 


3^x3^ 


20 


60 



Working Load, Diameter, and Weight of Ropes and 
Chains. (Klein's Weisbach, vol. iii, part 1, sec. 2, p. 561.) 

Hemp ropes: d = diam. of rope. Wire rope: d = diam. of wire, n -— 
number of wires, G = weight per running foot, k = permissible load in 
pounds per square inch of section, P — permissible load on rope or chain. 

Oval chains : d = diam of iron used ; inside dimensions of oval l.5d and 
2.6d. Each link is a piece of chain 2.6d long. G = weight of a single link == 
2.10d 3 lbs. ; G = weight per running foot = 9.73d 2 lbs. 





Hempen Rope. 


Wire Rope. 




Dry and Un tarred. 


Wet or Tarred. 




k (lbs.) = 
d (ins.) = 

P(lbs.) = 

G (lbs ) = 


1420 
0.03 VP 

1120d 2 = 2855(? 
1.28cZ2 = 0.00035P 


1160 
0.033 VP 

916d2 = 1975G 
1.54d2 == o.0005P 


17000 

0.0087 \f- 
v n 

13350nd 2 = 4590G 
2.91wd 2 =0.000218P 




Open-link Chain. 


Stud-link Chain. 


k (lbs. 
d (ins 
P(lbs 
G(lbs 


) = 

) = 
) = 
) = 


8500 
0.0087 VP 
13350d 2 = V660G 
9.73d 2 = 0.000737P 


11400 

0.0076 \/P 

17800^2 = 1660# 

10.65d 2 = 0.0006P 



Stud Chains 4/3 times as strong as open-link variety. [This is contrary to 
the statements of Capt. Beardslee, U. S. N., in the report of the U. S. Test 
Board. He holds that the open link is stronger than the studded link. See 
p. 308 ante]. 



340 



STRENGTH OF MATERIALS. 



STRENGTH AND WEIGHT OF WIRE ROPE, HEMPEN ROPE, AND 
CHAIN CABLES. (Klein's Weisbach.) 



Breaking Load 

in tons of 

2240 lbs. 


Kind of Cable. 


Girth of Wire Rope 

and of Hemp Rope 

Diameter of Iron 

of Chain, inches. 


Weight of One 

Foot In length. 

Pounds. 


1 Ton 


( Wire Rope 

< Hemp Rope 
( Chain 

( Wire Rope 
■I Hemp Rope 
{ Chain 
I Wire Rope 
•< Hemp Rope 
( Chain 

Wire Rope 
•< Hemp Rope 
( Chain 
( Wire Rope 
■< Hemp Rope 
( Chain 
( Wire Rope 

< Hemp Rope 
( Chain 

i Wire Rope 

•< Hemp Rope 

( Chain 

| Wire Rope 

■\ Hemp Rope 

( Chain 

i Wire Rope 

•< Hemp Rope 

( Chain 

( Wire Rope 

■< Hemp Rope 

( Chain 


1.0 
2.0 

H 

2.0 

5.0 

y z 

2.5 

7.0 
11/16 

3.0 

8.0 
13/16 

3.5 

9.0 
29/32 

4.0 
10.0 
31/32 

4.5 
11.0 

1.1/16 

5.0 
12.5 

1.3/16 

5.5 
14.0 

1.5/16 

6.0 

5.0 

1.7/16 


0.125 
0.177 


8 Tons 


0.500 
0.438 
978 


12 Tons 

16 Tons 


2.667 
0.753 
2.036 
4.502 
1.136 
2.365 


20 Tons 

24 Tons 


6.169 
1.546 
3.225 
7.674 
2.043 
4.166 


30 Tons .. 

36 Tons 


8.836 
2.725 
5.000 
10.335 
3.723 
5 940 


44 Tons 

54 Tons 


13.01 
4.50 
6.94 

16.00 
5.67 
7.92 

19.16 



Length sufficient to provide the maximum working stress : 

Hempen rope, dry and untarred 2855 feet. 

" " wet or tarred 1975 *' 

Wire rope 4590 " 

Open-link chain 1360 " 

Stud chain 1660 " 

Sometimes, when the depths are very great, ropes are given approximately 
the form of a body of uniform strength, by making them of separate pieces, 
whose diameters diminish towards the lower end. It is evident that by this 
means the tensions in the fibres caused by the rope's own weight can be 
considerably diminished. 

Rope for Hoisting or Transmission. Manila Rope 
(C. W. Hunt Company, New York.)— Rope used for hoisting or for trans- 
mission of power is subjected to a very severe test. Ordinary rope chafes 
and grinds to powder in the centre, while the exterior may look as though 
it was little worn. 

In bending a rope over a sheave, the strands and the yarns of these strands 
slide a small distance upon each other, causing friction, and wear the rope 
internally. 

The " Stevedore " rope used by the C. W. Hunt Co. is made by lubricating 
the fibres with plumbago, mixed with sufficient tallow to hold it in position. 
This lubricates the yarns of the rope, and prevents internal chafing and 
wear. After running a short time the exterior of the rope gets compressed 
and coated with the lubricant. 

In manufacturing rope, the fibres are first spun into a yarn, this yarn 
being twisted in a direction called "right hand." From 20 to 80 of these 
yarns, depending on the size of the rope, are then put together and twisted 
in the opposite direction, or "left hand," into a strand. Three of these 



STRENGTH OF ROPES. 341 

strands, for a 3-strand, or four for a 4-strand rope, are then twisted 
together, the twist being again in the "right hand " direction. When the 
strand is twisted, it untwists each of the threads, and when the three 
strands are twisted together into rope, it untwists the strands, but again 
twists up the threads. It is this opposite twist that keeps the rope in its 
proper form. When a weight is hung on the end of a rope, the tendency is 
for the rope to untwist, and become longer. In untwisting the rope, it 
would twist the threads up, and the weight will revolve until the strain of 
the untwisting strands just equals the strain of the threads being twisted 
tighter. In making a rope it is impossible to make these strains exactly 
balance each other. It is this fact that makes it necessary to take out the 
"turns" in a new rope, that is, untwist it when it is put at work. The 
proper twist that should be put in the threads has been ascertained approx- 
imately by experience. 

The amount of work that the rope will do varies greatly. It depends not 
only on the quality of the fibre and the method of laying up the rope, but 
also on the kind of weather when the rope is used, the blocks or sheaves 
over which it is run, and the strain in proportion to the strain put upon the 
rope. The principal wear comes in practice from defective or badly set 
sheaves, from excess of load and exposure to storms. 

The loads put upon the rope should not exceed those given in the tables, 
for the most economical wear. The indications of excessive load will be the 
twist coming out of the rope, or one of the strands slipping out of its proper 
position. A certain amount of twist comes out in using it the first day or 
two, but after that the rope should remain substantially the same. If it 
does not, the load is too great for the durability of the rope. If the rope 
wears on the outside, and is good on the inside, it shows that it has been 
chafed in running over the pulleys or sheaves. If the blocks are very small, 
it will increase the sliding of the strands and threads, and result in a more 
rapid internal wear. Rope made for hoisting and for rope transmission is 
usually made with four strands, as experience has shown this to be the most 
serviceable. 

The strength and weight of " stevedore " rope is estimated as follows: 

Breaking strength in pounds = 720 (circumference in inches) 2 ; 
Weight in pounds per foot = .032 (circumference in inches) 2 . 

The Technical Words relating to Cordage most frequently 
heard are: 

Yarn . —Fibres twisted together. 

Thread.— Two or more small yams twisted together. 

String.— The same as a thread but a little larger yarns. 

Strand.— Two or more large yams twisted together. 

Cord. — Several threads twisted together. 

Rope. — Several strands twisted together. 

Hawser.— A rope of three strands. 

Shroud-Laid.— A rope of four strands. ' 

Cable —Three hawsers twisted together. 

Yarns are laid up left-handed into strands. 

Strands are laid up right-handed into rope. 

Hawsers are laid up left-handed into a cable. 
A rope is : 

Laid by twisting strands together in making the rope. 

Spliced by joining to another rope by interweaving the strands. 

Whipped.— By winding a string around the end to prevent untwisting. 

Served.— When covered by winding a yarn continuously and tightly 
around it. 

Parceled.— By wrapping with canvas. 

Seized. — When two parts are bound together by a yarn, thread or string. 

Payed.— When painted, tarred or greased to resist wet. 

Haul. — To pull on a rope. 

Taut. — Drawn tight or strained. 

Splicing of Ropes. — The splice in a transmission rope is not only the 
weakest part of the rope but is the first part to fail when the rope is worn 
out. If the rope is larger at the splice, the projecting part will wear on the 
pulleys and the rope fail from the cutting off of the strands. The following 
directions are given for splicing a 4-strand rope. 

The engravings show each successive operation in splicing a \% inch 
manila rope. Each engraving was made from a full-size specimen. 



342 



STRENGTH OE MATERIALS. 




Fig. 81. 
Splicing of Ropes. 



SPLICING OF ROPES. 



343 



Tie a piece of twine, 9 and 10, around the rope to be spliced, about 6 feet 
from each end. Then unlay the strands of each end back to the twine. 

Butt the ropes together and twist each corresponding pair of strands 
loosely, to keep them from being tangled, as shown in Fig. 78. 

The twine 10 is now cut. and the strand 8 unlaid and strand 7 carefully laid 
in its place for a distance of four and a half feet from the junction. 

The strand (3 is next unlaid about one and a half feet and strand 5 laid in 
its place. 

The ends of the cores are now cut off so they just meet. 

Unlay strand 1 four and a half feet, laying strand 2 in its place. 

Unlay strand 3 one and a half feet, laying in strand 4. 

Cut all the strands off to a length of about twenty inches, for convenience 
in manipulation. 

The rope now assumes the form shown in Fig. 79 with the meeting points 
of the strands three feet apart. 

Each pair of strands is successively subjected to the following operation: 

From the point of meeting of the strands 8 and 7, unlay each one three 
turns; split both the strand 8 and the strand 7 in halves as far back as they 
are now unlaid and " whip " the end of each half strand with a small 
piece of twine. 

The half of the strand 7 is now laid in three turns and the half of 8 also 
laid in three turns. The half strands no.w meet and are tied in a simple 
knot, 11, Fig. 80, making the rope at this point its original size. 

The rope is now opened with a marlin spike and the half strand of 7 
worked around the half strand of 8 by passing the end of the half strand 7 
through the rope, as shown in the engraving, drawn taut and again worked 
around this half strand until it reaches the half strand 13 that was not laid 
in. This half strand 13 is now split, and the half strand 7 drawn through 
the opening thus made, and then tucked under the two adjacent sti'ands, as 
shown in Fig. 81. The other half of the strand 8 is now wound around the 
other half strand 7 in the same manner. After each pair of strands has 
been treated in this manner, the ends are cut off at 12, leaving them about 
four inches long. After a few days' wear they will draw into the body of the 
rope or wear off. so that the locality of the splice can scarcely be detected. 

Coal Hoisting. (C. W. Hunt Co.).— The amount of coal that can be 
hoisted with a rope varies greatly. Under the ordinary conditions of use 
a rope hoists from 5000 to 8000 tons. Where the circumstances are more 
favorable, the amounts run up frequently to 12,000 or 15,000 tons, occasion- 
ally to 20,000 and in one case 32,400 tons to a single fall. 

When ahoisting rope is first put in use, it is likely from the strain put upon 
it to twist up when the block is loosened from the tub. This occurs in the 
first day or two only. The rope should then be taken down and the 
"turns " taken out of the rope. When put up again the rope should give 
no further trouble until worn out. 

It is necessary that the rope should be much larger than is needed to bear 
the strain from the load. 

Practical experience for many years has substantially settled the most 
economical size of rope to be used which is given in the table below. 

Hoisting ropes are not spliced, as it is difficult to make a splice that will 
not pull out while running over the sheaves, and the increased wear to be 
obtained in this way is very small. 

Coal is usually hoisted with what is commonly called a " double whip; " 
that is, with a running block that is attached to the tub which reduces the 
strain on the rope to approximately one half the weight of the load hoisted. 
The following table gives the usual sizes of hoisting rope and the proper 
working strain: 

Stevedore Hoisting-rope. 
C. W. Hunt Co. 



Circumference of 
the rope in ins. 



4 

4^ 



Proper Working 


Nominal size of 


Approximate 


Strain on the 


Rope 


Coal tubs. Double 


Weight of a Coil, 


in lbs. 






whip. 


in lbs. 


350 




1/6 to 1/5 tons. 


360 


500 




1/5 


=a :: 


480 


650 




M 


650 


800 
1000 




S 


::« :: 


830 
960 



Hoisting rope is ordered by circumference, transmission rope by diameter. 



344 



STRENGTH OF MATERIALS. 



Weight and Strength of Manila Cordage. 

Dodge Manufacturing Co. 













o ^ 






11 


o g a 


Q P. 

a q 


T3 

a 

3 


Sg 


■Si a 


0) P« 
c o 


a 


Size. Dia 
in inche 


<»fa 5 


en fe w 
nil 

1*8 


«3 O 
.9 


.S3 5 

55 " 






c » 


£ S 


s* 


fa 


£ 1 


00"° A 


fa 


3/16 


12 


540 


50' 


1 5/16 


310 


16,000 


1' W 


H 


18 


780 


33' 4" 


1% 


346 


18,062 


1 8 


5/16 


24 


1,000 


25 


v/% 


390 


20,250 


1 6 


% 


30 


1,280 


20 


1 9/16 


435 


22,500 


1 5 


7/16 


37 


1,562 


17 8 


1% 


480 


25,000 


1 3 


M 


46 


2,250 


13 


1« 


581 


30,250 


1 


9/16 


65 


3,062 


9 3 


2 


678 


36,000 


10% 


% 


80 


4,000 


7 6 


m 


797 


42,250 


9 


% 

13/16 


98 


5,000 


6 


2 l A 


920 


49,000 


6% 


120 


6,250 


5 


2y 2 


1,106 


56,250 


m 


% 


142 


7,500 


4 3 


2% 


1,265 


64,000 


5^ 


1 


170 


9,000 


3 6 


Ws 


1,420 


72,250 


5 


1 1/16 


200 


10,500 


3 


3 


1,572 


81,000 


&A 


v/& 


230 


12,250 


2 7 


M 


1,760 


90,250 


4 


M 


271 


14,000 


2 3 


33/ 8 


1,951 


100,000 


3^ 



T. Spencer Miller (Enc/g Neivs, Dec. 6, 1890) gives the following table of 
breaking strength of mauila rope, which he considers more reliable than 
the strength computed by Mr. Hunt's formula, Breaking strength = 720 X 
(circumference in inches) 2 . Mr. Miller's formula is: Breaking weight lbs. = 
circumference 2 X a coefficient which varies from 900 for \#' to 700 for 2" 
diameter rope, as shown in the table. 



Diam. 
in. 


Circum- 
ference. 
in. 


Ultimate 

Strength. 

lbs. 


Coeffi- 
cient. 


Diam. 
in. 


Circum- 
ference, 
in. 


Ultimate 

Strength. 

lbs. 


Coeffi- 
cient. 




2 
2H 


2,000 
3,250 
4,000 
6,000 
7,000 
9,350 


900 
845 
820 
790 
780 
765 


m 

m 
m 

2 


5 
6 


10,000 
13,000 
15,000 
18,200 
21,750 
25,000 


760 
745 
735 
725 

712 
700 



For rope-driving Mr. Hunt recommends that the working strain should 
not exceed 1/20 of the ultimate breaking strain. For further data on ropes 
see " Rope-driving." 

Knots. — A great number of knots have been devised of which a few 
only are illustrated, but those selected are the most frequently used. In 
the cuts. Fig. 82, they are shown open, or before being drawn taut, in order 
to show the position of the parts. The names usually given to them are: 



A. Bight of a rope. 

B. Simple or Overhand knot. 

C. Figure 8 knot. 

D. Double knot. 

E. Boat knot. 

F. Bowline, first step. 

G. Bowline, second step. 
H. Bowline completed. 

I. Square or reef knot. 

J. Sheet bend or weaver's knot. 

K. Sheet bend with a toggle. 

L. Carrick bend. 

M. Stevedore knot completed. 

N. Stevedore knot commenced. 

O. Slip knot. 



P. Flemish loop. 

Q. ~ 

R. 



Chain knot with toggle. 

Half-hitch. 

Timber-hitch. 

Clove hitch. 

Rolling-hitch. 

Timber-hitch and half-hitch. 
W. Blackwall-hitch. 
X. Fisherman's bend. 

Round turn and half-hitch. 

Wall knot commenced. 
" " completed. 

Wall knot crown commenced. 
" " " completed. 



U. 



Y. 
Z. 

A A. 
BB. 

CC. 



345 



The principle of a knot is that no two parts, which would move in the 
same direction if the rope were to slip, should lay along side of and touch- 
ing each other. 

The bowline is one of the most useful knots, it will not slip, and after 
being strained is easily untied. Commence by making a bight in the rope, 
then put the end through the bight and under" the standing part as shown in 
O, then pass the end again through the bight, and haul tight. 

The square or reef knot must not be mistaken for the " granny " knot 
that slips under a strain. Knots H, K and M are easily untied after being 
under strain. The knot M is useful when the rope passes through an eye 
and is held by the knot, as it will not slip and is easily untied after being 
strained. 

ABC E 




Fig. 82.— Knots. 
The timber hitch S looks as though it would give way, but it will not; the 
greater the strain the tighter it will hold. The wall knot looks complicated, 
but is easily made by proceeding as follows: Form a bight with strand 1 
and pass the strand 2 around the end of it, and the strand 3 round the end 
of 2 and then through the bight of 1 as shown in the cut Z. Haul the ends 
taut when the appearance is as shown in AA. The end of the strand 1 is 
now laid over the centre of the knot, strand 2 laid over 1 and 3 over 2, when 
the end of 3 is passed through the bight of 1 as shown in BB. Haul all the 
strands taut as shown in CC. 



346 



STRENGTH OF MATERIALS. 



To Splice a Wire Rope.— The tools required will be a small marline 
spike, nipping cutters, and either clamps or a small hernp-rope sling with 
which to wrap around and untwist the rope. If a bench-vise is accessible 
it will be found convenient. 

In splicing rope, a certain length is used up in making the splice. An 
allowance of not less than 16 feet for ^ inch rope, and proportionately 
longer for larger sizes, must be added to the length of an endless rope in 
ordering. 

Having measured, carefully, the length the rope should be after splic- 
ing, and marked the points M and M', Fig. 83, unlay the strands from each 
end E and E' to M and M' and cut off the centre at M and M', and then: 

(1). Interlock the six unlaid strands of each end alternately and draw 
them together so that the points M and M' meet, as in Fig. 84. 

(2). Unlay a strand from one end, and following the unlay closely, lay into 
the seam or groove it opens, the strand opposite it belonging to the other 
end of the rope, until within a length equal to three or four times the length 
of one lay of the rope, and cut the other strand to about the same length 
from the point of meeting as at A, Fig. 85. 

(3). Unlay the adjacent strand in the opposite direction, and following the 
unlay closely, lay in its place the corresponding opposite strand, cutting the 
ends as described before at B, Fig. 85. 

There are now four strands laid in place terminating at A and B, with the 
eight remaining at M M\ as in Fig. 85. 

It will be well after laying each pair of strands to tie them temporarily at 
the points A and B. 

Pursue the same course with the remaining four pairs of opposite strands, 




NT 

Fig. 86. Fig. 87. 

Splicing Wire Rope. 
stopping each pair about eight or ten turns of the rope short of the preced- 
ing pair, and cutting the ends as before. 

We now have all the strands laid in their proper places with their respect- 
ive ends passing each other, as in Fig. 86. 

All methods of rope-splicing are identical to this point: their variety con- 
sists in the method of tucking the ends. The one given below is the one 
most generally practiced. 

Clamp the rope either in a vise at a point to the left of A, Fig. 86, and by a 
hand-clamp applied near A, open up the rope by untwisting sufficiently to 
cut the core at A, and seizing it with the nippers, let an assistant draw it 
out slowly, you following it closely, crowding the strand in its place until it 
is all laid in. Cut the core where the strand ends, and push the end back 
into its place. Remove the clamps and let the rope close together around it. 
Draw out the core in the opposite direction and lay the other strand in the 
centre of the rope, in the same manner. Repeat the operation at the five 
remaining points, and hammer the rope lightly at the points where the ends 
pass each other at A, A, B, B, etc.. with small wooden mallets, and the 
splice is complete, as shown in Fig. 87. 

If a clamp and vise are not obtainable, two rope slings and short wooden 
levers may be used to untwist and open up the rope. 

A rope spliced as above will be nearly as strong as the original rope and 
smooth everywhere. After running a few days, the splice, if well made, 
cannot be found except by close examination. 

The above instructions have been adopted by the leading rope manufac- 
turers of America, 



HELICAL STEEL SPRINGS. 347 



SPRINGS. 

Definitions. —A spiral spring is one which is wound around a fixed 
point or centre, and continually receding from it like a watch spring. A 
helical spring is one which is wound around an arbor, and at the same time 
advancing like the thread of a screw. An elliptical or laminated spring is 
made of flat bars, plates, or "leaves, 11 of regularly varying lengths, super- 
posed one upon the other. 

Laminated Steel Springs.— Clark (Rules, Tables and Data) gives 
the following from his work on Baihvay Machinenj, 1855: 

1.66L 3 . _ bthi m = 1.66£s . 

~ bthi ' S ~~ 11. 3L ; n ~ Abt* ' 

A. = elasticity, or deflection, in sixteenths of an inch per ton of load, . 
s = working strength, or load, in tons (2340 lbs.), 
L = span, when loaded, in inches, 
b = breadth of plates, in inches, taken as uniform, 
t = thickness of plates, in sixteenths of an inch, 
n = number of plates. 

Note.— The span and the elasticity are those clue to the spring when 
weighted. 

2 When extra thick back and short plates are used, they must be replaced 
by an equivalent number of plates of the ruling thickness, prior to the em- 
ployment of the first two formulae. This is found by multiplying the num- 
ber of extra thick plates by the cube of their thickness, and dividing by the 
cube of the ruling thickness. Conversely, the number of plates of the ruling 
thickness given by the third formula, required to be deducted and replaced 
by a given number of extra thick plates, are found by the same calculation. 
3. It is assumed that the plates are similarty and regularly formed, and 
that they are of uniform breadth, and but slightly taper at the ends. 
Reuleaux's Constructor gives for semi-elliptic springs: 
_ Snbh* , . 6P1 3 

P= -6T and f=Enbh*' 
S = max. direct fibre-strain in plate; b — width of plates; 
n = number of plates in spring; h = thickness of plates; 

I = one half lengtii of spring; / = deflection of end of spring; 

P = load on one end of spring; E — modulus of direct elasticity. 

The above formula for deflection can be relied upon where all the plates 
of the spring are regularly shortened; but in semi-elliptic springs, as used, 
there are generally several plates extending the full length of the spring, 
and the proportion of these long plates to the whole number is usually about 

5 5P1 3 
one fourth. In such cases / = ' , .. . (G-. R. Henderson, Trans. A. S, M. E., 
Enbh 3 ' ' 

vol. xvi.) 

In order to compare the formulae of Reuleaux and Clark we may make 
the following substitutions in the latter: s in tons — Pin lbs. -=- 1120; as = 
16/; L = 21; t — 16/i; then 

1.66 X 823 x P Pi 3 

A s = 1Q f = Amnv ii->fw,,;w,3 ' whence / - 



" 4096 X 1120 X «6/i 8 ' J 5,527,133' 

which corresponds with Reuleaux's formula for deflection if in the latter we 
take E = 33,162,800. 

P 256n67ia , _ 12.687n6fc a 

Also s = -rr^r = -t— ., whence P = , 

1120 11.3 X 21 I 

which corresponds with Reuleaux's formula for working load when Sin the 
latter is taken at 76,120. 

The value of Eis usually taken at 30,000,000 and S at S0,000, in which case 
Reuleaux's formulae become 

_ 13,333h6Zi 2 . . PI 3 

P — — — ■ and / 



5,000,000ii.6/t3' 



Helical Steel Springs. — Clark quotes the following from the report 
on Safety Valves (Trans. Inst. Engrs. and Shipbuilders in Scotland, 1874-5): 
„ d 3 X w 



348 SPRINGS. 

E = compression or extension of one coil in inches, 

d = diameter from centre to centre of steel bar constituting the spring, 

in inches, 
w = weight applied, in pounds, 
D = diameter, or side of the square, of the steel bar, in sixteenths of an 

inch, 
C = a constant, which may be taken as 22 for round steel and 30 for 

square steel. 

Note.— The deflection Efor one coil is to be multiplied by the number of 
free coils, to obtain the total deflection for a given spring. 

The relation between the safe load, size of steel, and diameter of coil, may 
be taken for practical purposes as follows: 






w —, for round steel; 
3 



V wd r, , , 

for square steel. 



Rankine's Machinery and Millwork, p. 390, gives the following: 
W _ cd* _ ,196/ri' . _ 12.566n/r 2 . 

v ~ 64nr 3 ' 1 ~ r ' *• ~ cd ' 

—- 1 = greatest safe sudden load. 

In which d is the diameter of wire in inches; c a co-efficient of transverse 
elasticity of wire, say 10,500,000 to 12,000,000 for charcoal iron wire and steel; 
r radius to centre of wire in coil; n effective number of coils; / greatest safe 
shearing stress, say 30,000; W any load not exceeding greatest safe load; 
v corresponding extension or compression; W 2 greatest safe load; and v 1 
greatest safe steady extension or compression. 

If the wire is square, of the dimensions d x d, the load for a given deflec- 
tion is greater than for a round wire of the diameter d in the ratio of 2.81 to 
1.96 or of 1.43 to 1, or of 10 to 7, nearly. 

Wilson Hartnell (Proc. Inst. M. E., 1882, p. 426), says: The size of a spiral 
spring may be calculated from the formula on page 304 of " Rankine's Use- 
ful Rules and Tables"; but the experience with Salter's springs has shown 
that the safe limit of stress is more than twice as great as there given, 
namely 60,000 to 70,000 lbs. per square inch of section with % inch wire, and 
about 50,000 with \& inch wire. Hence the work that can be done by 
springs of wire is four or five times as great as Rankine allows. 

For % inch wire and under, 

_, . , ' . .. 12,000 X (diam. of wire) 3 

Maximum load in lbs. = ~ ^ ^ : — - ; 

Mean radius of springs 

^ . . . . ., . ' „ . . . . 180,000 x (diam.)4 

Weight in lbs. to deflect spring 1 in. = — - — y - / J NQ . 

& Number of coils X (rad.) 3 

The work in foot-pounds that can be stored up in a spiral spring would 
lift it above 50 ft. 

In a few rough experiments made with Salter's springs the coefficient of 
rigiditv was noticed to be 12,600,000 to 13,700,000 with \i inch wire; 11,000,000 
for 11/32 inch; and 10,600,000 to 10,900,000 for % inch wire. 

Helical Springs.— J. Begtrup, in the American Machinist of Aug. 
18, 1892, gives formulas for the deflection and carrying capacity of helical 
springs of round and square steel, as follow: 

W = .3927-^^ 

™^ ,>, )• for round steel. 

F ~ 8 Ed* ' 



TT=.471 Sd3 



y for square steel. 



HELICAL SPRINGS. 



349 



W = carrying capacity in pounds, 

S = greatest tensile stress per square inch of material, 

d =' diameter of steel, 
D = outside diameter of coil, 
F — deflection of one coil, 
E — torsional modulus of elasticity, 
P = load in pounds. 

From these formulas the following table has been calculated by Mr. Beg- 
trup. A spring being made of an elastic material, and of such shape as to 
allow a great amount of deflection, will not be affected by sudden shocks or 
blows to the same extent as a rigid body, and a factor of safety very much 
less than for rigid constructions may be used; 

HOW TO USE THE TABLE. 

When designing a spring for continuous work, as a car spring, use a 
greater factor of safety than in the table; for intermittent working, as in 
a steam-engine governor or safety valve, use figures given in table; for 
square steel multiply line W by 1.2 and line F by .59. 

Example 1. — How much will a spring of %" round steel and 3" outside 
diameter carry with safety ? In the line headed D we find 3, and right un- 
derneath 473, which is the weight it will carry with safety. How many coils 
must this spring have so as to deflect 3" with a load of 400 pounds ? Assum- 
ing a modulus of elasticity of 12 millions we find in the centre line headed 
F the figure .0610; this is deflection of one coil for a load of 100 pounds; 
therefore .061 X 4 = .244" is deflection of one coil for 400 pounds load, and 3 
-7- .244 = 12J/2 is the number of coils wanted. This spring will therefore be 
4%" long when closed, counting working coils only, and stretch to 1%" . 

Example 2. — A spring 3J4" outside diameter of 7/16" steel is wound close; 
how much can it be extended without exceeding the limit of safety ? We 
find maximum safe load for this spring to be 702 pounds, and deflection of 
one coil for 100 pounds load .0405 inches; therefore 7.02 x .0405 = .284" is the 
greatest admissible opening between coils. We may thus, without know- 
ing the load, ascertain whether a spring is overloaded or not. 

Carrying Capacity and Deflection of Helical Springs of 
Round Steel. 

d = diameter of steel. D = outside diameter of coil. W = safe working 
load in pounds— tensile stress not exceeding 60,000 pounds per square inch. 
F = deflection by a load of 100 pounds of one coil, and a modulus of elasti- 
city of 10, 12 and 14 millions respectively. The ultimat e carrying capacity 
will be about twice the safe load. 



w • 


T) 


.25 


.50 


.75 


1.00 


1.25 


1.50 


1.75 


2.00 








o" -1 


W 


35 


15 


9 


7 


5 


4.5 


3.8 


3.3 








III 


\ 


.0276 


.3588 


1.433 


3.562 


7.250 


12.88 


20.85 


31.57 








f\ 


.0236 


.3075 


1.228 


3.053 


6.214 


11.04 


17.87 


27.06 










} 


.0197 


.2562 


1.023 


2.544 


5.178 


9.200 


14.89 


22.55 









- 


.50 


.75 


1.00 


1.25 


1.50 


1.75 


2.00 


2.25 


2.50 




§**-i 


W 


107 


65 


46 


36 


29 


25 


22 


19 


17 








[ 


.0206 


.0937 


.2556 


.5412 


.9856 


1.624 


2.492 


3.625 


5.056 








f] 


.0176 


.0804 


.2191 


.4639 


.8448 


1.392 


2.136 


3.107 


4.334 






I 

D 


.0147 


.0670 


.182 


.3866 


.7010 


1.160 

2.00 


1.780 


2.589 
2.50 


3.612 






5 


75 


1.00 


1.25 


1.50 


1.75 


2.25 


2 75 


3.00 




OO j.," 


W 


241 


167 


128 


104 


88 


75 


66 


59 


53 


49 




k 


1 


.0137 


.0408 


.0907 


.1703 


.2866 


.4466 


.6571 


.9249 


1.256 


1 660 




f\ 


.0118 


.0350 


.0778 


.1460 


.2457 


.3828 


.5632 


.7928 


1 077 


1 423 




^Q 


( 


.0098 


.0292 


.0648 


.1217 


.2048 


.3190 


.4693 


.6607 


.8975 


1.186 






D 


1.25 


1.50 


1.75 


2.00 


2.25 


2.50 


2.75 


3.00 


3.25 


3.50 




T 


W 


368 


294 


245 


210 


184 


164 


147 


134 


123 


113 




II 


1 


.0199 


.0389 


.0672 


.1067 


.1593 


.2270 


.3109 


.4139 


.5375 


6835 




v\ 


.0171 


.0333 


.0576 


.0914 


.1365 


.1944 


.2665 


.3548 


.4607 


.5859 






I 


.0142 


.0278 


.0480 


.0762 


.1137 


.1610 


.2221 


.2957 


.3839 


.4883 





350 



SPEINGS. 



Carrying Capacity and Deflection of Helical Springs of 
Round Steel.— (Continued). 



5 


D 


1.50 


1.75 


2.00 


2.25 


2.50 


2.75 


3.00 


3.25 


3.50 


3.75 


4.00 


2 


W 


605 


500 


426 


371 


329 


295 


267 


245 


226 


209 


195 


in 


( 


.0136 


.0242 


.0392 


.0593 


.0854 


.1187 


.1583 


.2066 


.2640 


.3312 


.4089 


II 


f\ 


.0117 


.0207 


.0336 


.0508 


.0732 


.1012 


.1357 


.1771 


.2263 


.2839 


.3505 


•8 


1 


.0097 


.0173 


.0280 


.0424 


.0610 
3.00 


.0853 


.1131 


.1476 


.1886 


.2366 


.2921 




2.00 


2.25 


2.50 


2.75 


3.25 


3.50 


3.75 


4.00 


4.25 


4.50 


II 


W 


765 


663 


589 


523 


473 


433 


398 


368 


343 


321 


30! 


\ 


.0169 


.0259 


.0377 


.0528 


.0711 


.0935 


.1200 


.1513 


.1874 


.2290 


.2761 


f\ 


.0145 


.0222 


.0323 


.0452 


.0610 


.0801 


.1029 


.1297 


.1006 


. 1963 


.2367 


is 


1 

D 


.0120 


.0185 


.0269 


.0376 


.0508 


.0668 
3.25 


.0858 


.1081 


.1338 


.1635 


.1972 


- 


2.00 


2.25 


2:50 


2.75 


3.00 


3.50 


3.75 


4.00 


4 50 


5.00 


2 


W 


1263 


1089 


957 


853 


770 


702 


644 


596 


544 


486 


432 


j> 


i 


.0081 


.0126 


.0186 


.0262 


.0357 


.0472 


.0617 


.0772 


.0960 


.1423 


.2016 


II 


f\ 


.0069 


.0108 


.0160 


.0225 


.0306 


.0405 


.0529 


.0661 


.0823 


.1220 


1728 


'e 


\ 


.0058 


.0090 


.0133 


.0187 


.0255 


.0337 


.0441 


.0551 


.0686 


.1017 


.1440 




D 


2.00 


2.25 


2.50 


2.75 


3.00 


3 25 


3.50 


3.75 


4.00 


4.50 


5.00 


& 


W 


1963 


1683 


1472 


1309 


1178 


1071 


982 


906 


841 


736 


654 


[ 


.0042 


.0067 


.0099 


.0141 


.0194 


.0259 


.0336 


.0427 


.0534 


.0796 


.1134 


II 


f\ 


.0036 


.0057 


.0085 


.0121 


.0167 


.0222 


.0288 


.0366 


.0457 


.0683 


.0972 


tt 


( 


.0030 


.0048 


.0071 


.0101 


.0139 


.0185 


.0240 


.0305 


.0381 


.0569 
5.00 


.0810 


- 


2.50 


2.75 


3.00 


3.25 


3.50 


3.75 


4.00 


4.25 


4.50 


5.50 


2 


W 


2163 


1916 


1720 


1560 


1427 


1315 


1220 


1137 


1065 


945 


849 


OS 


I 


.0056 


.0081 


.0112 


.0151 


.0197 


.0252 


.0316 


.0390 


.0474 


.0679 


.0935 


II 


f\ 


.0048 


.0070 


.0096 


.01.29 


.0169 


.0216 


.0271 


.0334 


.0406 


0582 


.0801 


■s 


} 

D 


.0040 


.0058 


.0080 


.0108 


.0141 


.0180 


.0225 


.0278 


.0339 


.0485 


.0668 




2.50 


2.75 


3.00 


3.25 


3.50 


3.75 


4.00 


4.25 


4.50 


5.00 


5.50 


S5 


W 


3068 


2707 


2422 


2191 


2001 


1841 


1704 


1587 


1484 


1315 


1180 


1 


.0034 


.0049 


.0068 


.0092 


.0121 


.0155 


.0196 


.0243 


.0297 


.0427 


.0591 


II 


F\ 


.0029 


.0042 


.0058 


.0079 


.0104 


.0133 


.0168 


.0208 


.0254 


.0366 


.0506 


•8 


I 

n 


.0024 


.0035 


.0049 


.0066 


.0086 


.0111 


.0140 


.0173 


.0212 
5.00 


.0305 


.0422 




3.00 


3.25 


3.50 


3.75 


4.00 


4.25 


4.50 


4.75 


5.50 


6.00 




w 


3311 


2988 


2723 


2500 


2311 


2151 


2009 


1885 


1776 


1591 


1441 


IIS 


[ 


.0043 


.0058 


.0077 


.0100 


.0127 


.0157 


.0193 


.0233 


.0279 


.0388 


.0522 


■«!-• 


jrl 


.0037 


.0050 


.0066 


.0086 


.0108 


.0135 


.0165 


.0200 


.0239 


.0333 


.0447 




J_ 


.0030 


.0042 


.0055 


.0071 


.0090 


.0112 


.0138 
4.50 


.0167 
4.75 


.0199 


.0277 
5.50 


.0373 




3.00 


3.25 


3.50 


3 75 


4.00 


4.25 


5.00 


6.00 


S 


W 


4418 


3976 


3615 


3313 


3058 


2840 


2651 


2485 


2339 


2093 


1893 


( 


.0028 


.0038 


.0051 


.0066 


.0084 


.0105 


.0129 


.0157 


.0189 


.0264 


.0356 




F \ 


.0024 


0033 


.0044 


.0057 


.0072 


.0090 


.0111 


.0135 


.0162 


.0226 


0305 


"8 


( 


.0020 


.0027 


.0036 


.0047 


.0060 


.0075 


.0093 


.0113 


.0135 


.0188 


.0254 




D 


3.50 


3.75 


4.00 


4.25 


4.50 


4.75 


5.00 


5.25 


5.50 


6.00 


6.50 


£ 


W 


6013 


5490 


5051 


4676 


4354 


4073 


3826 


3607 


3413 


3080 


2806 


A 


.0021 


.0027 


.0035 


.0045 


.0055 


.0067 


.0081 


.0097 


.0115 


.0156 


.0207 


ll 


.0018 


.0024 


.0030 


.0038 


.0047 


.0058 


.0070 


.0083 


.0098 


.0134 


.0177 


-8 


i 


.0015 


.0020 


.0025 


.0032 


.0039 


.0048 


.0058 


.0069 


.0082 


.0112 


.0148 




D 


3.50 


3.75 


4.00 


4.25 


4.50 


4.75 


5.00 


5.25 


5.50 


6.00 


6.50 


^ 


W 


9425 


8568 


7854 


7250 


6732 


6283 


5890 


5544 


5236 


4712 


4284 


II 


{ 


.0012 


.0016 


.0021 


.0026 


.0033 


.0041 


.0049 


.0059 


0071 


.0097 


.0129 


f\ 


.0010 


.0014 


.0018 


.0023 


.0028 


.0035 


.0043 


.0051 


.0061 


.0083 


.0111 




I 


.0008 


.0011 


.0015 


.0019 


.0023 


.0029 


.0035 


.0043 


.0051 


.0069 


.0092 



The formulae for deflection or compression given by Clark, Hartnell, and 
Begtrup, although very different in form, show a substantial agreement 
when reduced to the same form. Let d = diameter of wire in inches, D l = 
mean diameter of coil, n the number of coils, to the applied weight in 
pounds, and C a coefficient, then 



HELICAL SPRINGS. 351 

Cd* ' 
Cd* 



Compression or extension of one coil _ 



Weight in pounds to cause comp. or ext. of 1 in. _ 

The coefficient C reduced from Hartnell's formula is 8 X 180,000 =1,440,000; 
according to Clark, 16 4 X 22 = 1,441.792, and according to Begtrup (using 
12,000,000 for the torsional modulus of elasticity) = 12,000,000 -*- 8 = 1,500,000. 

Rankine's formula for greatest safe extension, v x = — '- — ■ may take 

the form v t = '' ™ * ■ if we use 30,000 and 12,000,000 as the values for / 

and c respectively. 

The several formulas for safe load given above may be thus compared, 
letting d = diameter of wire, and D x = mean diameter of coil, Rankine, 

.196/d3 T _ 3(d X 16)3 .39275^3 TT 
W = — * — ; Clark, W = ^—^ — - ; Begtrup, W = ~— ; Hartnell, 

12000d 3 1 

W — — . Substituting for / the value 30,000 given by Rankine, and for 

d 3 i73 

S, 60,000 as given by Begtrup, we have W = 11,760 — Rankine; 12,288 j- 

d 3 AS 1 x 

Clark; 23,562— Begtrup; 24,000 j- Hartnell. 

Taking from the Pennsylvania Railroad specifications the capacity when 
closed of the following springs, in which d — diameter of wire, D diameter 
outside of coil, D x = D — d, c capacity, H height when free, and h height 
when closed, all in inches. 

= \\i -Di = 1J4 c = 400 H = 9 h = 6 

3 2\i 1,900 8 5 

5M 5 2,100 7 4U 

5 4 8,100 ltfU 8 

8 m 10,000 9 5% 

4% 3M 16,000 4% S% 

d 3 
and substituting the values of c in the formula c — W = x — we find x, the 

d 3 X 

coefficient of — to be respectively 32,000; 38,000; 32,400; 24,888; 34,560; 

42,140, average 34,000. 

d 3 
Taking 12,000 as the coefficient of —according to Rankine and Clark for 

safe load, and 24,000 as the coefficient according to Begtrup and Hartnell, 
we have for the safe load on these springs, as we take one or the other co- 
efficient, 

T. 8. K. D. I. C. 

Rankine and Clark 150 600 1,012 3,000 3.750 5,400 lbs. 

Hartnell 300 1,200 2,024 6,000 7^500 10,800 " 

Capacity when closed, as above 400 1,900 2,100 8,100 10,000 16,000 " 

J. W. Cloud (Trans. A. S. M. E., v. 173) gives the following: 

„ Snd* _ . 32 PRH 

p =Tqr and /= -^^ ; 

P = load on spring; 

8 — maximum shearing fibre-strain in bar; 
d = diameter of steel of which spring is made ; 
B = l'adius of centre of coil; 
I = length of bar before coiling; 
G = modulus of shearing elasticity; 
/ = deflection of spring under load. 

Mr. Cloud takes S = 80,000 and G ~ 12,600,000. 

The stress in a helical spring is almost wholly one of torsion. For method 
of deriving the formulas for springs from torsional formula see Mr. Cloud's 
paper, above quoted. 



No. T. 


d = H 


S. 


H 


K. 


M 


D. 


l 


I. 


m 


a 


m 



352 



.ELLIPTICAL 


SPRINGS, SIZES, AND PROOF TESTS. 




Pennsylvania Railroad Specifications, 1889. 




a 


"3 


8* 




Tests. 




S.g 


o*§ 


fl 


O w 05 




Class. 








■ 


■§ "* 


•a 


p.y 


To stand ins. High. 


With Load 
of lbs. 




J 


5 


m 


















( 3% between bands. 


4800 


A, Triple 


40 


11*4 


3 x% 


3 


8 

(3M 


;; 


5500 

A. p. t.* 

6650 


C, Quadruple.. 


40 


l5i/ 2 


3 x% 


3 


1! 


, 


8000 

A. p. t.* 


A Triple 


36 


11% 


3 x% 


3 


11 


i (i 


6000 
8000 












( 5 bet. centre of eye 




E, Single 


40 


sin. 


3 x% 


3x11/32 


•< and top of leaf. 
(3 
2^ between bands. 


When free 

2350 

11,800 


F, Triple 


35 


11% 


3 x% 


3x11/32 


G, Double 


32 


% 


3 x% 


3 


W :: « 


When free 

8000 


if, Double 


36 


9^ 


3 x% 


4 


{1* - :: 


5400 
6000 


ft- j Double, I 
*' I 6 plates j 


22 


10% 


3^x3^ 


4^x11/32 


13/16 " 


13,800 


T \ Double, j 
■** 1 7 plates j 


22 


10% 


3^x% 


4^x11/32 


13/16 " 
(4 


15,600 

8000 


Jlf, Quadruple.. 


40 


16H 


3 x% 


3 


I 3 

2 


10,000 
A. p. t.* 



* A. p. t., auxiliary plates touching. 
PHOSPHOR-BRONZE SPRINGS. 

Wilfred Lewis (Engineers' Club, Philadelphia, 1887) made some tests with 
phosphor-bronze wire, .12 in. diameter, coiled in the form of a spiral spring, 
134 in. diameter from centre to centre, making 52 coils. 

This spring was loaded gradually up to a tension of 30 lbs., but as the load 
was removed it became evident that a permanent set had taken place. 
Such a spring of steel, according to the practice of the P. R. R., might be 
used for 40 lbs. A weight of 21 lbs. was then suspended so as to allow a 
small amount of vibration, and the length measured from day to day. In 30 
hours the spring lengthened from 20% inches to 213/g inches, and in 200 hours 
to 21J4 inches. It was concluded tliat 21 lbs. was too great for durability, and 
that probably 10 lbs. was as much as could be depended upon with safety. 

For a given load the extension of the bronze spring was just double the 
extension of a similar steel spring, that is, for the same extension the steel 
spring is twice as strong. 

SPRINGS TO RESIST TORSIONAL FORCE. 
(Reuleaux's Constructor.) 

Flat spiral or helical spring. .; P = ^ ~ ; / = R# = 12 , 



Round helical spring P = 



6 R 
Sn d 3 



- R&. 



Ebh 3 ' 
MPIR* 
' [v E d 4 ' 



it G d 4 
SPR*l b* + h* 
G b 3 h 3 ' 



Flat bar, in torsion P - 

P = force applied at end of radius or lever-arm R; & — angular motion at 
end of radius R; S = permissible maximum stress, = 4/5 of permissible 
stress in flexure; E — modulus of elasticity in tension; G — torsional modu- 
lus, = 2/5 E\ I — developed length of spiral, or length of bar; d — diameter 
of wire; b = breadth of flat bar; h = thickness, 



HELICAL SPRINGS FOR CARS AND LOCOMOTIVES. 353 



Ma 

Sao 

< r 



H a 

2* 

©3 
©X 

©o 

as 

©£ 

^ a 
co 



M 

S 



^ : 



: ^ : ^ 

. CO CO • coo to t- 


J> rf *> *^ fr- 


. CD 50 CO 
. t> i- lO TtiOO »o to 


•^"aj i^ts^ls 


et cc cj cj c3 


. 03 nj cScisSaScSa; 


•OO 'OOOO 
' 1- O O Of O o 
• i-i so • C> ©J CO lO 


OOO 
000 
000 
000 

800 


000 

000 

000 
000 
500 
700 
000 
500 


•TH^CO^ 


t)i e e to o 


• t- t- i- CO CO CO i-" QO 






00 o 



u 00 t-i jo co co iO -* to co od m e 



flo«xo!ioat-xoooc» o cs >c os -* c 



!3 . ^^ 



co m eo^io !> 



««r.MM t-i 1 



DCSOSCSOOCOcoOCO 



~ ■ ~ CO " i- — CO O -T X 0> ?> 'X) OS CO " 

o-*05Cicoi-coirtincoioj>-tf 






3 co e* eo ouo ! - o to h ?; o r 

%iococsc5eocooo£>t-»oiOTPiom!>-*« 



D tO CO CO « O 



S^SaCSOSSS S ^^'"S^^ 



•3 '« -4' 



MS 
C?c5 

dp 



354 RIVETED JOINTS. 



RIVETED JOINTS. 

Fairhairn's Experiments. (From Report of Committee on 
Riveted Joints, Proc. Inst. M. E., April, 1881.) 
The earliest published experiments on riveted joints are contained in the 
memoir by Sir VV. Fairbairn in the Transactions of the Royal Society. Mak- 
ing certain empirical allowances, he adopted the following ratios as ex- 
pressing the relative strength of riveted joints: 

Solid plate 100 

Double-riveted joint 70 

Single-riveted joint 56 

These well-known ratios are quoted in most treatises on riveting, and are 
still sometimes referred to as having a considerable authority. It is singular, 
however, that Sir W. Fairbairn does not appear to have been aware that the 
proportion of metal punched out in the line of fracture ought to be different 
in properly designed double and single riveted joints. These celebrated 
ratios would therefore appear to rest on a very unsatisfactory analysis of 
the experiments on which they were based. 

Loss of Strength in Punched Plates.— A report by Mr. W. 
Parker and Mr. John, made in 1878 to Lloyd's Committee, on the effect of 
punching and drilling, showed that thin steel plates lost comparatively little 
from punching, but that in thick plates the loss was very considerable. 
The following table gives the results for plates punched and not annealed 
or reamed: 

Thickness of Material of Loss of Tenacity, 

Plates. Plates. per cent. 

V A Steel 8 

% " 18 

^ " 26 

M " 33 

% Iron 18 to 23 

The effect of increasing the size of the hole in the die-block is shown in 
the following table: 

Total Taper of Hole Material of Loss of Tenacity due to 

in Plate, inches. Plates. Punching, per cent. 

1-16 Steel 17.8 

Vs " 12.3 

34 " (Hole ragged) 24.5 

The plates were from 0.675 to 0.712 inch thick. When %-in. punched holes 
were reamed out to 1% in. diameter, the loss of tenacity disappeared, and 
the plates carried as high a stress as drilled plates. Annealing also restores 
to punched plates their original tenacity. 

Strength of Perforated Plates. 

(P. D. Bennett, Eng'g, Feb. 12, 1886, p. 155.) 
Tests were made to determine the relative effect produced upon tensile 
strength of a flat bar of iron or steel: 1. By a %-inch hole drilled to the re- 
quired size; 2, by a hole punched J^ inch smaller and then drilled to the 
size of the first hole; and, 3, by a hole punched in the bar to the size of the 
drilled bar. The relative results in strength per square inch of original area 
were as follows: 

1. 
Iron. 

Unperforated bar 1.000 

Perforated by drilling 1 . 029 

" " punching and drilling. 1.030 
" " punching only 0.795 

In tests 2 and 4 the holes were filled with rivets driven by hydraulic pres- 
sure. The increase of strength per square inch caused by drilling is a phe- 
nomenon of similar nature to that of the increased strength of a grooved bar 
over that of a straight bar of sectional area equal to the smallest section of 
the grooved bar, Mr. Bennett's tests on an iron bar 0,84 in. diameter, 10 in. 



2. 


3. 


4. 


Iron . 


Steel. 


Steel. 


1.000 


1.000 


1.000 


1.012 


1.068 


1.103 


1.008 


1.059 


1.110 


0.894 


0.935 


0.927 



EFFICIENCY OF RIVETING BY DIFFERENT METHODS. o5o 



long, and a similar bar turned to 0.84 in. diameter at one point only, showed 
that the relative strength of the latter to the former was 1.323 to i.000. 

Riveted Joints.— Drilling versus Punching; of Holes. 

The Report of the Research Committee of the Institution of Mechanical 
Engineers, on Riveted Joints (1881), and records of investigations by Prof. 
A. B. W. Kennedy (1881, 1882, and 1885), summarize the existing information 
regarding the comparative effects of punching and drilling upon iron and 
steel plates. From an examination of the voluminous tables given in Pro- 
fessor Un win's Report, the results of the greatest number of the experi- 
ments made on iron and steel plates lead to the general conclusion that, 
while thin plates, even of steel, do not suffer very much from punching, yet 
in those of J^-inch thickness and upwards the loss of tenacity due to punch- 
ing ranges from 10$ to 23$ in iron plates, and from 11$ to 33$ in the case of 
mild steel. In drilled plates there is no appreciable loss of strength. It is 
possible to remove the bad effects of punching by subsequent reaming or 
annealing; but the speed at which work is turned out in these days is not 
favorable to multiplied operations, and such additional treatment is seldom 
practised. The introduction of a practicable method of drilling the plating 
of ships and other structures, after it has been bent and shaped, is a matter 
of great importance. If even a portion of the deterioration of tenacity can 
be prevented, a much stronger structure results from the same material and 
the same scantling. This has been fully recognized in the modern English 
practice (1887) of the construction of steam-boilers with steel plates; punch- 
ing in such cases being almost entirely abolished, and all rivet-holes being 
drilled after the plates have been bent to the desired form. 

Comparative Efficiency of Riveting done by Different 
Methods. 

The Reports of Professors Unwin and Kennedy to the Institution of Me- 
chanical Engineers (Proc. 1881, 1882, and 1885) tend to establish the four fol- 
lowing points: 

1. That the shearing resistance of rivets is not highest in joints riveted by 
means of the greatest pressure; 

2. That the ultimate strength of joints is not affected to an appreciable 
extent by the mode of riveting; and, therefore, 

3. That very great pressure upon the rivets in riveting is not the indispen- 
sable requirement that it has been sometimes supposed to be; 

4. That the most serious defect of hand-riveted as compared with machine- 
riveted work consists in the fact that in hand -riveted joints visible slip 
commences at a comparatively small load, thus giving such joints a low 
value as regards tightness, and possibly also rendering them liable to failure 
under sudden strains after slip has once commenced. 

The following figures of mean results, taken from Prof. Kennedy's tables 
(Proceedings 1885, pp. 218-225), give a comparative view of hand and hy- 
draulic riveting, as regards their ultimate strengths in joints, and the periods 
at which in both cases visible slip commenced. 



Total Breaking Load. 


Load at which Visible Slip began. 


Hand-riveting. 


Hydraulic Rivet- 
ing. 


Hand-riveting. 


Hydraulic Rivet- 
ing. 


Tons. 
86.01 

'82". 16 

U9.2 

m.6 


Tons. 

85.75 
77.00 
82.70 
78.58 
145.5 
140.2 
183.1 
183.7 


Tons. 
21.7 

25^6 

3L7 

25^6 


Tons. 
47.5 
35.0 
53.7 
54.0 
49.7 
46.7 
56.0 



In these figures hand-riveting appears to be r ither better than hydraulic 
riveting, as far as regards ultimate strength of joint; but is very much in- 
ferior to hydraulic work, in view of the small proportion of load borne by 
it before visible slip commenced. 



356 



RIVETED JOIffTS. 



Some of the Conclusions of the Committee of Research 
on Riveted Joints. 

(Proc. Inst. M. E., Apl. 1885.) 

The conclusions all refer to joints made in soft steel plate with steel 
rivets, the holes all drilled, and the plates in their natural state (unannealed). 
In every case the rivet or shearing area has been assumed to be that of the 
holes, not the nominal (or real) area of the rivets themselves. Also, the 
strength of the metal in the joint has been compared with that of strips 
cut from the same plates, and not merely with nominally similar material. 

The metal between the rivet-holes has a considerably greater tensile re- 
sistance per square inch than the imperforated metal. This excess tenacity 
amounted to more than 20$, both in %-inch and %-inch plates, when the 
pitch of the rivet was about 1.9 diameters. In other cases %-inch plate gave 
an excess of 15$ at fracture with a pitch of 2 diameters, of 10$ with a pitch 
of 3.6 diameters, and of 6.6$, with a pitch of 3.9 diameters; and ^-inch plate 
gave 7.8$ excess with a pitch of 2.8 diameters. 

In single-riveted joints it may be taken that about 22 tons per square inch 
is the shearing resistance of rivet steel, when the pressure on the rivets does 
not exceed about 40 tons per square inch. In double-riveted joints, with 
rivets of about % inch diameter, most of the experiments gave about 24 tons 
per square inch as the shearing resistance, but the joints in one series went 
at 22 tons. 

The ratio of shearing resistance to tenacity is not constant, but diminishes 
very markedly and not very irregularly as the tenacity increases. 

The size of the rivet heads and ends plays a most important part in the 
strength of the joints— at any rate in the case of single-riveted joints. An 
increase of about one third in the weight of the rivets (all this increase, of 
course, going to the heads and ends) was found to add about 8J^$ to the 
resistance of the joint, the plates remaining unbroken at the full shearing 
resistance of 22 tons per square inch, instead of tearing at a shearing stress 
of only a little over 20 tons. The additional strength is probably due to the 
prevention of the distortion of the plates by the great tensile stress in the 
rivets. 

The intensity of bearing pressure on the rivet exercises, with joints propor- 
tioned in the ordinary way, a very important influence on their strength. 
So long as it does not exceed 40 tons per square inch (measured on the pro- 
jected area of the rivets), it does not seem to affect their strength ; but pres- 
sures of 50 to 55 tons per square inch seem to cause the rivets to shear in 
most cases at stresses varying from 16 to 18 tons per square inch. For or- 
dinary joints, which are to be made equally strong in plate and in rivets, 
the bearing pressure should therefore probably not exceed 42 or 43 tons per 
square inch. For double-riveted butt-joints perhaps, as will be noted later, 
a higher pressure may be allowed, as the shearing stress may probably not 
be more than 16 or 18 tons per square inch when the plate tears. 

A margin (or net distance from outside of holes to edge of plate) equal to the 
diameter of the drilled hole has been found sufficient in all cases hitherto tried. 

To attain the maximum strength of a joint, the breadth of lap must be 
such as to prevent it from breaking zigzag. It has been found that the net 
metal measured zigzag should be from 30$ to 35$ in excess of that measured 
straight across, in order to insure a straight fracture. This corresponds to 
a diagonal pitch of 2/3 p -\- d/S, if p be the straight pitch and d the diam- 
eter of the rivet-hole. 

Visible slip or "give" occurs always in a riveted joint at a point very 
much below its breaking load, and by no means proportional to that load. 
A collation of the results obtained in measuring the slip indicates that it de- 
pends upon the number and size of the rivets in the joint, rather than upon 
anything else ; and that it is tolerably constant for a given size of rivet in a 
given type of joint. The loads per rivet at which a joint will commence to 
slip visibly are approximately as follows : 



Diameter of Rivet. 


Type of Joint. 


Riveting. 


Slipping Load per 
Rivet. 


% inch 
% " 

H " 

1 inch 

1 " 
1 " 


Single-riveted 
Double-riveted 
Double- riveted 
Single-riveted 
Double-riveted 
Double-riveted 


Hand 
Hand 

Machine 
Hand 
Hand 

Machine 


2.5 tons 
3.0 to 3.5 tons 
7 tons 

3.2 tons 

4.3 tons 

8 to 10 tons 



DOUBLE-RIVETED LAP-JOINTS. 



35? 



To find the probable load at which a joint of any breadth will commence 
to slip, multiply the number of rivets in the given breadth by the proper 
figure taken from the last column of the table above. It will be understood 
that the above figures are not given as exact; but they represent very well 
the results of the experiments. 

The experiments point to simple rules for the proportioning of joints of 
maximum strength. Assuming that a bearing pressure of 43 tons per square 
inch may be allowed on the rivet, and that the excess tenacity of the plate 
is 10% of its original strength, the following table gives the values of the ratios 
of diameter d of hole to thickness t of plate (d -+- t), and of pitch p to diam- 
eter of hole (p-=-cZ) in joints of maximum strength in %-inch plate. 

For Single-riveted Plates. 



Original Tenacity of 
Plate. 



Shearing Resistance of 
Rivets. 



Tons per 



28 



Lbs. per 



67,200 



67,200 
62,720 



Tons per 
sq. in. 



24 
24 



Lbs. per 



. in. 



49,200 
49,200 
53,760 
53,760 



Ratio. 
d-i-t 


Ratio. 
p -t-d 


2.48 
2.48 
2.28 
2.28 


2.30 
2.40 
2.27 
2.36 



Ratio. 
Plate Area 



0.667 
0.785 
0.713 
0.690 



This table shows that the diameter of the hole (not the diameter of the 
rivet) should be 2% times the thickness of the plate, and the pitch of the 
rivets 2% times the diameter of the hole. Also, it makes the mean plate area 
71% of the rivet area. 

If a smaller rivet be used than that here specified, the joint will not be of 
uniform, and therefore not of maximum, strength; but with any other size 
of rivet the best result will be got by use of the pitch obtained from the 
simple formula 

p=a— + d, 

where, as before, d is the diameter of the hole. 
The value of the constant a in this equation is as follows: 

For 30-ton plate and 22-ton rivets, a = 0.524 
" 28 " 22 " " 0.558 

" 30 " 24 " " 0.570 

" 28 " 24 " " 0.606 

d" 2 
Or, in the mean, the pitch p = 0.56 —r- -\- d. 

It should be noticed that with too small rivets this gives pitches often con- 
siderably smaller in proportion than 2% times the diameter. 

For double-riveted, lap-joints a similar calculation to that given 
above, but with a somewhat smaller allowance for excess tenacity, on 
account of the large distance between the rivet-holes, shows that for joints 
of maximum strength the ratio of diameter to thickness should remain pre- 
cisely as in single-riveted joints; while the ratio of pitch to diameter of hole 
should be 3.64 tor 30-ton plates and 22 or 24 ton rivets, and 3.82 for 28-ton 
plates with the same i ivets. 

Here, still more than in the former case, it is likely that thf prescribed 
size of rivet may often be inconveniently large. In this case the diameter 
of rivet should be taken as large as possible; and the strongest joint for a 
given thickness of plate and diameter of hole can then be obtained by using 
the pitch given by the equation 

p = a-j+d, 

where the values of the constant a for different strengths of plates and 
rivets may be taken as follows: 



358 



RIVETED JOINTS. 



+ d. 



Table of Proportions of DouMe-riveted Lap-joints, 

in which p = 

Original tenacity 
Thickness of of Plate, 

Plate. Tons per sq. in. 



Shearing Resist- 
ance of Rivets. 
Tons per sq. in. 

24 

24 

22 

22 

24 

24 

22 

22 



Value of Con- 
stant. 

a 
1.15 
1.22 
1.05 
1.12 
1.17 
1.25 
1.07 
1.14 



Practically, having assumed the rivet diameter as large as possible, 
can fix the pitch as follows, for any thickness of plate from '% to % inch: 



For 30-ton plate and 24-ton rivets { , 

" 28 " " " 22 " " \ J 



rt 2 
p - 1.06 -- + d 



In double-riveted butt-joints it is impossible to develop the full 
shearing resistance of the joint without getting excessive bearing pressure, 
because the shearing area is doubled without increasing the area on which 
the pressure acts. Considering only the plate resistance and the bearing 
pressure, and taking this latter as 45 tons per square inch, the best pitch 
would be about 4 times the diameter of the hole. We may probably say 
with some certainty that a pressure of from 45 to 50 tons per square inch on 
the rivets will cause shearing to take place at from 16 to 18 tons per square 
inch. Working out the equations as before, but allowing excess strength of 
only 5% on account of the large pitch, we find that the proportions of double- 
riveted butt-joints of maximum strength, under given conditions, are those 
of the following table: 

Double-riveted Butt-joints. 



Ratio 

P 

d 
3.85 
4.06 
4.03 
4.27 
4.20 



Practically, therefore, it may be said that we get a double-riveted butt-joint 
of maximum strength by making the diameter of hole about 1.8 times the 
thickness of the plate, and making the pitch 4.1 times the diameter of the 
hole. 

The proportions just given belong to joints of maximum strength. But in 
a boiler the one part of the joint, the plate, is much more affected by time 
than the other part, the rivets. It is therefore not unreasonable to estimate 
the percentage by which the plates might be weakened by corrosion, etc., 
before the boiler would be unfit for use at its proper steam-pressure, and to 
add correspondingly to the plate area. Probably the best thing to do in this 
case is to proportion the joint, not for the actual thickness of plate, but for 
a nominal thickness less than the actual by the assumed percentage. In 
this case the joint will be approximately one of uniform strength by the 
time it has reached its filial workable condition ; up to which time the joint 
as a whole will not really have been weakened, the corrosion only gradually 
bringing the strength of the plates down to that of rivets. 



riginal Ten- 


Shearing Re- 


Bearing 




acity 


sistance 


Pres- 


Ratio 


of Plate, 


of Rivets, 


sure, 


Tons per 


Tons per 


Tons per 


d 


sq. in. 


sq. in. 


sq. in. 


T 


30 


16 


45 


1.80 


28 


16 


45 


1.80 


30 


18 


48 


1.70 


28 


18 


48 


1.70 


30 


16 


50 


2.00 


28 


16 


50 


2.00 



KIVETED JOINTS. 



359 



Efficiencies of Joints. 

The average results of experiments by the committee gave: For double- 
riveted lap-joints in %-iuch plates, efficiencies ranging from 67.1$ to 81. 2$. 
For double-riveted butt-joints (in double shear) 61.4$ to 71.3$. These low re- 
sults were probably due to the use of very soft steel in the rivets. For single- 
riveted lap-joints of various dimensions the efficiencies varied from 54.8$ to 
60.8$. 

The experiments showed that the shearing resistance of steel did not in- 
crease nearly so fast as its tensile resistance. With very soft steel, for 
instance, of only 26 tons tenacity, the shearing resistance was about 80$ of 
the tensile resistance, whereas with very hard steel of 52 tons tenacity the 
shearing resistance was only somewhere about 65$ of the tensile resistance. 

Proportions of Pitch and Overlap of Plates to Diameter 
of Rivet-Hole and Thickness of Plate. 
(Prof. A. B. W. Kennedy, Proc. Inst. M. E„ April, 1885.) 
t = thickness of plate; 

d — diameter of rivet (actual) in parallel hole; 
p — pitch of rivets, centre to centre; 
s = space between lines of rivets; 
[I — overlap of plate. 
The pitch is as wide as is allowable without imparing the tightness of the 
joint under steam. 

For siugle-riveted lap-joints in the circular seams of boilers which have 
double-riveted longitudinal lap joints, 
d = t x 2.25; 

p — d x 2.25 =h5 (nearly); 
l=t x Q. 
For double-riveted lap-joints: 

d = 2.25*; 
p = 8t; 
s = 4.5*; 
I = 10.5*. 



Single-riveted Joints. 


Double-riveted Joints. 


* 


d 


P 


I 


* 


d 


V 


s 


i 


3-16 


7-16 


15-16 


W8 


3-16 


7-16 


V4 


Vs 


2 


H 


9-16 


m 


1% 


Va 


9-16 


2 


13-16 


m 


5-16 


11-16 


19-16 


m 


5-16 


11-16 


2^ 


M 


% 


13-16 


M 


2M 


% 


13-16 


3 


m 


4 


7-16 1 


2 3-16 


2% 


7-16 


1 


W* 


2 


4% 


Y2 UH 


V& 


3 


H 


% 


4 


2M 


5% 


9-16 1J4 


2 13-16 


W& 


9-16 


V4 


2y 2 



With these proportions and good workmanship there need be no fear of 
leakage of steam through the riveted joint. 

The net diagonal area, or area of plate, along a zigzag line of fracture 
should not be less than 30$ in excess of the net area straight across the 
joint, and 35$ is better. 

Mr. Theodore Cooper (E. R. Gazette, Aug. 22, 1890) referring to Prof. Ken- 
nedy's statement quoted above, gives as a sufficiently approximate rule for 
the proper pitch between the rows in staggered riveting, one half of the 
pitch of the rivets in a row plus one quarter the diameter of a rivet-hole. 

Apparent Excess in Strength of Perforated over Unper- 
forated Plates. (Proc. Inst. M. E., October, 1888.) 
The metal between the rivet-holes has a considerably greater tensile re- 
sistance per square inch than the imperforated metal. This excess tenacity 
amounted to more than 20$, both in %-inch and %-inch plates, when the 
pitch of the rivets was about 1.9 diameters. In other cases %-inch plate 
gave an excess of 15$ at fracture with a pitch of 2 diameters, of 10$ with a 
pitch of 3.6 diameters, and of 6.6$ with a pitch of 3.9 diameters; and %-inch 
plate gave 7.8$ excess with a pitch of 2.8 diameters, 



360 



RIVETED JOINTS. 



(1) The "excess strength due to perforation " is increased by anything 
which tends to make the stress in the plate uniform, and to diminish the 
effect of the narrow strip of metal at the edge of the specimen. 

(2) It is diminished by increase in the ratio of p/d, of pitch to diameter of 
hole, so that in this respect it becomes less as the efficiency of the joint 
increases 

(3) It is diminished by any increase in hardness of the plate. 

(4) For a given ratio p/d, of pitch to diameter of hole, it is also apparently 
diminished as the thickness of the plate is increased. The ratio of pitch to 
thickness of plate does not seem to affect this matter directly, at least 
within the limits of the experiments. 

Test of Double-riveted L<ap and Butt Joints. 
(Proc. Inst. M. E., October, 1888.; 
Steel plates of 25 to 26 tons per square inch T. S., steel rivets of 24.6 tons 
shearing-strength per square inch. 

Thickness of Diameter of Ratio of Pitch 



Kind of Joint. 



Lap.. 
Butt. . 
Lap... 



Plate. 



Lap . . 
Butt". 



Rivet-holes. 

0.8" 

0-7 



1.6 

1.3 



to Diameter. 
3.62 
3.93 
2.82 
3.41 
4.00 
3.94 
2.42 
3.00 
3.92 



Comparative 

Efficiency of 

Joint. 

75.2 

76.5 

68.0 

73.6 

72.4 

76.1 

63.0 

70.2 

76.1 



Some Rules which have been Proposed for the Diameter 
of the Rivet in Single Shear. (Iron, June 18, 1880.) 

Browne d = 2t (with double covers lj^tf) (1) 

Fairbairn d — 2t for plates less than % in. (2) 

" d — \)4,t for plates greater than % in. (3) 

Lemaitre d = 1.52 -f 0.16 (4) 

Antoine d = 1.1 Vt (5) 

Pohlig d - 2t for boiler riveting (6) 

" d = 'St for extra strong riveting (7) 

Redtenbacher d = 1.5*to2£ (8) 

Unwin d = %t + 5/16 to %t + % (9) 

" d = 1.2 Vt (10) 

The following table contains some data of the sizes of rivets used in 
practice, and the corresponding sizes given by some of these rules. 
Diameter of Rivets for Different Thicknesses of Plates. 





Diameter of Rivets, in inches. 


Thick- 
ness of 
plate. 
Inches. 




o 


P 


O «3 


p 




03 


03 

a 
< 


a 


d 
o 


5/16 

Va 

■ 7/16 

H 


98 

H 


Va 
Va 
Va 
13/16 


s 




% 
H 

Vs 
1 


21/32 
% 


% 

23/32 
13/16 
15/16 


Va 

11/16 
H 
% 


11/16 

H 

13/16 
Va 


H 

11/16 

s 


9/16 
Va 

11/16 

H 


I 

Va 


13/16 
Va 
Va 
15/16 




% 

13/16 

Va 




27/32 
15/16 
1 1/32 


1 

1 3/16 


13/16 
Va 

15/16 
15/16 


Va 

15/16 
1 
1 1/16 


Va 
Va 
Va 

l 


13/16 
Va 

15/16 
1 


l 
1 

l 


1 

1 3/16 








1 7/32 


m 


1 
1 
1 1/16 

m 


1 3/32 
1 3/16 

m 


l 


1 

i'i/ie 




1 

Wa 

m 



RIVETED JOIXTS. 



36i 



Strength of Double -riveted Seams, Calculated. — W. B. 

Rubles, Jr., in Pomer for June, 1890, gives tables of relative strength of 
rivets and parts of sheet between rivets in double-riveted seams, compared 
with strength of shell, based on the assumption that the shearing strength 
of rivets and the tensile strength of steel are equal. The following figures 
show the sizes in his tables which show the nearest approximation to equal- 
ity of strength of rivets and parts of plates between the rivets, together 
with the percentage of each relative to the strength of the solid plate. 



Ji 






Percentage of 


o « 






Percentage of 


■i — 


Pitch 


Size of 


Strength of 




Pitch 


Size of 


Strength of 


0/ p 


of 


Rivet- 


Plate. 




of 


Rivet- 


Plate. 


S ~- 


Rivets, 


holes, 






-5 aT 


Rivets, 


holes, 




oS 


inches. 


inches. 








inches. 


inches. 






Ha 






Rivets. 


Plate. 


frHfc 






Rivets. 


Plate. 


M 


m 


k 


.739 


.765 


7/16 


m 


H 


.734 


.728 


Va 


m 


9/16 


.795 


.775 


V/l« 


3% 


13/16 


.758 


.740 


Vi 


3^ 


% 


.785 


.800 


V/J6 


m 


% 


.758 


.759 


Va 


Ws 


11/16 


.819 


.810 


7/16 


4% 


15/16 


.765 


.773 


b/16 


V4 


9/16 


.749 


.735 


Vo 


m 


% 


.707 


.700 


5/16 


2% 


% 


.748 


.762 


Vo 


m 

4V 8 


13/16 


.721 


.718 


5/16 


ZVs 


11/16 


.761 


.780 


Vo 


Vs 


.740 


.731 


5/16 


Ws 


% 


.780 


.793 


v 


15/16 


.736 


.750 


% 


2H 


% 


.727 


.722 


V> 


1 


.761 


.758 


% 


m 


11/16 


.755 


.738 


9/16 


2¥a 


13/16 


.701 


.690 


% 


3^ 


% 


.754 


.760 


9/16 


3 


Vs 


.714 


.708 


% 


z% 


13/16 


.762 


.776 


9/16 


m 


15/16 


.727 


.722 


% 


m 


% 


.777 


.788 


9/16 


m 


1 


.745 


.733 


7/16 


Ws 


11/16 


.714 


.711 


9/16 


\y A 


1 1/16 


.742 


.750 



H. De B. Parsons (R. R. & Eng. Journal, 1890) holds that it is an error to 
assume that the shearing strength of the rivet is equal to the tensile strength. 
Also, referring to the apparent excess in strength of perforated over unper- 
forated plates, he claims that on account of the difficulty in properly match- 
ing the holes, and of the stress caused by forcing, as is too often the case 
in practice, this additional strength cannot *be trusted much more than 
that of friction. 

Adopting the sizes of iron rivets as generally used in American practice 
for steel plates from V| to 1 inch thick: the tensile strength of the plates as 
60,000 lbs.; the shearing strength of the rivets as 40,000 for single-shear and 
35,500 for double - shear, Mr. Parsons calculates the following table of 
pitches, so that the strength of the rivets against shearing will be approxi- 
mately equal to that of the plate to tear between rivet-holes. The diameter 
of the rivets has in all cases been taken at 1/16 in. larger than the nominal 
size, as the rivet is assumed to fill the hole under the power riveter. 

J&iveted Joints. 

Lap or Butt with Single Welt— Steel Plates and Iron Rivets. 



Thickness 

of 

Plates. 



Diameter 

of 

Rivets. 



1 1/8 



Single. 



1 3/16 
1 11/16 



1 11/16 
1% 



2 3/16 



Ws 
2 11/16 

m 

2 7/16 
2Vs 
2 7/16 



Efficiency. 



Single. 


Double. 


55.7% 


70. 0# 


52.7 


68.6 


49.0 


65.9 


43.6 


60.4 


42.0 


59.5 


38.6 


55.4 


38.1 


54.9 



362 



RIVETED JOINTS. 



Calculated Efficiencies Steel Plates and Steel Rivets.— 

The differences between the calculated efficiencies given in the two tables 
above are notable. Those given by Mr. Ruggles are probably too high, since 
he assumes the shearing strength of the rivets equal to the tensile"strength 
of the plates. Those given by Mr. Parsons are probably lower than will be 
obtained in practice, since the figure he adopts for shearing strength is 
rather low, and he makes no allowance for excess of strength of the perfo- 
rated over the imperforated plate. The following table has been calculated 
by th author on the assumptions that the excess strength of the perforated 
plate is 10$, and that the shearing strength of the rivets per square inch is 
four fifths of the tensile strength of the plate. If t = thickness of plate, 
d = diameter of rivet-hole, p = pitch, and T — tensile strength per square 
inch, then for single-riveted plates 



(p - d)t x i.ior = 



T&X 



„d* 



The coefficients .571 and 1.142 agree closely with the averages of those 
given in the report of the committee of the institution of Mechanical En- 
gineers, quoted on pages 357 and 358, ante. 





Diam. 


Pitch. 


Efficiency. 


yi 


Diam. 


Pitch. 


Efficiency. 




















a 


of 
Rivet- 


bio 


o> &J0 

2-5 


si 


2$ 


a 


of 
Rivet- 


<gi 


%i 


hi 


-2^ 


o 


hole. 


bti'Z 


3 © 


fc?i 


o 


hole. 


&c*3 


s"S 


c"S 


2"S 








O > 




c > 








o > 


'£, > 


2 > 


H 






ft s 


xg 


Q s 


H 




S | 


fi s 


"a 


°s 


in. 


in. 


in. 


in. 


% 


% 


in. 


in. 


in. 


in. 


% 


i 


3/16 


7/16 


1.020 


1.603 


57.1 


72.7 


l A 


U 


1.392 


2.035 


46.1 


63.1 


" 


l A 


1.261 


2.023 


60.5 


75.3 




% 


1.749 


2.624 


50.0 


66.6 


H 


y* 


1.071 


1.642 


53.3 


69*6 


i - 


1 


2.142 


3.284 


53.3 


70.0 




9/16 


1.285 


2.008 


56.2 


72.0 


" 


V& 


2.570 


4.016 


56.2 


72.0 


5/16 


9/16 


1.137 


1.712 


50.5 


67.1 


9/16 


I 


•1.321 


1.892 


43.2 


60.3 


" 


% 


1.339 


2.053 


53.3 


69.5 


" 


1.652 


2. -129 


47.0 


64.0 


" 


11/16 


1.551 


2.415 


55.7 


71.5 


'■ 


1 


2.015 


3.030 


50.4 


67.0 


% 


% 


1.218 


1.810 


48.7 


65.5 


" 


i% 


2.410 


3.6'.M 


53.3 


69.5 




% 


1.607 


2.463 


53.3 


69.5 


" 


134 


2.836 


4.422 


55.9 


71.5 


" 


Vs 


2.011 


3.206 


57.1 


72 7 


% 


% 


1.264 


1.778 


40.7 


57.8 


7/16 


% 


1.136 


1.647 


45.0 


62.0 




Vs 


1.575 


2.274 


44.4 


61.5 


" 


H 


1.484 


2.218 


49.5 


66.2 


" 


1 


1.914 


2.827 


47.7 


64.6 


" 


% 


1.869 


2.864 


53.2 


69.4 


" 


1U 


2.281 


3.438 


50.7 


67.3 




l 


2.305 


3.610 


56.6 


72.3 




m 


2,678 


4.105 


53.3 


69.5 



Riveting Pressure Required for Rridge and Roller 
Work. 

(Wilfred Lewis, Engineers' Club of Philadelphia, Nov., 1893.) 

A number of %-inch rivets were subjected to pressures between 10.000 and 
60.000 lbs. At 10,000 lbs. the rivet swelled and filled the hole without forming 
a head. At 20,000 lbs. the head was formed and the plates were slightly 
pinched. At 30.000 lbs. the rivet was well set. At 40,000 lbs. the metal in the 
plate surrounding the rivet began to stretch, and the stretching became 
more and more apparent as the pressure was increased to 50,000 and 60,000 
lbs. From these experiments the conclusion might be drawn that the pres- 
sure required for cold riveting was about 300,000 lbs. per square inch of rivet 
section. In hot riveting, until recently there was never any call for a pres- 
sure exceeding 60,000 lbs., but now pressures as high as 150,000 lbs. are not 
uncommon, and even 300,000 lbs. have been contemplated as desirable. 



SHEARING RESISTANCE OF RIVET IRON AND STEEL. 363 

Apparent Shearing Resistance of Rivet Iron and Steel. 

(Proc. Inst. M. E., 1879, Engineering, Feb. 20, 1880.) 

The true shearing resistance of the rivets cannot be ascertained from 
experiments on riveted joints (1), because the uniform distribution of the 
load to all the rivets cannot be insured; (2) because of the friction of the 
plates, which has the effect of increasing the apparent resistance to shear- 
ing in an element uncertain in amount. Probably in the case of single- 
riveted joints the shearing resistance is not much affected by the friction; 

Ultimate Shearing Stress 

Tons per sq in. Lbs. per sq. in. 

Iron, single shear (12 bars). . 24.15 54.096 \ rs -\ ay .- h . a 

" double shear (8 bars). . 22.10 49.504 j ^ iarke - 

kt 22.62 50.669 Barnaby. 

22.30 49.952 Rankine. 

" %-in. rivets 23.05 to 25.57 51.632 to 57.277 ) 

" %-in. rivets 24.32 to 27.94 54.477 to 62.362 V Riley. 

" mean value 25.0 56.000 j 

" %-in. rivets 19.01 42.582 Greig and Eyth. 

Steel 17 to 26 38.080 to 58.240 Parker. 

Landore steel, %-\n. rivets.. 31.67 to 33.69 70.941 to 75.466 ) 

" %-xa. rivets... 30.45 to 35.73 68 . 208 to 80 . 035 - Riley. 

" " mean\alue.. 33.3 74.592 ) 

Brown's steel 22.18 49.683 Greig and Eyth. 

Fairbairn's experiments show that a rivet is 6*^ weaker in a drilled than 
in a punched hole. By rounding the edge of the rivet-hole the apparent 
shearing resistance is increased 12%. Mr. Maynard found the rivets 4% 
weaker in drilled holes than in punched holes. But these results were 
obtained with riveted joints, and not by direct experiments on shearing. 
There is a good deal of difficulty in determining the true diameter of a 
punched hole, and it is doubtful whether in these experiments the diameter 
was very accurately ascertained. Messrs. Greig and Eyth's experiments 
also indicate a greater resistance of the rivets in punched holes than in 
drilled holes. 

If, as appears above, the apparent shearing resistance is less for double 
than for single shear, it is probably due to unequal distribution of the stress 
on the two rivet sections. 

The shearing resistance of a bar, when sheared in circumstances which 
prevent friction, is usually less than the tenacity of the bar. The following 
results show the decrease : 



Tenacity of 
Bar. 



Shearing 
Resistance. 



Harkort, iron - 

Lavalley, iron... 

Greig and Eyth, iron.. 
" " steel. 



16.5 
20.2 
19.0 
22.1 



0.62 
0.79 
0.85 
0.77 



In Wohler's researches (in 1870) the shearing strength of iron was found 
to be four-fifths of the tenacity. Later researches of Bauscbinger confirm 
this result generally, but they show that for iron the ratio of the shearing 
resistance and tenacity depends on the direction of the stress relatively to 
the direction of rolling. The above ratio is valid only if the shear is in a 
plane perpendicular to the direction of rolling, and if the tension is applied 
parallel to the direction of rolling. The shearing resistance in a plane 
parallel to the direction of rolling is different from that in a plane perpen- 
dicular to that direction, and again differs according as the plane of shear is 
perpendicular or parallel to the breadth of the bar. In the former case the 
resistance is 18 to 20$ greater than in a plane perpendicular to the fibres, or 
is equal to the tenacity. In the latter case it is only half as great as in a 
plane perpendicular to the fibres. 



364 



IRON AND STEEL. 



IRON AND STEEL. 



CLASSIFICATION OF IRON AND STEEL. 









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CAST IKON". 365 

CAST IRON. 

Grading of Pig Iron.— Pig iron is commonly graded according to its 
fracture, the number of grades varying in different districts. In Eastern 
Pennsylvania the principal grades recognized are known as No. 1 and 2 
foundry, gray forge or No. 3, mottled or No. 4, and white or No. 5. Inter- 
mediate grades are sometimes made, as No. 2 X, between No. 1 and No. 2, 
and special names are given to irons more highly silicized than No. 1, as 
No. 1 X, silver-gray, and soft. Charcoal foundry pig iron is graded by num- 
bers 1 to 5, but the quality is very different from the corresponding num- 
bers in anthracite and coke pig. Southern coke pig iron is graded into ten 
or more grades. Grading by fracture is a fairly satisfactory method of 
grading irons made from uniform ore mixtures and fuel, but is unreliable as 
a means of determining quality of irons produced in different sections or 
from different ores. Grading by chemical analysis, in the latter case, is the 
only satisfactory method. The following analyses of the five standard 
grades of northern foundry and mill pig irons are given by J. M. Hartman 
(Bull. I. & S.A., Feb., 1892): 

No. 1. No. 2. No. 3. No. 4. No. 4 B. No. 5. 

Iron 92.37 92.31 94.66 94.48 94.08 94.68 

Graphitic carbon . . 3.52 2.99 2.50 2.02 2.02 

Combined carbon.. .13 .37 1.52 1.98 1.43 3.83 

Silicon 2.44 2.52 .72 .56 .92 .41 

Phosphorus 1.25 1.08 .26 .19 .04 .04 

Sulphur 02 .02 trace .08 .04 .02 

Manganese 28 .72 .34 .67 2.02 .98 

Characteristics of These Irons. 

No. 1. Gray. — A large, dark, open-grain iron, softest of all the numbers 
and used exclusively in the foundry. Tensile strength low. Elastic limit 
low. Fracture rough. Turns soft and tough. 

No. 2. Gray.— A mixed large and small dark grain, harder than No. 1 iron, 
and used exclusively in the foundry. Tensile strength and elastic limit 
higher than No. 1. Fracture less rough than No. 1. Turns harder, less 
tough, and more brittle than No. 1. 

No. 3. Gray.— Small, gray, close grain, harder than No. 2 iron, used either 
in the rolling-mill or foundry. Tensile strength and elastic limit higher than 
No. 2. Turns hard, less tough, and more brittle than No. 2. 

No. 4. Mottled.— White background, dotted closely with small black spots 
of graphitic carbon; little or no grain. Used exclusively in the rolling-mill. 
Tensile strength and elastic limit lower than No. 3. Turns with difficulty; 
less tough and more brittle than No. 3. The manganese in the B pig iron 
replaces part of the combined carbon, making the iron harder and closing 
the grain, notwithstanding the lower combined carbon. 

No. 5. White. — Smooth, white fracture, no grain, used exclusively in the 
rolling mill. Tensile strength and elastic limit much lower than No. 4. Too 
hard to turn and more brittle than No. 4. 

Southern pig irons are graded as follows, beginning with the highest in 
silicon: Nos. 1 and 2 silvery, Nos. 1 and 2 soft, all containing over 3$ of 
silicon; Nos. 1, 2, and 3 foundry, respectively about 2 ?5#, 2 5^ and 2% silicon; 
No. 1 mill, or "foundry forge;" No. 2 mill, or gray forge; mottled; white. 

Good charcoal chilling iron for car w T heels contains, as a rule, 0.56 to 0.95 
silicon, 0.08 to 0.90 manganese, 0.05 to 0.75 phosphorus. The following is an 
analysis of a remarkably strong car wheel: Si, 0.734; Mn, 0.438; P. 0.428. 
S, 0.08» Graphitic C. 3.083; Combined C, 1,247; Copper, 0.029. The chill was 
very hard— *4 in - deep at root of flange, y% in. deep on tread. A good 
ordnance iron analyzed: Si. 0.30; Graphitic C, 2.20; Combined C, 1.70; P, 
0.44; Mn, 3 55 (?). Its specific gravity was 7.22 and tenacity 31,734 lbs. 
per sq. in 

Influence of Silicon, Phosphorus, Sulphur, and Man- 
ganese upon Cast Iron. — W. J. Keep, of Detroit, in several papers 
(Trans. A. I. M. E., 1889 to 1893), discusses the influence of various chemical 
elements on the quality of cast iron. From these the following notes have 
been condensed: 

Silicon.— Pig iron contains all the carbon that it could absorb during its 
reduction in the blast-furnace. Carbon exists in cast iron in two distinct 
forms. In chemical unjon, as " combined " carbon, it cannot be discerned, 
except as it may increase the whiteness of the fracture, in so-called white 



366 IRON AND STEEL. 

iron. Carbon mechanically mixed with the iron as graphite is visible, vary- 
ing in color from gray to black, while the fracture of the iron ranges from a 
light to a very dark gray. 

Silicon will expel carbon, if the iron, when melted, contains all the car- 
bon tbat it can hold and a portion of silicon be added. 

Prof. Turner concludes from his tests that the amount of silicon producing 
the maximum strength is about 1.80$. But this is only true when a white 
base is used. If an iron is used as a base which will produce a sound casting 
to begin with, each addition of silicon will decrease strength. Silicon itself is 
a weakening agent. Variations in the percentage of silicon added to a pig 
iron will not insure a given strength or physical structure, but these results 
will depend upon the physical properties of the original iron. 

After enough silicon has been added to cause solid castings, any further 
addition and consequent increase of graphite weakens the casting. The 
softness and strength given to castings by a suitable addition of silicon 
is, by a further increase of silicon, changed to stiffness, brittleness, and 
weakness. 

As strength decreases from increase of graphite and decrease of combined 
carbon, deflection increases; or, in other words, bending is increased by 
graphite. When no more graphite can form and silicon still increases, de- 
flection diminishes, showing that high silicon not only weakens iron, but 
makes it stiff. This stiffness is not the same strength-stiffness which is 
caused by compact iron and combined carbon. It is a brittle- stiffness. 

In pig irons which received their silicon while in the blast-furnace the 
graphite more easily separates, and the shrinkage is less than in any mix- 
ture. As silicon increases, shrinkage also increases. Silicon of itself in- 
creases shrinkage, though by reason of its action upon the carbon in ordi- 
nary practice it is truly said that silicon "takes the shrinkage out of cast- 
iron." The slower a casting crystallizes, the greater will be the quantity 
of graphite formed within it. 

Silicon of itself, however small the quantity present, hardens cast-iron; 
but the decrease of hardness from the change of the combined carbon to 
graphite, caused by the silicon, is so much more rapid than the hardening 
produced by the increase of silicon, that the total effect is to decrease hard- 
ness, until the silicon reaches from 3 to 5%. 

As practical foundry-work does not call for more than 3% of silicon, the 
ordinary use of silicon does reduce the hardness of castings; but this is pro- 
duced through its influence on the carbon, and not its direct influence on the 
iron. 

When the change from combined to graphite carbon has ceased to dimin- 
ish hardness, say at from 2/ to b% of silicon, the hardening by the silicon 
itself becomes more and more apparent as the silicon increases. 

Shrinkage and hardness are almost exactly proportional. When silicon 
varies, and other elements do not vary materially, castings with low shrink- 
age are soft; as shrinkage increases, the castings grow hard in almost, if 
not exactly, the same proportion. For ordinary foundry-practice the scale 
of shrinkage may be made also the scale of hardness, provided variations in 
sulphur, and phosphorus especially, are not present to complicate the re- 
sult. 

The term " chilling " irons is generally applied to such as, cooled slowly, 
would be gray, but cooled suddenly, become white either to a depth suffi- 
cient for practical utilization {e.g., in car-wheels) or so far as to be detrimen- 
tal. Many irons chill more or less in contact with the cold surface of the 
mould in which they are cast, especially if they are thin. Sometimes this 
is a valuable quality, but for general foundry purposes it is desirable to 
have all parts of a casting an even gray. 

Silicon exerts a powerful influence upon this property of irons, partially 
or entirely removing their capacity of chilling. 

When silicon is mixed with irons previously low in silicon the fluidity is 
increased. 

It is not the percentage of silicon, but the state of the carbon and the 
action of silicon through other elements, which causes the iron to be fiuid. 

Silicon irons have always had the reputation of imparting fluidity to other 
irons. This conies, no doubt, from the fact that up to 3% or 4% they increase 
the quantity of graphite in the resulting casting. 

From the statement of Prof. Turner, that the maximum strength occurs 
with just such a percentage of silicon, and his statement that a founder can, 
with silicon, produce just the quality of iron that he may need, and from 
his naming the composition of what he calls a typical foundry-iron, some 



INFLUENCE OP SILICON, ETC., UPON CAST IRON. 367 

founders have inferred that if they knew the percentages of silicon in their 
irons and in their ferro-silicon, they need only mix so as to get 2% of silicon 
in order to obtain, always and with certainty, the maximum strength. The 
solution of the problem is not so simple. Each of the irons which the foun- 
der uses will have peculiar tendencies, given them in the blast-furnace, 
which will exert their influence in the most unexpected ways. However, a 
white iron which will invariably give porous and brittle castings can be 
made solid and strong by the addition of silicon; a further addition of sili- 
con will turn the iron gray; and as the grayness increases the iron will grow 
weaker. Excessive silicon will again lighten the grain and cause a hard and 
brittle as well as a very weak iron. The only softening and shrinkage-les- 
sening influence of silicon is exerted during the time when graphite is being 
produced, and silicon of itself is not a softener or a lessener of shrinkage; 
but through its influence on carbon, and only during a certain stage, does it 
produce these effects. 

Phosphorus.— While phosphorus of itself, in whatever quantity present, 
weakens cast-iron, yet in quantities less than 1.5% its influence is n t suffi- 
ciently great to overbalance other beneficial effects, which are exerted 
before the percentage reaches 1%. Probably no element of itself weakens 
cast iron as much as phosphorus, especially when present in large quantities. 

Shrinkage is decreased when phosphorus is increased. All high-phosphorus 
pig irons have low shrinkage. Phosphorus does not ordinarily harden cast 
iron, probably for the reason that it does not increase combined carbon. 

The fluidity of the metal is slightly increased by phosphorus, but not to 
any such great extent as has been ascribed to it. 

The property of remaining long in the fluid state must not be confounded 
with fluidity, for it is not the measure of its ability to make sharp castings, 
or to run into the very thin parts of a mould. Generally speaking, the state- 
ment is justified that, to some extent, phosphorus prolongs the fluidity of 
the iron while it is filling the mould. 

The old Scotch irons contained about 1% of phosphorus. The foundry-irons 
which are most sought for for small and thiu castings in the Eastern States 
contain, as a general thing, over 1% of phosphorus. 

Certain irons which contain from 4% to 7% silicon have been so much used 
on account of their ability to soften other irons that they have come to be 
known as " softeners " and as lesseners of shrinkage. These irons are valu- 
able as carriers of silicon ; but the irons which are sold most as softeners 
and shrinkage-lesseners are those containing from 1% to 2% of phosphorus. 
We must therefore ascribe the reputation of some of them largely to the 
phosphorus and not wholly to the silicon which they contain? 

From y%% to \% of phosphorus will do all that can be done in a beneficial 
way, and all above that amount weakens the irou, without corresponding 
benefit. It is not necessary to search for phosphorus-irons. Most irons 
contain more than is needed, and the care should be to keep it within limits. 

Sulphur.— Only a small percentage of sulphur can be made to remain 
in carbonized iron, and it is difficult to introduce sulphur into gray cast iron 
or into any carbonized iron, although gray cast iron often takes from the 
fuel as much more sulphur as the iron originally contained. Percentages 
of sulphur that could be retained by gray cast iron cannot materially injure 
the iron except through an increase of shrinkage. The higher the carbon, 
or the higher the silicon, the smaller will be the influence exerted by 
sulphur. 

The influence of sulphur on all ca^t iron is to drive out carbon and 
silicon and to increase chill, to increase shrinkage, and, as a general thing, to 
decrease strength ; but if in practice sulphur will not enter such iron, we 
shall not have any cause to fear this tendency. In every-day work, however, 
it is found at times that iron which was gray when put into the cupola comes 
out white, with increased shrinkage and chill, and often with decreased 
strength. This is caused by decreased silicon, and can be remedied by an 
increase of silicon. 

Mr. Keep's opinion concerning the influence of sulphur, quoted above, is 
disagreed with by J. B. Nau (Iron Age, March 29, 1894). He says : 

"Sulphur, in whatever shape it may be present, has a deleterious influence 
on the iron. It has the tendency to render the iron white by the influence 
it exercises on the combination between carbon and iron. Pig iron contain- 
ing a certain percentage of it becomes porous and full of holes, and castings 
made from sulphurous iron are of inferior quality. This happens especially 
when the element is present in notable quantities. With foundry-iron con- 
taining as high as 0.1% of sulphur, castings of greater strength may be ob- 



368 IRON AND STEEL. 

tained than when no sulphur is present. Thus, in some tests on this element 
quoted by R. Akerman, it is stated that in the foundry-iron from Finspong, 
used in the manufacture of cannons, a percentage of 0.1$ to 0.14$ of sulphur 
in the iron increased its strength to a considerable extent. The percentage 
of sulphur found originally in the iron put in the cupola is liable to be 
further increased by part of the sulphur that is invariably found in the coke 
used. It is seldom that a coke with a small percentage of sulphur is found, 
whereas coke containing 1$ of it and over is very common. With such a 
fuel in the cupola, if no special precautions are resorted to, the percentage 
of sulphur in the metal will in most cases be increased." 

That the sulphur contents of pig iron may be increased by the sulphur 
contained in the coke used, is shown by some experiments in the cupola, 
reported by Mr. Nau. Seven consecutive heats were made. 

The sulphur content of the coke was 1$, and 11.7$ of fuel was added to the 
charge. 

Before melting, the silicon ranged from 0.320 to 0.830 in the seven heats ; 
after melting, it was from 0.110 to 0.534, the loss in melting being from .100 
to .375. The sulphur before melting was from .076 to .090, and after melting 
from .132 to .174, a gain from .044 to .098. 

From the results the following conclusions were drawn : 

1. In all the charges, without exception, sulphur increased in the pig iron 
after its passage through the cupola. In some cases this increase more 
than doubled the original amount of sulphur found in the pig iron. 

2. The increase of the sulphur contents in the iron follows the elimination 
of a greater amount of silicon from that same iron. A larger amount of 
limestone added to these charges would have produced a more basic cinder, 
and undoubtedly less sulphur would have been incorporated in the iron. 

3. This coke contained 1$ of sulphur, and if all its sulphur had passed into 
the iron there would have been an average increase of 0.12 of sulphur for 
the seven charges, while the real iDcrease in the pig iron amounted to only 
0.081. This shows that two thirds of the sulphur of the coke was taken up 
by the iron in its passage through the cupola. 

Manganese.— Manganese is a nearly white metal, having about the same 
appearance when fractured as white cast iron. Its specific gravity is 
about 8, while that of white cast iron, reasonably free from impurities, is 
but a little above 7.5. As produced commercially, it is combined with iron, 
and with small percentages of silicon, phosphorus, and sulphur. 

It is generally produced in the blast-furnace. If the manganese is under 
40$, with the remainder mostly iron, and silicon not over 0.50$, the alloy is 
called spiegeleisen, and the fracture will show flat reflecting surfaces, from 
which it takes its name. 

With manganese above 50$, the iron alloy is called ferro-manganese. 

As manganese increases beyond 50$, the mass cracks in cooling, and when 
it approaches 98$ the mass crumbles or falls in small pieces. 

Manganese combines with iron in almost any proportion, but if an iron 
containing manganese is remelted, more or less of the manganese will escape 
by volatilization, and by oxidation with other elements present in the iron. 
If sulphur be present, some of the manganese will be likely to unite with it 
and escape, thus reducing the amount of both elements in the casting. 

Cast iron, when free from manganese, cannot hold more than 4.50$ of car- 
bon, and 3.50$ is as much as is generally present ; but as manganese increases, 
carbon also increases, until we often find it in spiegel as high as 5$, and in 
ferro-manganese as high as 6$. This effect on capacity to hold carbon is 
peculiar to manganese. 

Manganese renders cast iron less plastic and more brittle. 

Manganese increases the shrinkage of cast iron. An increase of 1$ raised 
the shrinkage 26$. Judging from some test records, manganese does not 
influence chill at ad; but other tests show that with a given percentage of 
silicon the carbon may be a little more inclined to remain in the combined 
form, and therefore the chill may be a little deeper. Hence, to cause the 
chill to be the same, it would seem that the percentage of silicon should be 
a little higher with manganese than without it. 

An increase of 1$ of manganese increased the hardness 40$. If a hard 
chill is required, manganese gives it by adding hardness to the whole casting. 

J. B. Nau {Iron Age, March 29, 1894), discussing the influence of manga- 
nese on cast iron, says: 

Manganese favors the combination between carbon and iron. Its influ- 
ence, when present in sufficiently large quantities, is even great enough nor, 
only to keep the carbon which would be naturally found in pig iron com- 



TESTS OF CAST IROK. 369 

bined, but it increases the capacity of iron to retain larger amounts of car- 
bon and to retain it all in the combined state. 

Manganese iron is often used for foundry purposes when some chill and 
harduess of surface is required in the casting. For the rolls of steel-rail 
mills we always put into the mixture a large amount of mauganiferous iron, 
and the rolls so obtained alwaj r s presented the desired hardness of surface 
and in general a mottled structure on the outside. The inside, which al- 
ways cooled much slower, was gray iron. One of the standard mixtures that 
invariably gave good results was the following: 

50$ of foundry iron with 1.3$ silicon and 1.5$ manganese; 
35$ of foundry iron with 1$ silicon and 1.5$ manganese; 
15% steel (rail ends) with about 0.35$ to 0.40$ carbon. 
The roll resulting from this mixture contained about 1% of silicon and 1% 
of manganese. 

Another mixture, which differed but little from the preceding, was as 
follows: 
45% foundry iron with about 1.3$ silicon and 1.5$ manganese; 
30$ foundry iron with about 1% silicon and 1.5$ manganese; 
10$ white or mottled iron with about 0.5$ to 0.6$ Si. and 1.2$ Mn. 
15$ Bessemer steel-rail ends with about 0.35$ to 0.40$ C. and 0.6$ to 1$ Mn. 
The pis: iron used in the preceding mixtures contained also invariably 
from 1.5$ to 1.8$ of phosphorus, so that the rolls obtained therefrom carried 
about 1.3$ to 1.4$ of that element. The last mixture used produced rolls 
containing on the average 0.8$ to 1$ of silicon and 1$ of manganese. When- 
ever we tried to make those rolls from a mixture containing but 0.2$ to 0.3$ 
manganese our rolls were invariably of inferior quality, grayer, and con- 
sequently softer. Manganese iron cannot be used indiscriminately for 
foundry purposes. When greater softness is required in the castings man- 
ganese has to be avoided, but when hardness to a certain extent has to be 
obtained manganese iron can be used with advantage. 

Manganese decreases the magnetism of the iron. This characteristic in- 
creases with the percentage of manganese that enters into the composition 
of the iron. The iron loses all its magnetism when manganese reaches 25$ 
of its composition. This peculiarity has been made use of by French 
metallurgists to draw a clear line between spiegel and ferro-manganese. 
When the pig contains less than 25$ of manganese it is classified as spiegel, 
and when it contains more than 25 it$ is classified as ferro-manganese. For 
this reason manganese iron has to be avoided in castings of dynamo fields 
and other pieces belonging to electric machinery, where magnetic conduc- 
tibility is one of the first considerations. 

Irregular Distribution of Silicon in Pig Iron.— J. W. 
Thomas (Iron Age, Nov. 12, 1891) finds in analyzing samples taken from every 
other bed of a cast of pig iron that the. silicon varies considerably, the iron 
coming first from the furnace having generally the highest percentage. In 
one series of tests the silicon decreased from 2.040 to 1.713 from the first bed 
to the eleventh. In another case the third bed had 1.260 Si., the seventh 1.718, 
and the eleventh 1.101. He also finds that the silicon varies in each pig, be- 
ing higher at the point than at the butt. Some of his figures are: point of 
pig 2.328 Si., butt of same 2.157; point of pig 1.834, butt of same 1.787. 

Some Tests of Cast Iron. (G. Lanza, Trans. A. S. M. E., x., 187.)— 
The chemical analyses were as follows: 

Gun Iron, Common Iron, 
per cent. per cent. 

Total carbon 3.51 

Graphite 2.80 

Sulphur 0.133 0.173 

Phosphorus 0.155 0.413 

Silicon 1.140 1.89 

The test specimens were 26 inches long and square in section; those tested 
with the skin on being very nearly one inch square, and those tested with 
the skin removed being cast nearly one and one quarter inches square, and 
afterwards planed down to one inch square. 

Tensile Elastic ^dubis 

Strength. Limit. °^^ 

Unplaned common. 20.200 to 23,000 T. S. Av. = 22,066 6,500 13,194,233 

Planed common.... 20,300 to 20,800 " " =20,520 5,833 11,943,953 

Unplaned gun 27,000 to 28,775 " " =28,175 11,000 16,130,300 

Planed gun 29,500 to 31,000 " " = 30,500 8,500 15,932,880 



370 IttOH AND STEEL. 

The el istic limit is not clearly defined in cast iron, the elongations increas- 
ing faster than the increase of the loads from the beginning of the test. 
The modulus of elasticity is therefore variable, decreasing as the loads in- 
crease. For example, the following results of a test of common cast iron, 
reported by Prof. Lanza: 

T hs r>er sn in Elongation in Sets, Modulus of 

L,bs. per sq. in. 13 4 inches in Elasticity. 

1000 .0004 18,217,400 

2000 .0013 16,777,700 

3000 .0024 14,085,400 

4000 .0036 13,101,200 

5000 .0048 12,809,200 

6000 .0061 .0000 12,319,300 

8000 .0088 .0001 11,600.800 

10000 .0119 .0001 10,930 500 

12000 .0162 .0007 9,714,200 

CHEMISTRY OF FOUNDRY IRONS. 

(C. A. Meissner, Columbia College Qly, 1890; Iron Age, 1890.) 

Silicon is a very important element in foundry irons. Its tendency when 
not above 2^% is to cause the carbon to separate out as graphite, giving the 
casting the desired benefits of graphitic iron. Between 2\4>% and S}4% silicon 
is best adapted for iron carrying a fair proportion of low silicon scrap and 
close iron, for ordinarily no mixture should run below 1}4,% silicon to get good 
castings. 

From 3$ to h% silicon, as occurs in silvery iron, will carry heavy amounts of 
scrap. Castings are liable to be brittle, however, if not handled carefully 
as regards proportion uf scrap used. 

From 1% #to 2% silicon is best adapted for machine work; will give strong 
clean castings if nor, much scrap is used with it. 

Below \% silicon seems suited for drills and castings that have to stand 
great variations in temperature. 

Silicon has the effect of making castings fluid, strong, and open-grained ; 
also sound, by its tendency to separate the graphite from the total carbon, 
and consequent slight expansion of the iron on cooling, causing It to fill out 
thoroughly. Phosphorus, when high, has a tendency to make iron fluid, 
retain its heat longer, thereby helping to fill out all small spaces in casting. 
It makes iron brittle, however, when above %% in castings. It is excellent 
when high to use in a mixture of low-phosphorus irons, up to 1%% giving 
good results, but, as said before, the casting should be below %%. It has a 
strong tendency when above 1% in pig to make the iron less graphitic, pre- 
venting the separation of graphite. 

Sulphur in open iron seldom bothers the founder, as it is seldom present 
to any extent. The conditions causing open iron in the furnace cause low 
sulphur. A little manganese is an excellent antidote against sulphur in the 
furnace. Irons above 1% manganese seldom have any sulphur of any con- 
sequence. 

Graphite is the all-important factor in foundry irons; unless this is present 
in sufficient amount in the castiug, the latter will be liable to be poor. 
Graphite causes iron to slightly expand on cooling, makes it soft, tough and 
fluid. (The statement as to expansion on cooling is denied by VV. J. Keep.) 

Relation of the Appearance of Fracture to the Chemical 
Composition. — S. H. Chauvenet says. when run [from the blast-fur- 
nace] the lower bed is almost always close-grain, but shows practically the 
same analysis as the large grain in the rest of the cast. If the iron runs 
rapidly, the lower bed may have as large grain as any in the cast. If the 
iron runs rapidly for, say, six beds and some obstruction in the tap-hole 
causes the seventh bed to fill up slowly and sluggishly, this bed may be 
close-grain, although the eighth bed, if the obstruction is removed, will be 
open-grain. Neither the graphitic carbon nor the silicon seems to have any 
influence on the fracture in these cases, since by analysis the graphite and 
silicon is the same in each. The question naturally arises whether it would 
not be better to be guided by the analysis than by the fracture. The frac- 
ture is a guide, but it is not an infallible guide. Should not the open- and 
the close-grain iron from the same cast be numbered under the same grade 
when they have the same analysis ? 

Mr. Meissner had many analyses made for the comparison of fracture 



CHEMISTRY OF FOUNDRY IRONS. 



371 



with analysis, and unless the condition of furnace, whether the iron ran 
fast or slow, and from what part of pig bed the sample is taken, are known, 
the fracture is often very misleading. Take the following analyses: 





A. 


B. 


C. 


D. 


E. 


F. 




4.315 
0.008 
3.010 


4.818 
0.008 
2.757 


4.270 
0.007 
2.680 


3.328 
0.033 
2.243 


3.869 
0.006 
3.070 
0.108 


3.861 


Sulphur 

Graphitic car.. 


0.006 
3.100 
0.096 















A. Very close-grain iron, dark color, by fracture, gray forge. 

B. Open-grain, dark color, by fracture. No. 1. 

C. Very close-grain, by fracture, gray forge. 

D. Medium-grain, by fracture, No. 2, but much brighter and more open 
than A, C, or F. 

E. Very large, open-grain, dark color, by fracture, No. 1. 

F. Very close-grain, by fracture, gray forge. 

By comparing analyses A and B, or E and F, it appears that the close- 
grain iron is in each case the highest in graphitic carbon. Comparing A 
and E, the graphite is about the same, but the close-grain is highest in 
silicon. 

Analyses of Foundry Irons. (C. A. Meissner.) 
Scotch Irons. 



Name. 


Grade. 


Silicon. 


Phos- 
phorus. 


Manga- 
nese. 


Sul- 
phur. 


Graph- 
ite. 


Comb. 
Carton 


Summerlee 


1 
1 
1 

2 
1 
1 

1 
1 

2 


2.70 
2.47 
3.44 
2.70 
2.15 
■ 2.59 
1.70 
3.03 

4.00 


0.545 
0.760 
1.000 
0.810 
0.618 
0.840 
1.100 
1.200 

0.900 


1.80 
2.51 
1.70 
2.90 
2.80 
1.70 
1.83 
2.85 

3.41 


0.01 

0.015 

0.015 

0.02 

0.025 

0.010 

0.008 

0.010 


3.09 

2.00 
3.76 
3.75 
3.50 

1.78 


0.25 

0.80 
21 




3.75 




0.40 


Glengarnock 

Glengarnock said 
to carry % scrap 


0.90 




Description of Samples.— No. 1. Well known Ohio Scotch iron, almost 
silvery, but carries two-thirds scrag: made from part black-band ore. Very 
successful brand The high silicon gives it its scrap-carrying capacity. 

No. 2. Brier Hill Scotch castings, made at scale works; castings demand- 
ing more fluidity than strength. 



372 



IRON AND STEEL. 



No. 3. Formerly a famous Ohio Scotch brand, not now in the market 
Made mainly from black-band ore. 

No. 4. A good Ohio Scotch, very soft and fluid; made from black-band 
ore-mixture. 

Nos. 5a and 56. Brier Hill Scotch iron and casting; made for stove pur- 
poses; 350 lbs. of iron used to 150 lbs. scrap gave very soft fluid iron; worked 
well. 

No. 6a. Shows comparison between Summerlee (Scotch) (6a) and Brier Hill 
Scotch (6b). Drillings came from a Cleveland foundry, which found both 
irons closely alike in physical and working quality. 

No. 7. One of the best southern brands, very hard to compete with, owing 
to its general qualities and great regularity of grade and general working. 




Description of Samples.— No. 8. A famous Southern brand noted for fine 
machine castings. 

No. 9. Also a Southern brand, a very good machine iron. 

Nos. 10a and 106. Formerly one of the best known Ohio brands. Does not 
shrink; is very fluid and strong. Foundries having used this have reported 
very favorably on it. 

No. 11. Iron from Brier Hill Co., made to imitate No. 3 ; was stronger 
than No. 3; did not pull castings; was fluid and soft. 

No. 12. Copy of a very strong English machine iron. 

No. 13. A Pennsylvania iron, very tough and soft. This is partially Besse- 
mer iron, which accounts for strength, while high silicon makes it soft. 

No. 14. Castings made from Brier Hill Co.'s machine brand for scale works, 
very satisfactory, strong, soft and fluid. 

No. 15. Castings made from Brier Hill Co.'s one half machine brand, one 
half Scotch brand, for scale works, castings desired to be of fair strength, 
but very fluid and soft. 

No. 16a. Brier Hill machine brand made to compete with No. 3. 

No. 166. Castings (clothes-hooks) from same, said to have worked badly, 
castings being white and irregular. Analysis proved that some other iron 
too high in manganese had been used, and probably not well mixed. 

No. 17. A Pennsylvania iron, no shrinkage, excellent machine iron, soft 
and strong. 

No. 18. A very good quality Northern charcoal iron. 

"Standard Grades" of the Brier Hill Iron and Coal 

Company. 

Brier Hill Scotch Iron. — Standard Analysis, Grade Nos. 1 and 2. 

Silicon 2.00 to 3.00 

Phosphorus 0.50to0.75 

Manganese 2.00 to 2.50 

Used successfully for scales, mowing-machines, agricultural implements, 
novelty hardware, sounding-boards, stoves, and heavy work requiring no 
special strength. 



CHEMISTRY OF FOUNDRY IRONS. 



373 



Brier Hill Silvery Iron.— Standard Analysis, Grade No. 1. 

Silicon 3.50 to 5.50 

Phosphorus 1.00 to 1.50 

Manganese 2.00 to 2.25 

Used successfully for hollow-ware, car-wheels, etc., stoves, bumpers, and 
similar work, with heavy amounts of scrap in all cases. Should be mainly 
used where fluidity and no great strength is required, especially for heavy 
work. When used with scrap or close pig low in phosphorus, castings of 
considerable strength and great fluidity can be made 

Fairly Heavy Machine Iron.— Standard Analysis, Grade No. 1. 

Silicon 1.75 to 2.50 

Phosphorus 0.50to0.60 

Manganese 1.20 to 1.40 

The best iron for machinery, wagon-boxes, agricultural implements, 
pump-works, hardware specialties, lathes, stoves, etc., where no large 
amounts of scrap are to be carried, and where strength, combined with 
great fluidity and softness, are desired. Should not have much scrap with 
it. 

Regular Machine Iron. — Standard Analysis, Grade Nos. 1 and 2. 

Silicon.... 1.50to2.00 

Phosphorus 0. 30 to .50 

Manganese 0.80 to 1.00 

Used for hardware, lawn-mowers, mower and reaper works, oil-well 
machinery, drills, fine machinery, stoves, etc. Excellent for all small fine 
castings requiring fair fluidity, softness, and mainly strength. Cannot be 
well used alone for lai-ge castings, but gives good results on same when used 
with above mentioned heavy machine grade; also when used with the 
Scotch ia right proportion. Will carry but little scrap, and should be used 
alone for good strong castings. 

For Axles and Materials Requiring Great Strength, Grade No. 2. 

Silicon. 1.50 

Phosphorus 0.200 and less. 

Manganese 0.80 

This gave excellent results. 

A good neutral iron for guns, etc., will run about as follows: 

Silicon 1.00 

Phosphorus 0.25 

Sulphur 0.20 

Manganese none. 

It should be open No. 1 iron. 

This gives a very tough, elastic metal. More sulphur would make tough 
but decrease elasticity. 

For fine castings demanding elegance of design but no strength, phos- 
phorus to 3.00$ is good. Can also stand 1.50$ to 2.00$ manganese. For work 
of a hard, abrasive character manganese can run 2.00$ in casting. 
Analyses of Castings. 



Sample 
No. 


Silicon. 


Phos- 
phorus. 

1.400 
0.351 
0.327 
0.577 
0.742 
1.208 
0.418 
1.280 
0.879 
0.408 
0.660 
1.439 
0.900 
0.980 


Manganese 


Sulphur. 


Graphite. 


Comb. 
Carbon. 


31 


2.50 
0.85 
1.53 
1.84 
2.20 
2.50 
2.80 
3.10 
3.30 
2.88 
4.50 
3.43 
2.68 
1.90 


2.20 
0.92 
1.08 
1.04 
1.10 
1.16 
0.54 
1.14 
0.80 
1.10 
0.78 
0.90 
1.30 
1.20 








32 


0.030 
0.040 






33 
34a 


3.10 


0.58 


346 
34c 
















356 








35c 








35cZ 








35e 










0.025 






37a 






876 









374 IRON AND STEEL. 

No. 31. Sewing-machine casting, said to be very fluid and good casting. 
This is an odd analysis. I should say it would have been too hard and brit- 
tle, yet no complaint was made. 

No. 32. Very good machine casting, strong, soft, no shrinkage. 

No. 33. Drilliugs from an annealer-box that stood the heat very well. 

No. 34a. Drillings from door-hinge, very strong and soft. 

No. 346. Drilliugs from clothes-hooks,' tough and soft, stood severe ham- 
mering. 

No. 34c. Drillings from window-blind hinge, broke off suddenly at light 
strain. Too high phosphorus. 

No. 35a. Casting for heavy ladle support, very strong. 

Nos 356 and 35c. Broke after short usage. Phosphorus too high. Car- 
bumpers. 

No. 35o\ Elbow for steam heater, very tough and strong. 

No. 36. Cog-wheels, very good, shows absolutely no shrinkage. 

No. 37. Heater top network, requiring fluidity but no strength. 

No. 37a. Gray part of above. 

No. 376. White, honeycombed part of above. Probably bad mixing and 
got chilled suddenly. 

STRENGTH OF CAST IRON. 

Rankine gives the following figures: 

Various qualities, T. S 13,400 to 29,000, average 16,500 

Compressive strength 82,000 to 145,000, " 112,000 

Modulus of elasticity 14,000,000 to 22,900,000, " 17,000,000 

Specific Gravity and Strength. (Major Wade, 1856.) 

Third-class guns: Sp. Gr. 7.087, T. S. 20,148. Another lot: least Sp. Gr. 7.163, 
T. S. 22,402. 

Second-class guns: Sp. Gr. 7.154, T. S. 24,767. Another lot : mean Sp. Gr. 
7.302, T. S. 27,232. 

First class guns: Sp. Gr. 7.204, T. S. 28,805. Another lot: greatest Sp. Gr. 
7.402, T. S. 31,027. 

Strength of Charcoal Pig Iron. -Pig iron made from Salisbury 
ores, in furnaces at Wassaic and Millerton, N. Y., has shown over 40,000 lbs. 
T. S. per square inch, one sample giving 42,281 lbs. Muirkirk, Md., iron 
tested at the Washington Navy Yard showed: average for No. 2 iron, 21,601 
lbs. ; No. 3, 23,959 lbs. ; No. 4, 41,329 lbs. ; average density of No. 4, 7.336 (J. C. 
I. W., v. p. 44.) 

Nos. 3 and 4 charcoal pig iron from Chapinville, Conn., showed a tensile 
strength per square inch of from 34,761 lbs. to 41,882 lbs. Charcoal pig iron 
from i Shelby, Ala. (tests made in August, 1891), showed a strength of 
34,800 lbs. for No. 3; No. 4, 39,675 lbs.; No. 5, 46,450 lbs.; and a mixture of 
equal parts of Nos. 2, 3, 4. and 5, 41.470 lbs. (Bull. I. & 8. A.) 

Variation of Density and Tenacity of Gun-irons.— An in- 
crease of density invariably follows the rapid cooling of cast iron, and as a 
general rule the tenacity is increased by the same means. The tenacity 
generally increases quite uniformly with the density, until the latter ascends 
to some given point; after which an increased density is accompanied by a 
diminished tenacity. 

The turning-point of density at which the best qualities of gun-iron attain 
their maximum tenacity appears to be about 7.30. At this point of density, 
or near it, whether in proof-bars or gun-heads, the tenacity is greatest. 

As the density of iron is increased its liquidity when melted is diminished. 
This causes it to congeal quickly, and to form cavities in the interior of the 
casting. (Pamphlet of Builders 1 Iron Foundrv, 1893.) 

Specifications for Cast Iron tor the World's Fair Build- 
ings, 1892. — Except where chilled iron is specified, all castings shall be 
of tough gray iron, free from injurious cold-shuts or blow-holes, true to 
pattern, and of a workmanlike finish. Sample pieces 1 in. square, cast from 
the same heat of metal in sand moulds, shall be capable of sustaining on a 
clear span of 4 feet 6 inches a central load of 500 lbs. when tested in the 
rough bar. 

Specifications for Tests of Cast Iron in 12" B. Ii. Mortars* 
(Pamphlet of Builders Iron Foundry, 1893.)— Charcoal Gun Iron.— The tensile 
strength of the metal must average at each end at least 30,000 lbs. per 
square inch ; no specimen to be over 37,000 lbs. per square inch ; but one 
specimen from each end may be as low as 28,000 lbs. per square inch. The 



MALLEABLE CAST IROH. 375 

long extension specimens will not be considered in making up these aver- 
ages, but must show a good elongation and an ultimate strength, for each 
specimen, of not less than 24,000 lbs. The density of the metal must be such 
as to indicate that the metal has been sufficiently refined, but not carried so 
high as to impair the other qualities. 

Specifications for Grading Pig Iron for Car Wheels by 
Chill Tests made at the Furnace. (Penna. R. R. Specifications, 
1883.)— The chill cup is to be filled, even full, at about the middle of every 
cast from the furnace. The test-piece so made will be 7}4 inches long, 3^ 
inches wide, and 1% inches thick, and is to be broken across the centre when 
entirely cold. The depth of chill will be shown on the bottom of the test- 
piece, and is to be measured by the clean white portion to the point where 
gray specks begin to show in the white. The grades are to be by eighths of 
an inch, viz., %, %, %, ^, %%, %, etc., until the iron is mottled; the lowest 
grade being % of an inch in depth of chill. The pigs of each cast are to be 
marked with the depth of chill shown by its test-piece, and each grade 
is to be kept by itself at the furnace and in forwarding. 

Mixture of Cast Iron with Steel.— Car wheels are sometimes 
made from a mixture of charcoal iron, anthracite iron, and Bessemer 
steel. The following shows the tensile strength of a number of tests of 
wheel mixtures, the average tensile strength of the charcoal iron used being 
22,000 lbs.: 

lbs. per sq. in. 

Charcoal iron with 2)4% steel 22,467 

" 8%* steel 26,733 

" " " 6}4% steel and 6}4% anthracite 24,400 

" " " 714% steel and iy>% anthracite 28,150 

" 2y%% steel, 2%% wro't iron, and G*4% anth. .. 25,550 

41 " " 5 % steel, 5% wro't iron, and 10 % anth 26,500 

(Jour. C. I. W., hi. p. 184.) 

Cast Iron Partially Bessemerized.— Car wheels made of par- 
tially Bessemerized iron (blown in a Bessemer converter for 2>)4 minutes), 
chilled in a chill-test mould over an inch deep, just as a test of cold-blast 
charcoal iron for car wheels would chill. Car wheels made of this blown 
iron have run 250.000 miles. [Jour. C. I. W., vi. p. 77.) 

Bad Cast Iron.— On October 15, 1891, the cast-iron fly-wheel of a large 
pair of Corliss engines belonging to the Amoskeag Mfg. Co., of Manchester, 
N. H., exploded from centrifugal force. The fly-wheel was 30 feet diam- 
eter and 110 inches face, with one set of 12 arms, and weighed 116,000 lbs. 
After the accident, the rim castings, as well as the ends of the arms, were 
found to be full of flaws, caused chiefly by the drawing and shrinking of the 
metal. Specimens of the metal were tested for tensile strength, and varied 
from 15,000 lbs. per square inch in sound pieces to 1000 lbs. in spongy ones. 
None of these flaws showed on the surface, and a rigid examination of the 
parts before they were erected failed to give any cause to suspect their true 
nature. Experiments were carried on for some time after the accident in 
the Amoskeag Company's foundry in attempting to duplicate the flaws, but 
with no success in approaching the badness of these castings. 

MALLEABLE CAST IRON. 

Malleableized cast iron, or malleable iron castings, are castings made 
of ordinary cast iron which have been subjected to a process of decarboni- 
zation, which results in the production of a crude wrought iron. Handles, 
latches, and other similar articles, cheap harness mountings, plowshares, 
iron handles for tools, wheels, and pinions, and many small parts of ma- 
chinery, are made of malleable cast iron. For such pieces charcoal cast iron 
of the best quality (or other iron of similar chemical composition), should 
be selected. Coke irons low in silicon and sulphur have been used in place 
of charcoal irons. The castings are made in the usual way, and are then 
imbedded in oxide of iron, in the form, usually, of hematite ore, or in per- 
oxide of manganese, and exposed to a full red-heat for a sufficient length of 
time, to insure the nearly complete removal of the carbon. This decarboniza- 
tion is conducted in cast-iron boxes, in which the articles, if small, are 
packed in alternate layers with the decarbonizing material. The largest 
pieces require the longest time. The fire is quickly raised to the maximum 
temperature, but at the close of the process the furnace is cooled very 
slowly. The operation requires from three to five days with ordinary small 
castings, and may take two weeks for large pieces. 



376 



iROtf AKi) STEEL. 



Rules for Use of Malleable Castings, by Committee of Master 
Carbuilders' Ass'n, 1890. 

1. Never run abruptly from a heavy to a light section. 

2. As the strength of malleable cast iron lies in the skin, expose as much 
surface as possible. A star-shaped section is the strongest possible from 
which a casting can be made. For brackets use a number of thin ribs instead 
of one thick one. 

3. Avoid all round sections; practice has demonstrated this to be the 
weakest form. Avoid sharp angles. 

4. Shrinkage generally in castings will be 3/16 in. per foot. 
Strength, of Malleable Cast Iron.— Experiments on the strength 

of malleable cast iron, made in 1891 by a committee of the Master Car- 
builders' Association. The strength of this metal varies with the thickness, 
as the following results on specimens from 34 in. to iy 2 in. in thickness show: 



Dimensions. 


Tensile Strength. 


Elongation. 


Elastic Limit. 


in. in. 


lb. per sq. in. 


percent in 4 in. 


lb. per sq. in. 


1.52 by .25 


34,700 


2 


21,100 


1.52 " .39 


33,700 


2 


15,200 


1.53 " .5 


32,800 


2 


17,000 


1.53 " .64 


32,100 


2 


19,400 


2. " .78 


25,100 


M 


15,400 


1.54 " .88 


33,600 


m 


19,300 


1.06 " 1.02 


30,600 


i 


17,600 


1.28 " 1.3 


27,400 


i 




1.52 " 1.54 


28,200 


1H 





The low ductility of the metal is worthy of notice. The committee gives 
the following table of the comparative tensile resistance and ductility of 
malleable cast iron, as compared with other materials: 



Cast iron 

Malleable cast iron 

Wrought iron 

Steel castings 



Ultimate 

Strength, 

lb. per sq. in 



Comparative 

Strength ; 

Cast Iron 

= 1. 



20,000 
32,000 
50,000 
60,000 



Elongation 
Per Cent 
in 4 in. 



0.35 
2.00 
20.00 
10.00 



Comparative 

Ductility; 

Malleable 

Cast Iron 

= 1. 



0.17 
1 
10 



Another series of tests, reported to the Association in 1892, gave the 
following: 



Thick- 
ness. 


Width. 


Area. 


Elastic 
Limit. 


Ultimate 
Strength. 


Elongation 
in 8 in. 


in. 


in. 


sq. in. 


lb. per sq. 


lb. per sq. in. 


percent. 


.271 


2.81 


.7615 


23.520 


32,620 


1.5 


.293 


2.78 


.8145 


22,650 


28,160 


.6 


.39 


2.82 


1.698 


20,595 


32,060 


1.5 


.41 


2.79 


1.144 


20,230 


28,850 


1.0 


.529 


2.76 


1.46 


19,520 


27,875 


1.1 


.661 


2.81 


1.857 


18,840 


25,700 


.7 


.8 


2.76 


2.208 


18.390 


25,120 


1.1 


1.025 


2.82 


2.890 


18,220 


28,720 


1.5 


1.117 


2.81 


3.138 


17,050 


25,510 


1.3 


1.021 


2.82 


2.879 


18,410 


26,950 


1.3 . 



WROUGHT IRON". 



377 



WROUGHT IRON. 

Influence of Chemical Composition on the Properties 
of Wrought Iron. (Beaidslee on Wrought Iron and Chain Cables. 
Abridgement by W. Kent. Wiley & Sons, 1879.)— A series of 2000 tests of 
specimens from 14 brands of wrought iron, most of them of high repute, 
was made in 1877 by Capt. L. A. Beaidslee, U.S.N., of the United States 
Testing Board. Forty-two chemical analyses were made of these irons, 
with a view to determine what influence the chemical composition had 
upon the strength, ductility, and welding power. From the report of these 
tests by A. L. Holley the following figures are taken : 





Average 

Tensile 

Strength. 


Chemical Composition. 


Brand. 


S. 


P. 


Si. 


C. 


Mn. 


Slag. 


L 
P 
B 
J 

O 
C 


66,598 
54,363 
52,764 
51,754 
51,134 
50,765 


trace 

j 0.009 
1 0.001 

0.008 
j 0.003 
jO.005 
(0.004 
\ 0.005 

0.007 


j 0.065 
I 0.084 
0.250 
0.095 
0.231 
0.140 
0.291 
0.067 
0.078 
0.169 


0.080 
0.105 
0.182 
0.028 
0.156 
0.182 
0.321 
0.065 
0.073 
0.154 


0.212 
0.512 
0.033 
0.066 
0.015 
0.027 
0.051 
0.045 
0.042 
0.042 


0.005 
0.029 
0.033 
0.009 
0.017 
trace 
0.053 
0.007 
0.005 
0.021 


0.192 
0.452 

0.848 
1.214 

"6!678" 
1.724 
1.168 
0.974 



Where two analyses are given they are the extremes of two or more ana- 
lyses of the brand. Where one is given it is the only analysis. Brand L 
should be classed as a puddled steel. 

Order op Qualities Graded from No. 1 to No. 19. 



rand. 


■rensue 
Strength. 


reduction 
of Area. 


Elongation. 


Welding Power. 


L 


1 


18 


19 


most imperfect. 


P 


6 


6 


3 


badly. 


B 


12 


16 


15 


best. 


J 


16 


19 


18 


rather badly. 


O 


18 


1 


4 


very good. 


C 


bi . 19 


12 


16 






The reduction of area varied from 54.2 to 25.9 per cent, and the elonga- 
tion from 29..9 to 8.3 per cent. 

Brand O, the purest iron of the series, ranked No. 18 in tensile strength, 
but was one of the most ductile; brand B, fquite impure, was below the 
average both in strength and ductility, but was the best in welding power ; 
P, also quite impure, was one of the best in every respect except welding, 
while L, the highest in strength, was not the most pure, it had the least 
ductility, and its welding power was most imperfect. The evidence of the 
influence of chemical composition upon quality, therefore, is quite contra- 
dictory and confusing. The irons differing remarkably in their mechanical 
properties, it was found that a much more marked influence upon their 
qualities was caused by different treatment in rolling than by differences in 
composition. 

In regard to slag Mr. Holley says : " It appears that the smallest and 
most worked iron often has the most slag. It is hence reasonable to con- 
clude that an iron may be dirty and yet thoroughly condensed." 

In his summary of " What is learned from chemical analysis," he says : 
" So far, it may appear that little of use to the makers or users of wrought 
iron has been learned. . . . The character of steel can be surely pred- 
icated on the analyses of the materials; that of wrought iron is altered by 
subtle and unobserved causes " 

Influence of Reduction in Rolling from Pile to Rar on 
the Strength of "Wrought Iron.— The tensile strength of the irons 
used in Beardslee's tests ranged from 46,000 to 62,700 lbs. per sq. in., brand 
L, which was really a steel, not being considered. Some specimens of L 
gave figures as high as 70,000 lbs. The amount of reduction of sectional 



378 



IRON AND STEEL. 



4 


3 


2 


1 


% 


Ya 


80 


80 


72 


25 


9 


3 


15.7 


8.83 


4.36 


3.14 


2.17 


1.6 


46,322 


47,761 


48,280 


51,128 


52,275 


59,585 


23,430 


26,400 


31,892 


36,467 


39,126 






area in rolling the bars has a notable influence on the strength and elastic 
limit; the greater the reduction from pile to bar the higher the strength. 
The following are a few figures from tests of one of the brands : 

Size of bar, in. diam.: 
Area of pile, sq. in.: 
Bar per cent of pile : 
Tensile strength, lb.: 
Elastic limit, lb.: 

Specifications for Wrought Iron (F. H. Lewis, Engineers' Club 
of Philadelphia, 1891).— 1. All wrought iron must be tough, ductile, fibrous, 
and of uniform quality for each class, straight, smooth, free from cinder- 
pockets, flaws, buckles, blisters, and injurious cracks along the edges, and 
must have a workmanlike finish. No specific process- or provision of 
manufacture will be demanded, provided the material fulfils the require- 
ments of these specifications. 

2. The tensile strength, limit of elasticity, and ductility shall be deter- 
mined from a standard test-piece not less than J4 inch thick, cut from the 
full-sized bar, and planed or turned parallel. The area of cross-section shall 
not be less than Y 2 square inch. The elongation shall be measured after 
breaking on an original length of 8 inches. 

3. The tests shall show not less than the following results: 



For bar iron in tension 

For shape iron 

For plates under 36 in. wide. 
For plates over 36 in. wide . . 



Ultimate 

Strength, 

lbs per sq. 

inch. 



50,000 
48,000 
48,000 
46,000 



Limit of 

Elasticity, 

lbs. per sq. 

inch. 



26,000 
26,000 
26,000 
25,000 



Elongation in 
8 inches, 
per cent. 



12 
10 



4. When full-sized tension members are tested to prove the strength of 
their connections, a reduction in their ultimate strength of (500 x width of 
bar) pounds per square inch will be allowed. 

5. All iron shall bend, cold, 180 degrees around a curve whose diameter 
is twice the thickness of piece for bar iron, and three times the thickness 
for plates and shapes. 

6. Iron which is to be worked hot in the manufacture must be capable 
of bending sharply to a right angle at a working heat without sign of 
fracture. 

7. Specimens of tensile iron upon being nicked on one side and bent shall 
show a fracture nearly all fibrous. 

8. All rivet iron must be tough and soft, and be capable of bending cold 
until the sides are in close contact without sign of fracture on the convex 
side of the curve. 

Pennsylvania Railroad Specifications for merchant Bar 
Iron or Steel.— Miscellaneous merchant bar iron or steel for which no 
special specifications defining shapes and uses are issued, should have a 
tensile strength of 50,000 to 55,000 lbs. per square inch and an elongation of 
20$ in a section originally 2 inches long. 

No iron or steel will be accepted under this specification if tensile strength 
falls below 48,000 lbs. or goes above 60,000 lbs. per square inch, nor if elon- 
gation is less than 15$ in 2 inches, nor if it shows a granular fracture cover- 
ing more than 50$ of the fractured surface, nor if it shows any difficulty in 
welding. 

In preparing test-pieces from round or rectangular bars, they will be 
turned or shaped so that the tested sections may be the central portion of 
the bar, in all sizes up to 1% inches in any diametrical or side measurement. 
In larger sizes test-pieces will be made to fall about half-way from centre to 
circumference. 

Bars of iron y% in. thick or less, or tortured forms of iron, such as angle, tee 
or channel bars" will be accepted if tensile strength is above 45,000 lbs. and 
elongation above \2%; but the testing of such sizes and sections is optional, 



FORMULAE FOR UNIT STRAINS FOR IROls' A j\ r D STEEL. 379 



Specifications for Wrought Iron for the World's Fair 
Buildings. [Eny\j News, March 26. 1892.)— All iron to be used in the 
tensile members of open trusses, laterals, pins and bolts, except plate iron 
over 8 inches wide, and shaped iron, must show by the standard test-pieces 
a tensile strength in lbs. per square inch of : 

7,000 X area of original bar in sq. in. 
circumference of original harm inches' 
with an elastic limit not less than half the strength given by this formula, 
and an elongation of 20^ in 8 in. 

Plate iron 24 inches wide and under, and more than 8 inches wide, must 
show by the standard test-pieces a tensile strength of 48,000 lbs. per sq. in. 
with an elastic limit not less than 26,000 lbs. per square inch, and an elon- 
gation of not less than 12%. All plates over 24 inches in width must have a 
tensile strength not less than 46,000 lbs., with an elastic limit not less than 
26,000 lbs. per square inch. Plates from 24 inches to 36 inches in width must 
have au elongation of not less than 10$; those from 38 inches to 48 inches in 
width, 8$; over 48 inches in width, 5%. 

All shaped iron, flanges of beams and channels, and other iron not herein- 
before specified, must show by the standard test-pieces a tensile strength in 
lbs. per square inch of : 

7,000 X area of original bar 
1 circumference of original bar' 

with an elastic limit of not less than half the strength given by this formula, 
and an elongation of 15% for bars % inch and less in thickness, and of 12$ for 
bars of greater thickness. For webs of beams and channels, specifications 
for plates will apply. 

All rivet iron must be tough and soft, and pieces of the full diameter of 
the rivet must be capable of bending cold, until the sides are in close contact, 
without sign of fracture on the convex side of the curve. 

Stay-bolt Iron.— Mr. Vauclain, of the Baldwin Locomotive Works, 
at a meeting of the American Railway Master Mechanics' Association, in 
1892, says: Many advocate the softest iron in the market as the best for 
stay-bolts. He believed in an iron as hard as was consistent with heading 
the bolt nicely. The higher the tensile strength of the iron, the more vibra- 
tions it will stand, for it is not so easily strained beyond the yield-point. 
The Baldwin specifications for stay-bolt iron call for a tensile strength of 
50,000 to 52,000 lbs. per square inch, the upper figure being preferred, and 
the lower being insisted upon as the minimum. 

FORfflULi: FOK UNIT STRAINS FOR. IRON AND 

STEEL IN STRUCTURES. 

(F. H. Lewis, Engineers' (Jlub of Philadelphia, 1891.) 

The following formulae for unit strains per square inch of net sectional 
area shall be used in determining the allowable working stress in each mem- 
ber of the structure. (For definitions of soft and medium steel see Specifi- 
cations for Steel.) 

Tension Members. 



Floor-beam hangers or 
suspenders, forged 
bars 

Counter-ties 

Suspenders, hangei 
and counters, riveted 
members, net sec- 
tion 

Solid rolled beams , 

Riveted truss members 
and tension flanges 
of girders, net sec- 
tion 

Forged eyebars 

Lateral or cross-sec 
tion rods 



Wrought Iron. 



Will not be used 
6000 



5000 
8000 



7000(l + ™Hi) 
V max./ 

Will not be used 



Soft Steel. 



Will not be used 



5500 
8000 



8% greater than 
iron 



Will not be used 



Medium Steel. 



7000 
7000 



7000 
Will not be used 



9000(1+™^) 

V max. ' 

9000(l+™) 

V max./ 
/For eyebars\ 
V only, 17,000 ) 



380 



IRON AND STEEL. 



Shearing. 



On pins and shop rivets 

On field rivets 

In webs of girders.. 



Wrought Iron. 



6000 

4800 

Will not be used 



6600 
5200 
5000 



Medium Steel. 

7200 

Will not be used 

6000 



Bearing. 





Wrought Iron. 


Soft Steel. 


Medium Steel. 


On projected semi- 








intrados of main-pin 








holes 


12,000 


13,200 


14,500 


On projected semi-in- 








trados of rivet-holes* 


12,000 


13,200 


14,500 


On lateral pins 


15,000 


16,500 


18,000 


Of bed-plates on ma- 








sonry 


250 lbs. per sq. in. 







* Excepting that in pin-connected members taking alternate stresses, the 
bearing stress must not exceed 9000 lbs. for iron or steel. 
Bending. 
On extreme fibres of pins when centres of bearings are considered as 
points of application of strains: 

Wrought Iron, 15,000. Soft Steel, 16,000. Medium Steel, 17,000. 

Compression Members. 



Chord sections : 

Flat ends 

One flat and one pin end.. 

Chords with pin ends and 
all end-posts 

All trestle-posts 

Intermediate posts 

Lateral struts, and com- 
pression in collision 
struts, stiff suspenders 
and stiff chords. . . 



Wrought Iron. 



7 ooo(i + i^)- J 

\ max./ 

7000 (l-f^)- 35 * 

V max./ 

7000 (l+i™-)- 40 

V max./ 

700 o(l + ^)-35 
\ max./ 



10,500 - 50 - 



Soft Steel. 



Wo 

greater 
than 



Medium 
Steel. 



20£ 

greater 

than 



In which formulae I ~ length of compression member in inches, and r = 
least radius of gyration of member in inches. No compression member 
shall have a length exceeding 45 times its least width, and no post should be 
used in which I -+-r exceeds 125. 
Members S ubject to Alternate Tension and Compression. 



Wrought Iron. 



Soft Steel. 



Medium 
Steel. 



For compression only. . 
For the greatest stress . 



Use the formulae above 
^Jf « max. lesser ^ 

70001 1 - „ r— 

\ 2 max. greater^ 



20$ greater 
than iron 



Use the formula giving the greatest area of section. 

The compression flanges of beams and plate girders shall have the same 
cross-section as the tension flanges. 



FORMULAE FOB UNIT STRAINS FOR IRON AND STEEL. 381 

W. H. Burr, discussing the formulae proposed by Mr. Lewis, says: " Taking 
the results of experiments as a whole, I am constrained to believe that they 
indicate at least 15$ increase of resistance for soft-steel columns over those 
of wrought iron, with from 20% to 25% for medium steel, rather than 10$ and 
20$ respectively. 

" The high capacity of soft steel for enduring torture fits it eminently for 
alternate and combined stresses, and for that reason I would give it 15$ 
increase over iron, with about 22% for medium steel. 

"Shearing tests on steel seem to show that 15$ and 22$ increases, for the 
two grades respectively, are amply justified. 

" I should not hesitate to assign 15$ and 22$ increases over values for iron 
for bearing and bending of soft and medium steel as being within the safe 
limits of experience. Provision should also be made for increasing pin- 
shearing, bending and bearing stresses for increasing ratios of fixed to mov- 
ing loads " 

Maximum Permissible Stresses in Structural Materials 
used in Buildings. (Building Ordinances of the City of Chicago. 1893.) 
Cast iron, crushing stress: For plates, 15,000 lbs. per square inch; for lintels, 
brackets, or corbels, compression 13,500 lbs. per square inch, and tension 
3000 lbs. per square inch. For girders, beams, corbels, brackets, and trusses, 
16,000 lbs. per square inch for steel and 12,000 lbs. for iron. 

For plate girders : 

_. maximum bending moment in ft.-lbs. 
Flange area — 



CD. 
D = distance between centre of gravity of flanges in feet. 

c=r' 



j 13,500 for steel. 
1 10,000 for iron. 



maximum shear „ ( 10,000 for steel, 
Web area = . C = -j 6 ,000 for iron. 

For rivets in single shear per square inch of rivet area : 

Steel. Iron. 

If shop-driven. 9000 lbs. 7500 lbs. 

If field-driven 7500 " 6000 " 

For timber girders : 

b = breadth of beam in inches. 
d = depth of beam in inches. 
_ cbd* I = length of beam in feet. 

" — fT ' (160 for long-leaf yellow pine, 

c = <120 for oak, 

( 100 for white or Norway pine. 
Proportioning of Materials in the Memphis Bridge (Geo. 
S. Morison, Trans. A. S. C. E., 1893).— The entire superstructure of the Mem- 
phis bridge is of steel and it was all worked as steel, the rivet-holes being 
drilled in all principal members and punched and reamed in the lighter 
members. 

The tension members were proportioned on the basis of allowing the dead 
load to produce a strain of 20,000 lbs. per square inch, and the live load a 
strain of 10,000 lbs. per square inch. In the case of the central span, where 
the dead load was twice the live load, this corresponded to 15,000 lbs. total 
strain per square inch, this being the greatest tensile strain. 

The compression members were proportioned on a somewhat arbitrary 
basis. No distinction was made between live and dead loads. A maximum 
strain of 14,000 lbs. per square inch was allowed on the chords and other 
large compression members where the length did not exceed 16 times the 
least transverse dimension, this strain being reduced 750 lbs. for each addi- 
tional unit of length. In long compression members the maximum length 
was limited to 30 times the least transverse dimension, and the strains 
limited to 6,000 lbs. per square inch, this amount being increased by 200 lbs. 
for each unit by which the length is decreased. 

Wherever reversals of strains occur the member was proportioned to re- 
sist the sum of compression and tension on whichever basis (tension or 
compression) there would be the greatest strain per square inch ; and, in 
addition, the net section was proportioned to resist the maximum tension, 
and the gross section to resist the maximum compression. 

The floor beams and girders were calculated on the strain being limited to 
10,000 lbs. per square inch in extreme fibres. Rivet-holes in cover-plates and 
flanges were deducted. 



382 



IRON AND STEEL. 



The rivets of steel in drilled or reamed holes were proportioned on the 
basis of a bearing strain of 15,000 lbs. per square inch and a shearing strain 
of 7500 lbs. per square inch, and special pains were taken to get the double 
shear in as many rivets as possible. This was the requirement for shop 
rivets. In the case of field rivets, the number was increased one-half. 

The pins were proportioned on the basis of a bearing strain of 18,000 lbs. 
per square inch and a bending strain of 20,000 lbs. per square inch in ex- 
treme fibre, the diameters of the pins being never made more than one inch 
less than the width of the largest eye-bar attaching to them. 

The weight on the rollers of the expansion joint on Pier II is 40,000 lbs. 
per linear foot of roller, or 3,333 lbs. per linear inch, the rollers being 15 ins. 
in diameter. 

As the sections of the superstructure were unusually heavy, and the strains 
from dead load greatly in excess of those from moving load, it was thought 
best to use a slightly higher steel than is now generally used for lighter 
structures, and to work this steel without punching, all holes being drilled. 
A somewhat softer steel was used in the floor-system and other lighter 
parts. 

The principal requirements which were to be obtained as the results of 
tests on samples cut from finished material were as follows: 





Max. 




Ultimate 




Strength, 




lbs. per 




sq. inch. 


High-grade steel. 


78,500 


Eye-bar steel 


75,000 


Medium steel — 


72,500 


Soft steel 


63,000 



Min. 




Ultimate 


Min. Elastic 


Strength, 


Limit, lbs, 


lbs. per 


per sq. in. 


sq. inch. 




69,000 


40,000 


66,000 


38,000 


64.000 


37,000 


55,000 


30,000 



Min. per- 
centage of 
Elongation 
"1 inches . 



Min. Per- 
centage of 
Reduction 
at Fracture 



18 
20 
22 



40 
44 
50 



TENACITY OF METALS AT VARIOUS 
TEMPERATURES. 

The British Admiralty made a series of experiments to ascertain what loss 
of strength and ductility takes place in gun-metal compositions when raised 
to high temperatures. It was found that all the varieties of gun-metal 
suffer a gradual but not serious loss of strength and ductility up to a certain 
temperature, at which, within a few degrees, a great change takes place, 
the strength falls to about one half the original, and the ductility is wholly 
gone. At temperatures above this point, up to 500, there is little, if any, 
further loss of strength ; the temperature at which this great change and 
loss of strength takes place, although uniform in the specimens cast from 
the same pot, varies about 100° in the same composition cast at different 
temperatures, or with some varying conditions in the foundry process. 
The temperature at which the change took place in No. 1 series was ascer- 
tained to be about 370°, and in that of No. 2, at a little over 250°. Whatever 
may be the cause of this important difference in the same composition, the 
fact stated may be taken as certain. Rolled Muntz metal and copper arc 
satisfactory up to 500°, and may be used as securing-bolts with safety. 
Wrought iron, Yorkshire and remanufactured, increase in strength up to 
500°, but lose slightly in ductility up to 300°, where an increase begins and 
continues up to 500°, where it is still less than at the ordinary temperature 
of the atmosphere. The strength of Landore steel is not affected by temper- 
ature up to 500°, but its ductility is reduced more than one half. (Iron, Oct. 
6, 1877.) 

Tensile Strength of Iron and Steel at High Tempera- 
tures.— James E. Howard's tests (Iron Age, April 10, J8 C J0), shows that the 
tensile strength of steel diminishes as the temperature increases from 0° 
until a minimum is reached between 200° and 300° F., the total decrease 
being about 4000 lbs. per square inch in the softer steels, and from 6000 to 
8000 lbs. in steels of over 80,000 lbs. tensile strength. From this minimum point 
the strength increases up to a temperature of 400° to 650° F., the maximum 
being reached earlier in the harder steels, the increase amounting to from 
10,000 to 20,000 lbs. per square inch above the minimum strength at from 200o 



TENACITY OF METALS AT VAKIOUS TEMPERATURES. 383 

to 800°. From this maximum, the strength of all the steel decreases steadily 
at a rate approximating 10,000 lbs. decrease per 100° increase of tempera- 
ture. A strength of 20,000 lbs. per square inch is still shown by .10 C. steel 
at about 1000° F., and by .60 to 1.00 C. steel at about 1600° F. 

The strength of wrought iron increases with temperature from 0° up to a 
maximum at from 400 to 600° F., the increase being from 8000 to 10,000 lbs. 
per square inch, and then decreases steadily till a strength of only 6000 lbs. 
per square inch is shown at 1500° F. 

Cast iron appears to maintain its strength, with a tendency to increase, 
unt 1 900° is reached, beyond which temperature the strength gradually 
diminishes. Under the highest temperatures, 1500° to 1600° F., numerous 
cracks on the cylindrical surface of the specimen were developed prior to 
rupture. It is remarkable that cast iron, so much inferior in strength to the 
steels at atmospheric temperature, under the highest temperatures has 
nearly the same strength the high-temper steels then have. 

Strength of Iron and Steel Boiler-plate at High Tem- 
peratures. (Chas. Huston, Jour. F. I., 1877.) 

Average of Three Tests of Each. 

Temperature F. 68° 575° 925° 

Charcoal iron plate, tensile strength, lbs 55.366 63,080 65,343 

" " " contr. of area % 26 23 21 

Soft open-hearth steel, tensile strength, lbs 54,600 66,083 64,350 

" contr. % 47 38 33 

" Crucible steel, tensile strength, lbs 64,000 69,266 68,600 

" " " contr. % 36 30 21 

Strength of Wrought Iron and Steel at High Temper- 
atures. (Jour. F. L, cxii., 1881, p. 241.) Kollmann's experiments at Ober- 
hausen included tests of the tensile strength of iron and steel at tempera- 
tures ranging between 70° and 2000° F. Three kinds of metal were tested, 
viz., fibrous iron having an ultimate tensile strength of 52,464 lbs., an elastic 
strength of 38,280 lbs., and an elongation of 17.5$; fine-grained iron having 
for the same elements values of 56.892 lbs., 39,113 lbs., and 20$; and Bes- 
semer steel having values of 84,826 lbs., 55,029 lbs., and 14.5$. The mean 
ultimate tensile strength of each material expressed in per cent of that at 
ordinary atmospheric temperature is given in the following table, the fifth 
column of which exhibits, for purposes of comparison, the results of experi- 
ments carried on by a committee of the Franklin Institute in the years 
1832-36. 





Fibrous 


Fine-grained 


Bessemer 


Franklin 


Temperature 


Wrought 


Iron, 


Steel, 


Institute 


Degrees F. 


Iron, p. c. 


per cent. 


per cent. 


per cent. 





100.0 


100.0 


100.0 


96.0 


100 


100.0 


100.0 


100.0 


102.0 


200 


100.0 


100.0 


100.0 


105.0 


300 


97.0 


100.0 


100.0 


106.0 


400 


95.5 


100.0 


100.0 


106.0 


500 


92.5 


98.5 


98.5 


104.0 


600 


88.5 


95.5 


92.0 


99.5 


700 


81.5 


90.0 


68.0 


92.5 


800 


67.5 


77.5 


44.0 


75.5 


900 


44.5 


51.5 


36.5 


53.5 


1000 


26.0 


36.0 


31.0 


"36.0 


1100 


20.0 


30.5 


26.5 




1200 


18.0 


28.0 


22.0 




1300 


16.5 


23.0 


18.0 




1400 


13.5 


19.0 


15.0 




1500 


10.0 


15.5 


12.0 




1600 


7.0 


12.5 


10.0 




1700 


5.5 


10.5 


8.5 




1800 


4.5 


8.5 


7.5 




1900 


3.5 


7.0 


6.5 




2000 


3.5 


5.0 


5.0 





The Effect of Cold on the Strength of Iron and Steel.— 

The following conclusions were arrived at by Mr. Styffe in 1865 : 

(1) That the absolute strength of iron and steel is not diminished by 
cold, but that even at the lowest temperature which ever occurs in Sweden 
it is at least as great as at the ordinary temperature (about 60° F.). 



384 IRON" AND STEEL. 

(2) That neither in steel nor in iron is the extensibility less in severe cold 
than at the ordinary temperature. 

(3) That the limit of elasticity in both steel and iron lies higher in severe 
cold. 

(4) That the modulus of elasticity in both steel and iron is increased on 
reduction of temperature, and diminished on elevation of temperature ; but 
that these variations never exceed 0.05 % for a change of temperature of 1.8° 
F., and therefore such variations, at least for ordinary purposes, are of no 
special importance. 

Mr. C. P. Sandberg made in 1867 a number of tests of iron rails at various 
temperatures by means of a falling weight, since he was of opinion that, 
although Mr. Styffe's conclusions were perfectly correct as regards tensile 
strength, they migbt not apply to the resistance of iron to impact at low 
temperatures. Mr. Sandberg convinced himself that " the breaking strain " 
of iron, such as was usually employed for rails, " as tested by sudden blows 
or shocks, is considerably influenced by cold ; such iron exhibiting at 10° F. 
only from one third to one fourth of the strength which it possesses at 
84° F." Mr. J. J. Webster (Inst. C. E., 1880) gives reasons for doubting 
the accuracy of Mr. Sandberg's deductions, since the tests at the lower 
temperature were nearly all made with 21-ft. lengths of rail, while those at 
the higher temperatures were made with short lengths, the supports in 
every case being the same distance apart. 

W. H. Barlow (Proc. Inst. C. E.) made experiments on bars of wrought 
iron, cast iron, malleable cast iron, Bessemer steel, and tool steel. The bars 
were tested with tensile and transverse strains, and also by impact ; one 
half of them at a temperature of 50° F., and the other half at 5° F. The 
lower temperature was obtained by placing the bars in a freezing mixture, 
care being taken to keep the bars covered with it during the whole time of 
the experiments. 

The results of the experiments were summarized as follows : 

1. When bars of wrought iron or steel were submitted to a tensile strain 
and broken, their strength was not affected by severe cold (5° F.), but their 
ductility was increased about \% in iron and 3% in steel. 

2. When bars of cast iron were submitted to a transverse strain at a low 
temperature, their strength was diminished about 3% and their flexibility 
about 16%. 

3. When bars of wrought iron, malleable cast iron, steel, and ordinary 
cast iron were subjected to impact at a temperature of 5° F., the force re- 
quired to break them, and the extent of their flexibility, were reduced as 
follows, viz.: 

Reduction of Force Reduction of Flexi- 

of Impact, per cent. bility, per cent. 

Wrought iron, about 3 18 

Steel (best cast tool), about 3% 17 

Malleable cast iron, about 4)4 I 5 

Cast iron, about 21 not taken 

The experience of railways in Russia, Canada, and other countries where 
the winter is severe is that the breakages of rails and tires are far more 
numerous in the cold weather than in the summer. On this account a 
softer class of steel is employed in Russia for rails than is usual in more 
temperate climates. 

The evidence extant in relation to this matter leaves no doubt that the 
capability of wrought iron or steel to resist impact is reduced by cold. On 
the other hand, its static strength is not impaired by low temperatures. 

Effect of tow Temperatures on Strength of Railroad 
Axles. (Thos. Andrews, Proc. Inst. C. E., 1891.)— Axles 6 ft. 6 in. long 
between centres of journals, total length 7 ft. 3>£ in., diameter at middle 4J^ 
in., at wheel-sets b% in., journals 334 X 7 in. were tested by impact at temper- 
atures of 0° and 100° F. Between the blows each axle was half turned over, 
and was also replaced for 15 minutes in the water-bath. 

The mean force of concussion resulting from each impact was ascertained 
as follows : 

Let h = height of free fall in feet, w — weight of test ball, hiv = W — 
" energy," or work in foot-tons, x = extent of deflections between bearings, 

W hw 
then F (mean force) = — = — , 



DURABILITY OF IRON", CORROSION, ETC. 



385 



The results of these experiments show that whereas at a temperature of 
0° F. a total average mean force of 179 tons was sufficient to cause the 
breaking of the axles, at a temperature of 100° F. a total average mean 
force of 428 tons was requisite to produce fracture. In other words, the re- 
sistance to concussion of the axles at a temperature of 0° F. was only about 
42% of what it was at a temperature of 100° F. 

The average total deflection at a temperature of 0° F. was 6.48 in., as 
against 15.06 in. with the axles at 100° F. under the conditions stated; this 
represents an ultimate reduction of flexibility, under the test of impact, of 
about 57?S for the cold axles at 0° F., compared with the warm axles at 
100° F. 

EXPANSION OF IRON AND STEEL BY HEAT. 

James E. Howard, engineer in charge of the U. S. testing- machine at Wa- 
fertown, Mass., gives the following results of tests made on bars 35 inches 
long (/row Age, April 10, 1890): 





Marks. 


Chemical composition. 


Coefficient of 
Expansion. 


Metal. 


C. 


Mn. 


Si. 


Feby 
difference. 


Per degree 
F. per unit 
of length. 














.0000067302 


Steel 


la 
2a 
3a 
4a 
5a 
6a 
7a 
8a 
9a 
10a 


.09 
.20 
.31 
.37 
.51 

.11 

.81 
.89 
.97 


A i 

.57 
.70 
.58 
.93 

.58 
.56 

.57 
.80 


".bk" 
.07 
.08 
.17 
.19 
.28 


99.80 
99.35 
99.12 
98.93 
98.89 
98.43 
98.63 
98.46 
98.35 
97.95 


.0000067561 
.0000066259 


w 


.0000065149 


lt 


.0000066597 
.0000066202 


11 


.0000063891 


It 


.0000064716 


c< 


.0000062167 


(C 


.0000062335 


11 


.0000061700 




0000059261 














.0000091286 

















DURABILITY OF IRON, CORROSION, ETC. 

Durability of Cast Iron.— Frederick Graff, in an article on the 
Philadelphia water-supply, says that the first cast-iron pipe used there was 
laid in 1820. These pipes were made of charcoal iron, and were in constant 
use for 53 years. They were uncoated, and the inside was well filled with 
tubercles. In salt water good cast iron, even uncoated, will last for a cen- 
tury at least; but it often becomes soft enough to be cut by a knife, as is 
shown in iron cannon taken up from the bottom of harbors after long sub- 
mersion. Close-grained, hard white metal lasts the longest in sea water.— 
Eng'g News, April 23. 1887, and March 26. 1892. 

Tests of Iron after Forty Years' Service.— A square link 12 
inches broad, 1 inch thick and about 12 feet long was taken from the Kieff 
bridge, then 40 years old, and tested in comparison with a similar link which 
had been preserved in the stock-house since the bridge was built. The fol- 
lowing is the record of a mean of four longitudinal test -pieces, 1 X 1% X 8 
inches, taken from each link (Stahl und Eisen, 1890): 

Old Link taken New Link from 

from Bridge. Store-house. 

Tensile strength per square inch, tons 21 .8 22.2 

Elastic limit " " 11.1 11.9 

Elongation, per cent 14.05 13.42 

Contraction, per cent 17.35 18.75 

Durability of Iron in Bridges. (G-. Lindenthal, Eng'g, May 2, 
1884, p. 139.)— The Old Monongahela suspension bridge in Pittsburgh, built 
in 1845, was taken down in 1882. The wires of the cables were frequently 
strained to half of their ultimate strength, yet on testing them after 37 years' 



386 IRO^ AND STEEL. 

use they showed a tensile strength of from 72,700 to 100,000 lbs. per square 
inch. The elastic limit was from 67,100 to 78,600 lbs. per square inch. Re- 
duction at point of fracture, 35$ to 75$. Their diameter was 0.13 inch. 

A new ordinary telegraph wire of same gauge tested for comparison 
showed: T. S., of'100,000 lbs.; E. L., 81,550 lbs.; reduction, 57%. Iron rods 
used as stays or suspenders showed: T. S , 43,770 u> 49,720 lbs. per square 
inch; E. L., 26,380 to 29,200. Mr. Lindenthal draws these conclusions from 
his tests: 

" The above tests indicate that iron highly strained for a long number of 
years, but still within the elastic limit, and exposed to slight vibration, will 
not deteriorate in quality. 

" That if subjected to only one kind of strain it will not change its texture, 
even if strained beyond its elastic limit, for many yer.rs. It will stretch and 
behave much as in a testing-machine during a long test. 

" That iron will change its texture only when exposed to alternate severe 
straining, as in bending in different directions. If the bending is slight but 
very rapid, as in violent vibrations, the effect is the same." 

Corrosion of Iron Bolts.— On bridges over the Thames in London, 
bolts exposed to the action of the atmosphere and rain-water v\ ere eaten 
away in 25 years from a diameter of % in. to % in., and from % in. diameter 
to 5/16 inch, 

Wire ropes exposed to drip in colliery shafts are very liable to corrosion. 

Corrosion of Iron and Steel.— Experiments made at the Riverside 
Iron Works, Wheeling, W. Va., on the comparative liability to rust of iron 
and soft Bessemer steel: A piece of iron plate and a similar piece of steel, 
both clean and bright, were placed in a mixture of yellow loam and sand, 
with which had been thoroughly incorporated some carbonate of soda, nitrate 
of soda, ammonium chloride, and chloride of magnesium. The earth as 
prepared was kept moist. At the end of 33 days the pieces of metal were 
taken out, cleaned, and weighed, when the iron was found to have lost 0.84$ 
of its weight and the steel 0.72$. The pieces were replaced and after 28 days 
weighed again, when the iron was found to have lost 2.06$ of its original 
weight and the steel 1.79$. (Eng'g, June 26, 1891.) 

Corrosive Agents in the Atmosphere.— The experiments of F. 
Crace Calvert (Chemical Neivs, March 3, 1871) show that carbonic acid, in 
the presence of moisture, is the agent which determines the oxidation of 
iron in the atmosphere. He subjected perfectly cleaned blades of iron and 
steel to the action of different gases for a period of four months, with 
results as follows : 

Dry oxygen, dry carbonic acid, a mixture of both gases, dry and damp 
oxygen and ammonia: no oxidation. Damp oxygen: in three experiments 
one blade only was slightly oxidized. 

Damp carbonic acid: slight appearance of a white precipitate upon the 
iron, found to be carbonate of iron. Damp carbonic acid and oxygen: 
oxidation very rapid. Iron immersed in water containing carbonic acid 
oxidized rapidly. 

Iron immersed in distilled water deprived of its gases by boiling rusted 
the iron in spots that were found to contain impurities. 

Galvanic action is a most active agent of corrosion. It takes place 
when two metals, one electro-negative to the other, are placed in contact 
and exposed to dampness. 

Sulphurous acid (the product of the combustion of the sulphur in coal) is 
an exceedingly active corrosive agent, especially when the exposed iron is 
coated with soot. This accounts for the rapid corrosion of iron in railway 
bridges exposed to the smoke from locomotives. (See account of experi- 
ments by the author on action of sulphurous acid in Jour. Frank. Inst. , June, 
1875, p. 437.) An analysis of sooty iron rust from a railway bridge showed 
the presence of sulphurous, sulphuric, and carbonic acids, chlorine, and 
ammonia. Bloxam states that ammonia is formed from the nitrogen of the 
air during the process of rusting. 

Rustless Coatings for Iron and Steel.— Tinning, enamelling, lac- 
quering, galvanizing, electro-chemical painting, and other preservative 
methods are discussed in two important papers by M. P. Wood, in Trans. 
A. S. M. E., vols, xv and xvi. 

A Method of Producing an Inoxidizable Surface on 
iron and steel by means of electricity has been developed by M. A. de Meri- 
tens (Engineering.) The article to be protected is placed in a bath of ordi- 
nary or distilled water, at a temperature of from 158° to 176° F., and an 
electric current is sent through. The water is decomposed into its elements, 



DURABILITY OF IRON", CORROSION, ETC. 387 

Oxygen and hydrogen, and the oxygen is deposited on the metal, while the 
hydrogen appears at the other pole, which may either be the tank in which 
the operation is conducted or a plate of carbon or metal. The current has 
only sufficient electromotive force to overcome the resistance of the circuit 
and to decompose the water; for if it be stronger than this, the oxygren com- 
bines with the iron to produce a pulverulent oxide, which has no adherence. 
If the conditions are as they should be, it is only a few minutes after the 
oxygen appears at the metal before the darkening of the surface shows 
that the gas has united with the iron to form the magnetic oxide Fe 3 4 , 
which it is well known will resist the action of the air and protect the metal 
beneath it. After the action has continued an hour or two the coating is 
sufficiently solid to resist the scratch-brush, and it will then take a brilliant 
polish. 

If a piece of thickly rusted iron be placed in the bath, its sesquioxide 
(Fe 2 3 ) is rapidly transformed into the magnetic oxide. This outer layer 
has no adhesion, but beneath it there will be found a coating which is 
actually a part of the metal itself. 

In the early experiments M. de Meritens employed pieces of steel only, 
but in wrought and cast iron he was not successful, for the coating came oif 
with the slightest friction. He then placed the iron at the negative pol of 
the apparatus, after it had been already applied to the positive pole. Here 
the oxide was reduced, and hydrogen was accumulated in the pores of the 
metal. The specimens were then returned to the anode, when it was found 
that the oxide appeared quite readily and was very solid. But the result 
was not quite perfect, and it w r as not until the bath was filled with distilled 
water, in place of that from the public supply, that a perfectly satisfactory 
result was attained. 

Manganese Plating of Iron as a Protection from Rust. 
—According to the Italian Progresso, articles of iron can be protected 
against rust by sinking them near the negative pole of an electric bath com- 
posed of 10 litres of water, 50 grammes of chloride of manganese, and 200 
gramnies of nitrate of ammonium. Under the influence of the current the 
bath deposits on the articles a film of metallic manganese which prevents 
them from rusting. 

A Non-oxidizing Process of Annealing is described by H. 
P. Jones, in Eng'g Neivs, Jan. 2, 1892. The ordinary process of annealing, 
by means of which hard and brittle iron or steel is rendered soft and tough, 
consists in heating the metal to a good red-heat and then allowing it to cool 
gradually. While the metal is in a heated condition the surface becomes 
oxidized; and although for many classes of work this scale of oxide is of 
practical importance, yet in some cases it is very undesirable and even 
necessitates considerable expense in its removal. 

The new process uses a non-oxidizing gas, and is the invention of Mr. 
Horace K. Jones, of Hartford, Conn. The principal feature of this process 
consists in keeping the annealing-retort in communication with the gas 
holder or gas main during the entire process of heating and cooling, the j;as 
thus being allowed to expand back into the main, and being, therefore, 
kept at a practically constant pressure. 

The retorts used are made from wrought-iron tubes. The gas used is 
taken directly from the mains supplying the city with illuminating gas. It 
was noticed that if metal which had been blued or slightly oxidized was sub- 
jected to the annealing process it came out bright, the oxide being reduced 
by the action of the gas. Practical use has been made of this fact in deoxi- 
dizing metal. 

Comparative tests were made of specimens of metal annealed in illlumi- 
nating gas and of specimens annealed in nitrogen. The results of these tests 
were compared with the results of tests of specimens annealed in an open 
fire and cooled in ashes, and of specimens of the unannealed metal, and 
thus the relative efficiency of the gas process was determined. 

The specimens were made from steel wire .188 in. in diameter and were 
turned down to diameters of .156 and .150 in. Different lots of wire were 
tested in order to secure average results. The elongations were in each 
case referred to an original length of 1.15 ins. 

The difference in total per cent of elongation and in breaking load between 
the specimens annealed in nitrogen and those annealed in illuminating gas 
is very slight. The average results were as follows: 



388 



IRON AND STEEL. 





Gas used. 


No. 
Test 
Pieces . 


Breaking 

Load, 

lbs. per sq. in. 


Elongation. 




Total p. c. 


p. c. gained. 


A 

B 

C 

D 

E 

F 

G 

H 

I 


Nitrogen 
Illuminating 

Nitrogen 
Illuminating 

Nitrogen 
Illuminating 

Open fire 
Unannealed 
Unannealed 


4 
4 
4 
4 
5 
5 
8 
5 
5 


62,140 
63,140 
60.000 
60,400 
57,330 
57.070 
63,090 
97,120 
80,790 


29.12 
28.08 
28.00 
27.20 
30.88 
29.60 
26.76 
7.12 
8.80 


22.00 

20.86 
19.20 
18.40 
23.76 
22.48 
19.64 



Painting Wood and Iron Structures. (E. H. Brown, Eng'rs 
Club of Phila., Engineering Neivs, April 20, 1893.)— A paint consists of two 
portions— the pigment and the vehicle or binder. The pigment is a solid 
substance which is more or less finely ground, so as to be capable (when 
mixed with the vehicle) of being spread out in a thin layer or coating over 
the surface to be painted. The vehicle or binder is the liquid in which the 
pigment is mixed or ground, which serves to spread the pigment over the 
surface to be painted, and which also holds it to that surface. For ordinary 
painting the most generally used vehicle is linseed oil. 

Linseed oil possesses the peculiar property of drying by uniting with the 
oxygen of the air to form a tough, leather-like compound called lindxin. 

For painting on wood, zinc white has valuable pigment properties, but 
these seem to be most fully developed when this pigment is used in con- 
junction with white lead, and then to the best advantage when the mixture 
is used as a final coat over an elastic undercoating of white lead. So far no 
other white base has been discovered which possesses at the same time the 
other properties which render white lead valuable, namely, covering and 
spreading capacit3 r . 

Of the inert pigments, lampblack is probably the" most valuable. Being 
almost pure carbon, it is practically unchangeable except by fire. It ha^ the 
peculiar property of absorbing great quantities of linseed oil, and hence of 
spreading over a large surface. French ochre, an earth pigment containing 
more or less of the hydrated oxide of iron, possesses the property of absorb- 
ing a large quantity of oil, and hence has considerable spreading capacity, 
and also holds very firmly to any wooden surface to which it may be applied. 

The various mineral and metallic paints are almost all natural or artificial 
iron oxides. While these are cheap and useful for painting rough wooden 
structures they are sometimes really quite dangerous for application to iron 
work, because, instead of preventing oxidation, they are apt to further it. 

Coal tar is much used as a paint for the roughest class of work, both wood 
and iron; in the latter case especially for cast-iron pipes, smoke-stacks, and 
work to be buried underground. It has the nature both of a resin and an oil. 
It has the disadvantage of becoming exceedingly brittle by the action of cold, 
and softening at 115° F. Asphalt permits of somewhat wider range of temper- 
ature, but otherwise exhibits the same peculiarities. These substances, while 
they last, are probably the most valuable of paints, especially under water; 
but they are unfortunate in their tendency to flow or crawl on the surface 
to which they are applied, finally leaving the upper portions almost or quite 
bare. This is the case even under ground. 

Red lead has long been regarded as the best possible preservative for clean 
dry iron. But in order to be most effective, the iron must be perfectly clean 
and free from any suspicions of rust, and absolutely dry. Red lead should 
be perfectly pure and of the best and most careful preparation. That from 
any well-known corroding house may be depended upon for purity, but not 
always for quality. It is simply a red oxide of lead. The best type is orange 
mineral, which is made by roasting white lead. On account of its expense 
this is not so frequently used as it would deserve. Red lead proper is made 
directly from the metal, which is first oxidized to the yellow litharge, and 
then to the red oxide. This, however, does not give as good a paiut as that 
made from the scrap, settlings, and tailings of the white lead works. As red 
lead saponifies very quickly with linseed oil, it must be used within a few 
days after being ground, and, moreover, it is rather difficult to work. 



CHEMICAL COMPOSITION AND PHYSICAL CHARACTER. 389 

Hence there is great temptation to add some substance, such as whiting, to 
it in' order to make it work freer, as well as to cost less money for material. 

Before painting iron work it is essential that the iron itself should be ab- 
solutely free from rust. Rust has the peculiar property of spreading and 
extending from a centre, if there be the slightest chance to do so. Hence, a 
small amount of rust on the iron may grow under the surface of the paint, 
especially if it be true, as Dr. Dudley asserts, that linseed oii is permeable 
by air and moisture, and in time the paint will be flaked off by the rust un- 
derneath, thus gradually exposing the bare surface of the iron to the action 
of its destroying agent, oxygen in the presence of water. It is necessary 
to remove all the scale possible from wrought iron by means of stiff wire 
brushes, and then to remove the rust by a pickle of very dilute acid, which 
must afterward be thoroughly washed off before the paint is applied. The 
surface of the iron should be dry and at least moderately warmed before it 
is primed. The best method of painting a tin roof is to carefully remove 
all traces of oii or grease from the surface of the tin while it is yet bright 
with benzine; then to apply a coat of red lead and linseed oil, or the best 
quality of metallic paint, and to follow this with one or two coats of graphite 
paint. The graphite is almost unchangeable by atmospheric action, and is 
remarkably waterproof as well. 

Red Lead as a Preservative of Iron.— A. J. Whitney writes to 
Engineering News, August, 1891, that in 30 years' experience he has found 
red lead to be the best material for preserving iron under all circumstances. 

Quantity of Paint Required for a Given Surface. (M. P. 
Wood.)— Sq. ft. of surface -j- 200 = gallons of liquid paint for two coats; sq. 
ft. of surface h- 18 = lbs. of pure white lead for three coats. 

Qualities of Paints. — The Railroad and Engineering Journal, vols, 
liv and lv, 1890 and 1891, has a series of articles[on paint as applied to wooden 
structures, its chemical nature, application, adulteration, etc., by Dr. C. B. 
Dudley, chemist, and F. N. Pease, assistant chemist, of the Penna. R. R. 
They give the results of a long series of experiments on paint as applied to 
railway purposes. 

Graphite Paint. (M. P. Wood.)— Graphite, mixed with pure boiled 
linseed oil in which a small percentage of litharge, red lead, manganese, or 
other metallic salt has been added at the time of boiling to aid in the oxida- 
tion of the oil, forms a most effective paint for metallic surfaces, as well as 
for wood and fibrous substances. Wood surfaces protected by this paint, 
and exposed to the action of sea- water for a number of years, are found in 
a perfect state of preservation. 

STEEL. 

RELATION BETWREN THE CHEMICAL, COMPOSI- 
TION AND PHYSICAL CHARACTER OF STEEL. 

W. R. Webster (see Trans. A. I. M. E., vols, xxi and xxii, 1893-4) gives re- 
sults of several hundred analyses and tensile tests of basic Bessemer steel 
plates, and from a study of them draws conclusions as to the relation of 
chemical composition to strength, the chief of which are condensed as 
follows : 

The indications are that a pure iron, without carbon, phosphorus, man- 
ganese, silicon, or sulphur, if it could be obtained, would have a tensile 
strength of 34,750 lbs. per square inch, if tested in a %-inch plate. With 
this as a base, a table is constructed by adding the following hardening 
effects, as shown by increase of tensile strength, for the several elements 
named. 

Carbon, a constant effect of 800 lbs. for each 0.01$. 

Sulphur, " " 500 " " 0.01^. 

Phosphorus, the effect is higher in high-carbon than in low-carbon steels. 

With carbon hundreths % 9 10 11 12 13 14 15 16 17 

Each .01^ P has an effect of lbs. 900 1000 1100 1200 1300 1400 1500 1500 1500 

Manganese, the effect decreases as the per cent of manganese increases. 
( .00 .15 .20 .25 .30 .35 .40 .45 .50 .55 

Mn being per cent < to to to to to to to to to to 

( .15 .20 .25 .30 .35 .40 .45 .50 .55 .65 

Str'gth increases for .01 j* 240 240 220 200 180 160 140 120 100 100 lbs. 

Total incr. from Mn . . . 3600 4800 5900 6900 7800 8600 9'iOO 9900 10,400 1 1,400 



390 



STEEL. 



Silicon is so low in this steel that its hardening effect has not been con- 
sidered. 

With the above additions for carbon and phosphorus the following table 
has been constructed (abridged from the original by Mr. Webster). To the 
figures given the additions for sulphur and manganese should be made as 
above. 
Estimated Ultimate Strengths of Basic Bessemer Steel 

Plates. 

For Carbon, .06 to .24; Phosphorus, .00 to .10; Manganese and Sulphur, .00 in 

all cases. 



Carbon. 


.06 


.08 


.10 


.12 
44,950 


.14 


.16 .18 
48,300' 49,900 


.20 


53,100 


.24 


Phos. .005 


39,950 


41,550 


43,250 


46,650 


51,500 


54,700 


" .01 


-10,350 


41,950 


43,750 


45,550 


47,350 


49,050 50,650 


52,250 


53,850 


55,450 


" .02 


41,150 


42.750 


44,750 


46.750 


48,750 


50,550 52,150 


53.750 


55.350 


56,950 


" .03 


41,950 


43,550 


45,750 


47,950 


50.150 


52,050 53,650 


55, -,'50 


56,850 


58.450 


" .04 


42.750 


44,350 


46,750 


49,150 


51,550 


53.550 55,150 


50,750 


58,350 


59,950 


'• .05 


43,550 


45.150 


47,750 


50,350 


52,950 


55.050 56,650 


58,250 


59,850 


61.450 


" .06 


44.350 


45,950 


48,750 


51,550 


54,350 


56.550 58.150 


59.750 


61 350 


62.950 


" .0? 


45,150 


46,750 


49,750 


52,75C 


55,750 


58,050 59,650 


61,250 


62,850 


64.450 


" .08 


45,950 


47,550 


50,750 


53.950 


57,150 


59,550 61,150 


62,750 


64,350 


65,950 


" .09 


46,750 


48,350 


51,750 


55,150 


58,550 


61.050 62,650 


64,250 


65,850 


67,450 


" .10 


47,550 


49,150 


52,750 


56,35C 


59,95( 


62,550 64.150 


65,750 


67,350 


68,950 


.001 Phos = 


80 lbs. 


80 lbs. 


100 lb 


120 lb 


140 lb 


150 lb 150 lb 


150 1b 


150 1b 


1501b 



In all rolled steel the quality depends on the size of the bloom or ingot 
from which it is rolled, the work put oh it, and the temperature at which it 
is finished, as well as the chemical composition. 

The above table is based on tests of plates % inch thick and under 70 
inches wide; for other plates Mr. Webster gives the following corrections 
for thickness and width. They are made necessary only by the effect of 
thickness and width on the finishing temperature in ordinary practice. 
Steel is frequently spoiled by being finished at too high a temperature. 
Corrections for Size of Plates. 

Plates. Up to 70 ins. wide. Over 70 ins. wide. 

Inches thick. Lbs. Lbs. 

% and over. -2000 —1000 

11/16 



% 
9/16 



— 1750 


— 750 


— 1500 


— 500 


— 1250 


— 250 


— 1000 


— 


— 500 


± 500 





+ 1000 


+ 3000 


+ 5000 



Comparing the actual result of tests of 408 plates with the calculated 
results, Mr. Webster found the variation to range as in the table below. 
Summary of the Differences Between Calculated and 
Actual Results in 408 Tests of Plate Steel. 

In the first three columns the effects of sulphur were not considered; in 
the last three columns the effect of sulphur was estimated at 500 lbs. for 
each .01$ of S. 





"3 
53S 


•6 


1 


"3 


i 


i 






| 


A 


II 
a 


s 


si 


o ~ ^ 

«5g 




P 


w 


M 


P 




pq 


Per cent within 1000 lbs.. 


23.4 


32.1 


28.4 


24.6 


27.0 


26.0 


28.4 


2000 " .. 


40.9 


48.9 


45.6 


48.5 


54.9 


52.2 


55.1 


3000 " .. 


62.5 


71.3 


67.6 


67.8 


73.0 


70.8 


74.7 


4000 " .. 


75.5 


81.0 


78.7 


82.5 


85.2 


84.1 


89.9 


" 5000 " .. 


89.5 


91.1 


90.4 


93.0 


92.8 


92.9 


94.9 



STRENGTH OF BESSEMER AND OPEN-HEARTH STEELS. 391 



The last figure in the table would indicate that if specifications were drawn 
calling for steel plates not to vary more than 5000 lbs. T. S. from a specified 
figure (equal to a total range of 10,000 lbs.), there would be a probability of 
the rejection of 5% of the blooms rolled, even if the whole lot was made 
from steel of identical chemical analysis. In 1000 heats only 2% of the heats 
failed to meet the requirements of the orders on which they were graded; 
the loss of plates was much less than 1%, as one plate was rolled from each 
heat and tested before rolling the remainder of the heat. 

R. A. Hadfield (Jour. Iron 6b Steel Inst., No. 1, 1894) gives the strength of 
very pure Swedish iron, remelted and tested as cast, 20.1 tons (45,024 lbs.) 
per sq. in.; remelted and forged, 21 tons (47.040 lbs.). The analvsis of the 
cast bar was : C, 0.08; Si, 0.04; S, 0.02 : P. 0.02; Mn, 0.01; Fe. 99.82. 

Effect of Oxygen upon Strength of Steel.— A. Lantz, of the 
Peine works, Germany, in a letter to Mr. Webster, says: "We have found 
<luring the current year (1893) that oxygen plays an important role, till now 
little observed— such, indeed ,..th at given a like content of carbon, phospho- 
rus, and manganese in the blows, a blow with greater oxygen content gives 
a greater hardness and less ductility than a blow with less oxygen content. 1 ' 
The method used for determining: oxj^gen is that of Prof. Ledebur, given in 
Stahlund Eisen, May, 1892. p. 193. The variation in oxygen content may 
make a difference in strength of nearly one -half ton per square inch. 
(Jour. Iron 6b Steel Inst.. No. 1, 1894.) 

range of variation in strength of bessemer 

and Open-hearth steels. 

The Carnegie Sieel Co. in 1888 published a list of 1057 tests of Bessemer 
and open-hearth steel, from which the following figures are selected : 



Kind of Steel. 



(a) Bess, structural. . 

(b) " » .. 

(c) Bess, angles 

(d)O. H. fire-box.... 

(e) Tank 

(/) O. H. bridge 



Elastic -Limit. 



39,230 
39,970 
32,630 



Ultimate 
Strength. 



High't. Lowest 



73,540 
63,450 
62,790 
66.062 
69.940 



61,450 
65,200 
56,130 
50,350 
59.440 
63. 970 



Elongation 

per cent 
in 8 inches. 



High't. Lowest 



33.00 
30.25 
34.30 
36.00 
27.50 
30.00 



23.15 
26.25 
25.62 
19.25 



Requirements op Specifications. 

(a) Elastic limit, 35,000 ; tensile strength, 62,000 to 70,000 ; elong. 22fAn 8 in. 

(b) Elastic limit, 40,000 ; tensile strength, 67,000 to 75,000. 

(c) Elastic limit, 30,000 ; tensile strength, 56,000 to 64,000 ; elong. 20# in 8 in. 

(d) Tensile strength, 50,000 to 62,000 ; elong. 26% in 4 in. 

(e) Tensile strength, 60,000 to 65,000 ; elong. 18# in 8 in. 

(f) Tensile strength. 64,000 to 70,000 ; elong. 20% in 8 in. 

Strength of Open-hearth Structural Steel. (Pencoyd Iron 
Works.) — As a general rule, the percentage of carbon in steel determines its 
hardness and strength. The higher the carbon the harder the steel, the 
higher the tenacity, and the lower the ductility will be. The following list 
exhibits the average physical properties of good open-hearth steel : 



Percentage 
of Carbon. 



.10 
.15 
.20 
.25 
.30 
.35 
.40 



Ultimate 

Tenacity, 

lbs. per sq. in. 



57.000 
62,000 
67,000 
72,000 
77,000 
82,000 
87,000 



Elastic 

Limit, 

lbs. per sq. in. 



34,000 
37,000 
40,000 
43,000 
46,000 
49,000 
52,000 



Stretch in 
8 inches. 



28 per cent. 

26 

24 

22 



Reduction of 
Area, %. 



The coefficient of elasticity is practically uniform for all grades, and is 
the same as for iron, viz., 29,000,000 lbs. These figures form the average of 
a numerous series of tests from rolled bars, and can only serve as an ap- 



392 



proximation in single instances, when the variation from the average may- 
be considerable. Steel below .10 carbon should be capable of doubling flat 
without fracture, after being chilled from a red heat in cold water. Steel 
of .15 carbon will occasionally submit to the same treatment, but will 
usually bend around a curve whose radius is equal to the thickness of 
the specimen ; about 90$ of specimens stand the latter bending test without 
fracture. As the steel becomes harder its ability to endure this bending 
test becomes more exceptional, and when the carbon ratio becomes .20, 
little over 25$ of specimens will stand the last-described bending test. Steel 
having about .40$ carbon will usually harden sufficiently to cut soft iron 
and maintain an edge. 
Mehrtens gives the following tables in Stahl unci Eisen {Iron Age, April 20, 



Basic Bessemer Steel. 
680 Charges. 

Elastic Limit, Charges within 

pounds per Range, per cent 

sq. in. of total number. 

35.500 to 38,400 15.0 

38,400 to 39,800 31.6 

39,800 to 41,200 27.5 

41,200 to 42,700 16.0 

42,700 to 46,400 9.9 

Tensile Strength, Charges within 

pounds per Range, per cent 

sq. in. .of total number. 

55,600 to 56,900. 18.67 

56,900 to 58,300 38. 67 

58,300 to 59,700 23.53 

59,700 to 61,200 15.60 

61,200 to 62,300 3.53 

Structural Steel. 

Charges within 
Elongation. Range, per cent 
per cent. of total number. 
21 to 25 2.65 

25 to 26 8.53 

26 to 27 17.35 

27 to 28 26.76 

28 to 29... 23.68 

29 to 30 14.41 

30to32.5 6.62 

Rivet Steel. 
25.2 to 26 20.0 

26 to27 15.0 

27 to28 25.0 

28 to29 25.0 

29 to 29.8 15.0 



Basic Open-hearth Struc- 
tural Steel. 
489 Charges. 

Elastic Limit, Charges within 

pounds per Range, per cent 

sq. in. of total charges 

34,400 to 37.000 12.3 

37,000 to 38,400 15.6 

38,400 to 39,800 20.3 

39,800 to 41,200 17.4 

41,200 to 42,700 12.8 

42,700 to 44,100 11.4 

44,100 to 48,400 8.5 

Tensile Strength. 

,55,800 to 56,900 8.0 

56,900 to 58,300 26.4 

58,300 to 59,700 25.4 

59,700 to 61,200 19.6 

61,200 to 62,600 11.2 

62,600 to 65, 100 9.04 

Elongation, 
per cent. 

20to25 21.7 

25 to 26 7.7 

28 to 27 10.0 

27 to 28 .■ 11.0 

28 to 29 12.0 

29 to 30 13 3 

30 to 37.1 24.3 

Rivet Steel, 19 Charges. 

Tensile Strength. 

51,800 5.3 

51,900 to 53,300 .. 26.3 

53,300 to 54, 900 21.0 

54,900 to 56,300 21.0 

56,300 to 56,900 26.4 

Elongation all above 25 percent. 
In the basic Bessemer steel over 90$ was below 0.8 phosphorus, and all 
were below 0.10; manganese was below 0.6 in over 90$, and below 0.9 in all ; 
sulphur was below 0.05 in 84$, the maximum being 0.071; carbon was below 
0.10, and silicon below 0.01 in all. In the basic open-hearth steel phosphorus 
was below 0.06 in 96$, the maximum being 0.08; manganese below0.50 in 97$; 
sulphur below 0.07 in 88$, the maximum being .0.12. The carbon ranged 
from 0.09 to 0.14. 

Low Tensile Strength of Very Pure Steel.— Swedish nail-rod 
open-hearth steel, tested by the author in 1881, showed a tensile strength of 
only 42,591 lbs. per sq. in. A piece of American nail-rod steel showed 45,021 
lbs. per sq. in. Both steels contained about .10 carbon and .015 phosphorus, 
and were very low in sulphur, manganese, and silicon. The pieces tested 
were bars about 2 x % in. section. 

Ii©w Strength Due to Insufficient Work. (A. E. Hunt, 
Trans. A. I. M. E., 1886.)— Soft steel ingots, made jn the ordinary way for 
boiler plates, have only from 10,000 to 20,000 lbs. tensile strength per sq. in., 
an elongation of only about 10$ in 8 in., and a reduction of area of less than 
20$. Such ingots, properly heated and rolled down from 10 in. to y% in, 



STRENGTH OF BESSEMER AND OPEN-HEARTH STEELS. 393 



]longation 


Reduction 


in 8 in. 


of Area. 


Per cent. 


Per cent. 


27 


62 


25 


50 


22 


43 


26 


49 



thickness, will give from 55,000 to 65,000 lbs. tensile strength, an elongation 
in 8 in. of from 23$ to 33$, and a reduction of area of from 55$ to 70$. Any 
work stopping short of the above reduction in tnickness ordinarily yields in- 
termediate results in its tensile tests. 

Hardening of Soft Steel.— A. E. Hunt (Trans. A. I. M. E., 1883, vol. 
xii), says that soft steel, no matter how low in carbon, will harden to a cer- 
tain extent upon being heated red-hot and plunged into water, and that it 
hardens more when plunged into brine and less when quenched in oil. 

An illustration was a heat of open-hearth steel of 0.15$ carbon and 0.29$ of 
manganese, which gave the following results upon test-pieces from the same 
J4 in. thick plate. 

Maximum 

Load. 

lbs. per sq. in. 

Unhardened 55,000 

Hardened in water 74,000 

Hardened in brine 84,000 

Hardened in oil 67,700 

While the ductility of such hardened steel does not decrease to the extent 
that the increased tenacity would indicate, and is much superior to that of 
normal steel of the high tenacity, still the greatly increased tenacity after 
hardening indicates that there must be a considerable molecular change in 
the steel thus hardened, and that if such a hardening should be created 
locally in a steel plate, there must be very dangerous internal strains caused 
thereby. 

Effect of Cold. Rolling.— Cold rolling of iron and steel increases the 
elastic limit and the ultimate strength, and decreases the ductility. Major 
Wade's experiments on bars rolled and polished cold by Lauth's process 
showed an average increase of load required to give a slight permanent set 
as follows : Transverse, 162$; torsion, 130$; compression, 161$ on short 
columns \% in. long, and 64$ on columns 8 in. long; tension, 95$. The hard- 
ness, as measured by the weight required to produce equal indentations, 
was increased 50$; and it was found that the hardness was as great in the 
centre of the bars as elsewhere. Sir W. Fairbairn's experiments showed an 
increase in ultimate tensile strength of 50$, and a reduction in the elongation 
in 10 in. of from 2 in. or 20$, to 0.79 in. or 7.9$. 

Comparison of Tests of Full-size Eye-l>ars and Sample 

Test-pieces of Same Steel Used in the Memphis Bridge. 

(Geo. S. Morison, Trans. A. S. C. E„ 1893.) 





Full-Sized Eyebars, 




Sample Bars fr 


om Same Melts, 


Sectio 


is 10" w 


de X 1 to 2 3/16" 


thick. 




about 1 


in. area 




Reduc- 


Elongation. 


Elastic 


Max. 


Reduc- 


Elon- 


Elastic 


Max. 


tion of 






Limit, 


Load, 


tion, 


gation, 


Limit, 


Load, 


Area, 








p.c. 


Inches. 


p.c. 


lbs. per 


sq. in. 


p.c. 


p.c. 


lbs. per 


sq. in. 


39.6 


20.2 


16.8 


35.100 


67,490 


47.5 


27.5 


41,580 


73,050 


39.7 


26.6 


8.2 


37,680 


70,160 


52.6 


24.4 


42,650 


75,620 


44.4 


36.8 


11.8 


39,700 


65,500 


47.9 


28.8 


40,280 


70,280 


38.5 


38.5 


17.3 


33,140 


65,060 


47.5 


27.5 


41,580 


73,050 


40.0 


32.5 


13.5 


32,860 


65,600 


44.5 


20.0 


43,750 


75,000 


39.4 


36.8 


15.3 


31,110 


61.060 


42.7 


28.8 


42,210 


69,730 


34.6 


32.9 


13.7 


33,990 


63,220 


52.2 


28.1 


40,230 


69,720 


32.6 


13.0 


13.5 


29,330 


63,100 


48.3 


28.8 


38,090 


71,300 


7.3 


20 8 


6.9 


28,080 


55,160 


43.2 


24.2 


38,320 


70,220 


38.1 


28.9 


14.1 


29,670 


62.140 


59.6 


26.3 


40,200 


71,080 


31.8 


24.0 


11.8 


32,700 


65,400 


40.3 


25.0 


39,360 


69,360 


48.6 


39.4 


19.3 


30,500 


58,870 


40.3 


25.0 


40,910 


70,360 


10.3 


11.8 


12.3 


33,360 


73,550 


51.5 


25.5 


40,410 


69,900 


44.6 


32.0 


15.7 


32,520 


60,710 


43.6 


27.0 


40,400 


70,490 


46.0 


35.8 


14.9 


28,000 


58,720 


44.4 


29.5 


40,000 


66,800 


41.8 


23.5 


13.1 


32,290 


62,270 


42.8 


21.3 


40,530 


72,240 


41.2 


47.1 


15.1 


29,970 


58,680 


45.7 


27.0 


40,610 


70,480 



..-sized eye-bars was about 6 
the sample test-pieces. 



Tbe average st 
in., or about 12$ 1 



■ength of the f i 
2ss than that of 



394 STEEL. 

TREATMENT OF STRUCTURAL, STEEL. 

(James Christie, Trans. A. S. C. E., 1893.) 

Effect of Punching and Shearing.— There is no doubt that steel 
of higher tensile strength than is now accepted for structural purposes 
should not be punched or sheared, or that the softer material may contain 
elements prejudicial to its use however treated, but especially if punched. 
But extensive evidence is on record indicating that steel of good quality, in 
bars of moderate thickness and below or not much exceeding 80,000 lbs. 
tensile strength, is not any more, and frequently not as much, injured as 
wrought iron by the process of punching or shearing. 

The physical effects of punching and shearing as denoted by tensile test 
are for iron or steel: 

Reduction of ductility; elevation of tensile strength at elastic limit; reduc- 
tion of ultimate tensile strength. 

In very thin material the superficial disturbance described is less than in 
thick; in fact, a degree of thinness is reached where this disturbance prac- 
tically ceases. On the contrary, as thickness is increased the injury 
becomes more evident. 

The effects described do not invariably ensue; for unknown reasons there 
are sometimes marked deviations from what seems to be a general result. 

By thoroughly annealing sheared or punched steels the ductility is to a 
large extent restored and the exaggerated elastic limit reduced, the change 
being modified by the temperature of reheating and the method of cooling. 

It is probable that the best results combined with least expenditure can 
be obtained by punching all holes where vital strains are not transferred by 
the rivets; and by reaming for important joints where strains on riveted 
joints are vital, or wherever perforation may reduce sections to a minimum. 
The reaming should be sufficient to thoroughly remove the material dis- 
turbed by punching; to accomplish this it is best to enlarge punched holes 
at least % in diameter with the reamer. 

Riveting. — It is the current practice to perforate holes 1/16 in. larger 
than the rivet diameter. For work to be reamed it is also a usual require- 
ment to punch the holes from y% to 3/16 in. less than the finished diameter, 
the holes being reamed to the proper size after the various parts are 
assembled. 

It is also excellent practice to remove the sharp corner at both ends of 
the reamed holes, so that a fillet will be formed at the junction of the body 
and head of the finished rivets. 

The rivets of either iron or mild steel should be heated to a bright red or 
yellow heat and subjected to a pressure of not less than 50 tons per square 
inch of sectional area. 

For rivets of ordinary length this pressure has been found sufficient to 
completely fill the hole. If, however, tl e holes and the rivets are excep- 
tionally long, a greater pressure and a slower movement of the closing tool 
than is used for shorter rivets has been found advantageous in compelling 
the more sluggish flow of the metal throughout the longer hole. 

Welding.— No welding should be allowed on any steel that enters into 
structures. 

Upsetting.— Enlarged ends on tension bars for screw-threads, eyebars, 
etc., are formed by upsetting the material. With proper treatment and a 
sufficient increment of enlarged sectional area over the body of the bar the 
result is entirely satisfactory. The upsetting process should be performed 
so that the properly heated metal is compelled to flow without folding or 
lapping. 

Annealing.— The object of annealing structural steel is for the purpose 
of securing homogeneity of structure that is supposed to be impaired by un- 
equal heating, or by the manipulation necessarily attendant on certain pro- 
cesses. The objects to be annealed should be heated throughout to a 
uniform temperature and uniformly cooled. 

The physical effects of annealing, as indicated by tensile tests, depend on 
the grade of steel, or the amount of hardening elements associated with it; 
also on the temperature to which the steel is raised, and the method or rate 
of cooling the heated material. 

The physical effects of annealing medium-grade steel, as indicated by ten- 
sile test, are reported very differently by different observers, some claiming 
directly opposite results from others. It is evident, when all the attendant 
conditions are considered, that the obtained results must vary both in kind 
and degree. 



TREATMENT OF STRUCTURAL STEEL. 395 

The temperatures employed will vary from 1000° to 1500° F.: possibly even 
a wider range is used. In some cases the heated steel is withdrawn at full 
temperature from the furnace and allowed to cool in the atmosphere: in 
others the mass is removed from the furnace, but covered under a muffle, 
to lessen the free radiation; or. again, the charge is retained in the furnace, 
and the whole mass cooled with the furnace, and more slowly than by either 
of the other methods. 

The best general results from annealing will probably be obtained by in- 
troducing the material into a uniformly-heated oven in which the tempera- 
ture is not so high as to cause a possibility of cracking by sudden and 
unequal changing of temperature, then gradually raising the temperature 
of the material until it is uniformly about 1?00° F., then withdrawing the 
material after the temperature is somewhat reduced and cooling under 
shelter of a muffle, sufficiently to prevent too free and unequal cooling on 
the one hand or excessively slow cooling on the other. 

G. G. Mehrtens, Trans. A. S. C. E. 1S93. says : " A good mild steel can be 
worked as readily as wrought iron in the shop or the field, and even bear 
still harder treatment. It was, however, often thought necessary to require 
preliminary annealing to remove the initial strains due to rolling. The an- 
nealing is undoubtedly of great advantage to all steel above 64,000 lbs. 
strength per square inch, but it is questionable whether it is necessary in 
softer steels. The distortions due to heating cause trouble in subsequent 
straightening, especially of thin plates. It cannot be denied, however, that 
annealing produces greater toughness. 

"In a general way all unannealed mild steel for a strength of 56,000 to 
64.000 lbs. may be worked in the same way as wrought iron. Rough treat- 
ment or working at a blue heat must, however, be prohibited. Such treat- 
ment cannot be borne by wrought iron, although it does not suffer so-much 
as soft steel. Shearing is to be avoided, except to prepare rough plates, 
which should afterwards be smoothed by machine tools or files before using. 
Drifting is also to be avoided, because the edges of holes are thereby 
strained beyond the yield point. Reaming drilled holes is not necessary, 
particularly when sharp drills are used and neat work is done. A slight 
countersinking of the edges of drilled holes is all that is necessary. Work- 
ing the material while heated should be avoided as far as possible, and the 
engineer should bear this in mind when designing structures. Upsetting, 
cranking, and bending ought to be avoided, but when necessaiy the material 
should be annealed after completion: 

" The rivetiug of a mild-steel rivet should be finished as quickly as possible, 
before it cools to the dangerous heat. For this reason machine work is the 
best. There is a special advantage in machine work from the fact that the 
pressure can be retained upon the rivet until it has cooled sufficiently to 
prevent elongation and the rons^quenl loosening of the rivet." 

Punching and Drilling of Steel Plates. (Proc. Inst. M. E„ 
Aug. 1887. p. 3-'6.)— In Prof. Unwin's report the results of the greater num- 
ber of the experiments made on iron and steel plates lead to the general 
conclusion that, while thin plates, even of steel, do not suffer very much 
from punching, yet in those of ^ in. thickness and upwards the loss of te- 
nacity due to punching ranges from 10$ to 23? in iron plates and from 11$ to 
'13% in the case of mild steel. Mr. Parker found the loss of tenacity in steel 
plates to be as high as fully one third of the original strength of the plate. 
In drilled plates, on the contrary, there is no appreciable loss of strength. 
It is even possible to remove the bad effects of punching by subsequent 
reaming or annea'ing. 

"Working Steel at a Blue Heat.— Not only are wrought iron 'and 
steel much more brittle at a blue heat (i.e., the heat that would produce an ox- 
ide coating ranging from light straw to blue on bright steel, 430° to 600° F.), 
but while they are probably not seriously affected by simple exposure tob'ue- 
ness, even if prolonged, yet. if they be worked in this range of temperature 
they remain extremely brittle after cooling, and may indeed be more brittle 
than when at blueness; this last point, however, is not certain. (Howe, 
" Metallurgy of Steel," p. 534.) 

Tests by Prof. Krohn, for the German State Railways, show that working 
at blue heat has a decided influence on all materials tested, the injury done 
being greater on wrought iron and harder steel than on the softer steel. 
The fact that wrought iron is injured by working at a blue heat was reported 
by Stromeyer. (Engineering News. Jan. 9, 1892.) 

A practice among boiler-makers for guarding against failures due to work- 
ing at a blue heat consists in the cessation of work as soon as a plate which 



396 



had been red-hot becomes so cool that the mark produced by rubbing a 
hammer-handle or other piece of wood will not glow. A plate which is not 
hot enough to produce this effect, yet too hot to be touched by the hand, is 
most probablv blue-hot, and should under no circumstances be hammered 
or bent. (0. E. Stromever, Proc. Inst C. E. 1886.) 

Welding of Steel.— A. E. Hunt (A. I. M. E., 1892) says: I have never 
seen so-called " welded " pieces of steel pulled apart in a testing-machine or 
otherwise broken at the joint which have not shown a smooth cleavage- , 
plane, as it were, such as in iron would be condemned as an imperfect 
weld. My experience in this matter leads me to agree with the position 
taken by Mr. William Metcalf in his paper upon Steel in the Trans. A. S. 
C. E., vol. xvi., p. 301. Mr. Metcalf says, " I do not believe steel can be 
welded." 

INFLUENCE OF ANNEALING UPON MAGNETIC 
CAPACITY. 

Prof. D. E. Hughes (Eng'g, Feb. 8. 1884, p. 130) has invented a " Magnetic 
Balance," for testing the condition of iron and steel, which consists chiefly of 
a delicate magnetic needle suspended over a graduated circular index, and 
a magnet coil for magnetizing the bar to be tested. He finds that the fol- 
lowing laws hold with every variety of iron and steel : 

1. The magnetic capacity is directly proportional to the softness, or mo- 
lecular freedom. 

2. The resistance to a feeble external magnetizing force is directly as the 
hardness, or molecular rigidity. 

The magnetic balance shows that annealing not only produces softness in 
iron, .and consequent molecular freedom, but it entirely frees it from all 
strains previously introduced by drawing or hammering. Thus a bar of 
iron drawn or hammered has a peculiar structure, say a fibrous one, which 
gives a greater mechanical strength in one direction than another. This 
bar, if thoroughly annealed at high temperatures, becomes homogeneous in 
all directions, and has no longer even traces of its previous strains, provided 
that there has been no actual mechanical separation into a distinct series of 
fibres. 

Effect of Annealing upon tlie Magnetic Capacity of 
Different Wires; Tests by the Magnetic Balance. 



Description. 


Magnetic Capacity. 


Bright as sent. 


Annealed. 


Best Swedish charcoal iron, first variety. 
" " " " second " 
" third " 

Swedish Siemens-Martin iron 

Puddled iron, best best 

Bessemer steel, soft 

" " hard 

Crucible fine cast steel 


deg. on scale. 
230 
236 
275 
165 
212 
150 
115 
50 


deg. on scale. 
525 
510 
503 
430 
340 
291 
172 
84 



Crucible Fine Steel, Tempered. 



Bright-yellow heat, cooled completely in cold water. 

Yellow-red heat, cooled completely in cold water 

Bright yellow, let down in cold water to straw color. 

" " " " " " blue 

" ' ' cooled completely in oil 

" " let down in water to white 

Reheat, cooled completely in water 

"oil 

Annealed, " " "oil 



Magnetic 
Capacity. 



51 

58 
66 



SPECIFICATIONS FOR STEEL. 397 

SPECIFICATIONS FOR STEEL. 

Structural Steel.— There has been a change during the ten years from 
1880 to 1890, in the opinions of engineers, as to the requirements in specifica- 
tions for structural steel, in the direction of a preference for metal of low 
tensile strength and great ductility. The following specifications of differ- 
ent dates are given by A. E. Hunt and G. H. Clapp, Trans. A. I. M. E. 1890, 
xix, 926: 

Tension Members. 1879. 1881. 1882. 1885. 1887. 1888. 

Elastic limit 50,000 40@,45,000 40,000 40,000 40,000 38.000 

Tensile strength 80,000 70@,80,000 70,000 70,000 67@,?5,000 63@70.000 

Elongation in 8 in 12* 18* 18* 18* 20* 22* 

Reduction area 20* 30* 45* 42* 42* 45* 

Kind of steel O.H. O.H. or B. O.H. Not O.H. or B. O.H.or B. 

Compression Members : 

Elastic limit Same 50@55,000 50,000 50,000 Same as tension 

Tensile strength as 80@90,000 80,000 80,000 members. 

Elongation in 8 in ten- 12* 15* 15* " 

Reduction area sion. 20* 35* 35* " 

F. H. Lewis (Iron Age, Nov. 3, 1892) says: Regarding steel to be used under 
the same conditions as wrought iron, that is, to be punched without ream- 
ing, there seems to be a decided opinion (and a growing one) among engi- 
neers, that it is not safe to use steel in this way, when the ultimate tensile 
strength is above 65,000 lbs. The reason for this is, not so much because 
there is any marked change in the material of this grade, but because all 
steel, especially Bessemer steel, has a tendency to segregations of carbon 
and phosphorus, producing places in the metal which are harder than they 
normally should be. As long as the percentages of carbon and phosphorus 
are kept low, the effect of these segregations is inconsiderable; but when 
these percentages are increased, the existence of these hard spots in the 
metal becomes more marked, and it is therefore less adapted to the treat- 
ment to which wrought iron is subjected. 

There is a wide consensus of opinion that at an ultimate of 64,000 to 65,000 
lbs. the percentages of carbon and phosphorus (which are the two harden- 
ing elements) reach a point where the steel has a tendency to become tender, 
and to crack when subjected to rough treatment. 

A grade of steel, therefore, running in ultimate strength from 54,000 to 
62,000 lbs., or in some cases to 64,000 lbs., is now generally considered a 
proper material for this class of work. 

Millard Hunsicker, engineer of tests of Carnegie, Phipps & Co., writes as 
follows concerning grades of structural steel (Eng'g News, June 2, 1892): 

Grade of Steel.— Steel shall be of three grades— soft, medium, high. 

Soft Steel. — Specimens from finished material for test, cut to size speci- 
fied above, shall have an ultimate strength of from 54,000 to 62,000 lbs. per 
sq. in.; elastic limit one half the ultimate strength; minimum elongation of 
26* in 8 in.; minimum reduction of area at fracture 50*. This grade of 
steel to bend cold 180° flat on itself, without sign of fracture on the outside 
of the bent portion. 

Medium Steel.— Specimens from finished material for test, cut to size 
specified above, shall have an ultimate strength of 60,000 to 68,000 lbs. per 
sq. in.; elastic limit one half the ultimate strength; minimum elongation 20* 
in 8 in.; minimum reduction of area at fracture, 40*. This grade of steel 
to bend cold 180° to a diameter equal to the thickness of the piece tested, 
without crack or flaw on the outside of the bent portion. 

High Steel.— Specimens from finished material for test, cut to size speci- 
fied above, shall have an ultimate strength of 66 000 to 74.000 lbs. per sq. in. ; 
elastic limit one half the ultimate strength; minimum elongation. 18* in 8 
in. ; minimum reduction of area at fracture, 35*. This grade of steel to bend 
cold 180° to a diameter equal to three times the thickness of the test-piece, 
without crack or flaw on the outside of the bent portion. 

F. H. Lewis, Engineers' Club of Phila., 1891, gives specifications for struc- 
tural steel as follows: The phosphorus in acid open-hearth steel must be 
less than 0.10*, aud in all Bessemer or basic steel must be less than 0.08* 

The material will be tested in specimens of at least one half square inch 
section, cut from the finished material. Each melt of steel will be tested 
and each section rolled, and also widely differing gauges of the same section. 



398 STEEL. 

Requirements. Soft Steel. Medium Steel. 

Elastic limit, lbs. per sq. in. , at least 32,000 35,000 

Ultimate strength, lbs. per sq. in 54,000 to 62,000 60,000 to 70,000 

Elongation in 8 in., at least 25$ 20$ 

Reduction of area, per cent, at least 45$ 40$ 

In soft steel for web-plates over 36 in. wide the elongation will be reduced 
to 20$ and the reduction of area to 40$. 

It must bend cold 180 degrees and close down on itself without cracking 
on the outside. 

%-inch holes pitched % inch from a roll-finished or machined edge and 2 
inches between centres must not crack the metal; and %-inch holes pitched 
1% inches between centres and V/% inches from the edge must not split the 
metal between the holes. 

Medium steel must bend 180 degrees on itself around a lj^-inch round bar. 

Full-sized eye-bars, when tested to destruction, must show an ultimate 
strength of at least 56,000 lbs., and stretch at least 10$ in a length of 10 feet. 

A. E. Hunt, in discussing Mr. Lewis's specifications, advises a requirement 
as to the character of the fracture of tensile tests being entirely silky in 
sections of less than 7 square inches, and in larger sections the test specimen 
not to contain over 25$ crystalline or granular fracture. He also advises 
the drifting test as a requirement of both soft and medium steel; the require- 
ment being worded about as follows: " Steel to be capable of having a hole, 
punched for a %" rivet, enlarged by blows of a sledge upon a drift-pin 
until the hole (which in the first case should be punched \y^" from the roll- 
finish or machined edge) is 114" diameter in the case of soft steel, and 1*4'' 
diameter in the case of medium steel, without fracture." This drifting test 
is an excellent requirement, not only as a matter of record, but as a meas- 
ure of the ductility of the steel. 

H. H. Campbell, Trans. A. I. M. E. 1893, says: In adhering to the safest 
course, engineers are continually calling for a metal with lower phosphorus. 
The limit has been 0.10$; it is now 0.08$: soon it will be 0.06$; it should be 
0.04$. 

A. E. Hunt, Trans. A. I. M. E. 1892, says: Why should the tests for steel 
be so much more rigid than for iron destined for the same purpose ? Some 
of the reasons are as follows: Experience shows that the acceptable quali- 
ties of one melt of steel offer no absolute guarantee that the next melt to it, 
even though made of the same stock, will be equally satisfactory. 

Again, good wrought iron, in plates and angles, has a narrow range (from 
25,000 to 27,000 lbs.) in elastic limit per square inch, and a tensile strength of 
from 46,000 to 52,000 lbs. per square inch; whereas for steel the range in 
elastic limit is from 27,000 to 80,000 lbs., and in tensile strength from 4S,000 to 
120,000 lbs. per square inch, with corresponding variations in ductility. 
Moreover, steel is much more susceptible than wrought iron to widely vary- 
ing effects of treatment, by hardening, cold rolling, or overheating. 

It is now almost universally recognized that soft steel, if properly made 
and of good quality, is for many purposes a safe and satisfactory substitute 
for wrought iron, being capable of standing the same shop-treatment as 
wrought iron. But the conviction is equally general, that poor steel, or an 
unsuitable grade of steel, is a very dangerous substitute for wrought iron 
even under the same unit strains. 

For this reason it is advisable to make more rigid requirements in select- 
ing material which may range between the brittleness of glass and a duc- 
tilitv greater than that of wrought iron. 

Specifications for Steel for the World's Fair Buildings, 
Chicago, 1892.— No steel shall contain more than .08$ of phosphorus. 
From three separate ingots of each cast a round sample bar, not less than 
§4 in. in diameter, and having a length not less than twelve diameters be- 
tween jaws of testing machine, shall be furnished and tested by the manu- 
facturer. From these test-pieces alone the quality of the material in the 
steel works shall be determined as follows: 

All the test-bars must have a tensile strength of from 60.000 to 68,000 lbs. per 
square inch, an elastic limit of not less than half the tensile strength of the 
test-bar, an elongation of not less than 24$, and a reduction of area of not 
less than 40$ at the point of fracture. In determining the ductility, the elon- 
gation shall be measured after breaking on an original length of ten times 
th» shortest dimension of the test-piece 

Rivet steel shall have a tensile strength of from 52.000 to 58,000 lbs. per 
square inch, and an elastic limit, elongation, and reduction of area at the 



SPECIFICATIONS FOR STEEL. 



399 



point of fracture as stated above for test-bars, and be capable of bending 
double flat, without sign of fracture ou the convex surface of the bend. 

Boiler, Ship, and Tank Plates. W. F. Mattes (Iron Age, July 
9, 1893.1 recommends that the different qualities of steel plates be classified 
as follows : " __■ 



Tensile test, longitudinal 
coupon 

Elongation in 8-in. longitu- 
dinal coupon, percent 

Bending test, longitudinal 
coupon 

Bending test, transverse 
coupon 

Phosphorus limit 

Sulphur limit 

Surface inspection 



Tank. 



i Limit, 
\ 75,000 



0.15 

Easy. 



Ship. 



S 55,000 
1 to 65,000 

20 

Flat. - 
j Over 1 in, 
( diam. 

0.10 



i Careful. 



Shell. 



| 55,000 
I to 65,000 

Flat 
(Over 3^ in. 
1 diam. 

0.06 

0.065 
Close. 



Fire-box. 



j 55,000 
( to 60,000 



Flat. 
Flat. 

0.045 
0.05 
Rigid. 



A steel-manufacturing firm in Pittsburgh advertises six different grades 
of steel as follows : 
Extra fire-box. Fire-box. Extra flange. Flange. Shell. Tiink. 

The probable average phosphorus content in these grades is, respectively: 
.02 .03 .04 0.6 0.8 .10. 

Different specifications for steel plates are the following (1889) : 

United States Navy.— Shell : Tensile strength, 58,000 to 67,000 lbs. per sq. 
in. ; elongation, 22$ in 8-in. transverse section, 25$ in 8-in. longitudiualseciion. 

Flange : Tensile strength, 50,000 to 58,000 lbs.; elongation. 26$ in 8 inches. 

Chemical requirements : P. not over .035$ ; S. not over .040$. 

Cold-bending test : Specimen to stand being bent flat on itself. 

Quenching test : Steel heated to cherry-red. plunged in water 82° F., and 
to be bent around curve \Vo, times thickness of the plate. 

British Admiralty.— Tensile strength, 58,240 to 67,200 lbs.; elongation in 
8 in., 20$ ; same cold-bending and quenching tests as U. S. Navy. 

American Boiler -makers' 1 Association.— -Tensile strength, 55,000 to 65,000 
lbs.; elongation in 8 in., 20$ for plates % in. thick and under ; 22$ for plates 
% in. to % in. ; 25$ for plates % in. and over. 

Cold-bending test : For plates ^ in. thick and under, specimen must bend 
back on itself without fracture ; for plates over J^ in. thick, specimen must 
withstand bending 180° around a mandril, 1% times the thickness of the 
plate. 

Chemical requirements : P. not over .040$ ; S. not over .030$. 

American Shipmasters'' Association. — Tensile strength, 62,000 to 72,000 
lbs.; elongation, 16$ on pieces 9 in. long. 

Strips cut from plates, heated to a low red and cooled in water the tem- 
perature of which is 82° F., to undergo without crack or fracture being 
doubled over a curve the diameter of which does not exceed three times 
the thickness of the piece tested. 

Boiler Shell-plates, Front Tube-plate, and Butt-strips. 
(Penna. R. R., 1892.)— The metal desired is a homogeneous steel having a 
tensile strength of 60,000 lbs. per sq. in., and an elongation of 25$ in a 
section originally 8 in. long. These plates will not be accepted if the test- 
piece shows — 

1. A tensile strength of less than 55,000 lbs. per sq. in. ; 2. An elongation 
in section originally 8 in. long less than 20$ ; 3. A tensile strength over 
65.000 lbs. per sq. in. ; should, however, the elongatio» be 27$ or over, plates 
will not be rejected for high strength. 

Inside Fire-box Plates, including Back Tube-plate. 
(Penna. R. R. t 1892.)— The metal should show a tensile strength of 60,000 lbs. 
per sq. in., and an elongation of 28$ in a test section originally 8 in. long. 
Chemical Composition. Desired. Will be Rejected. 

Carbon 0.18 per cent. over 0.25, below 0.15 

Phosphorus, not above 0.03 " over 0.04 

Manganese, not above 0.40 " over 0.55 

Silicon, not above 0.02 " over 0.04 

Sulphur, not above 0.02 " over 05 

Copper, not above 0.03 " over 0.05 



400 STEEL. * 

These plates will not be accepted if the test-piece shows: 1. A tensile 
strength of less than 55,000 lbs. per sq. in. ; 2. An elongation in section 
originally 8 in. long, less than 2:2$ (20$ in plates 34 in ch thick) ; 3. A tensile 
strength over 65,000 lbs. per sq. in. (08,000 for plates 34 in. thick); should, 
however, the elongation be 30% or over, plates will not be rejected for high 
strength ; 4. Any single seam or cavity more than 34 in- long in either of the 
three fractures obtained on test for homogeneity, as described below. 

Homogeneity test : A portion of the test-piece is nicked with a chisel, or 
grooved on a machine, transversely about a sixteenth of an inch deep, in 
three places about 134 m « apart. The first groove should be made on one 
side, 134 m - from the square end of the piece; the second, 134 in. from 
it on the opposite side; and the third, 1J4 in. from the last, and on the 
opposite side from it. The test-piece is then put in a vise,' with the first 
groove about 34 ha, above the jaw, care being taken to hold it firmly. 
The projecting end of the test-piece is then broken off by means of a ham- 
mer, a number of light blows being used, and the bending being away 
from the groove. The piece is broken at the other two grooves in the same 
way. The object of this treatment is to open and render visible to the eye 
any seams due to failure to weld up, or to foreign interposed matter, or 
cavities due to gas bubbles in the ingot. After rupture, one side of each 
fracture is examined, a pocket lens being used if necessary, and the length 
of the seams and cavities is determined. The length of the longest seam or 
cavity determines the acceptance or rejection of the plate. 

Dr. C. B. Dudley, chemist of the Peuna. R. R. (Trans. A. I. M. E. 1892, vol. 
xx. p. 709), gives "as an example of the progressive improvement in specifi- 
cations the following : In the early days of steel boilers the specification in 
force called for steel of not less than 50,000 lbs. tensile strength and not less 
than 25$ elongation. Some metal was received having 75,000 lbs. tensile 
strength, and as the elongation was all right it was accepted ; but when those 
plates were being flanged in the boiler-shop they cracked and went to 
pieces. As a result, an upper limit of 65,000 lbs. tensile strength was 
established. 

Am. Ry. Master Mechanics'' Assn., 1894. — Same as Penna. R. R. Specifica- 
tions of 1892, including homogeneity test. 

Plate, Tank, and Sheet Steel. (Penna. R. R., 1888.*)— A test strip 
taken lengthwise of each plate, % in. thick and over, without annealing, 
should have a tensile strength of 60,000 lbs. per sq. in., and an elongation of 
25$ in a section originally 2 in. long. 

Sheets will not be accepted if the tests show the tensile strength less than 
55,000 lbs. or greater than 70,000 lbs. per sq. in., nor if the elongation falls 
below 20%. 

Steel Billets for Main and Parallel Rods. (Penna. R. R., 1884.) 
—One billet from each lot of 25 billets or smaller shipment of steel for main 
or parallel rods for locomotives will have a piece drawn from it under the 
hammer and a test-section will be turned down on this piece to % in. in 
diameter and 2 in. long. Such test-piece should show a tensile strength of 
85,000 lbs. and an elongation of 15$. 

No lot will be acceptable if the test shows less than 80,000 lbs. tensile 
strength or 12% e'ongatinn in 2 in. 

^Locomotive Spring Steel, (Penna. R. R., 1887.)— Bars which vary 
more than 0.01 in. in thickness, or more than 0.02 in. in width, from the size 
ordered, or which break where they are not nicked, or which, when properly 
nicked and held, fail to break square across where they are nicked, will be 
returned. The metal desired has the following composition: Carbon, 1.00$; 
manganese, 0.25$; phosphorus, not over 0.03$; silicon, not over 0.15$; sul- 
phur, not over 0.03$; copper, not over 0.03$. 

Shipments will not be accepted which show on analysis less than 0.90$ or 
over 1.10$ of carbon, or over 0.50$ of manganese, 0.05$ of phosphorus, 0.25$ 
of silicon, 0.05$ of sulphur, and 0.05$ of copper. 

Steel tor Locomotive Driving-axles. (Penna. R. R., 1883.)— 
Steel for driving-axles should have a tensile strength of 85,000 lbs. per sq. in. 
and an elongation of 15$ in section originally 2 in. long and % in. diameter, 
taken midway between centre and circumference of the axle. 

Axles will not be accepted if tensile strength is less than 80,000 lbs., nor if 
elongation is bplow 12$. 

Steel for Crank-pins. (Penna. R. R., 1886.)— Steel ingots for crank- 

* The Penna. R. R. specifications of the several dates given are still in force., 
July, 1894. 



SPECIFICATIONS FOR STEEL. 401 

pins must be swaged as per drawings. For each lot of 50 ingots ordered, 51 
must be furnished, from which one will be taken at random, and two pieces, 
with test sections % in. diameter and 2 in. long, will be cut from any part of 
it, provided that centre line of test-pieces falls \y% in. from centre line of in- 
got. Such test-pieces should have a tensile strength of 85,000 lbs. per sq. in. 
and an elongation of 15$. Ingots will not be accepted if the tensile strength 
is less than 80,000 lbs. nor if the elongation is below 12$. 

Dr. Chas. B. Dudley, Chemist of the P. R. R. (Trans. A. I. M. E. 1892), re- 
ferring to this specification, says : In testing a recent shipment, the piece 
from one side of the pin showed 88,000 lbs. strength and 22$ elongation, and 
the piece from the opposite side showed 106,000 lbs. strength and 14$ elonga- 
tion. Each piece was above the specified strength and ductility, but the 
lack of uniformity between the two sides of the pin was so marked that it 
was finally determined not to put the lot of 50 pins in use. To guard against 
trouble of this sort in future, the specifications are to be amended to require 
that the difference in ultimate strength of the two specimens shall not be 
more than 3000 lbs. 

Steel Car-axles. (Penna. R. R., 1891.)— For each 100 axles ordered 101 
must be furnished, from which one will be taken at random, and subjected 
to tests prescribed. 

Axles for passenger cars and passenger locomotive and tender trucks 
must be made of steel and be rough turned throughout. Two test-pieces 
will be cut from an axle, and the test sections of % in. diameter by 2 in. long 
may fall at any part of the axle provided that the centre line of the test- 
section is 1 in. from the centre line of ihe axle. Such test-pieces should have 
a tensile strength of 80.000 lbs. per sq. in. and an elongation of 20$. Axles 
will not be accepted if the tensile strength is less than 75,000 lbs. or the 
elongation below 15$, nor if the fractures are irregular. 

Axles for freight cars and freight-locomotive tender trucks must be made 
of steel, and will be subjected to the following test, wdiich they must stand 
without fracture : 

Axles 4 in. diameter at centre — Five blows at 20 ft. of a 1640-lb. weight, 
striking midway between supports 3 ft. apart; axle to be turned over after 
each blow. 

Axles 4% in. diameter at centre— Five blows at 25 ft. of a 1640-lb. weight, 
striking midway between supports 3 ft. apart: axles to be turned over after 
each blow. 

Steel for Rails.— P. H. Dudley (Trans. A. S. C. E. 1893) recommends 
the following chemical composition for rails of the weights specified : 

Weights per yard 60, 65, and 70 lbs. 75 and 80 lbs. 100 lbs. 

Carbon 45 to .55$ .50 to .60$ .65 to .75$ 

For all weights: Manganese, .80$ to 1.00$; silicon, .10$ to .15$; phos- 
phorus, not over .06$; sulphur, not over .07$. 

Carbon by itself up to or over 1$ increases the hardness andtensile strength 
of the iron rapidly, and at the same time decreases the elongation. The 
amount of carbon in the early rails ranged from 0.25 to 0.5 of 1$. while in 
recent rails and very heavy sections it has been increased to 0.5, 0.6, and 0.75 
of 1$. With good irons and suitable sections it can run from 0.55 to 0.75 of 
1$. according to the section, and obtain fine-grain tough rails with low 
phosphorus. 

Manganese is a necessary ingredient in the first place to take up the oxide 
of iron formed in the bath of molten metal during the blow. It also is of great 
assistance to check red shortness of the ingots during the first passes in 
the blooming train. In the early rails 0.4 to 0.5 of 1$ was sufficient when 
the ingots were hammered or the reductions in the passes in the trains were 
very much lighter than to day. With the more rapid rolling of recent years 
the manganese is very often increased to 1.25$ to 1.5$. It makes the rails 
hard with a coarse crystallization and with a decided tendency to brittleness. 
Rails high in manganese seem to flow quite easily, especiallj- under severe 
service or the use of sand, and oxidize rapidly in tunnels. From 0.80 to 1.00$ 
seems to be all that is necessary for good rolling at, the present time. 

Steel Rivets. (H. C. Torrance, Amer. Boiler Mfrs. Assn., 1890.)— The 
Government requirements for the rivets used in boilers of the cruisers built 
in 1890 are: For longitudinal seams, 58,000 to 67,000 lbs. tensile strength; 
elongation, not less than 26$ in 8 in., and all others a tensile strength of 
50,000 to 58,000 lbs., with an elongation of not less than 30$. They shall be 
capable of being flattened out cold under the hammer to a thickness of one 
half the diameter, and of being flattened oup hot to a thickness of one third 



402 STEEL. 



the diameter without showing cracks or flaws. The steel must not contain 
more than .035 of l^of phosphorus, nor more than .04 of 1% of sulphur. 

A lot of 20 succesive tests of rivet steel of the low tensile strength quality 
and 12 tests of the higher tensile strength gave the following results: 

Low Steel. Higher. 

Tensile strength, lbs. per sq. in.. . 51,230 to 54,100 59,100 to 61,850 

Elastic limit, lbs. per sq. in 31,050 to 33,190 32,080 to 33,070 

Elongation in 8 in., per cent 30.5 to 35 25 28.5 to 31.75 

Carbon, per cent 11 to .14 .16 to .18 

Phosphorus 027 to .029 .03 

Sulphur 033 to .035 .033 to .035 

The safest steel rivets are those of the lowest tensile strength, since they 
are the least liable to become hardened and fracture by hammering, or to 
break from repeated coucussive and vibratory strains to which they are 
subjected in practice. For calculations of the strength of riveted joints the 
tensile strength may be taken as the average of the figures above given, or 
52,665 lbs., and the shearing strength at 45,000 lbs. per sq. in. 

MISCELLANEOUS NOTES ON STEEL,. 

May Carbon be Burned. Out of Steel ?— Experiments made at 
the Laboratory of the Penna. Railroad Co. (Specifications for Springs, 1888) 
with the steel of spiral springs, show that the place from which the borings 
are taken for analysis has a very important influence on the amount of car- 
bon found. If the sample is a piece of the round bar, and the borings are 
taken from the end of this piece, the carbon is always higher than if the 
borings are taken from the side of the piece. It is common to find a differ- 
ence of 0.10$ between the centre and side of the bar-, and in some cases the 
difference is as high as 0.23$. Furthermore, experiments made with samples 
taken from the drawn out end of the bar show, usually, less carbon than 
samples taken from the round part of the bar, even though the borings may 
be taken out of the side in both cases. 

Apparently during the process of reducing the metal from the ingots to the 
round bar, with successive heatings, the carbon in the outside of the bar is 
burned out. 

" Recalescence " of Steel.— If we heat a bar of copper by a flame 
of constant strength, and note carefully the interval of lime occupied in 
passing from each degree to the next higher degree, we find that these in- 
tervals increase regularly, i.e., that the bar heats more and more slowly, as 
its temperature approaches that of the flame. If we substitute a bar of 
steel for one of copper, we find that these intervals increase regularly up to 
a certain point, when the rise of temperature is suddenly and in most cases 
greatly retarded or even completely arrested. After this the regular rise of 
temperature is resumed, though other like retardations ma3' recur as the 
temperature rises farther. So if we cool a bar of steel slowly the fall of 
temperature is greatly retarded when it reaches a certain point in dull red- 
ness. If the steel contains much carbon, and if certain favoring conditions 
be maintained, the temperature, after descending regularly, suddenly rises 
spontaneously very abruptly, remains stationary a while, and then rede- 
sceuds, This spontaneous reheating is known as " recalescence. 1 ' 

These retardations indicate that some change which absorbs or evolves 
heat occurs within the metal. A retardation while the temperature is rising 
points to a change which absorbs heat; a retardation during cooling points 
to some change which evolves heat. (Henry M. Howe, on " Heat Treatment 
of Steel," Trans. A. I. M. E., vol. xxii.) 

Effect of Nicking a Steel Bar.— The statement is sometimes made 
that, owing to the homogeneity of steel, a bar with a surface crack or nick 
in one of its edges is liable to fail by the gradual spreading of the nick, and 
thus break under a very much smaller load than a sound bar. With iron it 
is contended this does not occur, as this metal has a fibrous structure. Sir 
Benjamin Baker has, however, shown that this theory, at least so far as 
statical stress is concerned, is opposed to the facts, as he purposely made 
nicks in specimens of the mild steel used at the Forth Bridge, but found 
that the tensile strength of the whole was thus reduced by only about one 
torr per square inch of section. In an experiment by the Union Bridge Com- 
pany a full-sized steel counter-bar, with a screw-turned buckle connection, 
was tested under a heavy statical stress, and at the same time a weight 
weighing 1040 lbs. was allowed to drop on it from various heights. The bar 
was first broken by ordinary statical strain, and showed a breaking stress of 



MISCELLANEOUS NOTES ON" STEEL. 



403 



65,800 lbs. per square inch. The longer of the broken parts was then placed 
in the machine and put under the following loads, whilst a weight, as already- 
mentioned, was dropped on it- from various heights at a distance of five 
feet from the sleeve-nut of the turn-buckle, as shown below: 

Stress in pounds per sq. in 50,000 55,000 60,000 63,000 65,000 

ft. in. ft. in. ft. in. ft. in. ft. in. 

Height of fall 21 26 30 40 50 

The weight was then shifted so as to fall dirctly on the sleeve-nut, and the 
test proceeded as follows: 

Stress on specimen in lbs. per square inch 65,350 65,350 68,800 

ft. ft. ft. 

Height of fall 3 6 6 

It will be seen that under this trial the bar carried more than when origi- 
nally tested statically, showing that the nicking of the bar by screwing had 
not appreciably weakened its power of resisting shocks. — Eng , g Neics. 

Electric Conductivity of Steel.— Louis Campredon reports in Le 
Genie Civil the results of a series of experiments made to ascertain the rela- 
tions between electric resistance and chemical compositions of steel. The 
wires were No. 17, 3 mm. diameter. The results are given in the table below: 





Car- 




bon. 


1 


0.090 


2 


0.100 


3 


0.100 


4 


0.100 


5 


0.120 


6 


0.110 


7 


0.100 


8 


0.120 


9 


0.110 


10 


0.140 



0.020 
0.020 
0.020 
0.020 
0.030 
0.030 
0.020 
0.020 
0.030 
0.930 



Sulphur. 



0.050 
0.050 
0.060 
0.050 
0.070 
0.060 
0.070 
0.070 
0.060 
0.060 



Phos- 
phorus. 



0.030 
0.040 
0.040 
0.050 
0.050 
0.060 
0.040 
0.070 
0.060 
0.080 



Manga- 



0.210 
0.240 
0.260 
0.310 
0.330 
0.350 
0.400 
0.400 
0.490 
0.540 



0.410 
0.450 
0.480 
0.530 
0.600 
0.610 
0.630 
0.680 
0.750 
0.850 



Electric 
Resist - 



127.7 
133.0 
137.5 
140.3 
142.7 
144.5 
149.0 
150.3 
156.0 
173.0 



An examination of these series of figures shows that the purer and softer 
steel the better is its electric conductivity, and, furthermore, that manga- 
nese is the element which most influences the conductivity. 

Specific Gravity of Soft Steel. (W. Kent, Trans. A. I. M. E., xiv. 
585. »— Five specimens of boiler-plate of C. 0.14, P. 0.03 gave an average sp. 
gr. of 7.932, maximum variation 0.008. The pieces were first planed to re- 
move all possible scale indentations, then filed smooth, then cleaued in 
dilute sulphuric acid, and then boiled in distilled water, to remove all traces 
of air from the surface. 

The figures of specific gravity thus obtained by careful experiment on 
bright, smooth pieces of steel are, however, too high for use in determining 
the weights of rolled plates for commercial purposes. The actual average 
thickness of these plates is always a little less than is shown by the calipers, 
on account of the oxide of iron on the surface, and because the surface is 
not perfectly smooth and regular. A number of experiments on commercial 
plates, and comparison of other authorities, led to the figure 7.854 as the 
average specific gravity of open-hearth boiler-plate steel. This figure ;s 
easily remembered as being the same figure with change of position of the 
decimal point (.7854) which expresses the relation of the area of a circle to 
that of its circumscribed square. Taking the weight of a cubic foot of water 
at 62° F. as 62.36 lbs. (average of several authorities), this figure gives 489.775 
lbs. as the weight of a cubic foot of steel, or the even figure, 490 lbs., may be 
taken as a convenient figure, and accurate within the limits of the error of 
observation. 

A common method of approximating the weight of iron plates is to con- 
sider them to weigh 40 lbs. per square foot one iuch thick. Taking this 
weight and adding 2% gives almost exactly the weight of steel boiler-plate 
given above (40 X 12 X 1.02 = 489.6 lbs. per cubic foot). 

Occasional Failnres of Bessemer Steel.— G. H. Clapp and A. 
E. Hunt, in their paper on "The Inspection of Materials of Construction in 



404 STEEL. 

the United States " (Trans. A. I. M. E., vol. xix), say: Numerous instances 
could be cited to show the unreliability of Bessemer steel for structural pur- 
poses. One of the most marked, however, was the following: A 12-in. I-beam 
weighing 30 lbs. to the foot, 20 feet long, on being unloaded from a car 
broke in two about 6 feet from one end. 

The analyses and tensile tests made do not show any cause for the failure. 

The cold and quench bending tests of both the original %-in. round test- 
pieces, and of pieces cut from the finished material, gave satisfactory re- 
sults; the cold-bending tests closing down on themselves without sign of 
fracture. 

Numerous other cases of angles and plates that were so hard in places as 
to break off short in punching, or, what was worse, to break the punches, 
have come under our observation, and although makers of Bessemer steel 
claim that this is just as likely to occur in open-hearth as in Bessemer steel, 
we have as yet never seen an instance of failure of this kind in open-hearth 
steel having a composition such as C 0.25$, Mn 0.70$, P 0.80$. 

J. W. Wailes, in a paper read before the Chemical Section of the British 
Association for the Advancement of Science, in speaking of mysterious 
failures of steel, states that investigation shows that " these failures occur 
in steel of one class, viz., soft steel made by the Bessemer process.'" 

Segregation in Steel Ingots. (A. Pourcel, Trans. A. I. M. E. 1893.) 
— H. M. Howe, in his " Metallurgy of Steel,'' 1 gives a resume of observations, 
with the results of numerous analyses, bearing upon the phenomena of seg- 
regation. 

In 1881 Mr. Stubbs. of Manchester, showed the heterogeneous results of 
analyses made upon different parts of an ingot of large section. 

A test-piece taken 24 inches from the head of the ingot 7.5 feet in length 
gave by analysis very different results from those of a test-piece taken 30 
inches from the bottom. 

C. Mn. Si. S. P. 

Top 0.92 0.535 0.043 0.161 0.261 

Bottom 37 0.498 . 0.006 0.025 0.096 

Windsor Richards says he had often observed in test-pieces taken from 
different points of one plate variations of 0.05$ of carbon. Segregation is 
specially pronounced in an ingot in its central portion, and around the 
space of the piping. 

It is most observable in large ingots, but in blocks of smaller weight and 
limited dimensions, subjected to the influence of solidification as rapid as 
casting within thick walls will permit, it may still be observed distinctly. 
An ingot of Martin steel, weighing about 1000 lbs., and having a height of 
1.10 feet and a section of 10.24 inches square, gave the following: 

1. Upper section: C. S. P. Mn. 

Border 0.330 0.040 0.033 0.420 

Centre 0.530 0.077 0.057 0.430 

2. Lower section: C. S. P. Mn. 

Border 0.280 0.029 0.016 0.390 

Centre 0.290 0.030 0.038 0.390 

3. Middle section: C. S. P. Mn. 

Border „ 0.320 0.025 0.025 0.100 

Centre 0.320 0.048 0.048 0.4CH* 

Segregation is less marked in ingots of extra-soft metal cast in cast-iron 
moulds of considerable thickness. It is, however, still important, and ex- 
plains the difference often shown by the results of tests on pieces taken 
from different portions of a plate. Two samples, taken from the sound part 
of a flat ingot, one on the outside and the other in the centre, 7.9 inches from 
the upper edge, gave: 

• C. S. P. Mn. 

Centre 0.14 0.053 0.072 0.576 

Exterior 0.11 0.036 0.027 0.610 

Manganese is the element most uniformly disseminated in hard or soft 
steel. 

For cannon of large calibre, if we reject, in addition to the part cast in 
sand and called the masselotte (sinking-head), one third of the upper part 
of the ingot, we can obtain a tube practically homogeneous in composition, 
because the central part is naturally removed by the boring of the tube. 
With extra-soft steels, destined for ship- or boiler-plates, the solution for 
practically perfect homogeneity lies in the obtaining of a metal more closely 
deserving its name of extra-soft metal. 



STEEL CASTINGS. 405 

The injurious consequences of segregation must be suppressed by reduc- 
ing, as far as possible, the elements subject to liquation. 
Earliest Uses of Steel for Structural Purposes. (G. G. 

Mehrtens, Trans. A. S. C. E. 1893).— The Pennsylvania Railroad Company 
first introduced Bessemer steel in America in locomotive boilers in the year 
1863, but the steel was too hard and brittle for such use. The first plates 
made for steel boilers had a tenacity of 85,000 to 92,000 lbs. and an elongation 
of but 7$ to 10$. The results were not favorable, and the steel works were 
soon forced to offer a material of less tenacity and more ductility. The re- 
quirements were therefore reduced to a tenacity of 78,000 lbs. or less, and 
the elongation was increased to 15% or more. Even with this, between the 
years 1870 and 1S80, many explosions occurred and many careful examina- 
tions were made to determine their cause. It was found on examining the 
rivet-holes that there were incipient changes in the metal, many cracks 
around them, and points near them were corroded with rust, all caused by 
the shock, of tools in manufacturing. It was evident that the material 
was unsuitable, and that the treatment must be changed. In the beginning 
of 1878, Mr. Parker, chief engineer of the Lloyds, stated that there was then 
but one English steamer in possession of a steel boiler; a year later there 
were 120. In 1878 there were but five large English steamers built of steel, 
while in 1883 there were 116 building. The use of Bessemer steel in bridge- 
building was tried first on the Dutch State railways in 1863-61, then in Eng- 
land and Austria. In 1874 a bridge was built of Bessemer steel in Austria. 
The first use of cast steel for bridges was in America, for the St. Louis Arch 
Bridge and for the wire of the East River Bridge. These gave an impetus 
to the use of ingot metal, and before 1880 the Glasgow and Plattsmouth 
Bridges over the Missouri River were also built of ingot metal. Steel eye- 
bars were applied for the first time in the Glasgow Bridge. Since 1880 the 
introduction of mild steel in all kinds of engineering structures has steadily 
increased. 

STEEL CASTINGS. 

(E. S. Cramp, Engineering Congress, Dept. of Marine Eng'g, Chicago, 1893.) 

In 1891 American steel-founders had successfully produced a considerable 
variety of heavy and difficult castings, of which the following are the most 
noteworthy specimens: 

Bed-plates up to- 24,000 lbs.; stern-posts up to 54,000 lbs.; stems up to 
21,000 lbs. ; hydraulic cylinders up to 11,000 lbs. ; shaft-struts up to 32,000 lbs. ; 
hawse-pipes up to 7500 lbs. ; stern-pipes up to 8000 lbs. 

' The percentage of success in these classes of castings since 1890 has ranged 
from 65$ in the more difficult forms to 90$ in the simpler ones; the tensile 
strength has been from 62.000 to 78,000 lbs., elongation from 15$ to 25$. The 
best performance recorded is that of a guide, cast in January, 1893, which 
developed 84,000 lbs. tensile strength and 15.6$ elongation. 

The first steel castings of which anything is generally known were 
crossing-frogs made for the Philadelphia & Reading R. R. in July, 1867, by 
the William Butcher Steel Works, now the Midvale St<-el Co. The moulds 
were made of a mixture of ground fire-brick, black-lead crucible-pots 
ground fine, and fire-clay, and washed with a black-lead wash. The steel 
was melted in crucibles, and was about as hard as tool steel. The surface 
of these castings was very smooth, but the interior was very much honey- 
combed. This was before the days when the use of silicon was known for 
solidifying steel. The sponginess, which was almost universal, was a great 
obstacle to their general adoption.. 

The next step was to leave the ground pots out of the moulding mixture 
and to wash the mould with finely ground fire-brick. This was a great im- 
provement, especially in very heavy castings; but this mixture still clung so 
strongly to the casting that only comparatively simple shapes could be made 
with certainty. A mould made of such a mixture became almost as hard as 
fire-brick, and was such an obstacle to the proper shrinkage of castings, 
that, when at all complicated in shape, they had so great a tendency to 
crack as to make their successful manufacture almost impossible. By this 
time the use of silicon had been discovered, and the only obstacle in the way 
of making good castings was a suitable moulding mixture. This was ulti- 
mately found in mixtures having the various kinds of silica sand as the 
principal constituent. 

One of the most fertile sources of defects in castings is a bad design. 
Very intricate shapes can be cast successfully if they are so designed as to 



406 



STEEL. 



cool uniformly. Mr. Cramp says while he is not yet prepared to state that 
anything that can be cast successfully in iron can be cast in steel, indica- 
tions seem to point that way in all cases where it is possible to put on suit- 
able sinking-heads for feeding the casting. 

H. L. Gantt (Trans. A. S. M. E., xii. 710) says: Steel castings not only 
shrink much more than iron ones, but with less regularity. The amount of 
shrinkage varies with the composition and the heat of the metal; the hotter 
the metal the greater the shrinkage; and, as we get smoother castings from 
hot metal, it is better to make allowance for large shrinkage and pour the 
metal as hot as possible. Allow 3/10 or 14 in. per ft. in length 
for shrinkage, and J4 in. for finish on machined surfaces, except such as are 
cast '■up." Cope surfaces which are to be machined should, in large or 
hard castings, have an allowance of from % to % in. for finish, as a large 
mass of metal slowly rising in a mould is apt to become crusty on the sur- 
face, and such a crust is sure to be full of imperfections. On small, soft 
castings Y% in. on drag side and x A'vc\. on cope side will be sufficient. No core 
should have less than 34 in- fiuish on a side and very large ones should have 
as much as ^ in. on a side. Blow-holes can be entirely prevented in cast- 
ings by the addition of manganese and silicon in sufficient quantities; but 
both of these cause brittleness, and it is the object of the conscientious steel- 
maker to put no more manganese and silicon in his steel than is just suffi- 
cient to make it solid. The best results are arrived at when all portions of 
the castings are of a uniform thickness, or very nearly so. 

The following table will illustrate the effect of annealing on tensile 
strength and elongation of steel castings : 



Carbon. 


Unannealed. 


Annealed. 




Tensile Strength. 


Elongation. 


Tensile Strength. 


Elongation. 


.23$ 
.37 
.53 


68.738 
85,540 
90,121 


22. 40$ 
8.20 
2.35 


67.210 

82,228 
106.415 


31.40$ 
21.80 
9.80 



The proper annealing of large castings takes nearly a week. 

The proper steel for roll pinion*, hammer dies, etc., seems to be that con- 
taining about .60$ of carbon. Such castings, properly annealed, have woi;n 
well and seldom broken. Miscellaneous gearing should contain carbon .40$ 
to 60$, gears larger in diameter being softest. General machinery castings 
should, as a rule, contain less than .40$ of carbon, those exposed to great 
shocks containing as low at .20$ of carbon. Such castings will give a tensile 
strength of from 60,000 to 80,000 lbs. per sq. in. and at least 15$ extension in 
a 2 in. long specimen. Machinery and hull castings for war-vessels for the 
United States Navy, as well as carriages for naval guns, contain from .20$ to 
30$ of carbon. 

The following is a partial list of castings in which steel seems to be 
rapidly taking the place of iron: Hydraulic cylinders, crossheadsand pistons 
for large engines, roughing rolls, rolling-mill spindles, coupling-boxes, roll 
pinions, gearing, hammer-heads and dies, riveter stakes, castings for ships, 
car-couplers, etc. 

For description of methods of manufacture of steel castings by the Besse- 
mer, open-hearth, and crucible processes, see paper by P. G. Salom, Trans. 
A. I. M. E. xiv, 118. 

Specifications for steel castings issued by the U. S. Navy Department, 1889 
(abridged) : Steel for castings must be made by either the open-hearth or 
the crucible process, and must not show more than .06$ of phosphorus. All 
castings must be annealed, unless otherwise directed. The tensile strength 
of steel casiings shall be at least 60,000 lbs., with an elongation of at least 
15$ in 8 in. for all castings for moving parts of the machinery, and at least 
10$ in 8 in. for other castings. Bars 1 in. sq. shall be capable of bending 
cold, without fracture, through an angle of 90°, over a radius not greater 
than \% in. All castings must be sound, free from injurious roughness, 
sponginess, pitting, shrinkage, or other cracks, cavities, etc. 

Pennsylvania Railroad specifications, 1888: Steel castings should have a 
tensile strength of 70,000 lbs. per sq. in. and an elongation of 15$ in section 
originally 2 in. long. Steel castings will not be accepted if tensile strength 



MANGANESE, NICKEL, AND OTHER u ALLOY* 3 STEELS. 407 

falls below 60,000 lbs., nor if the elongation is less than 12$, nor if cast- 
ings have blow-holes and shrinkage cracks. Castings weighing 80 lbs. or 
more must have cast with them a strip to be used as a test-piece. The di- 
mensions of this strip must be % in. sq. by 12 in. long. 

MANGANESE, NICKEL, AND OTHER "ALLOY" 

STEELS. 

manganese Steel. (H. M. Howe, Trans. A. S. M. E., vol. xii.)— Man- 
ganese steel is an alloy of iron and manganese, incidentally, and probably 
unavoidably, containing a considerable proportion of carbon. 

The effect of small proportions of manganese on the hardness, strength, 
and ductility of iron is probably slight. The point at which manganese 
begins to have a predominant effect is not known : it may be somewhere 
about 2.5$. As the proportion of manganese rises above 2.5$ the strength 
and ductility diminish, while the hardness increases. This effect reaches a 
maximum with somewhere about 6$ of manganese. When the proportion 
of this element rises beyond 6$ the strength and ductility both increase, 
while the hardness diminishes slightly, (he maximum of both strength and 
ductility being reached with about 14$ of manganese. With this proportion 
the metal is still so hard that it is very difficult to cut it with steel tools. As 
the proportion of manganese rises above 15$ the ductility falls off abruptly, 
the strength remaining nearly constant till the manganese passes 18$, when 
it in turn diminishes suddenly. 

Steel containing from 4$ to 6.5$ of manganese, even if it have but 0.37$ of 
carbon, is reported to be so extremely brittle that it can be powdered under 
a hand-hammer when cold ; yet it is ductile when hot. 

Manganese steel is very free from blow-holes ; it welds with great diffi- 
culty; its toughness is increased by quenching from a yellow heat ; its elec- 
tric resistance is enormous, and very constant with changing temperature ; 
it is low in thermal conductivity. Its remarkable combination of great hard- 
ness, which cannot be materially lessened by annealing, and great tensile 
strength, with astonishing toughness and ductility, at once creates and 
limits its usefulness. The fact that manganese steel cannot be softened, 
that it ever remains so hard that it can be machined only with great diffi- 
culty, sets up a barrier to its usefulness. 

The following comparative results of abrasion tests of manganese and 
other steel were reported by T. T. Morrell : 

Abrasion by Pressure Against a Revolving Hardened-Steel Shaft. 

Loss of weight of manganese steel 1.0 

" blue-tempered hard tool steel 0.4 

" annealed hard tool steel 7.5 

" hardened Otis boiler-plate steel 7.0 

" annealed " " " 14.0 

Abrasion by an Emery-Wheel. 

Loss of weight of hard manganese-steel wheels 1.00 

softer " " 1.19 

" hardest carbon-steel Avheels 1.23 

" soft " " 2.85 

The hardness of manganese steel seems to be of an anomalous kind. The 
alloy is hard, but under some conditions not rigid. It is very hard in its 
resistance to abrasion ; it is not ahvays hard in its resistance to impact. 

Manganese steel forges readily at a yellow heat, though at a bright white 
heat it crumbles under the hammer. But it offers greater resistance to 
deformation, i.e., it is harder when hot, than carbon steel. 

The most important single use for manganese-steel is for the pins which 
hold the buckets of elevated dredgers. Here abrasion chiefly is to be 
resisted. 
Another important use is for the links of common chain-elevators. 
As a material for stamp-shoes, for horse-shoes, for the knuckles of an 
automatic car-coupler, manganese steel has not met expectations. 

Manganese steel has been regularly adopted for the blades of the Cyclone 
pulverizer. Some manganese-steel wheels are reported to have run over 
300.000 miles each without turning, on a New England railroad. 

Nickel Steel.— The remarkable tensile strength and ductility of nickel 
steel, as shown by the test-bars and the behavior of nickel-steel armor- 
plate under shot tests, are witness of the valuable qualities conferred upon 
steel by the addition of a few per cent of nickel. 



408 



STEEL. 



The following tests were made on nickel steels by Mr. Maunsel White of 
the Bethlehem Iron Company (Eng. <& M. Jour., Sept. 16, 1893.) : 







„ 


& 


Tensile 


Elastic 










Specimen 


B ■ 


"&rt 


Str'gth, 


Limit, 


p. c. 


p. c. 






from — 


5.2 
5 


B.S 
03 


lbs. per 
sq. in. 


lbs. per 
sq. in. 


ex. 


cont. 






Forged 
bars. * 


i .625 


4 


276,800 




2.75 


"Yof 


Special 


~" 


1 " 




246,595 




4.25 


treatment. 


03 


1 " 


" 


105,300 




19.25 


55.0 


Annealed. 


EQ 




f .564 


4 


142,800 


' 74.000 


13.0 


28.2 




% 




i n 




143.200 


74,000 


12.32 


27.6 




M . 


1^-in. 

round 

rolled bar.t 


1 » 


" 


117,600 


64,000 


17.0 


46.0 




■= 


1 ti 


" 


119,200 


65,000 


16.66 


42.1 




s 


* • < 


." 


91,600 
91,200 


51,000 
51,000 


22.25 
21.62 


53.2 
53.4 




eo 




" 


" 


85,200 
86,000 


53,000 
48,000 


21.82 
21.25 


49.5 

47.4 




rjH 




".798 


8 


115,464 


51,820 


36.25 


66.23 




£ 


lj^in. sq. 


" 




112,600 


60,000 


37.87 


62.82 






bar, rolled. $ 




" 


102,010 


39,180 


41.37 


69.59 


Annealed. 


1- 




" 


" 


102,510 


40,200 


44.00 


68.34 


" 




\500 


2 


114,590 


56,020 


47.25 


68.4 




i 1 


1-in. round 


" 




115.610 


59,080 


45.25 


62.3 




- 1 


bar, rolled. § 


- u 


" 


105,240 


45,170 


49.65 


72.8 


Annealed. 


gl 




- 


" 


106,780 


45,170 


55.50 


63.6 





* Forged from 6-in. ingot to % in. diam., with conical heads for holding. 

t Showing the effect of varying carbon. 

X Rolled down from 14-in. ingot to lJ4-in. square billet, and turned to size. 

§ Rolled down from 14 in. ingot to 1-in. round, and turned to size. 

Nickel steel has shown itself to be possessed of some exceedingly valuable 
properties; these are, resistance to cracking, high elastic limit, and homo- 
geneity. Resistance to cracking, a property to which the name of non fissi- 
bility has been given, is shown more remarkably as the percentage of nickel 
increases. Bars of 27$ nickel illustrate this property. A lJ4-in. square bar 
was nicked J4 '"• deep and bent double on itself without further fracture 
than the splintering off, as it were, of the nicked portion. Sudden failure or 
rupture of this steel would be impossible ; it seems to possess the toughness 
of rawhide with the strength of steel. With this percentage of nickel the 
steel is practically non-corrodible and non-magnetic. The resistance to 
cracking shown by the lower percentages of nickel is best illustrated in the 
many trials of nickel-steel armor. 

The elastic limit rises in a very marked degree with the addition of about 
3$ of nickel, the other physical properties of the steel remaining unchanged 
or perhaps slightly increased. 

In such places (shafts, axles, etc.) where failure is the result of the fatigue 
of the metal this higher elastic limit of nickel steel will tend to prolong in- 
definitely the life of the piece, and at the same time, through its superior 
toughness, offer greater resistance to the sudden strains of shock. 

Howe states that the hardness of nickel steel depends on the proportion 
of nickel and carbon jointly, nickel up to a certain percentage increasing 
the hardness, beyond this iessening it. Thus while steel with 2$ of nickel 
and 0.90$ of carbon cannot be machined, with less than 5$ nickel it can be 
worked cold readily, provided the proportion of carbon be low. As the 
proportion of nickel rises higher, cold-working becomes less easy. It forges 
easily whether it contain much or little nickel. 

The presence of manganese in nickel steel is most important, as it appears 
that without the aid of manganese in proper proportions, the conditions of 
treatment would not be successful. 

Tests of Nickel Steel.— Two heats of open-hearth steel were made by 
the Cleveland Rolling Mill Co., one ordinary steel made with 9000 lbs. each 
scrap and pig, and 165 lbs. ferro-manganese, the other the same with, the 
addition of 3$, or 540 lbs. of nickel. Tests of six plates rolled from each 
heat., 0.24 to 0.3 in. thick, gave results as follows : 

Ordinary steel, T. S. 52,500 to 56.500 ; E. L. 32,800 to 37,900 ; elong. 26 to 32$. 
Nickel steel, " 63,370 to 67,100 ; " 47,100 to 48,200 ; " 23J4 to 26$. 



MANGANESE, NICKEL, AND OTHER " ALLOY" STEELS. 409 

The nickel steel averages 31$ higher in elastic limit, 20$ higher in ultimate 
tensile strength, with but slight reduction in ductility. {Eng. & M. Jour., 
Feb. 25, 1893.) 

Aluminum Steel.— R. A. Hadfield (Trans. A. I. M. E. 1890) says : 
Aluminum appears to be of service as an addition to baths of molten iron or 
steel unduly saturated with oxides, and this in properly regulated steel 
manufacture should not often occur. Speaking generally, its role appears 
to be similar to that of silicon, though acting more powerfully. The state- 
ment that aluminum lowers the melting-point of iron seems to have no 
foundation in fact. If any increase of heat or fluidity takes place by the 
addition of small amounts of aluminum, it may be due to evolution of heat, 
owing to oxidation of the aluminum, as the calorific value of this metal is 
very high— in fact, higher than silicon. According to Berthollet, the con- 
version of aluminum to A1 2 3 equals 7900 cal. ; silicon to Si0 2 is stated as 7800. 

The action of aluminum maybe classed along with that of silicon, sulphur, 
phosphorus, arsenic, and copper, as giving no increase of hardness to iron, 
in contradistinction to carbon, manganese, chromium, tungsten, and nickel. 
Therefore, whilst for some special purposes aluminum may be employed in 
the manufacture of iron, at any rate with our present knowledge of its 
properties, this use cannot be large, especially when taking into considera- 
tion the fact of its comparatively high price. Its special advantage seems to 
be that it combines in itself the advantages of both silicon and manganese; 
but so long as alloys containing these metals are so cheap and aluminum 
dear, its extensive use seems hardly probable. 

J. E. Stead, in discussion of Mr. Hadfield's paper, said: Every one of our 
trials has indicated that aluminum can kill the most fiery steel, providing, 
of course, that it is added in sufficient quantity to combine with all the oxy- 
gen which the steel contains. The metal will then be absolutely dead, and 
will pour like dead-melted silicon steel. If the aluminum is added as metal- 
lic aluminum, and not as a compound, and if the addition is made just be- 
fore the steel is cast, 1/10$ is ample to obtain perfect solidity in the steel. 

Chrome Steel. (F. L. Garrison, Jour. F. I., Sept. 1891.)— Chromium 
increases the hardness of iron, perhaps also the tensile strength and elastic 
limit, but it lessens its weldibility. 

Ferro chrome, according to Berthier, is made by strongly heating the 
mixed oxides of iron and chromium in brasqued crucibles, adding powdered 
charcoal if the oxide of chromium is in excess, and fluxes to scorify the 
earthy matter and prevent oxidation. Chromium does not appear to give 
steel the power of becoming harder when quenched or chilled. Howe states 
that chrome steels forge more readily than tungsten steels, and when not 
containing over 0.5 of chromium nearly as well as ordinary carbon steels of 
like percentage of carbon. On the whole the status of chrome steel is not 
satisfactory. There are other steel alloys coming into use, which are so 
much better, that it would seem to be only a question of time when it will 
drop entirely out of the race. Howe states that many experienced chemists 
have found no chromium, or but the merest traces, in chrome steel sold in 
the markets. 

J. W. Langley (Trans. A. S. C. E. 1892) says: Chromium, like manganese, 
is a true hardener of iron even in the absence of carbon. The addition of 1% 
or 2$ of chromium to a carbon steel will make a metal which gets exces- 
sively hard. Hitherto its principal employment has been in the production 
of chilled shot and shell. Powerful molecular stresses result during cooling, 
and the shells frequently break spontaneously months after they are made. 

Tungsten Steel— Mushet Steel. (J. B. Nau, Iron Age. Feb. 11, 1892.) 
— By incorporating simultaneously carbon and tungsten in iron, it is possi- 
ble to obtain a much harder steel than with carbon alone, without danger of 
an extraordinary brittleness in the cold metal or an increased difficulty in 
the working of the heated metal. 

When a special grade of hardness is required, it is frequently the custom 
to use a high tungsten steel, known in England as special steel. A specimen 
from Sheffield, used for chisels, contained 9.3$ of tungsten, 0.7$ of silver, 
and 0.6$ of carbon. This steel, though used with advantage in its untem- 
pered state to turn chilled rolls, was not brittle; nevertheless it was hard 
enough to scratch glass. 

A sample of Mushet's special steel contained 8.3$ of tungsten and 1.73$ of 
manganese. The hardness of tungsten steel cannot be increased by the or- 
dinary process of hardening. 

The only operation that it can be submitted to when cold is grinding. It 
has to be given its final shape through hammering at a red heat, and even 






410 STEEL. 

then, when the percentage of tungsten is high, it has to be treated very 
carefully; and in order to avoid breaking it, not only is it necessary to reheat 
it several times while it is being hammered, but when the tool nas acquired 
the desired shape hammering must still be continued gently and with nu- 
merous blows until it becomes nearly cold. Then only can it be cooled en- 
tirely. 

Tungsten is not only emploj^ed to produce steel of an extraordinary hard- 
ness, but more especially to obtain a steel which, with a moderate hardness, 
allies great toughness, resistance, and ductility. Steel from Assailly, used 
for this purpose, contained carbon, 0.52%; silicon, 0.04$; tungsten, 0.3$; 
phosphorus, 0.04$; sulphur, 0.005$. 

Mechanical tests made by Styffe gave the following results : 

Breaking load per square inch of original area, pounds. . 172,424 

Reduction of area, per cent 0.54 

Average elongation after fracture, per cent 13 

According to analyses made by the Due de Luynes of ten specimens of the 
celebrated Oriental damasked steel, eight contained tungsten, two of them 
in notable quantities (0.518$ to 1$), while in all of the samples analyzed 
nickel was discovered ranging from traces to nearly 4$. 

Stein & Schwartz of Philadelphia, in a circular say : It is stated that 
tungsten steel is suitable for the manufacture of steel magnets, since it re- 
tains its magnetism longer than ordinary steel. Mr. Kniesche has made 
tungsten up to 98$ fine a specialty. Dr. Heppe, of Leipsig, has written a 
number of articles in German publications on the subject. The following 
instructions are given concerning the use of tungsten: In order to produce 
cast iron possessing great hardness an addition of one half to one and one 
half of tungsten is all that is needed. For bar iron it must be carried up to 
1$ to 2$, but should not exceed 2^$. For puddled steel the range is larger, 
but an addition beyond S}/ 2 % only increases the hardness, so that it is brought 
up to \y%% only for special tools, coinage dies, drills, etc. For tires 2^$ to 5$ 
have proved best, and for axles y% to 1^$. Cast steel to which tungsten has 
been added needs a higher temperature for tempering than ordinary steel, 
and should be hardened only between yellow, red, and white. Chisels made 
of tungsten steel should be drawn between cherry-red and blue, and stand 
well on iron and steel. Tempering is best done in a mixture of 5 parts of 
yellow rosin, 3 parts of tar, and 2 parts of tallow, and then the article is 
once more heated and then tempered as usual in water of about 15° C. 

Whitwortli Compressed Steel. (Proc. Inst. M. E.. May, 1887, p. 
167 )— In this system a gradually increasing pressure up to 6 or 8 tons per 
square inch is applied to the fluid ingot, and within half an hour or less 
after the application of the pressure the column of fluid steel is shortened 
\y% inch per foot or one eighth of its length; the pressure is then kept on for 
several hours, the result being that the metal is compressed into a perfectly 
solid and homogeneous material, free from blow-holes. 

In large gun-ring ingots during cooling the carbon is driven to the centre, 
the centre containing 0.8 carbon and the outer ring 0.3. The centre is bored 
out until a test shows that the inside of the ring contains the same percent- 
age of carbon as the outside. 

Compressed steel is made by the Bethlehem Iron Co. and the Carnegie 
. Steel Co. for armor-plate and for gun and other heavy forgings. 

CRUCIBLE STEEL. 

Selection of Grades by the Eye, and Effect of Heat Treat- 
ment. (J. W. Langley, Amer. Chemist, November, 1876.)— In 1874, Miller, 
Metcalf & Parkin, of Pittsburgh, selected eight samples of steel which were 
believed to form a set of graded specimens, the order being based on the 
quantity of carbon which they were supposed to contain. They were num- 
bered from one to eight. On analysis, the quantity of carbon was found to 
follow the order of the numbers, while the other elements present— silicon, 
phosphorus, and sulphur— did not do so. The method of selection is 
described as follows : 

The steel is melted in black-lead crucibles capable of holding about eighty 
pounds; when thoroughly fluid it is poured into cast-iron moulds, and when 
cold the top of the ingot is broken off, exposing a freshly-fractured surface. 
The appearance presented is that of confused groups of crystals, all appear- 
ing to have started from the outside and to have met in the centre; this 
general form is common to all ingots of whatever composition, but to the 
trained eye, and only to one long and critically exercised, a minute but in- 



CRUCIBLE STEEL. 



411 



describable difference is perceived between varying samples of steel, and 
tbis difference is now known to be owing almost wholly to variations in the 
amount of combined carbon, as the following table will show. Twelve sam- 
ples selected by the eye alone, and analyses of drillings taken direct from 
the ingot before it had been heated or hammered, gave results as below: 



Ingot 
Nos. 


Iron by 
Diff. 


Carbon. 


Diff. of 
Carbon. 


Silicon. 


Phos. 


Sulph. 


1 


99.614 
99.455 


.302 
.490 




.019 
.034 


.047 
.005 


.018 


2 


.188 


.016 


3 


99.363 


.529 


.039 


.043 


.047 


.018 


4 


99.270 


.649 


.120 


.039 


.030 


.012 


5 


99.119 


.801 


.152 


.029 


.035 


.016 


6 


99.086 


.841 


.040 


.039 


.024 


.010 


7 


99.044 


.S67 


.026 


.057 


.014 


.018 


8 


99.040 


.871 


.004 


.053 


.024 


.012 


9 


98.900 


.955 


.084 


.059 


.070 


.016 


10 


98 861 


1.005 


.050 


.088 


.034 


.012 


11 


98.752 


1.058 


.053 


.120 


.064 


.006 


12 


98.834 


1.079 


.021 


.039 


.044 


.004 



Here the carbon is seen to increase in quantity in the order of the num- 
bers, while the other elements, with the exception of total iron, bear no rela- 
tion to the numbers on the samples. The mean difference of carbon is .071. 
In mild steels the discrimination is less perfect. 

The appearance of the fracture by which the above twelve selections 
were made can only be seen in the coid ingot before any operation, except 
the original one of casting, has been performed upon 'it. As soon as it is 
hammered, the structure changes in a remarkable manner, so that all trace 
of the primitive condition appears to be lost. 

Another method of rendering visible to the eye the molecular and chemi- 
cal changes which go on in steel is by the process of hardening or temper- 
ing. When the metal is heated and plunged into water it acquires an 
increase of hardness, but a loss of ductility. If the heat to which the steel 
has been raised just before plunging is too high, the metal acquires intense 
hardness, but it is so brittle as to be worthless; the fracture is of a bright, 
granular, or sandy character. In this state it is said to be burned, and it 
cannot again be restored to its former strength and ductility by annealing; 
it is ruined for all practical purposes, but in just this state' it again shows 
differences of structure corresponding with its content in carbon. The 
nature of these changes can be illustrated by plunging a bar highly heated 
at one end and cold at the other into water, and then breaking it off in 
pieces of equal length, when the fractures will be found to show appear- 
ances characteristic of the temperature to which the sample was raised. 

The specific gravity of steel is influenced not only by its chemical analy- 
sis, but by the heat to which it is subjected, as is shown by the following 
table (densities referred to 60° F.): 

Specific gravities of twelve samples of steel from the ingot; also of six 
hammered bars, each bar being overheated at one end and cold at the 
other, in this state plunged into water, and then broken into pieces of 
equal length. 



Ingot 

Bar: 
*Burned 1. . 

2. . 

3... 

4... 


1 

7.855 


2 
7.836 


3 

7.841 

7.818 
7.814 
7.823 
7.826 
7.831 
7.844 


4 
7.829 

7.791 

7.811 
7.830 
7.849 
7.806 

7.824 


5 

7.838 


6 

7.824 

7.789 

7.784 
7.780 
7.808 
7.812 
7.829 


7 1 8 

7.819,7.818 

. ... 7.752 

7.755 

,7.758 

7.773 

17.790 

. ... |7.825 


9 

7.813 


10 

7.807 

7.744 

7.749 
7.755 
7.789 
7.812 
7.826 


11 
7.803 


12 

7.805 

7.690 
7.741 

7.769 
7.798 


5... 






7.811 


Cold 6... 






7.825 



* Order of samples from bar. 



412 



Effect of Heat on the Grain of Steel. (W. Metcalf,— Jeans on 
Steel, p. 642.) — A simple experiment will show the alteration produced in a 
high-carbon steel by different methods of hardening. If a bar of such steel 
be nicked at about 9 or 10 places, and about half an inch apart, a suitable 
specimen is obtained for the experiment. Place one end of the bar in a 
good fire, so that the first nicked piece is heated to whiteness, while the rest 
of the bar, being out of the fire, is heated up less and less as we approach 
the other end. As soon as the first piece is at a good white heat, which of 
course burns a high carbon steel, and the temperature of the rest of the bar 
gradually passes down to a very dull red, the metal should be taken out of 
the fire and suddenly plunged in cold water, in which it should be left till 
quite cold. It should then be taken out and carefully dried. An examina- 
tion with a file will show that the first piece has the greatest hardness, 
while the last piece is the softest, the intermediate pieces gradually passing 
from one condition to the other. On now breaking off the pieces at each 
nick it will be seen that very considerable and characteristic changes have 
been produced in the appearance of the metal. The first burnt piece is very 
open or crystalline in fracture; the succeeding pieces become closer and 
closer in the grain until one piece is found to possess that perfectly 
even grain and velvet-like appearance which is so much prized by experi- 
enced steel users. The first pieces also, which have been too much hard- 
ened, will probably be cracked; those at the other end will not be hardened 
through. Hence if it be desired to make the steel hard and strong, the 
temperature used must be high enough to harden the metal through, but 
not sufficient to open the grain. 

Changes in T 1 Him ate Strength and Elasticity due to 
Hammering, Annealing, and Tempering. (J. W. Langley, 
Trans. A. S. C. E. 1892.)— The following tabl« gives the result of tests made 
on some round steel bars, all from the same ingot, which were tested by 
tensile stresses, and also by bending till fracture took place: 





Treatment. 


si 

85c 


Carbon. 


p 
p 

s 

eg 

5 


2 33 a 


■a s 

eg 

~ 53 


a . 
W ■ 


a<j 


£3 

s 

a 


3 

o 


if 


-a u 

0) CD 

P5 a 


1 
2 
3 
4 


Cold-hammered bar 
Bar drawn black — 

Bar annealed 

Bar hardened and 
drawn black ; . 


153 

75 

175 

30 


1.25 
1 . 25 
1.31 

1.09 


.47 
.47 

.70 

.36 


.575 
.577 
.580 

.578 


92,420 
114,700 
68,110 

152,800 


141,500 
138,400 
98,410 

248,700 


2.00 
6.00 

10.00 

8.33 


2.42 

12.45 
11.69 

17.9 



The total carbon given in the table was found by the color test, which is 
affected, not only by the total carbon, but by the condition of the carbon. 

The analysis of the steel was: 

Silicon 242 Manganese 24 

Phosphorus 02 Carbon (true total carbon J by 

Sulphur 009 combustion) ... 1.31 

Heating Tool Steel. (Miller, Metcalf & Parkin, 1877.)— There are 
three distinct stages or times of heating: First, for forging; second, for 
hardening; third, for tempering. 

The first requisite for a good beat for forging is a clean fire and plenty of 
fuel, so that jets of hot air will not strike the corners of the piece; next, the 
fire should be regular, and give a good uniform heat to the whole part to be 
forged. It should be keen enough to heat the piece as rapidly as may be, 
and allow it to be thoroughly heated through, without being so fierce as to 
overheat the corners. 

Steel should not be left in the fire any longer than is necessary to heat it 
clear through, as " soaking " in fire is very injurious; and, on the other hand, 
it is necessary that it should be hot through, to prevent surface cracks. 

By observing these precautions a piece of steel may always be heated 
safely, up to even a bright yellow heat, when there is much forging to be 
done on it. 



CRUCIBLE STEEL. 413 

The best and most economical of welding fluxes is clean, crude borax, 
which should be first thoroughly melted and then ground to fine powder. 

After the steel is properly heated, it should be forged to shape as quickly 
as possible ; and just as the red heat is leaving the parts intended for cutting 
edges, these parts should be refined by rapid, light blows, continued until 
the red disappears. 

For the second stage of heating, for hardening, great care should be used: 
first, to protect the cutting edges and working parts from heating more 
rapidly than the body of the piece; next, that the whole part to be hardened 
be heated uniformly through, without any part becoming visibly hotter 
than the other. A uniform heat, as low as will give the required hardness, 
is the best for hardening. 

For every variation of heat, which is great enough to be seen, there will 
result a variation in grain, which maybe seen by breaking the piece: and 
for every such variation in temperature, there is a very good chance for a 
crack to be seen. Many a costly tool is ruined by inattention to this point. 
The effect of too high heat is to open the grain; to make the steel coarse. 
The effect of an irregular heat is to cause irregular grain, irregular strains, 
and cracks. 

As soon as the piece is properly heated for hardening, it should be 
promptly and thoroughly quenched in plenty of the cooling medium, water, 
brine, or oil, as the case may be. 

An abundance of the cooling bath, to do the work quickly and uniformly 
all over, is very necessary to good and safe work. 
To harden a large piece safely a running stream should be used. 
Much uneven hardening is caused by the use of too small baths. 
For the third stage of heating, to temper, the first important requisite is 
again uniformity. The next is time; the more slowly a piece is brought 
down to its temper, the better and safer is the operation. 

When expensive tools are to be made it is a wise precaution to try small 
pieces of the steel at different temperatures, so as to find out how low a heat 
will give the necessary hardness. The lowest heat is the best for any steel. 
Heating to Forge.— The trouble in the forge fire is usually uneven 
heat, and not too high heat. Suppose the piece to be forged has been put 
into a very hot fire, and forced as quickly as possible to a high yellow heat, 
so that it is almost up to the scintillating point. If this be done, in a few 
minutes the outside will be quite soft and in a nice condition for forging, 
while the middle parts will not be more than red-hot. Now let the piece be 
placed under the hammer and forged, and the soft outside will yield so 
much more readily than the hard inside, that the outer particles will be torn 
asunder, while the inside will remain sound. 

Suppose the case to be reversed and the inside to be much hotter than the 
outside; that is, that the inside shall be in a state of semi-fusion, while the 
outside is hard and firm. Now let the piece be forged, and the outside will 
be all sound and the whole piece will appear perfectly good until it is 
cropped, and then it is found to be hollow inside. 

In either case, if the piece had been heated soft all through, or if it had been 
only red-hot all through, it would have forged perfectly sound. 

In some cases a high heat is more desirable to save heavy labor but in 
every case where a fine steel is to be used for cutting purposes it must be 
borne in mind that very heavy forging refines the bars as they slowly cool, 
and if the smith heats such refined bars until they are soft, he raises the 
grain, makes them coarse, and he cannot get them fine again unless he has 
a very heavy steam-hammer at command and knows how "to use it well. 

Annealing. (Miller, Metcalf & Parkin.)— Annealing or softening is 
accomplished by heating steel to a red heat and then cooling it very slowly, 
to prevent it from getting hard again. 

The higher the degree of heat, the more will steel be softened, until the 
limit of softness is reached, when the steel is melted. 

It does not follow that the higher a piece of steel is heated the softer it 
will be when cooled, no matter how slowly it may be cooled; this is proved 
by the fact that an ingot is always harder than a rolled or hammered bar 
made from it. 

Therefore there is nothing gained by heating a piece of steel hotter than 
a good, bright, cherry-red: on the contrary, a higher heat has several dis- 
advantages: First. If carried too far, it may leave the steel actually harder 
than a good red heat would leave it. Second. If a scale is raised on the 
steel, this scale will be harsh, granular oxide of iron, and will spoil the tools 
used to cut it. Third. A high scaling heat continued for a little time 



414 



STEEL. 



changes the structure of the steel, makes it brittle, liable to crack in hard- 
ening, and impossible to refine. 

To anneal any piece of steel, heat it red-hot; heat it- uniformly and heat it 
through, taking care not to let the ends and corners get too hot. 

As soon as it is hot, take it out of the fire, the sooner the better, and cool 
it as slowly as possible. A good rule for heating is to heat it at so low a red 
that when the piece is cold it will still show the blue gloss of the oxide that 
was put there by the hamer or the rolls. 

Steel annealed in this way will cut very soft; it will harden very hard, 
without cracking; and when tempered it will be very strong, nicely refined, 
and will hold a keen, strong edge. 

Tempering. — Tempering steel is the act of giving it, after it has been 
shaped, the hardness necessary for the work it has to do. This is done by 
first hardening the piece, generally a good deal harder than is necessary, 
and then toughening it by slow heating and gradual softening until it is just 
right for work. 

A piece of steel properly tempered should always be finer in grain than 
the bar from which it is made. If it is necessary, in order to make the piece 
as hard as is required, to heat it so hot that after being hardened the grain 
will be as coarse as or coarser than the grain in the original bar, then the steel 
itself is of too low carbon for the desired work. 

If a great degree of hardness is not desired, as in the case of taps, and 
most tools of complicated form, and it is found that at a moderate heat the 
tools are too hard and are liable to crack, the smith should first use a lower 
heat in order to save the tools already made, and then notify the steelmaker 
that his steel is too high, so as to prevent a recurrence of the trouble. 

For descriptions of various methods of tempering steel, see "Tempering 
of Metals, 11 by Joshua Rose, in App. Cyc. Mech., vol. ii. p. 863; also, 
•'Wrinkles and Recipes," from the Scientific American. In both of these 
works Mr. Rose gives a "color scale, 11 lithographed in colors, by which the 
color to which the temper is to be drawn for different tools is shown. The 
following is a list of the tools in their order on the color scale, together with 
the approximate color and the temperature at which the color appears on 
brightened steel when heated in the air : 



Scrapers for brass; very pale yel- 
low, 430° F. 

Steel-engraving tools. 

Slight turning tools. 

Hammer faces. 

Planer tools for steel. 

Ivory-cutting tools. 

Planer tools for iron. 

Paper-cutters. 

Wood-engraving tools. 

Bone cutting tools. 

Milling-cutters; strata yellow, 460° F. 

Wire-drawing dies. 

Boring-cutters. 

Leather-cutting dies. 

Screw-cutting dies. 

Inserted saw-teeth. 

Taps. 

Rock-drills. 

Chasers. 

Punches and dies. 

Penknives. 

Reamers. 

Half-round bits. 

Planing and moulding cutters. 

Stone-cutting tools; brown yellow, 
500° F. 

Gouges. 



Hand-plane irons. 

Twist-drills. 

Flat drills for brass. 

Wood-boring cutters. 

Drifts. 

Coopers 1 tools. 

Edging cutters; light purple, 530° F. 

Augers. 

Dental and surgical instruments. 

Cold chisels for steel. 

Axes; dark purple, 550° F. 

Gimlets. 

Cold chisels for cast iron. 

Saws for bone and ivory. 

Needles. 

Firmer-chisels. 

Hack-saws. 

Framing-chisels. 

Cold chisels for wrought iron. 

Moulding and planing cutters to b3 

filed. 
Circular saws for metal. 
Screw-drivers. 
Springs. 
Saws for wood. 

Dark blue, 570° F. 

Pale blue, 610°. 

Blue tinged with green, 630°. 



FORCE, STATICAL MOMEKT, EQUILIBRIUM, ETC. 415 



MECHANICS. 

FORCE, STATICAL MOMENT, EQUILIBRIUM, ETC. 

Mechanics is the science that treats of the action of force upon bodies. 

A. Force is anything that tends to change the state of a body with respect 
to rest or motion. If a body is at.rest, anything that tends to put it in mo- 
tion is a force; if a body is in motion, anything that tends to change either 
its direction or its rate of motion is a force. 

A force should always mean the pull, pressure, rub, attraction (or repul- 
sion) of one body upon another, and always implies the existence of a simul- 
taneous equal and opposite force exerted by that other body on the first body, 
i.e., the reaction. In no case should we call anything a force unless we can 
conceive of it as capable of measurement by a spring-balance, and are able 
to say from what other body it comes. (I. P. Church.) 

Forces may be divided into two classes, extraneous and molecular: extra- 
neous forces act on bodies from without; molecular forces are exerted be- 
tween the neighboring particles of bodies. 

Extraneous forces are of two kinds, pressures and moving forces: pres- 
sures simply tend to produce motion; moving forces actually produce 
motion. Thus, if gravity act on a fixed bod3 r , it creates pressure; if on a free 
body, it px-oduces motion. 

Molecular forces are of two kinds, attractive and repellent: attractive 
forces tend to bind the particles of a body together; repellent forces tend 
to thrust them asunder. Both kinds of molecular forces are continually 
exerted between the molecules of bodies, and on the predominance of one 
or the other depends the physical state of a body, as solid, liquid, or gaseous. 

The Unit of Force used in engineering, by English writers, is the 
pound avoirdupois. (E'or some scientific purjjoses, as in electro-dynamics, 
forces are sometimes expressed in " absolute units." The absolute unit of 
force is that force which acting on a unit of mass during a unit of time pro- 
duces a unit of veloc ty; in English measures, that force which acting on 
the mass whose weight is one pound in London will in one second produce a 
velocity of one foot per second = 1 -*- 32.187 of the weight of the standard 
pound avoirdupois at London. In the French C. G. S. or centimetre-gramme 
second system it is the force which acting on the mass whose weight is one 
gramme'at Paris will produce in one second a velocity of one centimetre per 
second. This unit is called a " dyne " = 1/981 gramme at Paris.) 

Inertia is that property of a body by virtue of which it tends to continue 
in the state of rest or motion in which it may be placed, until acted on by 
some force. 

Newton's Laws of Motion.— 1st Law. If a body be at rest, it will 
remain at rest; or if in motion, it will move uniformly in a straight line till 
acted on by some force. 

2d Law. If a body be acted on by several forces, it will obey each as 
though the others did not exist, and this whether the body be at rest or in 
motion. 

3d Law. If a force act to change the state of a body with respect to rest 
or motion, the body will offer a resistance equal and directly opposed to the 
force. Or. to every action there is opposed an equal and opposite reaction. 

Graphic Representation of a Force.— Forces may be repre- 
sented geometrically by straight lines, proportional to the forces. A. force 
is given when we know its intensity, its point of application, and the direc- 
tion in which it acts. When a force is represented by a line, the length of the 
line represents its intensity; one extremity represents the point of applica- 
tion; and an arrow-head at the other extremity shows the direction of the 
force. 

Composition of Forces is the operation of finding a single force 
whose effect is the same as that of two or more given forces. The required 
force is called the resultant of the given forces. 

Resolution of Forces is the operation of finding two or more forces 
whose combined effect is equivalent to that of a given force. The required 
forces are called components of the given force. 

The resultant of two forces applied at a point, and acting in the same di- 
rection, is equal to the sum of the forces. If two forces act in opposite 
directions, their resultant is equal to their difference, and it acts in the 
direction of the greater. 



416 



MECHANICS. 




Fig. 88. 



If any number of forces be applied at a point, some in one direction and 
others in a contrary direction, their resultant is equal to the sum of those 
that act in one direction, diminished by the sum of those that act in the op- 
posite direction; or, the resultant is equal to the algebraic sum of the com- 
ponents. 

Parallelogram of Forces.— If two forces acting on a point be rep- 
resented in direction and intensity by adjacent sides of a parallelogram, 
their resultant will be represented by that diagonal of the parallelogram 
_ which passes through the point. Thus OR, Fig. 

H - — -^ K 88, is the resultant of OQ and OP. 

Polygon of Forces.— If several forces are 
applied at a point and act in a single plane, their 
resultant is found as follows: 

Through the point draw a line representing the 
first force ; through the extremity of this draw 
a line representing the second force; and so on, 
throughout the system; finally, draw a line from 
the starting-point to the extremity of the last line 
drawn, and this will be the resultant required. 

Suppose the body A, Fig. 89, to be urged in the directions Al, A2, A% A4, 
and A5 by forces which are to each other as the lengths of those lines. 
Suppose these forces to act successively and the body to first move from A 
to 1 ; the second force A2 then acts and finding the body at 1 would take it 
to 2'; the third force would then carry it to 3', the fourth to 4', and the fifth 
to 5'.' The line AW represents in magnitude and direction the resultant of 
all the forces considered. If there had 
been an additional force, Ax, in the group, 
the body would be returned by that force 
to its original position, supposing the 
forces to act successively, but if they had , 
acted simultaneously the body would never 2( 
have moved at all; the tendencies to mo- 
tion balancing each other. 

It follows, therefore, that if the several 
forces which tend to move a body can be 
represented in magnitude and direction 
by the sides of a closed polygon taken in 
order, the body will remain at rest; but if 
the forces are represented by the sides of 
an open polygon, the body will move and the direction will be represented 
by the straight line which closes the polygon 

Twisted Polygon.— The rule of the polygon of forces holds true even 
when the forces are not in one plane. In this case the lines Al, 1-2', 2'-3', 
etc form a twisted polygon, that is, one whose sides are not in one plane. 

Parallel oplpedon of Forces.-If three forces acting on a point be 
represented by three edges of a parallelopipedon which meet in a common 
point, their resultant will be represented by the diagonal of the parallelo- 
pipedon that passes through their common point. „,,.,, 

Thus OR, Fig. 90, is the resultant of OQ, OS, and OP. OM is the result- 
ant of OF and OQ. and OR is the resultant of OM and OS. 

Moment of a Force.— The mo- 




Fig. I 



ment of a force (sometimes called stat- 
ical moment), with respect to a point, 
is the product of the force by the per- 
pendicular distance from the point to 
the direction of the force. The fixed 
point is called the centre of mo- 
S 



b ^-< 



^rd 





Fig- 90, 



Fig, 91, 



F0RCE, STATICAL MOMENT, EQUILIBRIUM, ETC. 417 

merits ; the perpendicular distance is the lever-arm of the force; and the 
moment itself measures the tendency of the force to produce rotation about 
the centre of moments. 

If the force is expressed in pounds and the distance in feet, the moment 
is expressed in foot-pounds. It is necessary to observe the distinction be- 
tween foot-pounds of statical moment and foot-pounds of work or energy. 
(See Work.) 

In the bent lever, Fig. 91 (from Trautwine), if the weights n and m repre- 
sent forces, their moments about the point / are respectively n X «/ and 
m X fc. If instead of the weight m a pulling force to balance the weight 
n is applied in the direction bs, or by or bd, s, y, and d being the amounts of 
these forces, their respective moments are s x ft, y X fb, d X fh. 

If the forces acting on the lever are in equilibrium it remains at rest, and 
the moments on each side of / are equal, that is, n X af—m x fc, or s X ft, 
or y X fb, or d X hf. 

The moment of the resultant of any number of forces acting together in 
the same plane is equal to the algebraic sum of the moments of the forces 
taken separately. 

Statical Moment. Stability.— The statical moment of a body is 
the product of its weight by the distance of its line of gravity from some 
assumed line of rotation. The line of gravity is a vertical line drawn from 
its centre of gravity through the body. The stability of a body is that re- 
sistance which its weight alone enables it to oppose against forces tending 
to overturn it or to slide it along its foundation. 

To be safe against turning on an edge the moment of the forces tending to 
overturn it, taken with reference to that edge, must be less than the stati- 
cal moment. When a body rests on an inclined plane, the line of gravity 
being vertical, falls toward the lower edge of the body, and the condition of 
its not being overturned by its own weight is that the line of gravity must 
fall within this edge. In the case of an inclined tower resting on a plane 
the same condition holds— the line of gravity must fall within the base. The 
condition of stability against sliding along a horizontal plane is that the hor- 
izontal component of the force exerted tending to cause it to slide shall be 
less than the product of the weight of the body into the coefficient of fric- 
tion between the base of the body and its supporting plane. This coefficient 
of friction is the tangent of the angle of repose, or the maximum angle at 
which the supporting plane might be raised from the horizontal before the 
body "would beg-in to slide. (See Friction.) 

The Stability of a Dam against overturning about its lower edge 
is calculated by comparing its statical moment referred to that edge with 
the resultant pressure of the water against its upper side. The horizontal 
pressure on a square foot at the bottom of the dam is equal to the weight of 
a column of water of one square foot in section, and of a height equal to the 
distance of the bottom below water-level : or, if H is the height, the pressure 
at the bottom per square foot = 62.4 X iJlbs. At the water-level the pres- 
sure is zero, and it increases uniformly to the bottom, so that the sum of the 
pressures on a vertical strip one foot in breadth may be represented by the 
area of a triangle whose base is 62.4 x H and whose altitude is H, or 62 4i? 2 -4-2. 
The centre of gravity of a triangle being y§ of its altitude, the resultant of 
all the horizontal pressures may be taken as equivalent to the sum of the 
pressures acting at y%H, and the moment of the sum of the pressures is 
therefore 62.4 X H 3 -s- 6. 

Parallel Forces.— If two forces are parallel and act in the same direc- 
tion, their resultant is parallel to both, and lies between them, and the inten- 
sity of the resultant is equal to the sum of the intensities of the two forces. 
Thus in Fig. 91 the resultant of the forces n and m acts vertically down- 
ward at/, and is equal to n + m. 

If two parallel forces act at the extremities of a straight line and in the 
same direction, the resultant divides the line joining the points of application 
of the components, inversely as the components. Thus in Fig. 91, m : n :: 
af'.fc; and in Fig. 92, P : Q :: SN : SM. N 

The resultant of two parallel forces •" ^ 

acting in opposite directions is parallel / ] 

to both, lies without both, on the side S p' |C , ^- R 

and in the direction of the greater, /' i 

and its intensity is equal to the differ- M / \ > p 

ence of the intensities of the two ' L 

forces, Fig. 93, 



418 MECHANICS. 

Thus the resultant of the two forces Q and P, Fig. 93, is equal to Q - P = 

R. Of any two parallel forces and their 

N resultant each is proportional to the dis- 

q< -p tance between the other two; thus in both 

/ Figs. 92 and 93, P : Q : R : : SN : SM : MN. 

M-p — j %-p Couples.— If P and Q be equal and act 

/ i in opposite directions, R — 0; that is, they 

/ I have no resultant. Two such forces con- 

~ @L ! >r stitute what is called a couple. 

* C The tendency of a couple is to produce 

Fig. 93. rotation; the measure of this tendency, 

called the moment of the couple, is the 
product of one of the forces by the distance between the two. 

Since a couple has no single resultant, no single force can balance a 
couple. To prevent the rotation of a body acted on by a couple the applica- 
tion of two other forces is required, forming a second couple. Thus in Fig. 
94, Pand Q forming a couple, may be balanced 
by a second couple formed by R and S. The 
point of application of eitherP or £? may be a 
fixed pivot or axis. I P 

Moment of the couple PQ = P(c + b + a) = T 
moment of RS = Rb. Also, P + R = Q -f S. 

The forces R and S need not be parallel to P | < 

and Q, but if not, then their components parallel 
to PQ are to be taken Instead of the forces 
themselves. 

Equilibrium of Forces.— A system of 
forces applied at points of a solid body will be 
in equilibrium when they have no tendency to yS 

produce motion, either of translation or of rota- Fig. 94. 

tion. 

The conditions of equilibrium are : i. The algebraic sum of the compo- 
nents of the forces in the direction of any three rectangular axes must be 
separately equal to 0. 

2. The algebraic sum of the moments of the forces, with respect to any 
three rectangular axes, must be separately equal to 0. 

If the forces lie in a plane : 1. The algebraic sum of the components of the 
forces, in the direction of any two rectangular axes, must be separately 
equal to 0. 

2. The algebraic sum of the moments of the forces, with respect to any 
point in the plane, must be equal to 0. 

If a body is restrained by a fixed axis, as in case of a pulley, or wheel and 
axle, the forces will be in a equilibrium when the algebraic sum of the mo- 
ments of the forces with respect to the axis is equal to 0. 

CENTRE OF GRAVITY. 

The centre of gravity of a body, or of a system of bodies rigidly connected 
together, is that point about which, if suspended, all the parts will be in 
equilibrium, that is, there will be no tendency to rotation. It is the point 
through which passes the resultant of the efforts of gravitation on each of 
the elementary particles of a body. In bodies of equal heaviness through- 
out, the centre of gravity is the centre of magnitude. 

(The centre of magnitude of a figure is a point such that if the figure be 
divided into equal parts the distance of the centre of magnitude of the 
whole figure from any given plane is the mean of the distances of the centres . 
of magnitude of the several equal parts from that plane.) 

If a body be suspended at its centre of gravity, it will be in equilibrium in 
all positions. If it be suspended at a point out of its centre of gravity, it 
will swing into a position such that its centre of gravity is vertically beneath 
its point of suspension. 

To find the centre of gravity of any plane figure mechanically, suspend 
the figure by any point near its edge, and mark on it the direction of a 
plumb-line hung from that point ; then suspend it from some other point, 
and again mark the direction of tiie plumb-line in like manner. Then the 
centre of gravity of the surface will be at the point of intersection of the 
two marks of the plumb-line. 

The Centre of Gravity of Regular Figures, whether plane or 
solid, is the same as their geometrical centre ; for instance, a straight line, 



MOMENT OF INERTIA. 419 

parallelogram, regular polygon, circle, circular ring, prism, cylinder, 
sphere, spheroid, middle frustums of spheroid, etc. 

Of a triangle : On a line drawn from any angle to the middle of the op- 
posite side, at a distance of one third of the line from the side; or at the 
intersection of such lines drawn from any two angles. 

Of a trapezium or trapezoid: Draw the two diagonals, dividing it into 
four triangles. Draw lines joining the centres of gravity of opposite pairs 
of triangles, and their intersection is the centre of gravity. 

Of a sector of a circle : On the radius which bisects the arc, ^ — from the 

centre, c being the chord, r the radius, and I the arc. 

Of a semicircle: On the middle radius, .4244/- from the centre. 

Of a quadrant : On the middle radius, .6002r from the centre. 

Of a segment of a circle ; c 3 -^- \2a from the centre, c = chord, a = area. 

Of a parabolic surface : In the axis, 3/5 of its length from the vertex. 

Of a semi-parabola (surface) : 3/5 length of the axis from the vertex, and 
% of the semi-base from the axis. 

Of a cone or pyramid : In the axis, Y± of its length from the base. 

Of a paraboloid : In the axis, % of its length from the vertex. 

Of a cylinder, or regidar prism ; In the middle point of the axis. 

Of a frustum of a cone or pyramid : Let a — length of a line drawn from 
the vertex of the cone when complete to the centre of gravity of the base, and 
a' that portion of it between the vertex and the top of the frustum; then 
distance of centre of gravity of the frustum from centre of gravity of its 

_ a 3a' 3 

base-- - 4(a2 + aa / + a , 2) « 

For two bodies, fixed one at each end of a straight bar, the common 
centre of gravity is in the bar, at that point which divides the distance 
between ttieir respective centres of gravity in the inverse ratio of the 
weights. In this solution the weight of the bar is neglected. But it may 
be taken as a third body, and allowed for as in the following directions : 

For more than two bodies connected in one system: Find the common 
centre of gravity of two of them ; and find the common centre of these two 
jointly with a third body, and so on to the last body of the group. 

Another method, by the principle of moments : To find the centre of 
gravity of a system of bodies, or a body consisting of several parts, whose 
several centres are known. If the bodies are in a plane, refer their several 
centres to two rectangular co-ordinate axes. Multiply each weight by its 
distance from one of the axes, add the products, and divide the sum by the 
sum of the weights: the result is the distance of the centre of gravity from 
that axis. Do the same with regard to the other axis. If the bodies are 
not in a plane, refer them to three planes at right angles to each other, and 
determine the mean distance of the sum of the weights from each of the 
three planes. 

MOMENT OF INERTIA. 

The moment of inertia of the weight of a body with respect to an axis is 
the algebraic sum of the products obtained by multiplying the weight of 
each elementary particle by the square of its distance from the axis. If the 
moment of inertia with respect to any axis = 1, the weight of any element 
of the body = v, and its distance from the axis = r, we have / — 2(t«r 2 ). 

The moment of inertia varies, in the same body, according to the position 
of the axis. It is the least possible when the axis passes through the centre 
of gravity. To find the moment of inertia of a body, referred to a given 
axis, divide the body into small parts of regular figure. Multiply the weight 
of each part by the square of the distance of its centre of gravity froni the 
axis. The sum of the products is the moment of inertia. The value of the 
moment of inertia thus obtained will be more nearly exact, the smaller and 
more numerous the parts into which the body is divided. 

Moments op Inertia of Regular Solids.— Rod, or bar, of uniform thick- 
ness, with respect to an axis perpendicular to the length of the rod, 

1=1 TF (lT+ d2 )' (1 > 

W— weight of rod, 21 = length, d = distance of centre of gravity from axis. 
Thin circular plate, axis in its I T TTr /r 2 . ,„\ 

own plane, f T = W (j + d * /! ( 2 > 

r — radius of plate. 



420 MECHANICS. 

Circular plate,axis perpendicular / T TTr />" 2 , 7 „\ /AV 

to the plate, \ I = W \T + ^') (3) 

Circular ring, axis perpendicular I /> 2 + r' 2 V 

to its own plane, j * = ^ \ — g -f« 2 /' • • • • (4) 

j- and r' are the exterior and interior radii of the ring. 

Cylinder, axis perpendicular to) T TTr /r 2 , P , \ .... 

the axis of the cylinder, \ * = w \l[ + Y ^ '* ' ' ' ' ( ' 

r — radius of base, 21 — length of the cylinder. 

By making d — in any of the above formulse we find the moment of 
inertia for a parallel axis through the centre of gravity. 

The moment of inertia. 2ww* a ; numerically equals the weight of a body 
which, if concentrated at the distance unity from the axis of rotation, would 
require the same work to produce a given increase of angular velocity that the 
actual body requires. It bears the same relation to angular acceleration 
which weight does to linear acceleration (Rankine). The term moment of 
inertia is also used in regard to areas, as the cross-sections of beams under 
strain. In this case I = 2a?- 2 , in which a is any elementary area, and r its 
distance from the centre. (See Moment of Inertia, under Strength of Ma- 
terials, p. 247.) 

CENTRE AND RADIUS OF GYRATION. 

The centre of gyration, with reference to an axis, is a point at which, if 
the entire weight of a body be concentrated, its moment of inertia will re- 
main unchanged; or, in a revolving bodj% the point in which the whole 
weight of the body may be conceived to be concentrated, as if a pound of 
platinum were substituted for a pound of revolving feathers, the angular 
velocity and the accumulated work remaining: the same. The distance of 
this point from the axis is the radius of gyration. If W — the weight of a 
body, I = 2w 2 = its moment of inertia, and k = its radius of gyration, 

1 = WW = 2wr 2 ; k = a/ 2^.. 

The moment of inertia = the weight x the square of the radius of gyration. 

To fiud the radius of gyration divide the body into a considerable number 
of equal small parts— the more numerous the more nearly exact is the re- 
sult, — then take the mean of all the squares of the distances of the parts 
from the axis of revolution, and find the square root of the mean square. 
Or, if the moment of inertia is known, divide it by the weight and extract 
the square root. For radius of gyration of an area, as a cross-section of a 
beam, divide the moment of inertia of the area by the area and extract the 
square root. 

The radius of gyration is the least possible when the axis passes through 
the centre of gravity. This minimum radius is called the principal radius 
of gyration. If we denote it by k and any other radius of gyration by k', 
we have for the five cases given under the head of moment of inertia above 
the following values : 

<1) lengt 1 h, aXiS PerPent ° \* = l^/h V= |/f+^. 



<2) Circular plate, axis ) 
in its plane, j 



(3) Circular plate, axis > 
perpen. to plane, f 



(4) Circular ring, axis \ 
perpen. to plane, 



r /,.« 



i A-2 _i_ r /2 /: 



(5) Cylinder, axis per- !'*._■. A 2 . P . ,, _ /r* 
pen. to length, f f I + j' i/ T " 



CENTRES OP OSCILLATION AND OF PERCUSSION. 421 



Principal Radii of Gyration and Squares of Radii of 
Gyration. 

(For radii of gyration of sections of columns, see page 249.) 



Surface or Solid. 



Pavallelogram: | axis at its base 

height h J " mid-height 

Straight rod : ) • t ■, 

Rectangular prism: 

axes 2a, 2b, 2c, referred to axis 2a... 
Parallelopiped: length I, base 6, axis I 

at one end, at mid-breadth f 

Hollow square tube: 

out. side h, inn'r /</, axis mid-length. . 

very thin, side = h, " " 

Thin rectangular tube: sides b, h, I 
axis mid-length j 

Thincirc. plate: rad.r,diam.7j,ax. diam. 

Flat circ. ring: diams. h, h', axis diam. 

Solid circular cylinder: length I, } 
axis diameter at mid-length f 

Circular plate: solid wheel of uni- 
form thickness, or cylinder of any > 
length, referred to axis of cyl ) 

Hollow circ. cylinder, or flat ring:l 

1, length; R, r., outer and inner I 
radii. Axis, 1, longitudinal axis; | 

2, diam. at mid-length J 

Same: very thin, axis its diameter — 
" radius r; axis, longitud'l axis. . 

Circumf . of circle, axis its centre 

" •' " " " diam 

Sphere: radius r. axis its diam 

Spheroid : equatorial radius r, re- \ 

volving polar axis a — j 

Paraboloid : r = rad. of base, rev. i 

on axis f 

Ellipsoid: semi-axes a, b, c; re vol v- ( 

ing on axis 2a j 

Spherical shell: radii R, r, revolving (_ 
on its diam ) 

Same: very thin, radius r 

Solid cone: r — rad. of base, rev. on | 
axis j 



Square of R. 
Rad. of Gyration. jf Gyrat ion. 



.2886/i 
.5773? 

.2886/ 

.577 V& 2 + c 2 



.289 V41* + & 2 



.289 V/i 2 -f /t' 2 
.408/i 



y& 

J4 f/i 2 +/i' 2 



.289-V / / 2 4-3r 2 
.7071r 



.7071 VR* +- r 2 
59 i/ia + 3(JB a 4-r 9 ) 



.7071?- 
.6325r 
.6325r 

.5773r 

.4472 4/624 
.63254 



/R 5 - »;s 
6325 f W~^ 



.8165r 
.5477r 



i^/i 2 
l/12/i 2 

1/12/ 2 

(6 2 4- c 2 ) -4- : 
4/2 4- fo2 



(/I, 2 4- /l'2) h_ 12 

/i 2 -*-6 



CENTRES OF OSC1L.L.ATION AND OF PERCUSSION. 

Centre of Oscillation,,— If a body oscillate about a fixed horizontal 
axis, not passing through its centre of gravity, there is a point in the line 
drawn from the centre of gravity perpendicular to the axis whose motion 
is the same as it would be if the whole mass were collected at that point 
and allowed to vibrate as a pendulum about the fixed axis. This point is 
called the centre of oscillation. 

The Radius of Oscillation, or distance of the centre of oscillation 
from the point of suspension = the square of the radius of gyration h- dis- 
tance of the centre of gravity from the point of suspension or axis. The 
centres of oscillation and suspension are convertible. 

If a straight line, or uniform thin bar or cylinder, be suspended at one end, 
oscillating about it as an axis, the centre of oscillation is at % the length of 



422 MECHANICS. 

the rod from the axis. If the point of suspension is at ^ the length from 
the end, the centre of oscillation is also at % the length from the axis, that 
is, it is at the other end. In both cases the oscillation will be performed in 
the same time. If the point of suspension is at the centre of gravity, the 
length of the equivalent simple pendulum is infinite, and therefore the time 
of vibration is infinite. 

For a sphere suspended by a cord, r= radius, h = distance of axis of 
motion from the centre of the sphere, h' — distance of centre of oscillation 

2 ?- 2 
from centre of the sphere, I = radius of oscillation = 7i + h' = h -\ — — • 

5 h 

If the sphere vibrate about an axis tangent to its surface, It = r, and I = r 

+ 2/5r. If h = lOr, I = lOr + ~ 

Lengths of the radius of oscillation of a few regular plane figures or thin 
plates, suspended by the vertex or uppermost point. 

1st. Wheu the vibrations are flatwise, or perpendicular to the plane of the 
figure: 

In an isosceles triangle the radius of oscillation is equal to % of the height 
of the triangle. 

In a circle, % of the diameter. 

In a parabola, 5/7 of the height. 

2d. When the vibrations are edgewise, or in the plane of the figure: 

In a circle the radius of oscillation is % of the diameter. 

In a rectangle suspended by one angle, % of the diagonal. 

In a parabola, suspended by the vertex, 5/7 of the height, plus y§ of the 
parameter. 

In a parabola, suspended by the middle of the base, 4/7 of the height plus 
i^ the parameter. 

Centre of Percussion.— The centre of percussion of a body oscillat- 
ing about a fixed axis is the point at which, if a blow is struck by the body, 
the percussive action is the same as if the whole mass of the body were con- 
centrated at the point. This point is identical with the centre of oscillation. 

THE PENDULUM. 

A body of any form suspended from a fixed axis about which it oscillates 
by the force of gravity is called a compound pendulum. The ideal body 
concentrated at the centre of oscillation, suspended from the centre of sus- 
pension by a string without weight, is called a simple pendulum. This equi- 
valent simple pendulum has the same weight as the given body, and also 
the same moment of inertia, referred to an axis passing through the point 
of suspension, and it oscillates in the same time. 

The ordinary pendulum of a given length vibrates in equal times when the 
angle of the vibrations does not exceed 4 or 5 degrees, that is, 2° or 2V£° each 
side of the vertical. This property of a pendulum is called its isochronism. 

The time of vibration of a pendulum varies directly as the square root of 
the length, and inversely as the square root of the acceleration due to grav- 
ity at the given latitude and elevation above the earth's surface. 

If T — the time of vibration, I — length of the simple pendulum, g — accel- 
eration = 32.16, T = t i / -; since n is constant, Tec . At a given loca- 

y y Vg 

tion g is constant and Tec Vl. If Z be constant, then for any location 

1 . - 27 

Tec — -. If Tbe constant, gT 2 = irH; I oc g; g -- 
Vg 

the force of gravity at any place may be determined if the length of the 
simple pendulum, vibrating seconds, at that place is known. At New York 
this length is 39.1017 inches = 3.2585 ft., whence g = 32.16 ft. At London the 
length is 39.1393 inches. At the equator 39.0152 or 39.0168 inches, according 
to different authorities. 
Time of vibration of a pendulum of a given length at New York 



: t 



: Y 39.K 



_ Vi_ 

1017 6.253' 



t being in seconds and I in inches. Length of a pendulum having a given 
time of vibration, I — t 2 X 39.1017 inches. 



VELOCITY, ACCELERATION", FALLING BODIES. 423 

The time of vibration of a pendulum may be varied by the addition of a 
weight at a point above the centre of suspension, which counteracts the 
lower weight, and lengthens the period of vibration. By varying the height 
of the upper weight the time is varied. 

To find the weight of the upper bob of a compound pendulum, vibrating 
seconds, when the weight of the lower bob, and the distances of the weights 
from the point of suspension are given: 



19.1 + d) + d 2 ' 



W — the weight of the lower bob, w = the weight of the upper bob; D = 
the distance of the lower bob and d = the distance of the upper bob from 
the point of suspension, in inches. 

Thus, by means of a second bob, short pendulums may be constructed to 
vibrate as slowly as longer pendulums. 

By increasing w or d until the lower weight is entirely counterbalanced, 
the time of vibration may be made infinite. 

Conical Pendulum.— A weight suspended by a cord and revolving 
at a uniform speed in the circumference of a circular horizontal plane 
whose radius is r, the distance of the plane below the point of suspension be- 
ing h, is held in equilibrium by three forces — the tension in the cord, the cen- 
trifugal force, which tends to increase the radius r, and the force of gravity 
acting downward. If v = the velocity in feet per second, the centre of 
gravity of the weight, as it describes the circumference, g = 32.16, and r 
and h are taken in feet, the time in seconds of performing one revolution is 






PI = .8146/2. 

4.7T 2 



If t = 1 second, h - .8146 foot - 9.775 inches. 

The principle of the conical pendulum is used in the ordinary fly-ball 
governor for steam-engines. (See Governors.) 

CENTRIFUGAL. FORCE. 

A body revolving in a curved path of radius = E in feet exerts a force, 
called centrifugal force, F, upon the arm or cord which restrains it from 
moving in a straight line, or "flying off at a tangent." If W = weight of 
the body in pounds, N = number of revolutions per minute, V= linear 
velocity of the centre of gravity of the body, in feet per second, g = 32.16, 
then 

2nRN „ Wv* W\P WWRN* WRN* niWAmvirvTm itJ 

V = -60-; F = ~W = 22AQR = 36000 = ^933~ = • 00034 1°^^ 2 **• 

l£n = number of revolutions per second, F = 1. 2276 WRn*. 

(For centrifugal force in fly-wheels, see Fly-wheels.) 

VELOCITY, ACCELERATION, FALLING BODIES. 

Velocity is the rate of motion, or the distance passed over by a body in 
a given time. 

If s = space in feet passed over in t seconds, and v = velocity in feet per 
second, if the velocity is uniform, 

v = -; s = vt; t = -. 
t v 

If the velocity varies uniformly, the mean velocity v = 1 j~ 2 , in which 

^i is the velocity at the beginning and v? the velocity at the end of the time t. 

» = *Jp* <i) 

Acceleration is the change in velocity which takes place in a unit of 
time. Unit of acceleration = a = 1 foot per second in one second. For 
uniformly varying velocity, the acceleration is a constant quantity, and 

a - - 2 — t — H v % = Vi + at; Vi = v % ~at; t = — -. , . .(2) 



424 MECHANICS. 

If the body start from rest, v x = 0; then 

v = — ; v 2 = 2u ; a = -?- ; i> 2 = a*; i? 2 — a* = 0; * = - 2 -. 
Combining (1) and (2), we have 

If Vl = 0, 8 = ^t. 

Retarded Motion.— If the body start with a velocity v x and come to 
rest, v 2 = 0; then s = ~t. 
In any case, if the change in velocity is v, 

v. i; 2 a JO 

'=«* S= W S =^ 
For a body starting from or ending at rest, we have the equations 

v + at2 o » 

v = at; s = -c; s = — ; v 1 — 2as. 

Falling Bodies.- In the case of falling bodies the acceleration due 
to gravity is 32.16 feet per second in one second, = g. Then if v = velocity 
acquired' at the end of t seconds, or final velocity, and h - height or space 
in feet passed over in the same time, 

. . _ 2h 

v - gt = 32.16* = V2gh - 8.02 \/h = -j ', 

2 2g 64.32 2' 

~g "32.16 ~y gr "4.01 ~ v ' 

w = space fallen through in the Tth second = g(T — y£). 
Value of g.— The value of g increases with the latitude, and decreases 
with the elevation. At the latitude of Philadelphia, 40°, its value is 32.16. At 
the sea-level, Everett gives g = 32.173 - .082 cos 2 lat. -.000003 height in 
feet. 

Values of \/2g, calculated by an equation given by C. S. Pierce, are given 
in a table in Smith's Hydraulics, from which we take the following : 
Latitude...^.. 0° 10° 20° 30° 40° 50° 60° 

Value of V2g.. &0112 8.0118 8.0137 8.0165 8.0199 8.0235 8.0269 

The value of V2g decreases about .0004 for every 1000 feet increase in ele- 
vation above the sea-level. 

For all ordinary calculations for the United States, g is generally taken at 
32.16, and V2g at 8.02. In England g = 32.2, V%g = 8.025. Practical limit- 
ing values of g for the United States, according to Pierce, are : 

Latitude 49° at sea-level g — 32.186 

25° 10,000 feet above the sea g - 32.089 

From the above formula for falling bodies we obtain the following : 
During the first second the body starting from a state of rest (resistance 
of the air neglected) falls g -=- 2 = 16.08 feet ; the acquired velocity is g — 

at 2 
32.16 ft. per sec. ; the distance fallen in two seconds is h = -^ = 16.08 X 4 .= 

64.32 ft. ; and the acquired velocity is v = gt = 64.32 ft. The acceleration, or 
increase of velocity in each second, is constant, aud is 32.16 ft. per sec. Solv- 
ing the equations for difterent times, we find for 

Seconds,* 1 2 3 4 5 6 

Acceleration, g 32.16 X. 1 1 1 1 1 1 

Velocity acquired at end of time, v 32.16x1 2 3 4 .5 6 

Height of fall in each second, u... ... - 1 — X 1 3 5 7 9 11 

Total height of fall, h ^^ X 1 4 9 16 25 36 



/ 2h 



VELOCITY, ACCELERATION, FALLIKG BODIES. 425 



Fig. 95 represents graphically the velocity, space, etc., of a body falling for 
six seconds. The vertical line at the left is ■ 
the time in seconds, the horizontal lines 
represent one half the acquired velocities 
at the end of each second. The area of 
the small triangle at the top represents 
the height fallen through in the first 
second = y 2 g = 16.08 feet, and each of the 
other triangles is an equal space. The 
number of triangles between each pair of 
horizontal lines represents the height of 
fall in each second, and the number of 
triangles between any horizontal line and 
the top is the total height fallen during 16 
the time. The figures under h, tt, and v 
adjoining the cut are to be multiplied by 
16.08 to obtain the actual velocities and 25 
heights for the given times. 

Angular and Linear Velocity 
of a Turning Body.— Let r = radius of a 
turning body in feet, n = number of revo- 
lutions per minute, v = linear velocity of 
a point on the circumference in feet per second, 
per minute. 

bO 

Angular velocity is a term used to denote the angle through which any 

radius of a body turns in a second, or the rate at which any point in it 

having a radius equal to unit3 r is moving, expressed in feet per second. The 

unit of angular velocity is the angle which at a distance = radius from the 

centre is subtended by an arc equal to the radius. This unit angle = ; — 




Fig. 95. 
: velocity in feet 



degrees = 



2n X 57.3° = 360°, 
v 27m 
velocity, v = Ar, A = - — -^-. 



or the circumference. If A = angular 



Height Corresponding to a Given Acquired Velocity. 


>s 




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feet 


feet. 


feet 


feet. 


feet 


feet. 


feet 


feet. 


feet 


feet. 


feet 


feet. 


p. sec. 




p. sec. 




p. sec. 




p. sec. 




p. sec. 




p. sec. 




.25 


.0010 


13 


2.62 


34 


17.9 


55 


47.0 


76 


89.8 


97 


146 


.50 


.0039 


14 


3.04 


35 


19.0 


56 


48.8 


77 


92.2 


98 


149 


.75 


.0087 


15 


3.49 


36 


20.1 


57 


50.5 


78 


94.6 


99 


152 


1.00 


.016 


16 


3.98 


37 


21.3 


58 


52.3 


79 


97.0 


100 


155 


1.25 


.024 


17 


4.49 


38 


22.4 


59 


54.1 


80 


99.5 


105 


171 


1.50 


.035 


18 


5.03 


39 


23.6 


60 


56.0 


81 


102.0 


110 


188 


1.75 


.048 


19 


5.61 


40 


24.9 


61 


57.9 


82 


104.5 


115 


205 


2 


.062 


20 


6.22 


41 


26.1 


62 


59.8 


83 


107.1 


120 


224 


2.5 


.097 


21 


6.85 


42 


27.4 


63 


61.7 


84 


109.7 


130 


263 


3 


.140 


22 


7.52 


43 


28.7 


64 


63.7 


85 


112.3 


140 


304 


3.5 


.190 


23 


8.21 


44 


30.1 


65 


65.7 


S6 


115.0 


150 


350 


4 


.248 


24 


8.94 


45 


31.4 


66 


67.7 


87 


117.7 


175 


476 


4.5 


.314 


25 


9.71 


46 


32.9 


67 


69.8 


88 


120.4 


200 


622 


5 


.388 


26 


10.5 


47 


34.3 


68 


71 .9 


89 


123.2 


300 


1399 


6 


.559 


27 


11.3 


48 


35.8 


69 


74.0 


90 


125.9 


400 


2488 


7 


.761 


28 


12.2 


49 


37.3 


70 


76.2 


91 


128.7 


500 


3887 


8 


.994 


29 


13.1 


50 


38.9 


71 


78.4 


92 


131.6 


600 


5597 


9 


1.26 


30 


14.0 


51 


40.4 


72 


80.6 


93 


134.5 


700 


7618 


10 


1.55 


31 


14.9 


52 


42.0 


73 


82.9 


94 


137.4 


800 


9952 


11 


1.88 


32 


15.9 


53 


43.7 


74 


85.1 


95 


140.3 


900 


12593 


12 


2.24 


33 


16.9 


54 


45.3 


75 


87.5 


96 


143.3 


1000 


15547 



426 



MECHANICS. 



Falling Bodies 



Velocity Acquired by a Body Falling: a 
Given Height. 



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feet. 


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feet 


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feet 


feet. 


feet 


feet. 


feet 


feet. 


feet 




p. sec. 




p. sec. 




p. sec. 




p. sec. 




p. sec. 




p. sec. 


.005 


.57 


.39 


5.01 


1 20 


8.79 


5. 


17.9 


23. 


38.5 


72 


68.1 


.010 


.80 


.40 


5.07 


1 .22 


8.87 


.2 


18.3 


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38.9 


73 


68.5 


.015 


.98 


.41 


5.14 


1.24 


8.94 


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18.7 


24. 


39.3 


74 


69.0 


.020 


1.13 


.42 


5.20 


1.26 


9.01 


.6 


19.0 


.5 


39.7 


75 


69.5 


.025 


1.27 


.43 


5.26 


1.28 


9.08 


.8 


19.3 


25 


40.1 


76 


69.9 


.030 


1.39 


.44 


5.32 


1.30 


9.15 


6. 


19.7 


26 


40.9 


77 


70.4 


.035 


1.50 


.45 


5.38 


1.32 


9.21 


.2 


20.0 


27 


41.7 


78 


70.9 


.040 


1.60 


.46 


5.44 


1.34 


9.29 


.4 


20.3 


28 


42.5 


79 


'71.3 


.045 


1.70 


.47 


5.50 


1.36 


9.36 


.6 


20.6 


29 


43.2 


80 


71.8 


.050 


1.79 


.48 


5.56 


1.38 


9.43 


.8 


20.9 


30 


43.9 


81 


72.2 


.055 


1.88 


.49 


5.61 


1.40 


9.49 


7. 


21.2 


31 


44.7 


82 


72.6 


.060 


1.97 


.50 


5.67 


1.42 


9.57 


.2 


21.5 


32 


45.4 


83 


73.1 


.065 


2.04 


.51 


5.73 


1.44 


9 62 


.4 


21.8 


33 


46.1 


84 


73.5 


.070 


2.12 


.52 


5.78 


1.46 


9.70 


.6 


22.1 


34 


46.8 


85 


74.0 


.075 


2.20 


.53 


5.84 


1.48 


9.77 


.8 


22.4 


35 


47.4 


86 


74.4 


.080 


2.27 


.54 


5.90 


1.50 


9.82 


8. 


22.7 


36 


48.1 


87 


74.8 


.085 


2.34 


.55 


5.95 


1.52 


9.90 


.2 


23.0 


37 


48.8 


88 


75.3 


.090 


2.41 


.56 


6.00 


1.54 


9.96 


.4 


23.3 


38 


49.4 


89 


75.7 


.095 


2.47 


.57 


6.06 


1.56 


10.0 


.6 


23.5 


39 


50.1 


90 


76.1 


.100 


2.54 


.58 


6.11 


1.58 


10.1 


.8 


23.8 


40 


50.7 


91 


76.5 


.105 


2.60 


.59 


6.16 


1.60 


10.2 


9. 


24.1 


41 


51.4 


92 


76.9 


.110 


2.66 


.60 


6.21 


1.65 


10.3 


.2 


24.3 


42 


52.0 


93 


77.4 


.115 


2.72 


.62 


6.32 


1.70 


10.5 


.4 


24.6 


43 


52.6 


94 


77.8 


.120 


2.78 


.64 


6.42 


1.75 


10.6 


.6 


24.8 


44 


53.2 


95 


78.2 


.125 


2.84 


.66 


6.52 


1.80 


10.8 


.8 


25.1 


45 


53.8 


96 


78.6 


.130 


2.89 


.68 


6.61 


1.90 


11.1 


10. 


25.4 


46 


54.4 


97 


79.0 


.14 


3.00 


.70 


6.71 


2. 


11.4 


.5 


26.0 


47 


55.0 


98 


79.4 


.15 


3.11 


.72 


6.81 


2.1 


11.7 


11. 


26.6 


48 


55.6 


99 


79.8 


.16 


3.21 


!74 


6.90 


2.2 


11.9 


.5 


27.2 


49 


56.1 


100 


80.2 


.17 


3.31 


.76 


6.99 


2.3 


12.2 


12. 


27.8 


50 


56.7 


125 


89.7 


.18 


3.40 


.78 


7.09 


2.4 


12.4 


.5 


28.4 


51 


57.3 


150 


98.3 


.19 


3.50 


.80 


7.18 


2.5 


12.6 


13. 


28.9 


52 


57.8 


175 


106 


.20 


3.59 


.82 


7.26 


2.6 


12.9 


.5 


29.5 


53 


58.4 


200 


114 


.21 


3.68 


.84 


7.35 


2.7 


13.2 


14. 


30.0 


54 


59.0 


225 


120 


.22 


3.76 


.86 


7.44 


2.8 


13.4 


.5 


30.5 


55 


59.5 


250 


126 


.23 


3.85 


.88 


7.53 


2.9 


13.7 


15. 


81.1 


56 


60.0 


275 


133 


.24 


3.93 


.90 


7.61 


3. 


13.9 


.5 


31.6 


57 


60.6 


300 


139 


.25 


4.01 


.92 


7.69 


3.1 


14.1 


16. 


32.1 


58 


61.1 


350 


150 


.26 


4.09 


.94 


7.78 


3-2 


14.3 


.5 


32.6 


59 


61.6 


400 


160 


.27 


4.17 


.96 


7.86 


3.3 


14.5 


17. 


S3.1 


60 


62.1 


450 


170 


.28 


4.25 


.98 


7.94 


3.4 


14.8 


.5 


33.6 


61 


62.7 


500 


179 


.29 


4.32 


1.00 


8.02 


3.5 


15.0 


18. 


34.0 


62 


63.2 


550 


188 


.30 


4.39 


1.02 


8.10 


3.6 


15.2 


.5 


31.5 


63 


63.7 


600 


197 


.31 


4.47 


1.04 


8.18 


37 


15.4 


19. 


35.0 


64 


64.2 


700 


212 


.32 


4.54 


1.06 


8.26 


3-8 


15.6 


.5 


35.4 


65 


64.7 


800 


227 


.33 


4.61 


1.08 


8.34 


3-9 


15.8 


20. 


35.9 


66 


65.2 


900 


241 


.34 


4.68 


1.10 


8.41 


4- 


16.0 


.5 


36.3 


67 


65.7 


1000 


254 


.35 


4.74 


1.12 


8.49 


.2 


16.4 


81. 


36.8 


68 


66.1 


2000 


359 


.36 


4.81 


1.14 


8.57 


.4 


16.8 


.5 


37.2 


69 


66.6 


3000 


439 


.37 


4.88 


1.16 


8.64 


.6 


17.2 


22. 


37.6 


70 


67.1 


4000 


507 


.38 


4.94 


1.18 


8.72 


.8 


17.6 


.5 


38.1 


71 


67.6 


5000 


567 



Parallelogram of Velocities.— The principle of the composition 
and resolution of forces may also be applied to velocities or to distances 
moved in given intervals of time. Referring to Fig. 88, page 416, if a body 
at O has a force applied to it which acting alone would give it a velocity 
represented by OQ per second, and at the same time it is acted on by 




VELOCITY, ACCELERATION, FALLING BODIES. 427 

another force which acting alone would give it a velocity OP per second, 
the result of the two forces acting together for one second will carry it to 
R, OR being the diagonal of the parallelogram of OQ and OP, and the 
resultant velocity. If the two component velocities are uniform, the result- 
ant will be uniform and the line OR will be a straight line; but if either 
velocity is a varying one, the line will be a curve. Fig. 96 shows the 
resultant velocities, also the path traversed 
by a body acted on by two forces, one of 
which would carry it at; a uniform velocity 
over the intervals 1, 2, 3, B, and the other of 
which would carry it by an accelerated mo- 
tion over the intervals a, b, c, D in the same 
times. At the end of the respective inter- 
vals the body will be found at C,, C. 2 , C 3 , C, 
and the mean velocity during each interval 
is represented by the distances between 
these' points. Such a curved path is trav- 
ersed by a shot, the impelling force from 
the gun giving it a uniform velocity in the 
direction the gun is aimed, and gravity giv- 
ing it an accelerated velocity downward. 
The path of a projectile is a parabola. The 

distance it will travel is greatest when its initial direction is at an angle 45° 
above the horizontal. 

Mass— Force of Acceleration.-— The mass of a body, or the quantity 
of matter it contains, is a constant quantity, while the weight varies according 
to the variation in the force of gravity at different places. If g — the acceler- 
ation due to gravity, and w — weight, then the mass m — ~,io = mg. Weight 

here means the resultant of the force of gravity on the particles of a body, 
such as may be measured by a spriug-balance, or by the extension or 
deflection of a rod of metalloaded with the given weight. 

Force has been defined as that which causes, or tends to cause, or to 
destroy, motion. It may also be defined (Kennedy's Mechanics of Ma- 
chinery) as the cause of acceleration; and the unit of force as the force 
required to produce unit acceleration in a unit of free mass. 

Force equals the product of the mass by the acceleration, or f "'= ma. 

Also, if v = the velocity acquired in the time t, ft = mv; f — mv -*- t; the 
acceleration being uniform. 

The force required to produce an acceleration of g (that is, 32.16 ft. per 

sec.) in one second is / = mg = —g = w, or the weight of the body. Also, 
/ = ma = m 2 1 , in which r 2 is the velocity at the end, and v x the 

W (Vn — V-,) W 

velocity at the beginning of the time t, and/ = mg = = —a; 

— = -; or, the force required to give any acceleration to a body is to the 

weight of the body as that acceleration is to the acceleration produced by 
gravity. (The weight iv is the weight where g is measured.) 

Example.— Tension in a cord lifting a weight. A weight of 100 lbs. is 
lifted vertically by a cord a distance of 80 feet in 4 seconds, the velocity 
uniformly increasing from to the end of the time. What tension must be 
maintained in the cord? Mean velocity = v = 20 ft. per sec; final velocity 

= u„ = 2v - 40; accele-ation a — ^ = — = 10. Force / — ma = — = -oTTr x 

t 4 g o4.1\j 

10 = 31.1 lbs. This is the force required to produce the acceleration only; 
to it must be added the force required to lift the weight without accelera- 
tion, or 100 lbs., making a total of 131.1 lbs. 

The Resistance to Acceleration is the same as the force required to pro- 



Formulae for Accelerated Motion.— For cases of uniformly 
accelerated motion other than those of falling bodies, we have the formulae 

already given, / - - a, = H~* If tne body starts from rest, v x = 0, v 2 

9 9 *■ 



428 MECHANICS. 

= v, and/= — t, fgt -■ wv. We also have s = --. Transforming and sub- 
s' t -A 
stituting for g its value 32.16, we obtain 



/ = 


wv 2 
64.32s ~ 


wv 
32.16* "" 


IVS 

16708^ 




wv 2 _ 


16.08/f 2 


_ vt . 



' 64.32/ " 




- wv - _i_ i/— 
32.16/ ~ 4.01 \ f 



(^2 _ v 2\ 
- „ ) . 

(See also Work of Acceleration, under Work.) 

Motion on Inclined Planes.— The velocity acquired by a body 
descending an inclined plane by the force of gravity (friction neglected) is 
equal to that acquired by a body falling freely from the height of the plane. 

The times of descent down different inclined planes of the same height 
vary as the length of the planes. 

The rules for uniformly accelerated motion apply to inclined planes. If a 
is the angle of the plane with the horizontal, sin a = the ratio of the height 

to the length = - , and the constant accelerating force is g sin a. The final 

velocity at the end of t seconds is v — gt sin a. The distance passed over in 
t seconds is I = y% gt 2 sin a. The time of descent is 

I 



4.01 Yh 



MOMENTUM, VIS-VIVA. 

Momentum, or quantity of motion in a body, is the product of the mass 
by the velocity at any instant = mv — — v. 

Since the moving force = product of mass by acceleration, / = ma; and if 

the velocity acquired in t seconds = v, or a = -, / = -r- ; ft = mv; that is, 

the product of a constant force into the time in which it acts equals numer- 
ically the momentum. 

Since ft = mv, if t — 1 second mv — /. whence momentum might be de- 
fined as numerically equivalent to the number of pounds of force that will 
stop a moving body in 1 second, or the number of pounds of force which 
acting during 1 second will give it the given velocity. 

Vis-viva, or living force, is a term used by early writers on Mechanics 
to denote the energy stored in a moving body. Some defined it as the pro- 
duct of the mass into the square of the velocity, mv 2 , — —v 2 others as one 

half of this quantity or y 2 mv 2 , or the same as what is now known as energy. 
The term is now practically obsolete, its place being taken by the word 
energy. 

WORK, ENERGY, POWER. 

Work is the overcoming of resistance through a. certain distance. It is 
measured by the product of the resistance into the space through which it 
is overcome. It is also measured by the product of the moving force into 
the distance through which the force acts in overcoming the resistance. 
Thus in lifting a body from the earth against the attraction of gravity, the 
resistance is the weight of the body, and the product of this weight into the 
height the body is lifted is the work done. 

The Unit of Work, in British measures, is the foot-pound, or the 
amount of work done in overcoming a pressure or weight equal to one 
pound through one foot of space. 



WORK, ENERGY, POWER. 429 

The work performed by a piston in driving a fluid before it, or by a fluid 
in driving a piston before it, may be expressed in either of the following 
ways : 

Resistance X distance traversed 
= intensity of pressure x area x distance traversed ; 
=■ intensity of pressure X volume traversed. 

The work performed in lifting a body is the product of the weight of the 
body into the height through which its centre of gravity is lifted. 

If a machine lifts the centres of gravity of several bodies at once to heights 
either the same or different, the whole quantity of work performed in so 
doing is the sum of the several products of the weights and heights ; but 
that quantity can also be computed by multiplying the sum of all the 
weights into the height through which their common centre of gravity is 
lifted. (Rankine.) 

Power is the rate at which work is done, and is expressed by the quo- 
tient of the work divided by the time in which it is done, or by units of work 
per second, per minute, etc., as foot-pounds per second. The most common 
unit of power is the horse-jjoiver, established by James Watt as the power of 
a strong London draught-horse to do work during a short interval, and used 
by him to measure the power of his steam-engines. This unit is 33,000 foot- 
pounds per minute = 550 foot-pounds per second = 1,980,000 foot-pounds per 
hour. 

Expressions for Force, "Work, Power, etc. 

The fundamental conceptions in Dynamics are : 

Force, Time, Space, represented by the letters F, T, S. 

Velocity = space divided by time, V = — , if Fbe uniform. 

Work = product of force into space = JTS = W = FVT. (F uniform.) 

FS 
Power = rate of work = work divided by time = — - = P = product of 

force into velocity = FV. 

Power exerted for a certain time produces work; PT = FS = FVT = W. 

Effort is a name applied to a force which acts on a body in the direction 
of its motion. 

Resistance is that which is opposed to a moving force. It is equal and 
opposite force. 

Horse-power Hours, an expression for work measured as the 
product of a power into the time during which it acts = PT. Sometimes it 
is the summation of a variable power for a given time, or the average power 
multiplied by the time. 

Energy, or stored work, is the capacity for performing work. It is 
measured by the same unit as work, that is, in foot-pounds. It may be 
either -potential, as in the case of a body of water stored in a reservoir, 
capable of doing work by means of a water-wheel, or actual, sometimes 
called kinetic, which is the energy of a moving body. Potential energy is 
measured by the product of the weight of the stored body into the distance 
through which it is capable of acting, or by the product of the pressure it 
exerts into the distance through which that pressure is capable of acting. 
Potential energy may also exist as stored heat, or as stored chemical energy, 
as in fuel, gunpowder, etc., or as electrical energy, the measure of these 
energies being the amount of work that they are capable of performing. 
Actual energy of a moving body is the work which it is capable of performing 
against a retarding resistance before being brought to rest, and is equal to 
the work which must be done upon it to bring it from a state of rest to its 
actual velocity. 

The measure of actual energy is the product of the weight of the body 
into the height from which it must fall to acquire its actual velocity. If v = 
the velocitv in feet per second, according to the principle of falling bodies, 

v i 
h, the height due to the velocity = — , and if w = the weight, the energy = 



the mass into the square of the velocity = ^mv 2 . Since energy is the capacity 
for performing work, the units of work and energy are equivalent, or FS = 

l&nv 2 = — = ivh. Energy exerted = work done. 



430 MECHANICS. 

The actual energy of a rotating body whose angular velocity is A and 

moment of inertia 2wr 2 = / is -■— , that is, the product of the moment of 

2g 
inertia into the height due to the velocity, A, of a point whose distance from 

the axis of rotation is unity; or it is equal to — — , in which w is the weight of 

the bodv and v is the velocity of the centre of gyration. 

Work of Acceleration. -The work done in giving acceleration to a 
body is equal to the product of the force producing the acceleration, or of 
the resistance to acceleration, into the distance moved in a given time. This 
force, as already stated equals the product of the mass into the acceleration, 

or f = ma = ^-^ — - 1 . If the distance traversed in the time t = s, then 

9 t 

W Vn — Vj 

work — fs = 7 s. 

g . t 
Example.— What work is required to move a body weighing 100 lbs. hori- 
zontally a distance of 80 ft. in 4 seconds, the velocity uniformly increasing, 
friction neglected ? 
Mean velocity v = 20 ft. per second; final velocity = Vn — 2v — 40; initial 
, . , . Vn - v, 40 An £ io 100 

velocity Vi = 0; acceleration, a = -^— = — = 10; force = —a = ,— —^ x 

10 = 31.1 lbs. ; distance 80 ft. ; work = fs = 31 .1x80 = 2488 foot-pounds." 

The energy stored in the body moving at the final velocity of 40 ft. per 
second is 

„ 1 w „ 100 X 402 
y 2 mv-> = - - g v* = 2 x 3216 = 2488 foot-pounds, 

which equals the work of acceleration, 

IV Vn 10 Vn Vn, 1 io 

If a body of the weight W falls from a height H, the work of acceleration 
is simply WH, or the same as the work required to raise the body to the 
same height. 

Work of Accelerated Rotation.— Let A = angular velocity of a 
solid body rotating about an axis, that is, the velocity of a particle whose 
radius is unity. Then the velocity of a particle whose radius is r is v — Ar. 
If the angular velocity is accelerated from A] to^4 2 » the increase of the 
velocity of the particle is Vn — v 1 = r(A 1 - An), and the work of accelerating 
it is 

w v.;? - v^ _ wr* A^-AJ 
9 X 2 ~ g 2 ' 

in which w is the weight of the particle. 

The work of acceleration of the whole body is 



^l. 



\~ *9 



- X 2wr 2 . 



The term 2?cr 2 is the moment of inertia of the bodv. 

" Force of the Blow " of a Steam Hammer or Other Fall- 
ing Weight.— The question is often asked: "With what force does a 
falling hammer strike?" The question cannot be answered directly, and 
it is based upon a misconception or ignorance of fundamental mechanical 
laws. The energy, or capacity of doing work, of a body raised to a given 
height and let fall cannot be expressed in pounds, simply, but only in foot- 
pounds, which is the product of the weight into the height through which 
it fails, or the product of its weight -t- 64.32 into the square of the velocity, 
in feet per second, which it acquires after falling through the given height. 
If F = weight of the body, M its mass, g the acceleration due to gravity, 
S the height of fall, and v the velocity at the end of the fall, the energy in 
the body just before striking, is FS = i^M-u 2 = Wv* -f- 2g = Wv* -*- 64.32, 
which is the general equation of energy of a moving body. Just as the 
energy of the body is a product of a force into a distance, so the work it 
does when it strikes is not the manifestation of a force, which can be ex- 
pressed simply in pounds, but it is the overcoming of a resistance through 
a certain distance, which is expressed as the product of the average resist- 



WORK, EtfERGY, POWER. 431 

ance into the distance through which it is exerted. If a hammer weighing 
100 lbs. falls 10 ft., its energy is 1000 foot-pounds. Before being brought to 
rest it must do 1000 foot-pounds of work against one or more resistances. 
These are of various kinds, such as that due to motion imparted to the body 
struck, penetration against friction, or against resistance to shearing or 
other deformation, and crushing and heating of both the falling body and the 
body struck. The distance through which these resisting forces act is gen- 
erally indeterminate, and therefore the average of the resisting forces, 
which themselves generally vary with the distance, is also indeterminate. 
Impact of Bodies.— If two inelastic bodies collide, they will move on 
together as one mass, with a common velocity. The momentum of the com- 
bined mass is equal to the sum of the momenta of the two bodies before im- 
pact. If m x and m a are the masses of the two bodies and v x and v 2 their re- 
spective velocities before impact, and v their common velocity after impact, 
(" l i + vi a )v = m^Vx X m 2 v 2 , 

_ ffli^i -f" "*2 V 2 * 

- 2 , or, the velocity 
..„i + m 2 

of two inelastic bodies after impact is equal to the algebraic sum of their 
momenta before impact, divided by the sum of their masses. 

If two inelastic bodies of equal momenta impinge directly upon one an- 
other from opposite directions they will be brought to rest. 

Impact of Inelastic Bodies Causes a JLoss of Energy, and 
this loss is equal to the sum of the energies due to the velocities lost and 
gained by the bodies, respectively. 

\bm x v x * + ^m 2 r 2 2 - %{m x -f m 2 )u 2 = V z m x {v x - v) 2 -f- i^m 2 (t; 2 - u) 2 . 

In which v x — vis the velocity lost by m 1 and v - v 2 the velocity gained by m 2 . 
Example— Let m, — 10, m 2 = 8, v x = 12, v 2 — 15. 

If the bodies collide they will come to rest, for v = = 0. 

The enei'gy loss is 
3^10 X 144 + y 2 8 X 225 - ^ 18 X = ^10(12 - 0) 2 -f 3^8(15 - 0)« = 1620 ft. lbs. 

What becomes of the energy lost ? Ans. It is used doing internal work 
on the bodies themselves, changing their shape and heating them. 

For imperfectly elastic bodies, let e = the elasticity, that is, the ratio 
which the force of restitution, or the internal force tending to restore the 
shape of a body after it has been compressed, bears to the force of compres- 
sion; and let vi x and m 2 be the masses, v x and v 2 their velocities before im- 
pact, and v x 'v 2 ' their velocities after impact: then 

, _ niiVi -4- m 2 v 2 _ m 2 e{v x — v 2 ) _ 
1 — Hij -f m 2 vi x + m 2 ' 

, m x v x + m 2 v 2 , m x e(v x — v 2 ) 

Vn' = p 1 . 

m 1 -\-m 2 m x -\-m 2 

If the bodies are perfectly elastic, their relative velocities before and after 
impact are the same. That is : v x ' — v 2 ' = v 2 — v x . 

In the impact of bodies, the sum of "their momenta after impact is the 
same as the sum of their momenta before impact. 

m x v x ' -f m 2 v 2 ' — m x v x -f m 2 v 2 . 

For demonstration of these and other laws of impact, see Smith's Me- 
chanics; also, Weisbach's Mechanics. 
Energy of Recoil of Guns.— (Eng'g, Jan. 25, 1884, p. 72.) 
Let W — the weight of the gun and carriage; 
V = the maximum velocity of recoil; 
w = the weight of the projectile; 
v — the muzzle velocity of the projectile. 
Then, since the momentum of the gun and carriage is equal to the momen- 
tum of the projectile, we have WV = ivv, or V = wv -v- W. 

* The statement by Prof. W. D. Marks, in NystronTs Mechanics, 20th edi- 
tion, p. 454, that this formula is in error is itself erroneous. 



432 MECHANICS. 

Taking the case of a 10-inch gun firing a 400-lb. projectile with a muzzle 
velocity of 1400 feet per second, the weight of the gun and carriage being 22 
tons = 49,280 lbs., we find the velocity of recoil — 

Tr 1400X400 

V= — g — =11 feet per second. 



Now the energy of a body in motion is WV 2 -f- 2g. 

49 280 X ll 2 
Therefore the energy of recoil = — | ■ = 92,593 foot-pounds. 

400 X 1400 2 
The energy of the projectile is = 12,173,913 foot-pounds. 

Conservation of Energy.— No form of energy can ever be pro- 
duced except by the expenditure of some other form, nor annihilated ex- 
cept by being reproduced in another form. Consequently the sum total of 
energy in the universe, like the sum total of matter, must always remain 
the same. (S. Newcomb.) Energy can never be destroyed or lost; it can 
be transformed, can be transferred from one body to another, but no 
matter what transformations are undergone, when the total effects of the 
exertion of a given amount of energy are summed up the result will be 
exactly equal to the amount originally expended from the source. This law 
is called the Conservation of Energy. (Cotterill and Slade.) 

A heavy body sustained at an elevated position has potential energy. 
When it falls, just before it reaches the earth's surface it has actual or 
kinetic energy, due to its velocity. When it strikes it may penetrate the 
earth a certain distance or may be crushed. In either case friction results 
by which the energy is converted into heat, which is gradually radiated 
into the earth or into the atmosphere, or both. Mechanical energy and heat 
are mutually convertible. Electric energy is also convertible into heat or 
mechanical energy, and either kind of energy may be converted into the 
other. 

Sources of Energy.— The principal sources of energy on the earth's 
surface are the muscular energy of men and animals, the energy of the 
wind, of flowing water, and of fuel. These sources derive their energy 
from the rays of the sun. Under the influence of the sun's rays vegetation 
grows and wood is formed. The wood may be used as fuel under a steam 
boiler, its carbon being burned to carbonic acid. Three tenths of its heat 
energy escapes in the chimney and by radiation, and seven tenths appears 
as potential energy in the steam. In the steam-engine, of this seven tenths 
six parts are dissipated in heating the condensing water and are wasted ; 
the remaining one tenth of the original heat energy of the wood is con- 
verted into mechanical work in the steam-engine, which may be used to 
drive machinery. This work is finally, by friction of various kinds, or pos- 
sibly after transformation into electric currents, transformed into heat, 
which is radiated into the atmosphere, increasing its temperature. Thus 
all the potential heat energy of the wood is, after various transformations, 
converted into heat, which, mingling with the store of heat in the atmos- 
phere, apparently is lost. But the carbonic acid generated by the combus- 
tion of the wood is, again, under the influence of the sun's rays, absorbed 
by vegetation, and more wood may thus be formed having potential energy 
equal to the original. 

Perpetual Motion.— The law of the conservation of energs% than 
which no law of mechanics is more firmly established, is an absolute barrier 
to all schemes for obtaining by mechanical means what is called " perpetual 
motion," or a machine which will do an amount of work greater than the 
equivalent of the energy, whether of heat, of chemical combination, of elec- 
tricity, or mechanical energy, that is put into it. Such a result would be 
the creation of an additional store of energy in the universe, which is not 
possible by any human agency. 

The Efficiency of a Machine is a fraction expressing the ratio of 
the useful work to the whole work performed, which is equal to the energy 
expended. The limit to the efficiency of a machine is unity, denoting the 
efficiency of a perfect machine in which no work is lost. The difference 
between the energy expended and the useful work done, or the loss, is 
usually expended either in overcoming friction or in doing work on bodies 
surrounding the machine from which no useful work is received. Thus in 
an engine propelling a vessel part of the energy exerted in the cylinder 



AHIMAL POWER. 



433 



does the useful work of giving motion to the vessel, and the remainder is 
spent in overcoming the friction of the machinery and in making currents 
and eddies in the surrounding water. 

ANIMAL, POWER. 

Work of a Man against Known Resistances. (Rankine.) 



Kind of Exertion. 



R, 

lbs. 



ft. per 
sec. 



T" 

3600 

(hours 

per 

day). 



RV, 
ft.-lbs. 
per sec. 



RVT, 

ft.-lbs. 
per day. 



Raising his own weight up 
stair or ladder 

Hauling up weights with rope, 
and lowering the rope un 
loaded 

Lifting weights by hand 

Carrying weights up-stairs 
and returning unloaded 

Shovelling up earth tc 
height of 5 ft. 3 in 

Wheeling earth in barrow up 
slope of 1 in 12, % horiz. 
veloc. 0.9 ft. per sec. and re- 
turning unloaded 

Pushing or pulling horizon- 
tally (capstan or oar) 



8. Turning a crank or winch . 



, Working pump . . 
). Hammering .... 



143 

40 
44 

143 

6 

132 

26.5 
12.5 
18.0 
20.0 
13.2 
15 



0.75 
0.55 



0.13 
1.3 



2.0 
5.0 
2.5 
14.4 
2.5 



72.5 

30 
24.2 

18.5 

7.8 



53 
62.5 



2,088,000 



648,000 
522,720 



280,800 



356,400 
1,526,400 



2min. 
10 



1,296,000 



1,188,000 
480,000 



Explanation.— R, resistance; V, effective velocity = distance through 
which R is overcome -h total time occupied, including the time of moving 
unloaded, if any; T", time of working, in seconds per day; T" -i- 3600, same' 
time, in hours per day; RV, effective power, in foot-pounds per second; 
RVT, daily work. 

Performance of a Man in Transporting Loads 
Horizontally. (Rankin e.) 



Kind of Exertion. 



11. Walking unloaded, transport- 

ing his own weight 

12. Wheeling load L in 2-whld. 

barrow, return unloaded. . 

13. Ditto in 1-wh. barrow, ditto.. 

14. Travelling with burden 

15. Carrying burden, returning 
unloaded 

16. Carrying burden, for 30 sec- 
onds only 







T 


LV, 


L, 
lbs. 


V, 
ft. -sec. 


3600 


lbs. 


(hours 
per 


con- 
veyed 






day). 


1 foot. 


140 


5 


10 


700 


224 


1% 


10 


373 


132 


m 


10 


220 


90 


%H 


7 


225 


140 


Ws 


6 


233 


(252 










-h26 


11.7 




1474.2 


1 o 


23.1 








LVT, 
lbs. con- 
veyed 
1 foot. 



25,200,000 



428,000 
920,000 
670,000 



5,032,800 



Explanation.— L, load; V, effective velocity, computed as before; T'', 
time of working, in seconds per day; T" -i- 3600, same time in hours per day; 
LV, transport per second, in lbs. conveyed one foot; LVT, daily transport. 



434 



MECHANICS. 



In the first line only of each of the two tables above is the weight of the 
man taken into account in computing the work done. 
Clark says that the average net daily work of an ordinary laborer at a 
pump, a winch, or a crane may be 
taken at 3300 foot-pounds per minute, 
or one- tenth of a horse-power, for 8 
hours a day; but for shorter periods 
from four to five times this rate may 
be exerted. 

Mr. Glynn says that a man may 
exert a force of 25 lbs. at the handle 
of a crane for short periods; but that 
for continuous work a force of 15 lbs. 
is all that should be assumed, moving 
through 220 feet per minute. 

Man-wheel.— Fig. 97 is a sketch 
of a very efficient man-power hoist- 
ing-machine which the author saw in 
Berne, Switzerland, in 1889. The face 
of the wheel was wide enough for 
three men to walk abreast, so that 
nine men could work in it at one time. 




Work of a Horse against a Known Resistance. 


(Rankine.) 


Kind of Exertion. 


R. 


V. 


T. 
3600 


RV. 


RVT. 


1. Cantering and trotting, draw- 

ing a light railway carriage 
(thoroughbred) 

2. Horse drawing cart or boat, 

walking (draught-horse) 

3. Horse drawing a gin or mill, 


1 min.22^ 
< mean 30^| 
( max. 50 

120 

100 
66 


1 14% 

3.6 

3.0 
6.5 


4. 

8 

8 


447^ 

432 

300 
429 


6,444,000 

12,441,600 

8,640,000 
6,950,000 


4. Ditto, trotting 



Explanation.— R, resistance, in lbs.; V, velocity, in feet per second; T" 
-*- 3600, hours work per day; RV, work per second; RVT, work per day. 

The average power of a draught-horse, as given in line 2 of the above table, 
being 432 foot-pounds per second, is 432/550 = 0.785 of the conventional value 
assigned by Watt to the ordinary unit of the rate of work of prime movers. 
It is the mean of several results of experiments, and may be considered the 
average of ordinary performance under favorable circumstances. 

Performance of a Horse in Transporting Loads 
Horizontally. (Rankine.) 



Kind of Exertion. 


L. 


V. 


T. 


LV. 


LVT. 


5. Walking with cart, always 


1500 
750 

1500 
270 
180 


3.6 

7.2 

2.0 
3.6 
7.2 


10 

4^ 

10 

10 


5400 
5400 

3000 
972 
1296 


194,400,000 


6. Trotting, ditto 

7. Walking with cart, going load- 

ed, returning empty; V, 


87,480,000 

108,000,000 
34,992,000 
32,659,200 


8. Carrying burden, walking 

9. Ditto, trotting 



Explanation.— L, load in lbs.; V, velocity in feet per second; T-^3600, 
working hours per day; LV, transport per second; LVT, transport per day. 

This table has reference to conveyance on common roads only, and those 
evidently in bad order as respects the resistance to traction upon them. 

Horse Gin.— In this machine a horse works less advantageously 
than in drawing a carriage along a straight track. In order that the best 



ELEMENTS OF MACHINES. 435 

possible results may be realized with a horse-gin, the diameter of the cir- 
cular track in which the horse walks should not be less than about forty 
feet. 

Oxen, Mules, Asses.— Authorities differ considerably as to the power 
of these animals. The following may he taken as an approximative com- 
parison between them and draught-horses (Rankine): 

Ox. — Load, the same as that of average draught-horse; best velocity and 
work, two thirds of horse. 

Mule. — Load, one half of that of average draught-horse; best velocity, 
the same with horse: work one half. 

Ass.— Load, one quarter that of average draught-horse; best velocity the 
same; work one quarter. 

Reduction of Draught of Horses by Increase of Grade 
of Roads. (Engineering Record, Prize Essays on Roads, 1892.)— Experi- 
ments on English roads by Gay frier & Parnell: 

Calling load that can be drawn on a level .100: 

On a rise of 1 in 100. 1 in 50. 1 in 40. 1 in 30. 1 in 26. 1 in 20. 1 in 10. 

A horse can draw only 90. 81. 72. 64. 54. 40. 25. 

The Resistance of Carriages on Roads is (according to Gen. 
Morin>given approximately by the following empirical formula: 

W 
R = ~[aA-b(u- 3.28)]. 

In this formula R = total resistance; r = radius of wheel in inches; W = 
gross load; u = velocity in feet per second; while a and b are constants, 
whose values are: For good broken-stone road, a — A to .55, b = .024 to .026; 
for paved roads, a = .27, b ~ .0684. 

Rankine states that on gravel the resistance is about double, and on 
sand five times, the resistance on good broken-stone roads. 

ELEMENTS OF MACHINES. 

The object of a machine is usually to transform the work or mechanical 
energy exerted at the point where the machine receives its motion into 
work at the point where the final resistance 
is overcome. The specific end may be to 
change the character or direction of mo- 
tion, as from circular to rectilinear, or vice 
versa, to change the velocity, or to overcome 
a great resistance by the application of a 
moderate force. In all cases the total energy 
exerted equals the total work done, the latter 
including the overcoming of all the f fictional Fig. 98. 
resistances of the machine as well as the use- 
ful work performed. No increase of power 
can be obtained from any machine, since this 
is impossible according to the law of conser- 
vation of energy. In a f rictionless machine the I B 

product of the force exerted at the driving- 
point into the velocity of the driving-point, 
or the distance it moves in a given interval 
of time, equals the product of the resistance 
into the distance through which the resist- 
ance is overcome in the same time. Fig. 99. 

The most simple machines, or elementary 
machines, are reducible to three classes, viz., 
the Lever, the Cord, and the Inclined Plane. 

The first class includes every machine con- 
sisting of a solid body capable of revolving g 
on an axis, as the Wheel and Axle. 

The second class includes every machine in 
which force is transmitted by means of flexi- 
ble threads, ropes, etc., as the Pulley. 

The third class includes every machine in jr IG jqo. 

which a hard surface inclined to the direc- 
tion of motion is introduced, as the Wedge and the Screw. 

A Lever is an inflexible rod capable of motion about a fixed point, 
called a fulcrum. The rod may be straight or bent at any angle, or curved. 

It is generally regarded, at first, as without weight, but its weight may be 



A C B 

p fj Ow 



Ow 



436 MECHANICS. 

considered as another force applied in a vertical direction at its centre of 
gravity. 

The arms of a lever are the portions of it intercepted between the force, 
P, and fulcrum, C, and between the weight, W, and fulcrum. 

Levers are divided into three kinds or orders, according to the relative 
positions of the applied force, weight, and fulcrum. 

In a lever of the first order, the fulcrum lies between the points at which 
the force and weight act. (Fig. 98.) 

In a lever of the second order, the weight acts at a point between the 
fulcrum and the point of action of the force. (Fig. 99.) 

In a lever of the third order, the point of action of the force is between 
that of the weight and the fulcrum. (Fig. 100.) 

In all cases of levers the relation between the force exerted or the pull, 
P, and the weight lifted, or resistance overcome, W, is expressed by the 
equation P X AC = W X BC, in which AC is the lever-arm of P, and BC 
is the lever-arm of W, or moment of the force = the moment of the resist- 
ance. (See Moment.) 

In cases in which the direction of the force (or of the resistance) is not at 
right angles to the arm of the lever on which it acts, the " lever-arm" is the 
length of a perpendicular from the fulcrum to the line of direction of the 
force (or of the resistance). W : P : : A C : BC, or, the ratio of the resistance to 
the applied force is the inverse ratio of their lever-arms. Also, if Vw is the 
velocity of W, and Vp is the velocity of P, W : F : : Vp : Vw, and Px Vp 
= Wx Vw. 

If Sp is the distance through which the applied force acts, and Sw is the 
distance the weight is lifted or through which the resistance is overcome, 
W : P : : Sp : Sw; Wx Sw= PXSp, or the weight into the distance it is lifted 
equals the force into the distance through which it is exerted. 

These equations are general for all classes of machines as well as for 
levers, it being understood that friction, which in actual machines increases 
the resistance, is not at present considered. 

Tlie Bent Lever.— In the bent lever (see Fig. 91, page 416) the lever- 
arm of the weight m is cf instead of bf. The lever is in equilibrium when 
n X of = in x cf, but it is to be observed that the action of a bent lever may 
be very different from that of a straight lever. In the latter, so long as the 
force and the resistance act in lines parallel to each other, the ratio of the 
lever-arms remains constant, although the lever itself changes its inclina- 
tion with the horizontal. In the bent lever, however, this ratio changes: 
thus, in the cut, if the arm bf is depressed to a horizontal direction, the dis- 
tance cf lengthens while the horizontal projection of af shortens, the latter 
becoming zero when the direction of af becomes vertical. As the arm af 
approaches the vertical, the weight in which may be lifted with a given 
force s is very great, but the distance through which it may be lifted is 
very small. In all cases the ratio of the weight m to the weight n is the in- 
verse ratio of the horizontal projection of their respective lever-arms. 

Tlie Moving Strut (Fig. 101) is similar to the bent lever, except that 
one of the arms is missing, and that the force and the resistance to be 
overcome act at the same end of the 
single arm. The resistance in the 
case shown in the cut is not the 
weight W, but its resistance to 
being moved, R, which may be sim- 
ply that due to its friction on the 
horizontal plane, or some other op- 
posing force. When the angle be- 
tween the strut and the horizontal 
plane changes, the ratio of the 
resistance to the applied force 
changes. When the angle becomes 
very small, a moderate force will 
Fig. 101. overcome a very great resistance, 

which tends to become infinite as 
the angle approaches zero. If a = the angle, P X sin a — R x versin a. 
If a. = 5 degrees, sin a — 0.8716, versin a = .00381, R — 23 R, nearly. 

The stone-crusher (Fig. 102) shows a practical example of the use of two 
moving struts. 

The Toggle-joint is an elbow or knee-joint consisting of two bars so 
connected that they may be brought into a straight line and made to pro- 
duce great endwise pressure when a force is applied to bring them into this 




ELEMENTS OF MACHINES. 



437 



position. It is a case of two moving struts placed end to end, the moving 
force being applied at their point of junction, in a direction at right angles 
to the direction of the resistance, the other end of one of the struts resting 
against a fixed abutment, and that of the other against the body to be 
moved. If a — the angle each strut makes with the straight line joining the 
points about which their outer ends rotate, the ratio of the resistance 
to the applied force is R : P : : sin a : 2 versin a; 2R versin a = P sin a. The 





Fig. 102. 



Fig. 103. 



ratio varies when the angle varies, becoming infinite when the angle 
becomes zero. 

The toggle-joint is used where great resistances are to be overcome 
through very small distances, as in stone-crushers (Fig. 103). 

The Inclined. Plane, as a mechanical element, is supposed perfectly 
hard and smooth, unless friction be considered. It assists in sustaining a 
heavy body by its reaction. This reaction, however, being normal to the 
plane, cannot entirely counteract the weight of the body, which acts verti- 
cally downward Some other force must therefore 
be made to act upon the body, in order that it may 
be sustained. 

If the sustaining force act parallel to the plane 
(Fig. 104), the force is to the weight as the height of 
the plane is to its length, measured on the incline. 

If the force act parallel to the base of the plane, 
the power is to the weight as the height is to the 
base. 

If the force act at any other angle, let i = the 
angle of the plane with the horizon, and e = the 
angle of the direction of the applied force with the 
angle of the plane. P : W :: sin i : cos e; P X cos e 




Fig. 104. 



W sin i. 

Problems of the inclined plane may be solved by the parallelogram of 
forces thus : 

Let the weight Wbe kept at rest on the incline by the force P, acting in 
i he line bP, parallel to the plane. Draw the vertical line ba to represent 
the weight ; also bb' perpendicular to the plane, and complete the parallelo- 
gram b'c. Then the vertical weight ba is the resultant of bb', the measure of 
support given by the plan » to the weight, and be, the force of gravity tend- 
ing to draw the weight down the plane. The force required to maintain 
the weight in equilibrium is represented by this force be. Thus the force 
and the weight are in the ratio of be to ba. Since the triangle of forces abe 
is similar to the triangle of the incline ABC, the latter may be substituted 
for the former in determining the relative magnitude of the forces, and 

P : W : : be : ab : : BC : AB. 

The Wedge is a pair of inclined planes united by their bases. In the 
application of pressure to the head or butt end of the wedge, to cause it to 
penetrate a resisting body, the applied force is to the resistance as the 
thickness of the wedge is to its length. Let t be the thickness, I the length, 
IF the resistance, and P the applied force or pressure on the head of the 

wedge. Then, friction neglected, P : W : : t : I; P = — r— ; W — — . 

The Screw is an inclined plane wrapped around a cylinder in such a 
way that the height of the plane is parallel to the axis of the cylinder If 
the screw is formed upon the internal surface of a hollow cylinder, it is 
usually called a nut. When force is applied to raise a weight or overcome 
a resistance by means of a screw and nut, either the screw or the nut may 



438 



MECHANICS. 



be fixed, the other being movable. The force is generally applied at the end 
of a wrench or lever-arm, or at the circumference of a wheel. If r = radius 
of the wheel or lever-arm, andp = pitch of the screw, or distance between 
threads, that is, the height of the inclined plane 
for one revolution of the screw, P = the applied 
force, andW= the resistance overcome, then, neg- 
lecting resistance due to friction, 2nr X P = Wp ; 
W = 6.283Pr -f- p. The ratio of P to W is thus 
independent of the diameter of the screw. In 
actual screws, much of the power transmitted is 
lost through friction. 

The Cam is a revolv- 
ing inclined plane. It may 
be either an inclined plane 
wrapped around a cylin- 
der in such a way that the 
height of the plane is ra- 
dial to the cylinder, such 
as the ordinary lifting- 
cam, used in stamp-mills 
(Fig. 105), or it may be an inclined plane curved edgewise, and rotating in a 
plane parallel to it's base (Fig. 106). The relation of the weight to the applied 
force is calculated in the same manner as in the case of the screw. 




Fig. 105. 




Fig. 106. 



(\ 



A., 




Pulleys or Blocks.— P = force applied, or pull ; W = weight lifted 
or resistance. In the simple pulley A (Fig. 107) the point P on the pulling 
rope descends the same amount that the weight is lifted, therefore P = W. 
In B and C the point P moves twice as far as the weight is lifted, there- 
fore W — 2P. In B and C there is one movable block, and two plies of the 
rope engage with it. In D there are three sheaves in the movable block, 
each with two plies engaged, or six in all. Six plies of the rope are there- 
fore shortened by the same amount that the weight is lifted, and the point 
P moves six times as far as the weight, consequently W = 6P. In general, 
the ratio of W X.Q Pis equal to the number of plies of the rope that are 
shortened, and also is equal to the number of plies that eugage the lower 
block. If the lower block has 2 sheaves and the upper 3, the end of the rope 
is fastened to a hook in the top of the lower block, and then there are 5 
plies shortened instead of 6, and W — 5P. If V = velocity of W. and v = 
velocity of P, then in all cases VW = vP, whatever the number of sheaves 
or their arrangement. If the hauling rope, at the pulling end. passes first 
around a sheave in the upper or stationary block, it makes no difference in 
what direction the rope is led from this block to the point at which the pull 
on the rope is applied ; but if it first passes around the movable block, it is 
necessary that the pull be exerted in a direction parallel to the line of action 
of the resistance, or a line joining the centres of the two blocks, in order to 
obtain the maximum effect. If the rope pulls on the lower block at an 
angle, the block will be pulled out of the line drawn between the weight 
and the upper block, and the effective pull will be less than the actual pull 



ELEMENTS OF MACHINES. 



439 




on the rope in the ratio of the cosine of the angle the pulling rope makes 
with the vertical, or line of action of the resistance, to unity. 

Differential Pulley. (Fig. 108.)— Two pulleys, i?aud C, of different 
radii, rotate as one piece about a fixed axis, A. An end- 
less chain, BDECLKH, passes over both pulleys. The 
rims of the pulleys are shaped so as to hold the chain and 
prevent it from slipping. One of the bights or loops in 
which the chain hangs, DE, passes under and supports the 
running block F. The other loop or bight, HKL, hangs 
freely, and is called the hauling part. It is evident that 
the velocity of the hauling part is equal to that of the 
pitch-circle of the pulley B. 

In order that the velocity-ratio may be exactly uniform, 
the radius of the sheave F should be an exact mean be- 
tween the radii of B and C. 

Consider that the point B of the cord BD moves through 
an arc whose length = AB, during the same time the 
point C or the cord CE will move downward a distance = 
AC. The length of the bight or loop BDEC will be 
shortened by AB — AC, which will cause the pulley F to 
be raised half of this amount. If P = the pulling force on 
the cord HK, and W the weight lifted at F, then P X 
AB = W X M(AB - AC). 

To calculatethe length of chain required for a differential 
pulley, take the following sum: Half the circumference of 
A + half the circumference of B -j- half the circumference 
of F -\- twice the greatest distance of F from A + the 
least length of loop HKL. The last quantity is fixed 
according to convenience. 
The Differential Windlass (Fig. 109) is identical in principle 
with the differential pulley, the difference in con- 
struction being that in the differential windlass the 
running block hangs in the bight of a rope whose two 
parts are wound round, and have their ends respec- 
tively made fast to two barrels of different radii, 
which rotate as one piece about the axis A. The dif- 
ferential windlass is little used in practice, because 
Hof the great length of rope which it requires. 
The Differential Screw (Fig. 110) is a com- 
pound screw of different pitches, in which the 
threads wind the same way. N t and. N 2 are the two 
nuts; SjSj, the longer-pitched thread; S 2 S 2 . the 
shorter-pitched thread: in the figure both these 
threads are left-handed. At each turn of the screw 
the nut N-2 advances relatively to iV 2 through a dis- 
tance equal to the difference of the pitch. The use 
of the differential screw is to combine the slowness 
of advance due to a fine pitch with the strength of thread which can be 
obtained by means of a coarse pitch only. 

A Wheel and Axle, or Windlass, resembles two pulleys on one axis, 
having different diameters. If a weight be lifted bj 7 means of a rope wound 
over the axle, the force being applied at the 
rim of the wheel, the action is like that of a 
lever of which the shorter arm is equal to 
the radius of the axle plus half the thick- 
ness of the rope, and the longer arm is 
equal to the radius of the wheel. A wheel 
and axle is therefore sometimes classed 
as a perpetual lever. If P = the applied force, D = 

W = the weight lifted, and d the diameter of the axle -J- the diameter of 
the rope, PD = Wd. 

Toothed-wheel Gearing is a combination of two or more wheels 
and axles (Fig. 111). If a series of wheels and pinions gear into each other, 
as in the cut, friction neglected, the weight lifted, or resistance over- 
come, is to the force applied inversely as the distances through which 
they act in a given time. If R, R x , R 2 be the radii of the successive wheels, 
measured to the pitch-line of the teeth, and r, r x , r 2 the radii of the cor- 
responding pinions, Pthe applied force, and W the weight lifted, Px 




V 



Fig. 109. 



Fig. 110. 
: diameter of the wheel, 



440 



MECHANICS. 



K X . R* X R 9 = W x r x r x X r 2 , or the applied force is to the weight 
as the product of the radii of the pinions is to the product of the radii of 
the wheels; or, as the product of the numbers expressing the teeth in 
each pinion is to the product of the numbers expressing the teeth in each 

Endless Screw, or Worm-gear. (Fig. Ii2.)-This gear is com- 
monly used to convert motion at high speed into motion at very slow* 





Fig. 111. 



'Fig. 112. 



speed. When the handle P describes a complete circumference, the -pitch- 
line of the cog-wheel moves through a distance equal to the pitch of the 
screw, and the weight Wis lifted a distance equal to the pitch of the screw 
multiplied by the ratio of the diameter of the axle to the diameter of the 
pitch-circle of the wheel. The ratio of the applied force to the weight 
lifted is inversely as their velocities, friction not being considered; but the 
friction in the worm-gear is usually very great, amounting sometimes to 
three or four times the useful work done. 

If v = the distance through which the force Pacts in a given time, say 1 
second, and V= distance the weight W is lifted in the same time, r = 
radius of the crank or wheel through which Pacts, t = pitch of the screw, 
and also of the teeth on the cog-wheel, d = diameter of the axle, 

and D = diameter of the pitch-line of the cog-wheel, v = — — - — 

XV; V=vXtd-r- 6.283rd. Pv = WV+ friction. 



STRESSES IN FRAMED STRUCTURES. 

Framed structures in general consist of one or more triangles, for the 
reason that the triangle is the one polygonal form whose shape cannot be 
changed without distorting one of its sides. Problems in stresses of simple 
framed structures may generally be solved either by the application of the 
triangle, paralellogram, or polygon of forces, by the principle of the lever, 
or by the method of moments. We shall give a few examples, referring the 
student to the works of Burr, Dubois, Johnson, and others for more elabo- 
rate treatment of the subject. 

1. A Simple Crane. (.Figs. 113 and 114.) — ^4 is a fixed mast, B a brace or 
boom, T a tie, and P the load. Required the strains in B and T. The weight 
P, considered as acting at the end of the boom, is held in equilibrium by 
three forces: first, gravity acting downwards; second, the tension in T; and 
third, the thrust of B. Let the length of the line p represent the magnitude 
of the downward force exerted by the load, and draw a parallelogram with 
sides bt parallel, respectively, to B and T, such that pis the diagonal of the 
parallelogram. Then b and t are the components drawn to the same scale 
as p, p being the resultant. Then if the length p represents the load, t is 
the tension in the tie, and b is the compression in the brace. 

Or. more simply. T, B, and that portion of the mast included between them 
or A' may represent a triangle of forces, and the forces are proportional to 
the length of the sides of the triangle; that is, if the height of the triangle .4' 
= t he load, then B = the compression in the brace, and T = the tension in the 

T 
tie; or if P = the load in pounds, the tension in T = P x — , , and the com- 



STRESSES IK FRAMED STRUCTURES. 



441 



pression in B = P X 



B 



Also, if a = the angle the inclined member makes 



with the mast, the other member being horizontal, and the triangle being 
right-angled, then the length of the inclined member = height of the tri- 
angle x secant a, and the strain in the inclined member = P secant a. Also, 
the strain in the horizontal member = P tan a. 

The solution by the triangle or parallelogram of forces, and the equations 
Tension in 2' = P x T/A', and Compression ini? = P X B/A', hold true even 
if the triangle is not right-angled, as in Fig. 115; but the trigonometrical rela- 




Fig. 113. 



Fis. 114. 



Fig. 115. 



tions above given do not hold, except in the case of a right-angled triangle. 
It is evident that as A' decreases, the strain in both Tand B increases, tend- 
ing to become infinite as A' approaches zero. If the tie Tis not attached to 
the mast, but is extended to the ground, as shown in the dotted line, the 
tension in it remains the same. 

2. A Guyed Crane or Derrick. (Fig. 116.)— The strain in B is, as 
before, PxB/A', A' being that portion of the vertical included between B and 
T, wherever Tmay be attached to A. If, however, the tie Tis attached to B 
beneath its extremity, there may be in addition a bending strain in B due to 
a tendency to turn about the point of attachment of Tas a fulcrum. 

The strain in T may be calculated by the principle of moments. The mo- 
ment of P is Pc, that is, its weight X its perpendicular distance from the 
point of rotation of B on the mast. The moment of the strain on T is the 
product of the strain into the perpendicular distance from the line of its 



«.__F— 




direction to the same point of rotation of B, or Td. The strain in T there- 
fore = Pc-i- d. As d decreases the strain on T increases, tending to infin- 
ity as d approaches zero. 

The strain on the guy-rope is also calculated by the method of moments. 
The moment of the load about the bottom of the mast O is, as before, Pc. 
If the guy is horizontal the strain in it is Fand its moment is Ff, and F — 
Pcs-f. If it is inclined, the moment is the strain G X the perpendicular 
distance of the line of its direction from O, or Gg, and G — Pc-i- g. 

The guy-rope having the least strain is the horizontal one F, and the strain 



442 



MECHANICS. 




in G = the strain in F X the se- 
cant of the angle between F and 
G. As G is made more nearly- 
vertical g decreases, and the 
strain increases, becoming infi- 
nite when g — 0. 

3. Shea r-p oles with 
Guys. (Fig. 109.)— Resultant of 
strain in both masts = P X BD 
-i-BC. Resultant strain in both 
guys=Px AB-*-BC. The strain 
on each mast (or guy) will be half 
the above, multiplied by the se- 
cant of half the angle the masts 
Fig. 117. <br guys) make with each other. 

Two Diagonal Braces and a Tie-rod. (Fig. 118.)— Suppose the 
braces are used to sustain a single load P. Compressive stress on AD = 

y%P X -jg \ on CA- y % P x ■—. This is true only if CB and BD are of equal 

length, in which case ^ of P is supported by each abutment C and D. If 
they are unequal in length (Fig. 119), then, 
by the principle of the lever, find the re- 
actions of the abutments P^ and P Q . If P 
is the load applied at the point B on the 
lever 'CD, the fulcrum being D, then R x X 
CD = P X BD and P a X CD = P X BC; 
R S =PXBD+CD; P 2 = P X PC-h CD. 
The strain on AC = P x X AC-i-AB, and 
on AD = P a X AD -*- AB. 

The strain on the tie = Ri X CB -j- AB 
= R*XBD + AB. 

A When CB = BD, R t = R^, the strain 

p> ^<K R on CB and BD is the same, whether 

Jr 1 ^-<^^Txv 4 2 the Draces are °f equal length or 

(9\\v I not ' and is e Q ual to H p X V&CD + AB. 

^^y \^J ^Zs. ! If the braces support a uniform load, 

Xs!_ as a pair of rafters, the strains caused 

by such a load are equivalent to that 

caused by one half of the load applied 

at the centre. The horizontal thrust 

of the braces against each other at the 

apex equals the tensile strain in the tie. 

King-post Truss or Bridge. (Fig. 120.)— If the load is distributed 
over the whole length of the truss, the effect is the same as if half the load 
were placed at the centre, the other half being carried by the abutments. Let 
P — one half the load on the truss, then 
tension in the vertical tie AB = P. Com- 
pression in each of the inclined braces = 
%P X AD -*- AB. Tension in the tie CD 
- y$P X BD-i- AB. Horizontal thrust of 
inclined brace AD at. D = the tension in 
the tie. If W = the total load on one truss 
uniformly distributed, I = its length and 
d = its depth, then the tension on the hor- 
Wl 
8rT 
Inverted King-post Truss. (Fig. 121.)— If P = a load applied at 

^». ^«~ v.„i.p r^e r. .. — ,-i- ™i„ j; ... -i ___x_ ■■ >° n n ; ~_ a r> T> 





Fig. 119. 




Fig. 120. 



B, or one half of a uniformly dis 




buted load, then compression on AB = P 
(the floor-beam CD not being considered 
to have any resistance to a slight bending). 
Tension on AC or AD = *AP X AD + AB. 
Compression on CD = y>P X BD -f- AB. 

Queen-post Truss. (Fig. 123.)-If 
uniformly loaded, and the queen-posts di- 
vide the length into three equal bays, the 
load may be considered to be divided into 
three equal parts, two parts of which, P x 
andP 2 , are concentrated at the panel joiuts 



STRESSES IK FRAMED STRUCTURES. 



443 



and the remainder is equally divided between the abutments and supported 
by them directly. The two parts P x and P 2 only are considered to affect 
the members of the truss. Strain in 
the vertical ties BE and CF each 
equals P x or P 2 . Strain on AB and 
CD each = P t x CD + CF. Strain 
on the tie AE or EF ov ED = F X 
FD h- CF. Thrust on BC = tension 
on EF. 

For stability to resist heavy un- 
equal loads the queen-post truss 
should have diagonal braces from 
B to Pand from Cto E. 

Inverted Queen-post 
Truss. (Fig;. 123.) — Compression 
on EB and FC each = P. Tension 
on AB and CD each = P X AB + 
EB. Compression on AE or EF or 
FD = PX AE -^- EB. Tension on 
BC = compression on EF. For sta- 
bility to resist unequal loads, ties 
should be run from C to E and from 
Fig. 123. BtoF. 

Burr Truss of Five Panels. (Fig. 124. )— Four fifths of the load may 
be taken as concentrated at the points E, JKT, L and F, the other fifth being 

B G H C 





®t <& © 



Fig. 124. 



supported directly by the two abutments. For the strains in BA and CD 
the truss may be considered as a queen-post truss, with the loads P x , P 2 
concentrated at Pand the loads P 3 , P 4 concentrated at F. Then, compres- 
sive strain on AB = (P 1 + P 2 ) X AB^-BE. The strain on CD is the same if 
the loads and panel lengths are equal. The tensile strain on BE or CF = 
P l + P 2 . That portion of the truss between E and Pmay be considered as 
a smaller queen-post truss, supporting the loads P 2 , P 3 at K and L. The 
strain on EG or HF = P 2 X EG -f- GK. The diagonals GL and KH receive no 
strain unless the truss is unequally loaded. The verticals GK and HL each 
receive a tensile strain equal to P 2 or P 3 . 

For the strain in the horizontal members: BG and CH receive a thrust 
equal to the horizontal component of the thrust in AB or CD, — (Pj + P 2 ) 
X tan angle ABE, or (Pj + P 2 ) X AE-i-BE. GH receives this thrust and 
also, in addition, a thrust equal to the horizontal component of the thrust in 
EG or HF, or, in all, (P x -4- P 2 + P 3 )X AE-i-BE. 

The tension in AE or FD equals the thrust in BG or HC, and the tension 
in EK. KL. and LF equals the thrust in GH. 

Pratt or Whipple Truss. (Fig. 125.)— In this truss the diagonals are 
ties, and the verticals are struts or columns. 

Calculation by the method of distribution of strains: Consider first the 
load P x . The truss having six bays or panels, 5/6 of the load is transmitted 
to the abutment H, and 1/6 to the abutment O, on the principle of the lever. 
As the five sixths must be transmitted through J A and AH, write on these 
members the figure 5. The one sixth is transmitted successively through 
JC, CK, KD, DL, etc., passing alternately through a tie and a strut. Write 
on these members, up to the strut GO inclusive, the figure 1. Then consider 
the load P 2 , of which 4/6 goes to AH and 2/6 to GO. Write on KB, BJ, J A, 
and AH the figure 4, and on KD, DL, LE, etc., the figure 2. The load P a 



444 



MECHANICS. 



transmit 3/6 in each direction; write 3 on each of the members through 
which this stress passes, and so on for all the loads, when the figures on the 
several members will appear as on the cut. Adding them up, we have the 
following totals : 

j A J BH BK CJ CL DK DM EL EN FM FO GN 
15 10 1 6 3 3 6 1 10 15 
, AH BJ CK DL EM FN GO 
! 15 10 7 6 7 10 15 



Tension on diagonals -j 

Compression on verticals ■ 



Each of the figures in the first line is to be multiplied by 1/6PX. secant of 
angle HAJ, or 1/6P X A J-r- AH, to obtain the tension, and each figure in the 
lower line is to be multiplied by 1/6P to obtain the compression. The diag- 
onals HB and FO receive no strain. 




6 6 6 5 6 



P 3 P4 

Fig. 125. 

It is common to build this truss with a diagonal strut at HB instead of the 
post HA and the diagonal AJ; in which case 5/6 of the load Pis carried 
through JB and the strut BH, which latter then receives a strain = 15/6P X 
secant of HBJ. 

The strains in the upper and lower horizontal members or chords increase 
from the ends to the centre, as shown in the case of the Burr truss. AB 
receives a thrust equal to the horizontal component of the tension in AJ, or 
15/6PX tan AJB. BC receives the same thrust + the horizontal component 
of the tension in BK, and so on. The tension in the lower chord of each panel 
is the same as the thrust in the upper chord of the same panel. (For calcu- 
lation of the chord strains by the method of moments, see below.) 

The maximum thrust or tension is at the centre of the chords and is equal 
WL 
to — -, in which W is the total load supported by the truss, L is the length, 

and D the depth. This is the formula for maximum stress in the chords 
of a truss of any form whatever. 

The above calculation is based on the assumption that all the loads P lr P 2 , 
etc., are equal. If they are unequal the value of each has to be taken into 
account in distributing the strains. Thus the tension in AJ, with unequal 
loads, instead of being 15 X 1/6 P secant would be sec X (5/6?! + 4/6 P 2 + 
3/6 P 3 + 2/6 P 4 4- 1/6 P 5 .) Each panel load, P x etc., includes its fraction of 
the weight of the truss. 

General Formula for Strains in Diagonals and Verticals. 
—Let n— total number of panels, x — number of any vertical considered 
from the nearest end, counting the end as 1, r = rolling load for each panel, 
P = total load for each panel, 



Strain on verticals 



(n— x)-\-(n— x) 2 — (x- 



2n 



l)-Ha?-l) a TT . r(a;-l)+(a:-l) a 
+ 2n 



For a uniformly distributed load, leave out the last term, 
[r(x-l)-\-\a>— l)?]-=r2M. 

Strain on principal diagonals = strain on verticals X secant 0, that is 
secant of the angle the diagonal makes with the vertical. 

Strain on the counterbraces : The strain on the counterbrace in the first 
panel is 0, if the load is uniform. On the 2d, 3d, 4th, etc., it is P secant 

X — , , — — — — , etc., P being the total load in one panel. 



STRESSES IN FRAMED STRUCTURES. 



445 



Strain in the Chords— Method of Moments. -Let the truss be 
uniformly loaded, the total load acting on it = W. Weight supported at 
each end, or reaction of the abutment = W/2. Length of the truss = L. 
Weight on a unit of length = W/L. Horizontal distance from the nearest 
abutment to the point (say Jl/in Fig. 125) in the chord where the strain is to 
be determined = x. Horizontal strain at that point (tension on the lower 
chord, compression in the upper) = H. Depth of the truss = D. By the 
method of moments we take the difference of the moments, about the point 
M, of the reaction of the abutment and of the load between and the abut- 
ments, and equate that difference with the moment of the resistance, or of 
the strain in the horizontal chord, considered with reference to a point in 
the opposite chord, about which the truss would turn if the first chord were 
severed at M. 

The moment of the reaction of the abutment is Wx/2. The moment of 
the load from the abutment to M is W/Lx X the distance of its centre of 
gravity from M, which is x/2, or moment = Wx 2 -h2L. Moment of the stress 

• 4.1. i, * riri Wx Wx* , TT Wy x*\ „ 

in the chord = HD = — — ~, whence H — — / x - — V If x — or L, 

WL~ 
H = 0. If x = L/2, H = -=-, which is the horizontal strain at the middle 

of the chords, as before given. 

The Howe Truss. (Fig. 126.)— In the Howe truss the diagonals are 
struts, and the verticals are ties. The calculation of strains may be made 




in the same method as described above for the Pratt truss. 

The Warren Girder. (Fig. 127.)— In the Warren girder, or triangular 
truss, there are no vertical struts, and the diagonals may transmit either 




Fig. 187. 

tension or compression. The strains in the diagonals may be calculated by 
the method of distribution of strains as in the case of the rectangular truss. 
On the principle of the lever, the load P t being 1/10 of the length of the 
span from the line of the nearest support a, transmits 9/10 of its weight to a 
and 1/10 to g. Write 9 on the right hand of the strut la. to represent the 
compression, and 1 on the right hand of 16, 2c, 3d. etc., to represent com- 
pression, and on the left hand of 62, c3, etc., to represent tension. The load P 2 
transmits 7/10 of its weight to a and 3/10 to g. Write 7 on each member from 
2 to a and 3 on each member from 2 to g, placing the figures representing 
compression on the right hand of the member, and those representing 
tension on the left. Proceed in the same manner with all the loads, then 



446 



MECHANICS. 



sum up the figures on each side of each diagonal, and write the difference 
of each sum beneath, and on the side of the greater sum, to show whether 
the difference represents tension or compression. The results are as follows: 
Compression, la, 25; 2b, 15; 3c, 5; 3d, 5; 4e, 15; 5a, 25. Tension, 16, 15; 2c, 
5: 4d, 5; he. 15. Each of these figures is to be multiplied by 1/10 of one of 
the loads as P 1 , and by the secant of the angle the diagonals make with a 
vertical line. 

The strains in the horizontal chords may be determined by the method of 
moments as in the case of rectangular trusses. 

Roof-truss.— Solution by Method of Moments. — The calculation of 
strains in structures by the method of statical moments consists in taking a 
cross-section of the structure at a point where there are not more than 
three members (struts, braces, or chords). 

To find the strain in either one of these members take the moment about 
the intersection of the other two as an axis of rotation. The sum of the 
moments of these members must be if the structure is in equilibrium. 
But the moments of the two members that pass through the point of refer- 
ence or axis are both 0, hence one equation containing one unknown quan- 
tity can be found for each cross-section. 







A 






R 




/ \ 








/ 


P 3 


S C 






y 


E^ 


i \ 




SJSP 


/r ^ 


^^ 


X 


! \ 


20 




/ 


1 ^»^*^ 






V \ 






/ 


jj^ - 


^^V Z 




i^, x 






/^ 


25 ^" , ^*^ 


12.5 ^S 


15 


ljj.5 \ 


/ z 


9 


A 


\ 


* >- 




k 


D 



Fig. 128. 



In the truss shown in Fig. 128 take a cross-section at ts, and determine the 
strain in the three members cut by it, viz., CE, ED, and DF. Let X = force 
exerted in direction CE, Y = force exerted in direction DE, Z = force ex- 
erted in direction FD. 

For X take its moment about the intersection of F and Z at D = Xx. For 
Y take its moment about the intersection of Xand Z at A — Yy. For Z take 
its moment about the intersection of X and Y aUE — Zz. Let z = 15, x — 
18.6, y - 38.4. AD = 50, CD = 20 ft. Let Pj, P 2 , P 3 , P 4 be equal loads, as 
shown, and 3}4 P the reaction of the abutment A. 

The sum of all the moments taken about D or A or E will be when the 
structure is at rest. Then - Xx + 3.5P X 50 - P 3 X 12.5 - P 2 X 25 - P, X 
37.5 = 0. 

The +. signs are for moments in the direction of the hands of a watch or 
"clockwise " and — signs for the reverse direction or anti-clockwise. Since 
P= P, = Pz - P 3 , - 18.6X+ 175P - 75P = 0; - 18.6X = - 100P; X = 

100P-^18.6 = 5.376P. 
- Yy + P 3 X 37.5 + P 2 X 25 f P, X 12.5 = 0; 38.4F = 75P; Y = 75P-=- 38.4 

= 1 953P. 
-Zz + 3.5P X 37.5 - P, X 25 - P 2 X 12.5 - P 3 X = 0; lhZ = 93.75P; Z = 
6. 25 P. 

In the same manner the forces exerted in the other members have been 
found as follows: EG = 6.73P; GJ = 8.07P; J A = 9.42P; JH = 1.35P; GF = 
1.59P; AH= 8.75P; HF = 7.50P. 

The Fink Roof-truss. (Fig. 129.)— An analysis by Prof. P. H. Phil- 
brick (Van N. Mag., Aug. 1880) gives the following results: 



STRESSES IN FRAMED STRUCTURES. 
C 



447 




D 

Fig. 129. 
W = total load on roof; 
iV = No. of panels on both rafters; 
W/N = P = load at each joint 6, d, /, etc.; 
V = reaction at A = % W = y 2 NP = 4P; 
AD = S; AC=L; CD = D; 
*i> ^2> ^3 = tension on De, eg, gA, respectively; 
c x , c 2 , c 3 , c 4 = compression on Cb, fed, df, and/4. 



Strains in 

1, or De = t x = 2PS -4- D; 

2, " eg = t 2 = SPS -i- D; 
" gr4 = # 3 = 7/2PS-=-D; 

4, " 4/ = c 4 = 7/2PL -f- D; 

5, » fd = c 3 = 7/2PL/D -PD/L; 
" " dfe = c 3 = 7/2PL/D -2PD/L; 



7, or &C = d =7/2 Pi/D - 3 PD/L: 

8, " fee or/g = PS-r-L; 

9, " de = 2PS + L; 

10, " cd or dg = %PS -=- D; 

11, " ec = PS^-D; 

12, " cC - 3/2 PS -v- D. 



Example.— Given a Fink roof -truss of span 64 ft., depth 16 ft., with four 
panels on each side, as in the cut; total load 32 tons, or 4 tons each at the 
points/, d, b, C, etc. (and 2 tons each at A and B, which transmit no strain 
to the tr uss mem bers). Here W — 32 tons, P = 4 tons, S = 32 ft., D = 16 
ft., L = YS" 2 + D 2 = 2.236 X D. L -*- D = 2.23( 
S + L= .8944 



D 2 = 2.236 X D. _L -s- D = 2.236, D -*- £ = .4472, flf -s- D = 2, 
The strains on the numbered members then are as follows: 



1, 2X4X2 

2, 3X4X2 

3, 7/2 X 4 X 2 =28 

4, 7/2 X 4 X 2.236 = 31.3 

5, 31.3-4 X .447 =29.52 

6, 31.3- 8 X .447 =27.72 



tons; 



7, 3 


1.3- 12 X .447 =25.94 tons 


8, 


4 X .8944 = 3.58 " 


9, 


8 X .8944 = 7.16 " 


10, 


2X2 =4 


11, 


4X2=8 


12, 


6 X 3 =12 



448 



HEAT. 

THERMOMETERS. 

The Fahrenheit thermometer is generally used in English-speaking coun- 
tries, and the Centigrade, or French thermometer, in countries that use the 
metric system. In many scientific treatises in English, however, the Centi- 
grade temperatures are also used, either with or without their Fahrenheit 
equivalents. The Reaumur thermometer is used in Russia, Sweden, Turkey, 
and Egypt. (Clark.) 

In the Fahrenheit thermometer the freezing-point of water is taken at 32°, 
and the boiling-point of water at mean atmospheric pressure at the sea- 
level, 14.7 lbs. per sq. in., is taken at 212°, the distance between these two 
points being divided into 180°. In the Centigrade and Reaumur thermometers 
the freezing-point is taken at 0°. The boiling-point is 100° in the Centigrade 
scale, and 80° in the Reaumur. 

1 Fahrenheit degree — 5/9 deg. Centigrade = 4/9 deg. Reaumur. 

1 Centigrade degree = 9/5 deg. Fahrenheit = 4/5 deg. Reaumur. 

1 Reaumur degree = 9/4 cleg. Fahrenheit = 5/4 deg. Centigrade. 

Temperature Fahrenheit = 9/5 X temp. C. + 32° = 9/4 R. -f- 32°. 

Temperature Centigrade = 5/9 (temp. F. — 32°) = 5/4 R. 

Temperature Reaumur = 4/5 temp C. = 4/9 (F. — 32°). 

Mercurial Thermometer, (Rankine. S. E., p. 234.)— The rate of 
expansion of mercury with rise of temperature increases as the temperature 
becomes higher ; from which it follows, tnat if a thermometer showing the 
dilatation of mercury simply were made to agree with an air thermometer 
at 32° and 212°, the mercurial thermometer would show lower temperatures 
than the air thermometer between those standard points, and higher tem- 
peratures beyond them. 

For example, according to Regnanlt, when the air thermometer marked 
350° C. (- 662° F.), the mercurial thermometer would mark 362.16° C. (= 
683.89° F.), the error of the latter being in excess 12.16° C. (= 21.89° F.). 

Actual mercurial thermometers indicate intervals of temperature propor- 
tional to the difference between the expansion of mercury and that of glass. 

The inequalities in the rate of expansion of the glass (which are very 
different for different kinds of glass) correct, to a greater or less extent, the 
errors arising from the inequalities in the rate of expansion of the mercury. 

For practical purposes connected with heat engines, the mercurial ther- 
mometer made of common glass may be considered as sensibly coinciding 
with the air-thermometer at all temperatures not exceeding 500° F. 

PYROMETRY. 

Principles Used in Various Pyrometers.— Contraction of clay 
by heat, as in the Wedgwood pyrometer used by potters. Not accurate, as 
the contraction varies with the quality of the clay. 

Expansion of air, as in the air-thermometers, Wiborgh's pyrometer, Ueh- 
ling and Steinhart's pyrometer, etc. 

Specific heat of solids, as in the copper-ball, platinum-ball, and fire-clay 
pyrometers. 

Relative expansion of two metals or other substances, as copper and iron, 
as in Brown's and Bulkley's pyrometers, etc. 

Melting-points of metals, or other substances, as in approximate deter- 
minations of temperature by melting pieces of zinc, lead, etc. 

Measurement of strength of a thermo-electric current produced by heat- 
ing the junction of two metals, as in LeCliatelier"s pyrometer. 

Changes in electric resistance of platinum, as in the Siemens pyrometer. 

Time required to heat a weighed quantity of water enclosed in a vessel, 
as in the water pyrometer. 

Thermometer for Temperatures up to 800° F.— Mercury with 
compressed nitrogen in the tube above the mercury. Made by Queen & Co., 
Philadelphia. 



TEMPERATURES, CENTIGRADE AND 44Q 

FAHRENHEIT. — * 



c. 


F. 


C 


F. 


C. 


F. 


C. 


F. 


C. 


F. 


C. 


F. 


C. 


F. 


-40 


-40. 


26 


78.8 


92 


197.6 


158 


316.4 


224 


435.2 


290 


554 


950 


1742 


—39 


-38.2 


.'7 


80.6 


93 


199.4 


159 


318.2 


225 


437. 




572 


960 


1760 


-38 


-36.4 


28 


82.4 


94 


201.2 


160 


320. 


220 


438.8 


310 


590 


970 


1778 


-37 


—34 6 


29 


84.2 


95 


203. 


161 


321.8 


227 


440.6 


320 


608 


980 


1796 


-36 


-32.8 


30 


80. 


90 


204.8 


162 


323.6 


228 


442.4 


330 


626 


990 


1814 


-35 


-31. 


31 


87.8 


97 


206. (j 


163 


325.4 


229 


444.2 


3l<) 


644 


1000 


1832 


-34 


—29.2 


32 


89.6 


98 


208.4 


164 


327.2 


230 


446. 


350 


662 


1010 


1850 


-33 


-27.4 


33 


91.4 


99 


210.2 


165 


329. 


231 


447.8 


300 


680 


1020 


1868 


-32 


-25.6 


34 


93.2 


100 


212. 


166 


330.8 


232 


449.6 


370 


698 


1030 


1886 


-31 


-23.8 


35 


95. 


101 


213.8 


167 


332.6 


233 


451.4 


380 


716 


1040 


1904 


-30 


-22. 


36 


96.8 


102 


215.(j 


168 


334.4 


234 


453.2 




734 


1050 


1922 


-29 


-20.2 


37 


98.6 


103 


217.4 


169 


336.2 


235 


455. 


400 


752 


1060 


1940 


—28 


-18.4 


38 


100.4 


101 


219.2 


170 


338. 


230 


456.8 


410 


770 


1070 


1958 


-27 


-16.6 


39 


102.2 


105 


221. 


171 


339.8 


237 


458.6 


420 


788 


1080 


1976 


-26 


-14.8 


40 


104. 


100 


-.22.8 


172 


341.6 


238 


460.4 


430 


806 


1090 


1994 


-25 


-13. 


41 


105.8 


107 


224.6 


173 


343.4 


239 


462 2 


440 


824 


1100 


2012 


—24 


-11.2 


42 


107.6 


108 


226.4 


174 


345.2 


240 


464. 


450 


842 


1110 


2030 


-23 


- 9.4 


43 


109.4 


109 


228.2 


175 


347. 


241 


465.8 


460 


860 


1120 


2048 


-22 


— 7.6 


44 


111.2 


no 


230. 


176 


348.8 


242 


467.6 


470 


878 


1130 


2066 


-21 


— 5.8 


45 


113. 


111 


231.8 


177 


350.6 


243 


469.4 


480 


896 


1140 


2084 


-20 


- 4. 


46 


114.8 


112 


233 6 


178 


352.4 


244 


471.2 


490 


914 


1150 


2102 


-19 


- 2.2 


47 


116.6 


113 


235.4 


179 


354.2 


245 


473. 


500 


932 


1160 


2120 


-18 


- 0.4 


48 


118.4 


114 


237.2 


180 


356. 


240 


474.8 


510 


950 


1170 


2138 


-17 


+ 1.4 


49 


120.2 


115 


239. 


181 


357.8 


247 


476.6 




968 


1180 


2156 


-16 


3.2 


50 


122. 


110 


240.8 


182 


359.6 


248 


478.4 




986 


1190 


2174 


—15 


5. 


51 


123.8 


117 


242.6 


183 


361.4 


249 


480.2 


540 


1004 


1200 


2192 


-14 


6.8 


52 


125.6 


118 


244.4 


184 


363.2 


250 


482. 


550 


1022 


1210 


2210 


-13 


8.6 


53 


127.4 


119 


246.2 


185 


365. 


251 


483.8 




1040 


1220 


V 


-12 


10.4 


54 


129.2 


120 


248. 


186 


366.8 


252 


485.6 


570 


1058 


1230 


2246 


-11 


12.2 


55 


131. 


121 


249.8 


187 


368.6 


253 


487.4 




1076 


1240 


2264 


-10 


14. 


56 


132.8 


122 


251.Q 


188 


370.4 


254 


489.2 




1094 


1250 




- 9 


15.8 


57 


134.6 


123 


253.4 


189 


372.2 


255 


491. 


000 


1112 


1260 


2300 


- 8 


17.6 


58 


136.4 


124 


255 . 2 


190 


374. 


256 


492.8 


610 


1130 


1270 


2318 


7 


19.4 


59 


138.2 


125 


257 '. 


191 


375.8 


257 


494.6 




1148 


1280 


2336 


- 6 


21.2 


GO 


140. 


120 


258.3 


192 


377.6 




496.4 




1166 


1290 




- 5 


23. 


01 


141.8 


127 


260.6 


193 


379.4 


259 


498.2 


040 


1184 


1300 


2372 


- 4 


24.8 


62 


143.6 


128 


262.4 


194 


381.2 


260 


500. 




1202 


1310 


2390 


- 3 


26.6 


03 


145.4 


129 


264.2 


195 


383. 


201 


501.8 




1220 


1320 


2408 


- 2 


28.4 


04 


147.2 


130 


266. 


196 


384.8 


202 


303.6 




1238 


1330 


2426 


— 1 


30.2 


05 


149. 


131 


267.8 


197 


386.6 


203 


5D5rT4 




1256 


1340 


2444 





32. 


00 


150.8 


132 


269.6 


198 


388.4 


204 


507.2 


1274 


1350 


2462 


+ 1 


33.8 


67 


152.6 


133 


271.4 


199 


390.2 


205 


509. 


700 


1292 


1360 


2480 


2 


35.6 


08 


154.4 


134 


273.2 


200 


392. 


260 


510.8 


710 


1310 


1370 


2498 


3 


37.4 


09 


156 


2 


135 


275. 


201 


393.8 


267 


512.6 




132S 


1380 


2516 


4 


39.2 


70 


158 




130 


276.8 


202 


395.6 




514.4 




1346 


1390 


2534 


5 


41. 


71 


159 


8 


137 


278.6 


203 


397.4 


209 


516.2 


741 


1364 


1400 


2552 


6 


42.8 


72 


101 


6 


li 


280.4 


204 


399.2 


270 


518. 


750 


1382 


1410 


2570 


7 


44.6 


73 


103 


4 


139 


282.2 


205 


401. 


271 


519.8 


7 6i 


1400 


1420 


2588 


8 


46.4 


74 


165 


2 


140 


284. 


206 


402.8 




521.6 


770 


1418 


1430 


2606 


9 


48.2 


75 


167 




141 


285.8 


207 


404.6 




523.4 


78C 


1436 


1440 


2624 


10 


50. 


70 


168 


8 


142 


287.6 


20S 


406.4 


274 


525.2 




1454 


1450 


2642 


11 


51.8 


77 


170 


6 


143 


289.4 


209 


408.2 


275 


527. 




1472 


1460 


2660 


12 


53.6 


78 


172 


4 


144 


291.2 


210 


410. 


276 


528.8 


810 


1490 


1470 


2678 


13 


55.4 


79 


174 


2 


145 


293. 


211 


411.8 


277 


530.6 


820 


1508 


1480 


2696 


14 


57.2 


80 


170 




146 


294.8 


212 


413.6 


278 


532.4 




1526 


1490 


2714 


15 


59. 


81 


177 


8 


147 


296.6 


213 


415.4 


279 


534.2 


810 


1544 


1500 




16 


60.8 


82 


179 


6 


14S 


298.4 


214 


417.2 




536. 


iS5i 


1562 


1510 


2750 


17 


62.6 




181 


4 


149 


300.2 


215 


419. 


281 


537.8 


800 


1580 


1520 


2768 


18 


64.4 


84 


183 


2 


150 


302. 


216 


420.8 


■:s2 


539.6 




1598 


1530 


2786 


19 


66.2 


85 


18.5 




151 


303.8 


217 


422.6 


2S3 


541.4 




1616 


1540 


2804 


20 


68. 


80 


186 


8 


152 


305.6 


218 


424.4 


284 


543.2 




1034 


1550 


2822 


21 


69.8 


87 


188.6 


153 


307.4 


219 


426.2 


285 


545. 




1052 


1600 


2912 


22 


71.6 




190.4 


154 


309.2 


220 


428. 




546.8 


910 


1670 


1650 


3002 


23 


73.4 




192.2 


155 


311. 


221 


429.8 


287 


548.6 




1688 


1700 


3092 


24 


75.2 


90 


194. 


150 


312.8 


222 


431.6 


2SS 


550.4 


930 


1706 


1750 


3182 


25 


77. 


91 


195.8 


157 


314.6 


226 


433.4 


289 


552.2 


940 1724 


1800 


3272 



AKf) fEMPERATURES, FAHRENHEIT AND 

^ uv CENTIGRADE. 



F. 


C. 


F. 


C. 


F. 


C. 


F. 


c. 


F. 


c. 


F. 


C. 


F. 


c. 


-40 


—40. 


2G 


— 3.3 


92 


33.3 


158 


70. 


224 


106.7 


290 


143.3 


3G0 


182.2 


—39 


-39.4 


27 


— 2.8 


93 


33.9 


159 


70.6 




107.2 




143.9 


370 


187.8 


—38 


—38.9 


28 


— 2 2 


94 


34.4 


160 


71.1 




107.8 


2! 


144.4 




193.3 


-37 


-38.3 


29 


— 1.7 


95 


35. 


161 


71.7 




108.31293 


145. 




198.9 


-36 


-37.8 


30 


— 1.1 


96 


35.6 


162 


72.2 


. 


108.9 




145.6 


400 


204.4 


-35 


—37.2 


31 


— 0.6 


97 


36.1 


163 


72.8 


. 


109.4 


295 


146.1 


41C 


210. 


-34 


-36.7 


32 


0. 


98 


36.7 


164 


73.3 




110. 




146.7 




215.6 


-33 


-36.1 


33 


-f 0.6 


99 


37.2 


165 


73.9 




110.6 


297 


147.2 




221.1 


-32 


—35.6 


34 


1.1 


100 


37.8 


166 


74.4 




111.1 


29f 


147.8 


410 


226.7 


-31 


-35. 


35 


1.7 


101 


38.3 


167 


75. 




111.7 


1 


148.3 




2322 


—30 


—34.4 


36 


2.2 


102 


38.9 




75.6 


234 


112.2 


300 


148.9 


. 


237.8 


—29 


—33.9 


37 


2.8 


103 


39.4 


169 


76.1 




112.8 


301 


149.4 


470 


243.3 


—28 


-33.3 


38 


3.3 


104 


40. 


170 


76.7 




113.3 


302 


150. 


480 


248.9 


—27 


-32.8 


39 


3.9 


105 


40.6 


171 


77.2 




113.9 




150.6 


490 


254.4 


—26 


-32.2 


40 


4.4 


106 


41.1 


172 


77 8 


!! 


114.4 


301 


151.1 


500 


260. 


—25 


-31.7 


41 


5. 


107 


41.7 


173 


78.3 


239 


115. 


305 


151.7 


510 


265.6 


—24 


-31.1 


42 


5.6 


108 


42.2 


174 


78.9 




115.6 


306 


152.2 




271.1 


-23 


-30.6 


43 


6.1 


109 


42.8 


175 


79.4 


241 


116.1 


307 


152.8 


530 


276.7 


-22 


—30. 


44 


6.7 


110 


43.3 


176 


80. 




116.7 


3'X 


153.3 


540 


282.2 


-21 


—29.4 


45 


7.2 


111 


43.9 


177 


80.6 


243 


117.2 


30! 


153.9 


550 


287.8 


-20 


-28.9 


46 


7.8 


112 


44.4 


178 


81.1 


244 


117.8 


310 


154.4 


5G0 


293.3 


-19 


—28.3 


47 


8.3 


113 


45. 


179 


81.7 


245 


118.3 


311 


155. 


570 


298.9 


-18 


—27.8 


48 


8.9 


114 


45.6 


180 


82.2 


24 ( 


118.9 


312 


155.6 


580 


304.4 


-17 


—27.2 


49 


9.4 


115 


46.1 


181 


82.8 


247 


119.4 


313 


156.1 


590 


310. 


-16 


—26.7 


50 


10. 


116 


46.7 


182 


83.3 




120. 


314 


156.7 


600 


315.6 


—15 


—26.1 


51 


10.6 


117 


47.2 


183 


83.9 


24'. 


120.6 


315 


157.2 


61.C 


321.1 


—14 


—25.6 


52 


11.1 


118 


47.8 


184 


84.4 


;-.:-; 


121.1 


316 


157.8 


62C 


326.7 


—13 


—25. 


53 


11.7 


119 


48.3 


185 


85. 


251 


121.7 


317 


158.3 


G3C 




—12 


—24.4 


54 


12.2 


120 


48.9 


186 


85.6 




122.2 


318 


158.9 


64C 


337.8 


-11 


-23.9 


55 


12.8 


121 


49.4 


187 


86.1 


2.>-: 


122.8 


319 


159.4 


650 


; 


—10 


—23.3 


56 


13.3 


122 


50. 




86.7 


. 


123.3 


320 


160. 


660 


348.9 


- 9 


—22.8 


57 


13.9 


123 


50.6 


1S9 


87.2 


255 


123.9 


321 


160.6 


670 


354.4 


- 8 


—22.2 


58 


14.4 


124 


51.1 


190 


87.8 


256 


124.4 


322 


161.1 


680 


360. 


— 7 


—21.7 




15. 


125 


51.7 


191 


88.3 


25? 


125. 


323 


161.7 


691 


365 6 


— 6 


—21.1 


60 


15.6 


126 


52.2 


192 


88.9 


25.' 


125.6 


324 


162.2 




371.1 


— 5 


-20.6 


61 


16.1 


127 


52.8 


193 


89.4 


259 


126.1 


325 


162.8 


710 


376.7 


- 4 


—20. 


62 


16.7 


128 


53.3 


194 


90. 


2fi! 


126.7 


326 


163.3 


720 


382.2 


— 3 


—19.4 


63 


17.2 


129 


53.9 


195 


90.6 


261 


127.2 


327 


163.9 


730 


387.8 


— 2 


-18.9 


64 


17.8 


130 


54.4 


196 


91.1 




127.8 


328 


164.4 


740 


393.3 


— 1 


—18 3 


65 


18.3 


131 


55. 


197 


91.7 




128.3 


329 


165. 


750 


398.9 





-17.8 


66 


18.9 


132 


55.6 


198 


92.2 


264 


128.9 


330 


165.6 


760 


404.4 


+ 1 


—17.2 


67 


19.4 


133 


56.1 


199 


92.8 


265 


129.4 


331 


166.1 


770 


410. 


2 


-16.7 


68 


20. 


134 


56.7 


0(5 


93.3 




130. 


332 


166.7 


7S0 


415.6 


3 


-16.1 


69 


20.6 


135 


57.2 


201 


93.9 


267 


130.6 


333 


167.2 


790 


421.1 


4 


—15.6 


70 


21.1 


136 


57.8 


202 


94.4 


26! 


131.1 


334 


167.8 


SOO 


426.7 


5 


—15. 


71 


21.7 


137 


58.3 


203 


95. 




131.7 


335 


168.3 


810 


432.2 


6 


—14.4 


72 


22.2 


138 


58.9 


204 


95.6 


270 


132.2 


336 


168.9 




437.8 


7 


-13.9 


73 


22.8 


139 


59.4 


205 


96.1 


271 


132.8 


337 


169.4 


830 


443.3 


8 


—13.3 


74 


23.3 


140 


60. 


06 


96.7 




133.3 


338 


170. 


840 


448.9 


9 


—12.8 


75 


23.9 


141 


60.6 


207 


97.2 




133.9 


339 


170.6 


850 


454.4 


10 


—12.2 


76 


24.4 


142 


61.1 




97.8 


274 


134.4 


340 


171.1 


860 


460. 


11 


-11.7 


77 


25. 


143 


61.7 


209 


98.3 


275 


135. 


341 


171.7 


870 


465.6 


12 


—11.1 


78 


25.6 


144 


62.2 


210 


98.9 




135.6 


342 


172.2 


880 


471.1 


13 


—10.6 


79 


26.1 


145 


62.8 


211 


99.4 




136.1 


343 


172.8 


890 


476.7 


14 


-10. 


80 


26.7 


146 


63.3 




100. 




136.7 


344 


173.3 


900 


482.2 


15 


— 9.4 


81 


27.2 


147 


63.9 




100.6 




137.2 


345 


173.9 


910 


487.8 


16 


— 8.9 


82 


27.8 


148 


64.4 


214 


101.1 




137.8 


346 


174.4 


920 


493.3 


17 


— 8.3 




28.3 


149 


65. 


215 


101.7 




138.3 


347 


175. 


930 


498.9 


18 


- 7.8 




28.9 


150 


65.6 




102.2 




138.9 


341 


175.6 


940 


504.4 


19 


— 7.2 




29.4 


151 


66.1 


217 


102.8 




139.4 


m 


176.1 


950 


510. 


20 


— 6.7 


86 


30. 


152 


66.7 


218 


103.3 




140. 


350 


176.7 


960 


515.6 


21 


— 6.1 


87 


30.6 


153 


67.2 




103.9 




140.6 


351 


177.2 


970 


521.1 


22 


- 5.6 




31.1 


154 


67.8 




104.4 




141.1 




177.8 


80 


526.7 


23 


— 5. 


89 


31.7 


155 


68.3 


221 


105. 




141.7 


353 


178.3 


990 


532.2 


24 


— 4.4 


00 


32.2 


156 


68.9 


222 


105.6 




142.2 


354 


178.9 


1000 


537.8 


25 


— 3.9 


91 


32.8 


157 


69.4 


223 


106.1 




142.8 


355 


179.4 


1010 


543.3 



PYROMETRY. 451 

Platinum or Copper Ball Pyrometer.— A weighed piece of 
platinum, copper, or iron is allowed to remain in the furnace or heated 
chamber till it has attained the temperature of its surroundings. It is then 
suddenly taken out and dropped into a vessel containing water of a known 
weight and temperature. The water is stirred rapidly and its maximum 
temperature taken. Let W — weight of the water, w the weight of the ball, 
t = the original and T the final heat of the water, and S the specific heat of 
the metal ; then the temperature of fire may be found from the formula 



*." wS ^- 

For a fuller description, by J. C. Hoadley, see Trans. A. S. M. E., vi, 702. 
The mean specific heat of platinum above 32° is .03333 or l/30th that of water, 
and it increases with the temperature, the increase being about .000305 for 
each 100° F. 

For accuracy corrections are required for variations in the specific heat of 
the water and of the metal at different temperatures, for loss of heat by 
radiation from the metal during the transfer from the furnace to the water, 
and from the apparatus during the heating of the water; also for the heat- 
absorbing capacity of the vessel containing the water. 

Fire-clay or fire-brick may be used instead of the metal ball. 

lie Chatelier's Tliermo-electric Pyrometer. — For a very full 
description see paper by Joseph Struthers, School of Mines Quarterly, vol. 
xii, 1891; also, paper read by Prof. Roberts-Austen before the Iron and Steel 
Institute, May 7, 1891. 

The principle upon which this pyrometer is constructed is the measure- 
ment of a current of electricity produced by heating a couple composed of 
two wires, one platinum and the other platinum with \0% rhodium— the cur- 
rent produced being measured by a galvanometer. 

The composition of the gas which surrounds the couple has no influence 
on the indications. 

When temperatures above 2500° F. are to be studied, the wires must have 
an isolating support and must be of good length, so that all parts of a fur- 
nace can be reached. 

For a Siemens furnace, about llt£ feet is the general length. The wires 
are supported in an iron tube, J^ inch interior diameter and held in place by 
a cylinder of refractory clay having two holes bored through, in which the 
wires are placed. The shortness of time (five seconds) allows the tempera- 
ture to be taken without deteriorating the tube. 

Tests made by this pyrometer in measuring furnace temperatures under 
a great variety of conditions show that the readings of the scale uncorrected 
are always within 45° F. of the correct temperature, and in the majority of 
industrial measurements this is sufficiently accurate. Le Chatelier's py- 
rometer Is sold by Queen & Co., of Philadelphia. 

Graduation of I*e Chatelier's Pyrometer.— W. C. Roberts- 
Austen in his Researches on the Properties of Alloys, Proc. Inst. M. E. 1892, 
says : The electromotive force produced by heating the thermo-junction 
to any given temperature is measured by the movement of the spot of light 
on the scale graduated in millimetres. A formula for converting the divi- 
sions of the scale into thermometric degrees is given by M. Le Chatelier; but 
it is better to calibrate the scale by heating the thermo-junction to temper- 
atures which have been very carefully determined by the aid of the air- 
thermometer, and then to plot the curve from the data so obtained. Many 
fusion and boiling-points have been established by concurrent evidence of 
various kinds, and are now very generally accepted. The following table 
contains certain of these : 

Water boils. 

Lead melts. 

Mercury boils. 

Zinc melts. 

Sulphur boils. 

Aluminum melts. 

Selenium boils. 

The Temperatures Developed in Industrial Furnaces.— 
M. Le Chatelier states that by means of his pyrometer he has discovered 
that the temperatures which occur in melting steel and in other industrial 
operations have been hitherto overestimated. 



Deg. F. 


Deg. C. 


212 


100 


618 


326 


676 


358 


779 


415 , 


838 


448 ' 


1157 


625 


1229 


665 



Deg. F. 


Deg. C. 


1733 


945 


Silver melts. 


1859 


1015 


Potassium sul- 
phate melts 


1913 


1045 


Gold melts. 


1929 


1054 


Copper melts. 


2732 


1500 


Palladium melts. 


3227 


1775 


Platinum melts. 



452 HEAT. 

M. Le Cliatelier finds the melting heat of white cast iron 1135° (2075 6 F.), 
and that of gray cast iron 1220° (22J8° F.). Mild steel melts at 1475° (2687° 
F.), semi-mild at 1455° (2651° F.), and hard steel at 1410° (2570° F.). The 
furnace for hard porcelain at the end of the baking has a heat of 1370° 
(2498° F.). The heat of a normal incandescent lamp is 1800° (3272° F.), but 
it may be pushed to beyond 2100° (3812° F.). 

Prof. Roberts-Austen (Recent Advances in Pyrometry, Trans. A. I. M. E., 
Chicago Meeting, 1M?3) gives an excellent description of modern forms of 
pyrometers. The following are some of his temperature determinations. 

Gold-melting, Royal Mint. 

Degrees. Degrees. 

Centigrade. Fahr. 

Temperature of standard alloy, pouring into moulds. . . . 1180 2156 
Temperature of standard alloy, pouring into moulds (on 

a previous occasion, by thermo-couple) 1147 2097 

Annealing blanks for coinage, temperature of chamber.. 890 1634 

Silver-melting, Royal Mint. 

Temperature of standard alloy, pouring into mould 980 1796 

Ten-ton Open-hearth Furnace, Woolwich Arsenal. 

Temperature of steel. 0.3$ carbon, pouring into ladle 1645 2993 

Temperature of steel, 0.3$ carbon, pouring into large 

mould 1580 2876 

Reheating furnace, Woolwich Arsenal, temperature of 

interior 930 1706 

Cupola furnace, temperature of No. 2, cast-iron pouring 

into ladle 1600 2912 

The following determinations have been effected by M. Le Chatelier: 

Bessemer Process. 

Six- ton Converter. 

Degrees. Degrees. 
Centigrade Fahr. 

A. Bath of slag 1580 2876 

B. Metal in ladle 1640 2984 

C. Metal in ingot mould 1580 2876 

D. Ingot in reheating furnace 1200 2192 

E. Ingot under the hammer 1080 1976 

Open-hearth Furnace (Siemens). 
Semi-Mild Steel. 

A. Fuel gas near gas generator 720 1328 

B. Fuel gas entering into bottom of regenerator chamber 400 752 

C. Fuel gas issuing from regenerator chamber 1200 2192 

Air issuing from regenerator chamber 1000 1832 

Chimney Gases. 

Furnace in perfect condition 300 590 

Open-hearth Furnace. 

End of the melting of pig charge 1420 2588 

Completion of conversion 1500 2732 

Molten Steel. 
In the ladle— Commencement of casting 1580 2876 

End of casting 1490 2714 

Inthemoulds 1520 2768 

For very mild (soft) steel the temperatures are higher by 50° C. 

Siemens Crucible or Pot Furnace. 

1000° C, 2912° F. 

Rotary Puddling Furnace. 

Degrees C. Degrees F. 

Furnace ... 1340-1230 2444-224(1 

Puddled ball— End of operation 1330 2420 



PYltOMETllY. 453 

Blast-furnace (6 ray -Bessemer Pig). 

Opening in face of tuyere 1930 3506 

Molten metal — Commencement of fusion 1400 2552 

End, or prior to tapping 1570 2858 

Hoffman Red-brick Kiln. 
Burning temperatures 1100 2012 

The Wihorgh Air-pyrometer. (E. Trotz, Trans. A. I. M. E. 
1892.)— The inventor using the expansion-coefficient of air, as determined 
by Gay-Lussac, Dulon, Rudberg, and Regnaulfc, bases his construction on 
the following theory : If an air-volume, V, enclosed in a porcelain globe 
and connected through a capillary pipe with the outside air, be heated to 
the temperature T (which is to be'determined) and thereupon the connection 
be discontinued, aud there be then forced into the globe containing V 
another volume of air V of known temperature t, which was previously 
under atmospheric pressure H, the additional pressure /i, due to the addi- 
tion of the air-volume V to the air-volume V, can be measured by a ma- 
nometer. But this pressure is of course a function of the temperature 2'. 
Before the introduction of V, we have the two separate air-volumes, Fat 
the temperature T and V at the temperature f, both under the atmospheric 
pressure H. After the forcing in of V into the globe, we have, on the 
contrary, only the volume V ol the temperature T, but under the pressure 
H+h. 

The Wiborgh Air-pyrometer is adapted for use at blast-furnaces, smelt ing- 
works, hardening and tempering furnaces, etc., where determinations of 
temperature from 0° to 2400° F. are required. 

Seger's Fire-clay Pyrometer. (H. M. Howe, Eng. and Mining 
Jour., June 7, 1890.)— ProlVssor Seger uses a series of slender triangular 
fire-clay pyramids, about 3 inches high and % inch wide at the base, aud 
each a little less fusible than the next : these he calls "normai pyramids" 
( u normal-kegel "). When the series is placed in a furnace whose temper- 
ature is gradually raised, one after another will bend over as its range of 
plasticity is reached ; and the temperature at which it has bent, or "wept," 
so far that its apex touches the hearth of the furnace or other level surface 
on which it is standing, is selected as a point on Seger's scale. These points 
may be accurately determined by some absolute method, or they may 
merely serve to give comparative results. Unfortunately, these pyramids 
afford no indications when the temperature is stationary or falling. 

Mesure and Nonel's Pyrometric Telescope. (Ibid.)— Mesure 
and Nouel's pyrometric telescope gives us an immediate determination of 
the temperature of incandescent bodies, and is therefore much better 
adapted to cases where a great number of observations are to be made, and 
at short intervals, than Seger's. Such cases arise in the careful heating of 
steel. The little telescope, carried in the pocket or hung from the neck, can 
be used by foreman or heater at any moment. 

It is based on the fact that a plate of quartz, cut at right angles to the 
axis, rotates the plane of polarization of polarized light to a degree nearly 
inversely proportional to the square of the length of the waves; and, 
further, on the fact that while a body at dull redness merely emits red 
light, as the temperature rises, the orauge, yellow, green, and blue waves 
successively appear. 

If, now, such a plate of quartz is placed between two Nicol prisms at 
right angles, "a ray of monochromatic light which passes the first, or 
polarizer, and is watched through the second, or analyzer, is not extin- 
guished as it was before interposing the quartz. Part of the light passes 
the analyzer, and, to again extinguish it, we must turn one of the Nicols a 
certain angle," depending on the length of the waves of light, and hence on 
the temperature of the incandescent object which emits this light. Hence 
the angle through which we must turn the analyzer to extinguish the light 
is a measure of the temperature of the object observed. 

The instrument is made by Ducietet, of Paris, in two sizes ; cost, $20 and 
$25. 

The Uehling and Steinbart Pyrometer. (For illustrated descrip- 
tion see Engineering, Aug. 24, 1894.)— The action of the pyrometer is based 
on a principle which involves the law of the flow of gas through minute 
apertures in the following manner : If a closed tube or chamber be supplied 
with a minute inlet and a minute outlet aperture and air be caused by a 
constant suction to flow in through one and out through the other of these 
apertures, the tension iu the chamber between the apertures will vary with 



454 HEAT. 

the difference of temperature between the inflowing and outflowing air. If 
the inflowing air be made to vary with the temperature to be measured, 
and the outflowing air be kept at a certain constant temperature, then the 
tension in the space or chamber between the two apertures will be an exact 
measure of the temperature of the inflowing air, and hence of the tem- 
perature to be measured. 

In operation it is necessary that the air be sucked into it through the first 
minute aperture at the temperature, to be measure:!, through the second 
aperture at a lower but constant temperature, and that the suction be of a 
constant tension. The first aperture is therefore located in the end of a 
platinum tube in the bulb of a porcelain tube over which the hot blast 
sweeps, or inserted into the pipe or chamber containing the gas whose tem- 
perature is to be ascertained. 

The second aperture is located in a coupling, surrounded by boiling water, 
and the suction is obtained by an aspirator aud regulated by a column of 
water of constant height. 

The tension in the chamber between the apertures is indicated by a 
manometer. 

The Air-thermometer. (Prof. R. C. Carpenter, Eng'g News, Jan. 5, 
1893.) — Air is a perfect thermometric substance, and if a given mass of air 
be considered, the product of its pressure and volume divided by iis 
absolute temperature is in every case constant. If the volume of air 
remain constant, the temperature will vary with the pressure; if the 
pressure remain constant the temperature will vary with the volume. As 
the former condition is more easily attained air-thermometers are usually 
constructed of constant volume, in which case the absolute temperature 
will vary with the pressure. 

If we denote pressure by p and p', the corresponding absolute temper- 
atures by T and T', we should have 

T 
p:p'::T:T' and T'=p'—.- 

* p 

The absolute temperature Tis to be considered in every case 460 higher 
than the thermometer-reading expressed in Fahrenheit degrees. From the 
form of the above equation, if the pressure be corresponding to a known 
absolute temperature, 2'can be found. The quotient is a constant which may 
be used in all determinations with the instrument. The pressure on the 
instrument can be expressed in inches of mercury, and is evidently the 
atmospheric pressure b as shown by a barometer, plus or minus an addi- 
tional amount h shown by a manometer attached to the air thermometer. 

That is, in general, p = b x h. 

The temperature of 32° F. is fixed as the point of melting ice, in which 
case 2' = 460 X 32 = 492° F. This temperature can be produced by sur- 
rounding the bulb in melting ice and leaving several minutes, so that the 
temperature of the confined air shall acquire that of the surrounding ice. 
When the air is at that temperature, note the reading of the attached 
manometer h, and that of a barometer; the sum will be the value of p cor- 
responding to the absolute temperature of 492° F. The constant of the 
instrument, K — 492 — p, once obtained, can be used in all future determina- 
tions. 

High Temperatures judged by Color.— The temperature of a 
body can be approximately judged by the experienced eye unaided, and 
M. Pouillet has constructed a table, which has been generally accepted,- 
giving the colors and their corresponding temperature as below: 

Deg. C. Deg. F. 

Deep orange heat. . . 1100 2021 

Clear orange heat.. 1200 2192 

White heat 1300 2372 

Bright white heat.. 1400 2552 

) 1500 2732 

Dazzling white heat > to to 

J 1600 2912 



Deg. C. 


Deg. F 


Incipient red heat.. 525 


977 


Dull red heat 700 


1292 


Incipient cherry-red 




heat 800 


1472 


Cherry-red heat 900 


1652 


Clear cherry - red 




heat 1000 


1832 



The results obtained, however, are unsatisfactory, as much depends on 
the susceptibility of the retina of the observer to light as well as the degree 
of illumination under which the observation is made. 



QUANTITATIVE MEASUREMENT OF HEAT. 



455 



A bright bar of iron, slowly heated in contact with air, assumes the fol- 
lowing tints at annexed temperatures (Claudel): 



Cent. Fahr. 

Yellow at 225 437 

Orange at 243 473 

Red at 265 509 

Violetat.. 277 531 



Cent. Fahr. 

Indigo at 288 550 

Blue at 293 559 

Green at 332 630 

' ' Oxide-gray " 400 752 



BOILING POINTS AT ATMOSPHERIC PRESSURE. 

14.7 lbs. per square inch. 



Ether, sulphuric 100° F. 

Carbon bisulphide . 118 

Ammonia 140 

Chloroform 140 

Bromine 145 

Wood spirit . 150 

Alcohol 173 

Benzine 176 

Water 212 



Average sea- water 213.2° F. 

Saturated brine 226 

Nitricacid 248 

Oil of turpentine 315 

Phosphorus 554 

Sulphur 570 

Sulphuric acid 590 

Linseed oil 597 

Mercury 676 



The boiling points of liquids increase as the pressure increases. The boil- 
ing point of water at any given pressure is the same as the temperature of 
saturated steam of the same pressure. (See Steam.) 

MELTING-POINTS OF VARIOUS SUBSTANCES. 

The following figures are given by Clark (on the authority of Pouillet, 
Claudel, and Wilson), except those marked *, which are given by Prof. Rob- 
erts-Austen in his description of the Le Chatelier pyrometer. These latter 
are probably the most reliable figures. 

Sulphurous acid - 148° F. Alloy, 1 tin, 1 lead.. 370 to 466° F. 

Carbonic acid — 108 

Mercury - 39 

Bromine -f- 9.5 

Turpentine 14 

Hyponitric acid 16 

Ice 32 

Nitro-glycerine 45 

Tallow 92 

Phosphorus 112 

Acetic acid. 113 

Stearine 109 to 120 

Spermaceti 120 

Margaric acid 131 to 140 

Potassium 136 to 144 

Wax 142 to 154 

Stearic acid 158 

Sodium 194 to 208 

Alloy, 3 lead, 2 tin, 5 bismuth 199 

Iodine 225 

Sulphur 239 

Alloy, \y z tin, 1 lead 334 



Tin 442 to 446 

Cadmium 442 

Bismuth 504 to 507 

Lead 608 to 618* 

Zinc 680 to 779* 

Antimony 810 to 1150 

Aluminum 1 157* 

Magnesium 1200 

Calcium Full red heat. 

Bronze 1692 

Silver 1733* to 1873 

Potassium sulphate 1859* 

Gold 1913* to 2282 

Copper 1929* to 1996 

Cast iron, white... 1922 to 2075* 
gray 2012 to 2786 2228* 

Steel 2372 to 2532 

" hard 2570*; mild, 2687* 

Wrought iron 2732 to 2912 

Palladium 2732* 

Platinum 3227* 



For melting-point of fusible alloys, see Alloys. 

Cobalt, nickel, and manganese, fusible in highest heat of a forge. Tung- 
sten and chromium, not fusible in forge, but soften and agglomerate. Plati- 
num and iridium, fusible only before the oxyhydrogen blowpipe. 

QUANTITATIVE MEASUREMENT OF HEAT. 

Unit of Heat.— The British unit of heat, or British thermal unit 
(B. T. U.), is that quantity of heat which is required to raise the temperature 
of 1 lb. of pure water 1° Fahr., at or near 39°. 1 F., the temperature of maxi- 
mum density of water. 

The French thermal unit, or calorie, is that quantity of heat which is re- 
quired to raise the temperature of 1 kilogramme of pure water 1° Cent., at or 
about 4° C, which is equivalent to 39°. 1 F. 

1 French calorie = 3.968 British thermal units; 1 B. T. U. = .252 calorie. 
The " pound calorie " is sometimes used by English writers; it is the quaes- 



456 



tity of heat required to raise the temperature of 1 lb. of water 1° C. 1 lb. 
calorie = 2.2046 B. T. U. = 5/9 calorie. The heat of combustion of carbon, to 
C0 2 , is said to be 8080 calories. This figure is used either for French calories or 
for pound calories, as it is the number of pounds of water that can be raised 
1° O. by the complete combustion of 1 lb. of carbon, or the number of 
kilogrammes of water that can be raised 1° C. by the combustion of 1 kilo. 
of carbon; assumiug in each case that all the heat generated is transferred 
to the water. 

The Mechanical Equivalent of Heat is the number of foot- 
pounds of mechanical "energy equivalent to one British thermal unit, heat 
and mechanical energy being mutually convertible. Joule's experiments, 
1843-50, gave the figure 772, which is known as Joule's equivalent. More re- 
cent experiments by Prof. Rowland (Proc. Am. Acad. Arts and Sciences,. 
1880; see also Wood's Thermodynamics) give higher figures, and the most 
probable average is now considered to be 778. 

1 heat-unit is equivalent to 778 ft.-lbs. of energy. 1 ft. lb. = 1/778 =.0012852 
heat-units. 1 horse-power = 33,000 ft.-lbs. per minute = 2545 heat-uuits per 
hour = 42,416 4- per minute = .70694 per second. 1 lb. carbon burned to CO a 
= 14,544 heat-units. 1 lb. C. per H.P. per hour = 2545 -{-[14544 = 17i# efficiency 
(.174986). 

Heat of Combustion of Various Substances in Oxygen. 



Authority. 



Heat-units. 


Cent. 


Fahr. 


( 34,462 


62,032 


\ 33,808 


60,854 


( 34,342 


61,816 


28,732 


51,717 


( 8,080 


14,544 


{ 7,900 


14,220 


8,137 


14,647 


7,859 


14,146 


7,861 


14,150 


7,901 


14,222 


2,473 


4,451 


I 2,403 


4,325 


{ 2,431 


4,376 


( 2,3S5 


4,293 


5,607 


10,093 


( 13,120 


23,616 


{ 13,108 


23,594 


{ 13,063 


23,513 


111,858 


21,344 


■{ 11,942 


21,496 


(11,957 


21,523 


j 10,102 
1 9,915 


18,184 


17,847 



Hydrogen to liquid water at 0° C 

" to steam at 100° C 

Carbon (wood charcoal) to carbonic 
acid, CO a ; ordinary temperatures. 

Carbon, diamond to C0 2 . . 

" black diamond to C0 2 

" graphite to C0 2 

Carbon to carbonic oxide, CO 

Carbonic oxide to CO a , per unit of CO 

CO to C0 2 per unit of C = 2}£ X 2403 

Marsh-gas, Methane, CH 4 to water 

and CO a 

defiant gas, Ethylene, C a H 4 to 
water and CO a . 

Benzole gas, C 6 H„ to water and CO^ 



Favre and Silbermaun. 

Andrews. 

Thomsen. 

Favre and Silbermann. 

Andrews. 
Berthelot. 



Favre and Silbermann. 

Andrews. 

Thomsen. 

Favre and Silbermann. 

Thomsen. 

Andrews. 

Favre and Silbermann. 



Andrews. 
Thomsen. 



Favre and Silbermann. 



In burning 1 pound of hydrogen with 8 pounds of oxygen to form 9 pounds 
of water, the units of heat evolved are b2,032 (Favre and S.); but if the 
resulting product is not cooled to the initial temperature of the gases, 
part of the heat is rendered latent in the steam. The total heat of 1 lb. 
of steam at 212° F. is 1146.1 heat-units above that of water at 32°, and 
9 X 1146 1 = 10,315 heat-units, which deducted from 62,032 gives 51,717 as the 
heat evolved by the combustion of 1 lb. of hydrogen and 8 lbs. of oxygen at 
32° F. to form steam at 212° F. 

By the decomposition of a chemical compound as much heat is absorbed 
or rendered latent as was evolved when the compound was formed. If 1 lb. 
of carbon is burned to C0 2 , generating 14,544 B.T.U., and the C0 2 thus formed 
is immediately reduced to CO in the presence of glowing carbon, by the 
reaction C0 2 + C = 2CO, the result is the same as if the 2 lbs. C had been 
burned directly to 2CO, generating 2 X 4451 = 8902 heat-units; consequently 
14,544 — 8902 = 5642 heat-units have disappeared or become latent, and the 



SPECIFIC HEAT. 



457 



"unburning " of C0 2 to CO is thus a cooling operation. (For heats of com- 
bustion of various fuels, see Fuel.) 

SPECIFIC HEAT. 

Thermal Capacity.— The thermal capacity of a body is the quantity 
of heat required to raise its temperature one degree. The ratio of the heat 
required to raise the temperature of a given substance one degree to that 
required to raise the temperature of water one degree from the temperature 
of maximum density 39.1 is commonly called the specific heat of the sub- 
stance. Some writers object to the term as being an inaccurate use of the 
words " specific " and " heat." A more correct name would be " coefficient 
of thermal capacity." 

Determination of Specific Heat.— Method by Mixture.— The 
body whose specific beat is to be determined is raised to a known tempera- 
ture, and is then immersed in a mass of liquid of which the weight, specific 
heat, and temperature are known. When both the body and the liquid 
have attained the same temperature, this is carefully ascertained. 

Now the quantity of heat lost by the body is the same as the quantity of 
heat absorbed by the liquid. 

Let c, w, and t be the specific heat, weight, and temperature of the hot 
body, and c', w', and V of the liquid. Let T be the temperature the mix- 
ture assumes. 

Then, by the definition of specific heat, c X w X (t - T) = heat-units lost 
by the hot body, and c' X w' x (T - t') = heat-units gained by the cold 
liquid. If there is no heat lost by radiation or conduction, these must be 
equal, and 

av(t - T) = c'iv'(T- t') or c = C '™' l . T ~ *'\ 

Specific Heats of Various Substances. 

The specific heats of substances, as given by different authorities, show 
considerable lack of agreement, especially in the case of gases. 

The following tablesgive the mean specific heats of the substances named 
according to Regnaulr. (From Rontgen's Thermodynamics, p. 134.) These 
specific heats are average values, taken at temperatures which usually come 
under observation in technical application. The actual specific heats of all 
substances, in the solid or liquid state, increase slowly as the body expands 
or as the temperature rises. It is probable that the specific heat of a body 
when liquid is greater than when solid. For many bodies this has been 
verified by experiment. 

Solids. 

Steel (soft) 0.1165 

Steel (hard) 1175 

Zinc 0.0956 

Brass 0.0939 

Ice... 0.5040 

Sulphur 0.2026 

Charcoal 0.2410 

Alumina 0.1970 

Phosphorus 0.1887 



Antimony 0.0508 

Copper 0.0951 

Gold 0.0324 

Wrought iron 0.1138 

Glass 0.1937 

Cast iron 0. 1298 

Lead 0.0314 

Platinum . 0324 

Silver 0.0570 

Tin 0.0562 



Water 1.0000 

Lead (melted) 0.0402 

Sulphur " 0.2340 

Bismuth " 0.0308 

Tin " 0.0637 

Sulphuric acid 0.3350 



Liquids. 



Mercury 0.0333 

Alcohol (absolute) 0.7000 

Fusel oil 0.5640 

Benzine 0.4500 

Ether 0.5034 



458 HEAT. 



Gases. 
Constant Pressure Constant Volume. 

Air 0.23751 0.16847 

Oxygen 0.21751 0.15507 

Hydrogen 3.40900 2.41226 

Nitrogen 0.24380 0.17273 

Superheated steam 0.4805 0.346 

- Carbonic acid 0.217 0.1535 

Olefiant'Gas (CH a ) 0.404 0.173 

Carbonic oxide 0.2479 0.1758 

Ammonia 0.508 0.299 

Ether....... 0.4797 0.3411 

Alcohol 0.4534 0.3200 

Acetic acid. , 0.4125 

Chloroform.. 0.1567 

In addition to the above, the [following are given by other authorities. 
(Selected from various sources.) 

Metals. 



Platinum at 32° F 0333 

(increased .000305 for each 100° F.) 
Cadmium. .. . 0567 



Copper, 32° to 212° F . .094 

" 32° to 572° F 1013 

Zinc 32°to212°F 0927 

32° to 572° F 1015 

Nickel ,. .1086 

Aluminum, 0° F. to melting- 
point (A. E. Hunt) 0.2185 

Other Solids. 



Wrought iron (Petit & Dulong). 

32° to 212° 1098 

" 32° to 39:.'° 115 

" 32° to 572° 1218 

32° to 662° 1255 

Wrought iron (J. C. Hoadley, 
A. S. M. E., vi. 713), 

Wrought iron, 32° to 200° 1129 

32° to 600° 1327 

32° to 2000° 2619 



Brickwork and masonry, about. .20 

Marble 210 

Chalk 215 

Quicklime 217 

Magnesian limestone 217 

Silica 191 

Corundum 198 

Stones generally 2 to 22 



Coal 20to241 

Coke 203 

Graphite 202 

Sulphate of lime 197 



Soda 231 

Quartz .188 

River sand 195 



Woods. 

Pine (turpentine) 467 I Oak 570 

Fir 650 | Pear 500 



Alcohol, density .793 622 

Sulphuric acid, density 1.87 335 

1.30 661 

Hydrochloric acid 600 



Liquids. 



Olive oil 310 

Benzine 393 

Turpentine, density .872 472 

Bromine 1.111 

At Constant At Constant 

Pressure. Volume. 

Sulphurous acid 1553 .1246 

Light carburetted hydrogen, marsh gas (CH 4 ). .5929 .4683 

Blast-furnace gases 2277 

Specific Heat of Salt Solution. (Schuller.) 

Per cent salt in solution 5 10 15 20 25 

Specific heat 9306 .8909 .8606 .8490 .8073 

Specific Heat of Air.— Regnault gives for the mean value 

Between — 30° C. and + 10° C 0.23771 

0°C. " 100° C .0.23741 

0°C. " 200°C 0.23751 

Hanssen uses 0.1686 for the specific heat of air at constant volume. The 
value of this constant has never been found to any degree of accuracy by 
direct experiment. Prof. Wood gives 0.2375 -4- 1.406 = 0.1689. The ratio of 



EXPANSION BY HEAT. 



459 



the specific heat of a fixed gas at constant pressure to the sp. ht. at con- 
stant volume is given as follows by different writers (Eng'g, July 12, 1889): 
Regnault, 1.3953; Moll and Beck, 1.4085; Szathmari, 1.4027; J. Macfarlane 
Gray, 1.4. The first three are obtained from the velocity of sound in air. The 
fourth is derived from theory. Prof. Wood says: The value of'the ratio for 
air, as found in the days of La Place, was 1.41, and we have 0.2377 h- 1.41 
= 0.1686, the value used by Clausius, Hanssen, and many others. But this 
ratio is not definitely known. Rankine in his later writings used 1.408, and 
Tait in a recent work gives 1.404, while some experiments gives less than 
1.4 and others more than 1.41. Prof. Wood uses 1.406. 

Specific Heat of Gases.— Experiments by Mallard and Le Chatelier 
indicate a continuous increase in the specific heat at constant volume of 
steam, CO a , and even of the perfect gases, with rise of temperature. The 
variation is inappreciable at 100° C, but increases rapidly at the high tem- 
peratures of the gas-engine cylinder. (Robinson's Gas and Petroleum 
Engines.) 

Specific Heat and Latent Heat of Fusion of Iron and 
Steel. (H. H. Campbell, Trans. A. I. M. E., xix. 181.) 

Akerman. Troilius. 

Specific heat pig iron, to 1200° C 0.16 

1200tol800°C 0.21 

Oto 1500° C 0.18 

1500tol800°C 0.20 

Calculating by both sets of data we have : 

Akerman. Troilius. 

Heating from to 1800° C . . 318 330 calories per kilo. 

Hence probable value is about 325 calories per kilo. 

Specific heat, steel (probably high carbon) (Troilius) 1175 

" soft iron " .1081 

Hence probable value solid rail steel 1125 

" " " melted rail steel 1275 

Akerman. Troilius. 
Latent heat of fusion, pig iron, calories per kilo. .46 

44 " gray pig 33 

" *' " white pig 23 

From which we may assume that the truth is about : Steel, 20 ; pig iron, 30. 

EXPANSION BY HEAT. 

In the centigrade scale the coefficient of expansion of air per degree is 
0.003665 = 1/273; that is, the pressure being constant, the volume of a perfect 
gas increases 1/273 of its volume at 0° C. for every increase in temperature 
of PC. In Fahrenheit units it increases 1/491.2 = .002036 of its volume at 
32° F. for every increase of 1° F. 

Expansion of Gases by Heat from 32° to 212° F. (Regnault.) 



Hydrogen 

Atmospheric air. . 

Nitrogen 

Carbonic oxide . . . 

Carbonic acid 

Sulphurous acid , 



Increase in Volume, 
Pressure Constant. 
Volume at 32° Fahr. 
= 1.0, for 



0.3661 



o.;: 



70 



0.3670 
0.3669 
0.3710 
0.3903 



0.002034 
0.002039 
0.002039 
0.002038 
0.002061 
0.002168 



Increase in Pressure, 
Volume Constant. 
Pressure at 32° 
Fahr. = 1.0, for 



0.3667 
0.3665 
0.3668 
0.3667 
0.3688 
0.3845 



002037 
002036 
002039 
002037 
002039 
002136 



If the volume is kept constant, the pressure varies directly as the a 
temperature. 



460 



Lineal Expansion of Solids at Ordinary Temperatures. 

(British Board of Trade; from Clark ) 



For 
1° Fahr. 



For 

1° Cent. 



Coef- 
ficient 

of 
Expan- 
sion 
from 
32° to 
212° F. 



Accord- 
ing to 
Other 

Author- 
ities. 



Length = 1 Length=l 



Aluminum (cast) . . 

Antimony (crj-st.) 

Brass, cast 

" plate 

Brick 

Bronze (Copper, 17; Tin, 2^; Zinc 1) 

Bismuth 

Cement, Portland (mixed), pure ... . 
Concrete : cement, mortar, and pebbles 

Copper 

Ebonite 

Glass, English flint 

thermometer 

" hard 

Granite, gray, dry 

" red, dry 

Gold, pure 

Iridium, pure 

Iron, wrought 

" cast 

Lead 

Magnesium 

Marbles, various -j JJ om 

j from 



.00001234 
.00000627 
.00000957 
.00001052 
.00000306 
.00000986 
.00000975 
.00000594 
.00000795 
.00000887 
.00004278 
.00000451 
.00000499 
.00000397 
.00000438 
.00000498 
.00000786 
.00000356 
.00000648 
.00000556 
.00001571 



Masonry, brick J 

Mercury (cubic expansion) 

Nickel 

Pewter 

Plaster, white 

Platinum 

Platinum, 85 per cent I 
Iridium, 15 " "{"•■•■ 

Porcelain , 

Quartz, parallel to major axis, t 0° to 

40° C 

Quartz, perpendicular to major axis. 

t0° to 40° C 

Silver, pure 

Slate 

Steel, cast 

" tempered 

Stone (sandstone), dry .. 

Rauville 

Tin 



Wedgwood wan 

Wood, pine 

Zinc 

Zinc, 8 | 

Tin, If 



.00000308 
.00000786 
.00000256 
.00000494 
.00009984 
.00000695 
.00001129 
.00000922 
.00000479 
.00000453 
.00000200 

.00000434 

.00000788 
.00001079 
.00000577 
.00000636 
00000689 
.00000652 
.00000417 
.00001163 
.00000489 
.00000276 
.00001407 
.00001496 



.00002221 
.00001129 
.00001722 
.00001894 
.00000550 
.00001774 
.00001755 
.00001070 
.00001430 
.00001596 
.00007700 
.00000812 
.00000897 
.00000714 
.00000789 
.00000897 
.00001415 
.00000641 
.00001166 
.00001001 
.00002828 



.00000554 
.00001415 
.00000460 
.00000890 
.00017971 
.00001251 
.00002033 
.00001660 
.00000863 
.00000815 
.00000360 
.00000781 

.00001419 
.0000194: 
.00001038 
.00001144 
.00001240 
.00001174 
.00000750 
.00002094 
.00000881 
.00000496 
.00002532 

.00002692 



.002221 
.001129 
.001722 
.001894 
.000550 
.001774 
.001755 
.001070 
.001430 
.001596 
007700 
.000812 
.000897 
.000714 
.000789 
.000897 
.001415 
.000641 
.001166 
.001001 



000554 
.001415 
.000460 
.000890 
.017971 
.001251 
.002033 
.001660 
.000863 
.000815 



.000781 

.001419 
.001943 
.001038 
.001144 
.001240 
.001174 
.00U750 
.002094 
.000881 
.000496 
.002532 



Cubical expansion, or expansion of volume = linear expansion x c 



LATENT HEATS OF FUSION. 461 

Absolute Temperature— Absolute Zero.— The absolute zero of a 
gas is a theoretical consequence of the law of expansion by heat, assuming 
that it is possible to continue the cooling of a perfect gas until its volume is 
diminished to nothing. 

If the volume of a perfect gas increases 1/273 of its volume at 0° C. for 
every increase of temperature of 1° C, and decreases 1/278 of its volume for 
every decrease of temperature of 1° C, then at - 273° C. the volume of the 
imaginary gas would be reduced to nothing. This point — 273° C. or 491.2° 
F. below the melting-point of ice on the air thermometer, or 492.66° F. be- 
low on a perfect gas thermometer = — 459.2° F. (or — 460.66°), is called the 
absolute zero; and absolute temperatures are temperatures measured, on 
either the Fahrenheit or centigrade scale, from this zero. The freezing 
point, 32° F., corresponds to 491.2° F. absolute. If p be the pressure and 
v the volume of a gas at the temperature of 32° F. = 491.2° on the absolute 
scale = T , and p the pressure, and v the volume of the same quantity of 
gas at any other absolute temperature T, then 

pv _ T _ t + 459.2 pv _ p v 
p^v ~ T ~ 491.2" ; ~T ~ ~1\ ' 

The value of p v -*- T for air is 53.37, and pv - 53.37T, calculated as fol- 
lows bv Prof. Wood: 

A cubic foot of dry air at 32° F. at theSsea-level weighs 0.080728 lb. The 

volume of one pound is v = ■ n o n ^ 9 ^ — 12.387 cubic feet. The pressure per 

square foot is 2116.2 lbs. 

p v _ 2116.2 X 12.387 _ 26214 _ 
~T^ ~ 491.13 - 4903 ~ 5d ' d7 ' 

The figure 491.13 is the number of degrees that the absolute zero is below 
the melting-point of ice, by the air thermometer. On the absolute scale, 
whose divisions would be indicated by a perfect gas thermometer, the cal- 
culated value approximately is 492.66, which would make pv = 53.21 T. Prof. 
Thomson considers that - 273.1° C. = — 491.4° F., is the most probable value 
of the absolute zero. See Heat in Ency. Brit. 

Expansion of Liquids from 32° to 212° F.— Apparent ex- 
pansion in glass (Clark). Volume at 212°, volume at 32° being 1: 

Water 1.0466 Nitric acid 1.11 

Water saturated with salt 1.05 Olive and linseed oils 1.08 

Mercury.. 1 .0182 Turpentine and ether 1 .07 

Alcohol!...; 1.11 Hydrochlor. and sulphuric acids 1.06 

For water at various temperatures, see Water. 
For air at various temperatures, see Air. 

LATENT HEATS OF FUSION AND EVAPORATION. 
Latent Heat means a quantity of heat which has disappeared, having 
been employed to produce some change other than elevation of temperature. 
By exactly reversing that change, the quantity of heat which has dis- 
appeared is reproduced. Maxwell defines it as the quantity of heat which 
must be communicated to a body in a given state in order to convert it into 
another state without changing its temperature. 

Latent Heat of Fusion.— When a body passes from the solid to the 
liquid state, its temperature remains stationary, or nearly stationary, at a 
certain melting point during the whole operation of melting; and in order 
to make that operation go on, a quantity of heat must be transferred to the 
substance melted, being a certain amount for each unit of weight of the 
substance. This quantity is called the latent heat of fusion. 

When a body passes from the liquid to the solid state, its temperature 
remains stationary or nearly stationary during the whole operation of freez- 
ing; a quantity of heat equal to the latent heat of fusion is produced in the 
body and rejected into the atmosphere or other surrounding bodies. 

The following are examples in British thermal units per pound, as given 
by Rankine: 

qnhstanoes Melting Latent Heat 

Substances. Points. of Fusion. 

Ice (according to Person) 32 142.65 

Spermaceti 56 148 

Beeswax 140 175 

Phosphorus 177 9.06 

Sulphur... 405 16.86 

Tin 426 500 



462 HEAT. 

Prof. Wood considers 144 heat units as the most reliable value for the 
latent heat of fusion of ice. Box gives only 26.6 for tin. Clements gives 233 
for cast iron. 

Latent Heat of Evaporation.— When a body passes from the 
solid or liquid to the gaseous state, its temperature during the operation 
remains stationary at a certain boiling point, depending on the pressure of 
the vapor produced ; and in order to make the evaporation go on, a quantity 
of heat must be transferred to the substance evaporated, whose amount for 
each unit of weight of the substance evaporated depends on the temperature. 
That heat does not raise the temperature of the substance, but disappears 
in causing it to assume the gaseous state, and it is called the latent heat of 
evaporation. 

When a body passes from the gaseous state to the liquid or solid state, its 
temperature remains stationary, during that operation, at the boiling-poiut 
corresponding to the pressure of the vapor: a quantity of heat equal to the 
latent heat of evaporation at that temperature is produced in the body; and 
in order that the operation of condensation may go on, that heat must be 
transferred from the body condensed to some other body. 

The following are examples of the latent heat of evaporation in British 
thermal units, of one pound of certain substances, when the pressure of the 
vapor is one atmosphere of 14.7 lbs. on the square inch: 

Q„v, danno Boiling-point under Latent Heat in 

bubstance. one atm Fahr British units. 

Water 212.0 965.7 (Regnault.) 

Alcohol , 172.2 364.3 (Andrews.) 

Ether 95.0 162.8 

Bisulphide of carbon 114.8 156.0 

The latent heat of evaporation of water at a series of boiling-points ex 
tending from a few degrees below its freezing-point up to about 375 degrees 
Fahrenheit has been determined experimentally by M. Regnault. The re- 
sults of those experiments are represented approximately by the formula, 
in British thermal units per pound, 

I nearly = 1091.7 - 0.7(£ - 32°) = 965.7- 0.7(£ - 212°). 

The Total Heat of Evaporation is the sum of the heat which 
disappears in evaporating one pound of a given substance at a given tem- 
perature (or latent heat of evaporation) and of the heat required to raise its 
temperature, before evaporation, from some fixed temperature up to the 
temperature of evaporation. The latter part of the total heat is called the 
sensible heat. 

In the case of water, the experiments of M„ Regnault show that the total 
heat of steam from the temperature of melting ice increases at a uniform 
rate as the temperature of evaporation rises. The following is the formula 
in British thermal units per pound: 

h = 1091.7 + 0.305(f - 32°). 

For the total heat, latent heat, etc., of steam at different pressures, see 
table of the Properties of Saturated Steam. For tables of total heat, latent 
heat, and other properties of steams of ether, alcohol, acetone, chloroform, 
chloride of carbon, and bisulphide of carbon, see Rontgen's Thermodynam- 
ics (Dubois's translation.) For ammonia and sulphur dioxide, see Wood's 
Thermodynamics; also, tables under Refrigerating Machinery, in this book. 

EVAPORATION AND DRYING. 

In evaporation, the formation of vapor takes place on the surface; in boil- 
ing, within the liquid : the former is a slow, the latter a quick, method of 
evaporation. 

If we bring an open vessel with water under the receiver of an air-pump 
and exhaust the air the water in the vessel will commence to boil, and if we 
keep up the vacuum the water will actually boil near its freezing-point. The 
formation of steam in this case is due to the heat which the water takes out 
of the surroundings. 

Steam formed under pressure has the same temperature as the liquid in 
which it was formed, provided the steam is kept under the same pressure. 

By properly coolirjg the rising steam from boiling water, as in the multiple- 
effect evaporating' systems, we can regulate the pressure so that the water 
b ils at low temperatures. 



EVAPORATION. 463 

Evaporation of Water in Reservoirs.— Experiments at the 
Mount Hope Reservoir, Rochester, N. Y., in 1891, gave the following results: 

July. Aug. Sept. Oct. 

Meau temperature of air in shade 70.5 70.3 68.7 53.3 

"' u " water in reservoir 68.2 70.2 66.1 54.4 

" humidity of air, per cent 67.0 74.6 75.2 74.7 

Evaporation in inches during month 5.59 4.93 4.05 3.23 

Rainfall in inches during month 3.44 2.95 1.44 2.16 

Evaporation of Water from Open Channels, (Flynn's 
Irrigation Canals and Flow of Water.)— Experiments from 1881 to 1885 in 
Tulare County, California, showed an evaporation from a pan in the river 
equal to an average depth of one eighth of an inch per day throughout the 
year. 

When the pan was in the air the average evaporation was less than 3/16 
of an inch per day. The average for the month of August was 1/3 inch per 
day, and for March and April 1/12 of an inch per day. Experiments in 
Colorado show that evaporation ranges from .088 to .16 of an inch per day 
during the irrigating season. 

In Northern Italy the evaporation was from 1/12 to 1/9 inch per day, while 
in the south, under the influence of hot winds, it was from 1/6 to 1/5 inch 
per day. 

In the hot season in Northern India, with a decidedly hot wind blowing, 
the average evaporation was y% inch per day. The evaporation increases 
with the temperature of the water. 

Evaporation by the Multiple System.— A multiple effect is a 
series of evaporating vessels each having a steam chamber, so connected 
that the heat of the steam or vapor produced in the first vessel heats the 
second, the vapor or steam produced in the second heats the third, and so 
on. The vapor from the-last vessel is condensed in a condenser. Three 
vessels are generally used, in which case the apparatus is called a Triple 
Effect. In evaporating in a triple effect the vacuum is graduated so that the 
liquid is boiled at a constant and low temperature. 

Resistance to Roiling 1 .— Brine. (Rankine.)— The presence in a 
liquid of a substance dissolved in it (as salt in water) resists ebullition, and 
raises the temperature at which the liquid boils, under a given pressure; but 
unless the dissolved substance enters into the composition of the vapor, the 
relation between the temperature and pressure of saturation of the vapor 
remains unchanged. A resistance to ebullition is also offered by a vessel of 
a material which attracts the liquid (as when water bcils in a glass vessel), 
and the boiling take place by starts. To avoid the errors which causes of 
this kind produce in the measurement of boiling-points, it is advisable to 
place the thermometer, not in the liquid, but in the vapor, which shows the 
true boiling-point, freed from the disturbing effect of the attractive nature 
of the vessel. The boiling-point of saturated brine under one atmosphere 
is 226° Fahr., and that of weaker brine is higher than the boiling-point of 
pure water by 1.2° Fahr., for each 1/32 of salt that the water contains. 
Average sea-water contains 1/32; and the brine in marine boilers is not suf- 
fered to contain more than from 2/32 to 3/32. 

Methods of Evaporation Employed in the Manufacture 
of Salt. (F. E. Engelhardt, Chemist Onondaga Salt Springs; Report for 
1889.)— 1. Solar heat— solar evaporation. 2. Direct fire, applied to the heat- 
ing surface of the vessels containing brine— kettle and pan methods. 3. The 
steam-grainer system— steam-pans, steam-kettles, etc. 4. Use of steam and 
a reduction of the atmospheric pressure over the boiling brine— vacuum 
system. 

When a saturated salt solution boils, it is immaterial whether it is done 
under ordinary atmospheric pressure at 2^8° F., or under four atmospheres 
with a temperature of 320° F., or in a vacuum under 1/10 atmosphere, the 
result will always be a fine-grained salt. 

The fuel consumption is stated to be as follows: By the kettle method, 40 
to 45 bu. of salt evaporated per ton of fuel, anthracite dust burned on per- 
forated grates; evaporation, 5.53 lbs. of water per pound of coal.. By the 
pan method, 70 to 75 bu. per ton of fuel. By vacuum pans, single effect, 86 
lbs. per ton of anthracite dust (2000 lbs.). With a double effect nearly 
double that amount can be produced. 



464 



Heat. 



Solubility of Common Salt in Pure Water. (Andrese.) 



Temp, of brine, F 

100 parts water dissolve parts . . 
100 parts brine contain salt — 



52 50 
5.63 35.69 
5.27 26.30 



56 104 140 176 
3.03 36.32 37.06 38.00 
3.49 26.64 27.04 27.54 



According to Poggial, 100 parts of water dissolve at 229.66° F., 40.35 parts 
of salt, or in per cent of brine, 28.749. Gay Lussac found that at 229.72° F., 
100 parts of pure water would dissolve 40.38 parts of salt, in per cent of 
brine, 28.764 parts. 

The solubility of salt at 229° F. is only 2.5$ greater than at 32°. Hence we 
cannot, as in the case of alum, separate the salt from the water by allowing 
a saturated solution at the boiling point to cool to a lower temperature. 

Solubility of Sulpbate of Lime in Pure "Water. (Marignac.) 

Temperature F. degrees. 
Parts water to dissolve I 

1 part gypsum f 

Parts water to dissolve 1 ( 

part anhydrous CaS0 4 ) 

In salt brine sulphate of lime is much more soluble than in pure water. 
In the evaporation of salt brine the accumulation of sulphate of lime tends 
to stop the operation, and it must be removed from the pans to avoid waste 
of fuel. 

The average strength of brine in the New York salt districts in 1889 was 
69.38 degrees of the salinometer. 

Strength of Salt Brines.— The following table is condensed from 
one given in U. S. Mineral Resources for 1888, on the authority of Dr. 
Englehardt. 

Relations between Salinometer Strength, Specific Gravity, 
Solid Contents, etc., of Brines of Different Strengths. 



32 


64.5 


89.6 


100.4 


105.8 


127.4 


186.8 


212 


415 


386 


371 


368 


370 


375 


417 


452 


525 


488 


470 


466 


468 


474 


528 


572 













«H 


, ^ 


T3 










CD 

2 

5b 

CD 
T3 

U 
CD 

"S 

s 

o 


1 

W) 

a 


be 
o 

s 


H 

o 

a 


o . 
o c 

^ g 

o.S 

IS 


o8«w 

'" CD . 
'ob'S « 


£ . 

$$ 

CD O 

I 5 

o u 


"* £ O 


of coal required t 
oduce a bushel o 
It, 1 lb. coal evapo 
ting 6 lbs. of water 


** CO 

til 

cc cS O 
03 &« 




Sg 


c$ 


ft 


® 


0^ 


S5S 


^a 


P %<A 


w ^g; g 


p& o 




to 


PQ 


U2 


Ph 


£ 


P4 


o 


Ph 


J 


PQ 


1 


.26 


1.002 


.265 


8.34? 


.022 


2,531 


21,076 


3,513 


.560 


2 


.52 


1.003 


.530 


8.356 


.044 


1,264 


10,510 


1,752 


1.141 


4 


1.04 
1.56 


1.007 
1.010 


1.060 
1.590 


8.389 
8.414 


.088 
.133 


629.7 

418.6 


5,227 
3,466 


871.2 
577.7 


2.295 


6. 




3.462 


8. 




2.08 
2.60 
3.12 
3.64 
4.16 


1.014 
1.017 
1.021 
1.025 
1028 


2.120 
2 650 
3.180 
3.710 
4.240 


8.447 
8.472 
8.506 
8.539 
8.564 


.179 
.224 
.270 
.316 
.364 


312.7 
249.4 
207.0 
176.8 
154.2 


2,585 
2,057 
1,705 
1,453 
1,265 


430.9 
342.9 

284.2 
242.2 
210.8 


4.641 


10 


5 833 


12 


7.038 


14.. 


8.256 


16. 




9.488 


is 




4.68 


1.032 


4.77C 


8.597 


.410 


136.5 


1,118 


186.3 


10.73 


20 




5.20 


1.035 


5.300 


8.622 


.457 


122.5 


1,001 


176.8 


11.99 


30 




7.80 


1.054 


7.950 


8.781 


.698 


80.21 


648.4 


108.1 


18.51 


40 




10.40 
13.00 


1.073 
1.093 


10.600 

13.250 


8.939 
9.105 


.947 

1.206 


59.09 
46.41 


472.3 
366.6 


78.71 
61.10 


25.41 


50. 




32 73 


fi() 




15 60 


1.114 


15.900 


9.2*0 


1.475 


37.94 


296.2 


49.36 


40.51 


70. 




18.20 


1.136 


18.550 


9.464 


1.755 


31.89 


245.9 


40.98 


48.80 


80 


20.80 
23.40 


1.158 
1.182 


21.200 
23.850 


9.647 
9.847 


2.045 
2.348 


27.38 
23.84 


208.1 

178.8 


31.69 
29.80 


57 65 


90 


67 11 


100 


26.00 


1.205 


26.500 


10.039 


2.660 


21.04 


155.3 


25.88 


77.26 









EVAPORATION. 465 

Concentration of Sugar Solutions,,* (From " Heating: and Con- 
centrating Liquids by Steam," by John G. Hudson; The Engineer, June 13, 
1890.)— In the early stages of the process, when the liquor is of low density, the 
evaporative duty will be high, say two to three (.British) gallons per square 
foot of heating surface with 10 lbs. steam pressure, but will gradually tall to 
an almost nominal amount as the final stage is approached. As a generally 
safe basis for designing, Mr. Hudson takes an evaporation of one gallon per 
hour for each square foot of gross heating surface, with steam of the pres- 
sure of about 10 lbs. 

As examples of the evaporative duty of a vacuum pan when performing 
the earlier stages of concentration, during which all the heating surface 
can be employed, he gives the following: 

Coil Vacuum Pan.— 4% in. copper coils, 528 square feet of surface; 
steam in coils, 15 lbs.; temperature in pan, 141° to 148°; density of feed, 25° 
Beaume, and concentrated to 31° Beaum6. 

First Trial. — Evaporation at the rate of 2000 gallons per hour = 3.8 gallons 
per square foot; transmission, 376 units per degree of difference of tem- 
perature. 

Second Trial.— Evaporation at the rate of 1503 gallons per hour = 2.8 gal- 
lons per square foot; transmission, 265 units per degree. 

As regards the total time needed to work up a charge of massecuite from 
liquor of a given density, the following figures, obtained by plotting the 
results from a large number of pans, form a guide to practical working. 
The pans were all of the coil type, some with and some without jackets, 
the gross heating surface probably averaging, and not greatly differiug 
from, .25 square foot per gallon capacity, and the steam pressure 10 lbs. per 
square inch. Both plantation and refining pans are included, making 
various grades of sugar: 

Density of Feed (degs. Beaum6). 
10° 15° 20° 25° 30° 
Evaporation required per gallon masse- 
cuite discharged 6.123 3.6 2.26 1.5 .97 

Average working hours required per 

charge 12. 9. 6^ 5. 4. 

Equivalent average evaporation per hour 
per square foot of gross surface, as- 
suming .25 sq. ft. per gallon capacity.. 2.04 1.6 1.39 1.2 .97 
Fastest working hours required per 

charge 8.5 5.5 3.8 2.75 2.0 

Equivalent average evaporation per 
hour per square foot 2.88 2.6 2.38 2.18 1.9 

The quantity of heating steam needed is practically the same in vacuum 
as in open pans. The advantages proper to the vacuum system are pri- 
marily the reduced temperature of boiling, and incidentally the possibility 
of using heating steam of low pressure. 

In a solution of sugar in water, each pound of sugar adds to the volume 
of the water to the extent of .061 gallon at a low density to .0638 gallon at 
high densities. 

A Metnod of Evaporating by Exhaust Steam is described 
by Albert Stearns in Trans. A. S. M. E., vol. viii. A pan 17' 6" x 11' x 1' 6", 
fitted with cast-iron condensing pipes of about 250 sq. ft. of surface, evapo- 
rated 120 gallons per hour from clear water, condensing only about one half 
of the steam supplied by a plain slide-valve engine of 14" x 32" cylinder, 
making 65 revs, per min., cutting off about two thirds stroke, with steam at 
75 lbs. boiler pressure. 

It was found that keeping the pan-room warm and letting only sufficient 
air in to carry the vapor up out of a ventilator adds to its efficiency, as the 
average temperature of the water in the pan was only about 165° F. 

Experiments were made with coils of pipe in a small pan, first with no 
agitator, then with one having straight blades, and lastly with troughed 
blades; the evaporative results being about the proportions of one, two, and 
three respectively. 

In evaporating liquors whose boiling point is 220° F., or much above that 
of water, it fs found that exhaust steam can do but little more than bring 
them up to saturation strength, but on weak liquors, syrups, glues, etc., it 
should be very useful. 

* For other sugar data see Bagasse as Fuel, under Fuel. 



466 HEAT. 

Drying in Vacuum, — An apparatus for drying grain and other sub- 
stances in vacuum is described by Mr. Emil Passburg in Proc. Inst. Mech. 
Engrs., 1889. The three essential requirements for a successf I and eco- 
nomical process of drying are: 1. Cheap evaporation of the moisture; 
2. Quick drying at a low temperature; 3. Large capacity of the apparatus 
employed. 

The removal of the moisture can be effected in either of two ways: either 
by slow evaporation, or by quick evaporation— that is, by boiling. 

Sloio Evaporation.— The principal idea carried into practice in machines 
acting by slow evaporation is to bring the wet substance repeatedly into 
contact with the inner surfaces of the apparatus, which are heated by 
steam, while at the same time a current of hot air is also passing through 
the substances for carrying off the moisture. This method requires much 
heat, because the hot-air current has to move at a considerable speed in 
order to shorten the drying process as much as possible; consequently a 
great quantity of heated air passes through and escapes unused. As a car- 
rier of moisture hot air cannot in practice be charged beyond half its full 
saturation; and it is in fact considered a satisfactory result if even this 
proportion be attained. A great amount of heat is here produced which is 
not used; while, with scarcely half the cost for fuel, a much quicker re- 
moval of the water is obtained by heating it to the boiling point. 

Quick Evaporation by Boiling.— This does not take place until the water 
is brought up to the boiling point and kept there, namely, 212° F., under 
atmospheric pressure. The vapor generated then escapes freely. Liquids 
are easily evaporated in this way, because by their motion consequent on 
boiling the heat is continuously conveyed from the heating surfaces through 
the liquid, but it is different with solid substances, and many more difficul- 
ties have to be overcome, because convection of the heat ceases entirely in 
solids. The substance remains motionless, and consequently a much 
greater quantity of heat is required than with liquids for obtaining the 
same results. 

Evaporation in Vacuum— All the foregoing disadvantages are avoided if 
the boiling-point of water is lowered, that is, if the evaporation is carried 
out under vacuum. 

This plan has been successfully applied in Mr. Passburg's vacuum drying 
apparatus, which is designed to evaporate large quantities of water con- 
tained in solid substances. 

The drying apparatus consists of a top horizontal cylinder, surmounted 
by a charging vessel at one end, and a bottom horizontal cylinder with a 
discharging vessel beneath it at the same end. Both cylinders are encased 
in steam-jackets heated by exhaust steam. In the top cylinder works a re- 
volving cast-iron screw with hollow blades, which is also heated by exhaust 
steam. The bottom cylinder contains a revolving drum of tubes, consisting 
of one large central tube surrounded by 24 smaller ones, all fixed in tube- 
plates at both ends; this drum is heated by live steam direct from the boiler. 
The substance to be dried is fed into the charging vessel through two man- 
holes, and is carried along the top cylinder by the screw creeper to the back 
etid, where it drops through a valve into the bottom cylinder, in which it is 
lifted by blades attached to the drum and travels forwards in the reverse 
direction: from the front end of the bottom cylinder it falls into a discharg- 
ing vessel through another valve, having by this time become dried. The 
vapor arising during the process is carried off by an air-pump, through a 
dome and air-valve on the top of the upper cylinder, and also through 
a throttle-valve on the top of the lower cylinder; both of these valves are 
supplied with strainers. 

As soon as the discharging vessel is filled with dried material the valve 
connecting it with the bottom cylinder is shut, and the dried charge taken 
out without impairing the vacuum in the apparatus. When the charging 
vessel requires replenishing, the intermediate valve between the two cylin- 
ders is shut, and the charging vessel filled with a fresh supply of wet mate- 
rial; the vacuum still remains unimpaired in the bottom cylinder, and has 
to be restored only in the top cylinder after the charging vessel has been 
closed again. 

In this vacuum the boiling-point of the water contained in the wet mate- • 
rial is brought down as low as 110° F. The difference between this tempera- 
ture and that of the heating surfaces is amply sufficient for obtaining good 
results from the employment of exhaust steam for heating all the surfaces 
except the revolving drum of tubes. The water contained in the solid sub- 
stance to be dried evaporates as soon as the latter is heated to about 110° F,£ 



RADIATION OF HEAT. 467 

and as long as there is any moisture to be removed the solid substance is 
not heated above this temperature. 

Wet grains from a brewery or distillery, containing from 75$£ to 78% of 
water, have by this drying process been converted in some localities from 
a worthless incumbrance into a valuable food-stuff. The water is removed 
by evaporation only, no previous mechanical pressing being resorted to. 

At Messrs. Guinness's brewery in Dublin two of these machines are em- 
ployed. In each of these the top cylinder is 20' 4" long and 2' 8" diam., and 
the screw working inside it makes 7 revs, per min.; the bottom cylinder is 
19' 2" long and 5' 4" diam.. and the drum of the tubes inside it makes 5 revs, 
per min. The drying surfaces of the two cylinders amount together to a 
total area of about 1000 sq. ft., of which about 40% is heated by exhaust steam 
direct from the boiler. There is only one air-pump, which is made large 
enough for three machines; it is horizontal, and has only one air-cylinder, 
which is double-acting, 17% in. diam. and 17% in. stroke; and it is driven at 
about 45 revs, per min. As the result of about eight months' experience, the 
two machines have been drying the wet grains from about 500 cwt. of malt 
per day of 24 hours. 

Roughly speaking, 3 cwt. of malt gave 4 cwt. of wet grains, and the latter 
yield 1 cwt. of dried grains; 500 cwt. of malt will therefore yield about 670 
cwt. of wet grains, or 335 cwt. per machine. The quantity of water to be 
evaporated from the wet grains is from 75$ to 78% of their total weight, or 
say about 512 cwt. altogether, being 256 cwt. per machine. 

RADIATION OF HEAT. 

Radiation of heat takes place between bodies at all distances apart, and 
follows the laws for the radiation of light. 

The heat rays proceed in straight lines, and the intensity of the rays 
radiated from any one source varies inversely as the square of their distance 
from the source. 

This statement has been erroneously interpreted by some writers, who 
have assumed from it that a boiler placed two feet above a fire would re- 
ceive by radiation only one fourth as much heat as if it were only one foot 
above The law refers only to the emanation of heat rays in all directions 
in radial lines from a single point. When the radiation is from a multitude 
of points, as from the surface of a fire or flame, the rays from the several 
points cross each other and cause, the intensity at moderate distances to be 
much greater than the law of inverse squares would indicate. Moreover, in 
the case of boiler furnaces the side walls reflect those rays that are received 
at an angle — following the law of optics, that the angle of incidence is equal 
to the angle of reflection,— with the result that the intensity of heat two feet 
above the fire is practically the same as at one foot above, instead of only 
one-fourth as much. 

The rate at which a hotter body radiates heat, and a colder body absorbs 
heat, depends upon the state of the surfaces of the bodies as well as on their 
temperatures. The rate of radiation and of absorption are increased by 
darkness and roughness of the surfaces of the bodies, and diminished by 
smoothness and polish. For this reason the covering of steam pipes and 
boilers should be smooth and of a light color: uncovered pipes and steam- 
cylinder covers should be polished. 

The quantity of heat radiated by a body is also a measure of its heat- 
absorbing power, under the same circumstances. When a polished body is 
struck by a ray of heat, it absorbs part of the heat and reflects the rest. 
The reflecting power of a body is therefore the complement of its absorbing 
power, which latter is the same as its radiating power. 

The relative radiating and reflecting power of different bodies has been 
determined by experiment, as shown in the table below, but as far as quan- 
tities of heat are concerned, says Prof. Trowbridge (Johnson's Cyclopaedia, 
art. Heat), it is doubtful whether anything further than the said relative 
determinations can, in the present state of our knowledge, be depended 
upon, the actual or absolute quantities for different temperatures being still . 
uncertain. The authorities do not even agree on the relative radiating 
powers. Thus, Leslie gives for tin plate, gold, silver, and copper the figure 
12, which differs considerably from the figures in the table below, given by 
Clark, stated to be on the authority of Leslie, De La Provostaye and De- 
sains, and Melloni, 



468 



Relative Radiating and Reflecting Power of Different 
Substances. 



°&c 


&c 


uBc 


•9ri 


ajz <d 




lo £ 






<6 ° 


'•5&P* 


ojPh 


«<H 


tf 


P3 




100 





100 





100 





98 


2 


93 to 98 


7 to 2 


90 


10 


85 


15 


72 


28 


27 


73 


25 


75 


23 


77 


23 


77 



° bo 

.S t- 2 



P3 



Lampblack 

Water 

Carbonate of lead . . . 

Writing-paper 

Ivory, jet, marble. . . 

Ordinary glass 

Ice 

Gum lac 

Silver-leaf on glass. . 

Cast iron, bright pol- 
ished 

Mercury, about 

Wrought iron, pol- 
ished 



Zinc,polished 

Steel, polished 

Platinum, polished.. 
" in sheet . . 

Tin 

Brass, cast, dead 
polished 

Brass, bright pol- 
ished 

Copper, varnished . . 
" hammered.. 

Gold, plated 

'• on polished 
steel 

Silver, polished 
bright 



81 
83 



93 
86 
93 
95 



Experiments of Dr. A. M. Mayer give the following: The relative radia- 
tions from a cube of cast iron, having faces rough, as from the foundry, 
planed, " drawfiled,' 1 and polished, and from the same surfaces oiled, areas 
below (Prof. Thurston, in Trans. A. S. M. E., vol. xvi.) : 



Surface. 



Bough 

Planed 

Drawfiled. 
Polished... 



Oiled. 



100 
60 
49 



Dry. 



It here appears that the oiling of smoothly polished castings, as of cylin- 
der-heads of steam-engines, more than doubles the loss of heat by radiation, 
while it does not seriously affect rough castings. 

CONDUCTION AND CONVECTION OF BEAT, 

Conduction is the transfer of heat between two bodies or parts of a 

body which touch each other. Internal conduction takes place between the 

parts of one continuous body, and external conduction through the surface 

of contact of a pair of distinct bodies. 

The rate at which conduction, whether internal or external, goes on, 
being proportional to the area of the section or surface through which it 
takes place, may be expressed in thermal units per square foot of area per 
hour 

Internal Conduction varies with the heat conductivity, which de- 
pends upon the nature of the substance, and is directly proportional to the 
difference between the temperatures of the two faces of a layer, and in- 
versely as its thickness. The reciprocal of the conductivity is called the 
internal thermal resistance of the substance. If r represents this resistance, 
x the thickness of the layer in inches, T' and Tthe temperatures on the two 
faces, and q the quantity in thermal units transmitted per hour per square 



T 



foot of area, q = — 



(Rankine.) 



Peclet gives the following values of r 

Gold, platinum, sil ver . 001 6 

Copper .0018 

Iron 0.0043 

Zinc 0.0045 



Lead . 0090 

Marble 0.0716 

Brick 0.1500 



CONDUCTION AND CONVECTION OF HEAT. 



469 



Relative Heat-conducting Power of Metals. 

(* Calvert, & Johnson ; t Weidemann & Franz ) 
Silver = 1000. 



*C. & J. 

.. 1000 



Metals, 

Silver 

Gold 981 

Gold, with 1% of 

silver 840 

Copper, rolled 845 

Copper, cast 811 

Mercury 677 

Mercury, with 1.25$ 

of tin 412 

Aluminum 665 

Zinc, rolled 641 

Zinc: 

cast vertically 628 

cast horizontally. . 608 



tW. & F. 
1000 
532 



Metals. 

Cadmium 

Wrought iron 

Tin 

Steel 

Platinum 

Sodium 

Cast iron 

Lead 

Autimony : 

cast horizontally. . 

cast vertically. . . 
Bismuth 



*C. & J. 

.... 577 
... 436 
.... 422 
.... 397 



119 
145 
116 
84 



215 
192 
61 



Influence of a Non-metallic Substance in Combination on the 
Conducting Power of a Metal. 



Influence of carbon on iron : 

Wrought iron 

Steel 

Cast iron 



Influence of arsenic on copper : 

Cast copper 811 

397 Copper with 1% of arsenic 570 

359 " with .5% of arsenic 669 

" with .25^ of arsenic. ... 771 

Steam-pipe Coverings. 

(Experiments by Prof. Ordway, Trans. A. S. M. E., v,73; also Circular No. 27 
of Boston Mfrs. Mutual Fire Ins. Co., 1890.) 
It will be observed that several of the incombustible materials are nearly 
as efficient as wool, cotton, and feathers, with which they may be compared 
in the following table. The materials which may be considered wholly 
free from the danger of being carbonized or ignited by slow contact with 
pipes or boilers are printed in Roman type. Those which are more or less 
liable to be carbonized are printed in italics. 

Table I. 





Pounds of 








Water 


Solid 


■gS 




heated 


Matter in 




Substance 1 inch thick. Heat applied, 


10° F., per 


1 square 


£ o 


310° F. 


hour. 


foot 1 inch 


p— ■ 




through 


thick, parts 


^u 




1 square 


in 1000. 






foot. 




<j a 




8.1 
9.6 


56 
50 


944 




950 


3. Carded cotton wool 


10.4 


20 


980 


4. Hair felt 


10.3 


185 


815 




9.8 


56 


944 


6. Compressed lampblack 


10.6 


244 


756 


7. Cork charcoal 


11.9 


53 


947 


8. White-pine charcoal 


13.9 


119 


881 




35.7 


506 


494 




12.4 
42.6 


23 

285 


977 




715 


12. Light carbonate of magnesia 


13.7 


60 


940 


13. Compressed carbonate of magnesia 


15.4 


150 


850 


14. Loose fossil-meal 


14.5 


60 


940 


15. Crowded fossil-meal 


15.7 


112 


888 




20.6 


253 


747 




30.9 


368 


632 


18. Fine asbestos 


49.0 


81 


919 




48.0 
62.1 



527 


1000 


20. Sand 


471 



470 



Covering. 



Pounds of Water 
heated 10° F., 
per hour, by 
1 square foot. 



21. Best slag-wool 

22. Paper 

23. Blotting-paper wound tight 

24. Asbestos paper wound tight 

25. Cork strips bound on , 

26. Straw rope wound spirally 

27. Loose rice chaff 

28. Paste of fossil-meal with hair 

29. Paste of fossil-meal with asbestos . . 

30. Loose bituminous-coal ashes 

31. Loose anthracite-coal ashes 

32. Paste of clay and vegetable fibre . 




Professor Ordway's report says: Careful experiments have been made 
with various non-conductors, each used in a mass one inch thick, placed on 
a flat surface of iron kept heated by steam to 310° Fahr. Table I gives the 
amount of heat transmitted per hour through each kind of non-conductor 
one inch thick, reckoned in pounds of water heated 10° Fahr., the unit of area 
being one square foot of covering. 

The substances given in Table II were actually tried as coverings for 
two-inch steam-pipe, but for convenience of comparison the results have 
been reduced by calculation to the same terms as in Table I. 

Later experiments have given results for still air which differ little from 
those of Nos. 3, 4, and 6. In fact the bulk of matter in the best non-conduc- 
tors is relatively too small to have any specific effect, except to entrap the 
air and keep it stagnant. These substances keep the air still by virtue of 
the roughness of their fibres or particles. The asbestos, No. 18, had smooth 
fibres, which could not prevent the air from moving about. 

Later trials with an asbestos of exceedingly fine fibre have made a some- 
what better showing, but asbestos is really one of the poorest non-conduc- 
tors. By reason of its fibrous character it may be used advantageously 
to hold together other incombustible substances, but the less the better. 
We have made trials of two samples of a " magnesia covering," consisting 
of carbonate of magnesia with a small percentage of good asbestos fibre. 
One transmitted heat which, reduced to the terms of Table I, would amount 
to 15 lbs.: the denser one gave 20 lbs. The former contained 250/1000 
of solid matter; the latter 396/1000. 

Any suitable substance which is used to prevent the escape of steam 
heat should not be less than one inch thick. 

Any covering should be kept perfectly dry. for not only is water a good 
carrier of heat, but it has been found that still water conducts heat about 
eight times as rapidly as still air. 

Heat-conducting Power of Covering Materials. 

(J. J. Coleman, Eiig'g, Sept. 5, 1884, p. 237.) 
Experiments were made by filling a 10-in. cube with ice, surrounding it 
with the different materials to be tested, and noting the quantity of ice 
melted per hour with each insulator. 



Charcoal 140 

Sawdust 163 

Gas works breeze 230 

Wood and air-space 280 



The relative results were as follows 
Silicate cotton (mineral wool) . . . 100 

Hair felt 117 

Cotton wool 122 

Sheep's wool 136 

Iuf usur.al earth 136 

The Rate of External Conduction through the bounding stir- 
face between a solid body and a fluid is approximately proportional to the 
difference of temperature, when that is small; but when that difference is 
considerable the rate of conduction increases faster than the simple ratio 
of that difference. (Rankine,) 



COHDUCTIOtf AKD CONVECTION OF HEAT. 471 

If r, as before, is the coefficient of internal thermal resistance, e and e' the 
coefficient of external resistance of the two surfaces, x the thickness of the 
plate, and V and Tthe temperatures of the two fluids in contact with the 

T — T 

two surfaces, the total thermal resistance is q = — : — — . According to 

e+e' + rx 

Peclet, e + e' = =-=., in which the constants A and B have 

A\\ -\- B{1 — 1 )J 
the following values : 

B for polished metallic surfaces 0028 

B for rough metallic surfaces and for non-metallic surfaces. . .0037 

A for polished metals, about 90 

A for glassy and varnished surfaces 1 .34 

A for dull metallic surfaces 1.58 

A for lamp-black 1 .78 

When a metal plate has a liquid at each side of it, it appears from experi- 
ments by Peclet that B = .058, A = 8.8. 

The results of experiments on the evaporative power of boilers agree very 
well with the following approximate formula for the thermal resistance of 
boiler plates and tubes : 

which gives for the rate of conduction, per square foot of surface per liour, 
(T - T) 2 
S= a ' 

This formu'a is proposed by Rankine as a rough approximation, near 
enough to the truth for its purpose. The value of a lies between 160 and 200. 

Convection, or carrying of heat, means the transfer and diffusion of 
the heat in a fluid mass by means of the motion of the particles of that 
mass. 

The conduction, properly so called, of heat through a stagnant mass of 
fluid is very slow in liquids, and almost, if not wholly, inappreciable in 
gases. It is only by the continual circulation and mixture of the particles of 
the fluid that uniformity of temperature can be maintained in the fluid 
mass, or heat transferred between the fluid mass and a solid body. 

The free circulation of each of the fluids which touch the side of a solid 
plate is a necessary condition of the correctness of Rankine's formulae for 
the conduction of heat through that plate; and in these formulae it is im- 
plied that the circulation of each of the fluids by currents and eddies is such 
as to prevent any considerable difference of temperature between the fluid 
particles in contact with one side of the solid plate and those at considerable 
distances from it. 

When heat is to be transferred by convection from one fluid to another, 
through an intervening layer of metal, the motions of the two fluid masses 
should, if possible, be in opposite directions, in order that the hottest par- 
ticles of each fluid may be in communication with the hottest particles of 
the other, and that the minimum difference of temperature between the 
adjacent particles of the two fluids may be the greatest possible. 

Thus, in the surface condensation of steam, by passing it through metal 
tubes immersed in a current of cold water or air, the cooling fluid should be 
made to move in the opposite direction to the condensing steam. 

Transmission of Heat, through Solid Plates, from 
"Water to Water. (Clark, S.E.). — M. Peclet found, from experiments 
made with plates of wrought iron, cast iron, copper, lead, zinc, and tin, 
that when the fluid in contact with the surface of the plate was not circu- 
lated by artificial means, the rate of conduction was the same for different 
metals and for plates of the same metal of different thicknesses. But 
when the water was thoroughly circulated over the surfaces, and when 
these were perfectly clean, the quantity of transmitted heat was inversely 
proportional to the thickness, and directly as the difference in temperature 
of the two faces of the plate. When the metal surface became dull, the 
rate of transmission of heat through all the metals was very nearly the 
same. 

It follows, says Clark, that the absorption of heat through metal plates is 
more active whilst evaporation is in progress— when the circulation of the 
water is more active— than while the water is being heated up to the boiling 
point. 



L_ 



472 



HEAT. 



Transmission from Steam to Water,— M. Peclet's principle is 
supported by the results of experiments made in 1867 by Mr. Isherwood on 
the conductivity of different metals. Cylindrical pots, 10 inches in diameter, 
21*4 inches deep inside, and % inch, J4 inch, and % inch thick, turned and 
bored, were formed of pure copper, brass (GO copper and 40 zinc), rolled 
wrought iron, and remelted cast iron. They were immersed in a steam 
bath, which was varied from 220° to 320° F. Water at 21*: was supplied to 
the pots, which were kept filled. It was ascertained that the rate of evapora- 
tion was in the direct ratio of the difference of the temperatures inside and 
outside of the pots; that is, that the rate of evaporation per degree of 
difference of temperatures was the same for all temperatures; and that the 
rate of evaporation was exactly the same for different thicknesses of the 
metal. The respective rates of conductivity of the several metals were as 
follows, expressed in weight of water evaporated from and at 212° F. per 
square foot of the interior surface of the pots per degree of difference of 
temperature per hour, together with the equivalent quantities of heat-units: 
Water at 212°. Heat-units. Ratio. 

Copper 6651b. 642.5 1.00 

Brass 577" 556.8 .87 

Wrought iron 387" 373.6 .58 

Cast iron 327" 315.7 .49 

Whitham, "Steam Engine Design," p. 283, also Trans. A. S. M. E. ix, 425, in 
using these data in deriving a formula for surface condensers calls these 
figures those of perfect conductivity, and multiplies them by a coefficient 
C, which he takes at 0.323, to obtain the efficiency of condenser surface in 
ordinary use, i.e., coated with saline and greasy deposits. 

Transmission of Heat from Steam to Water through 
Coils of Iron Pipe.— H. G. C. Kopp and F. J. Meystre (Stevens Indi- 
cator, Jan., 1894), give an account of some experiments on transmission of 
heat through coils of pipe. They collate the results of earlier experiments 
as follows, for comparison: 





6 


Steam Con- 


Heat trans- 






<2 


densed per 


mitted per 








Square foot per 


square foot per 




. 


w 


degree differ- 


degree differ- 




-2 


«M 


ence of temper- 


ence of temper- 






o 


ature per hour. 


ature per hour. 


Remarks. 


a 


05 










.5*> 


O 6C-S 


fc 










cS 3 


££ 3 


"efiE-i 


£-X3H 




W 




£ o 


>c3o 




> a . 








ffia 


Ht-ft 


ffiw 


W ^m 




Laurens 


Copper coils... 
2 Copper coils. 


.292 


.981 
1.20 


315 


974 
1120 




Havrez.. 


Copper coil . . . 


.268 


1.26 


280 


1200 




Perkins. 


Iron coil 




.24 
.22 




215 

208.2 


j Steam pressure 
1 = 100. 
( Steam pressure 
\ = 10. 


Box 


Iron tube .... 


.235 
.196 

.206 




230 
207 
210 






Havrez.. 


Cast-iron boil- 
















.077 


.105 


82 


100 











From the above it would appear that the efficiency of iron surfaces is less 
than that of copper coils, plate surfaces being far inferior. 

In all experiments made up to the present time, it appears that the tem- 
perature of the condensing water was allowed to rise, a mean between the 
initial and final temperatures being accepted as the effective temperature. 
But as water becomes warmer it circulates more rapidly, thereby causing 
the water surrounding the coil to become agitated and replaced bj cooler 
water, which allows more heat to be transmitted. 



CONDUCTION" AND CONVECTION OF HEAT. 



473 



Again, in accepting the mean temperature as that of the condensing me- 
dium, the assumption is made that the rate of condensation is in direct pro- 
portion to the temperature of the condensing water. 

In order to correct and avoid any error arising from these assumptions 
and approximations, experiments were undertaken, in which all the condi- 
tions were constant during each test. 

The pressure was maintained uniform throughout the coil, and provision 
was made for the free outflow of the condensed steam, in order to obtain 
at all times the full efficiency of the condensing surface. The condensing 
water was continually stirred to secure uniformity of temperature, which 
was regulated by means of a steam-pipe and a cold-water pipe entering the 
tank in which the coil was placed. 

The following is a condensed statement of the results 

Heat Transmitted per Square Foot of Cooling Surface, per Degree 
of Difference of Temperature. (British Thermal Units.) 



Temperature 
of Condens- 
ing Water. 


1-in. Iron Pipe; 

Steam inside, 

60 lbs. Gauge 

Pressure. 


1^ in. Pipe; 

Steam inside, 

10 lbs. 

Pressure. 


\y 2 in. Pipe; 

Steam inside, 

10 lbs. 

Pressure. 


1^ in. Pipe; 

Steam inside, 

60 lbs. 

Pressure. 


80 
100 
120 
140 
160 
180 
200 


265 
269 
272 

277 
281 
299 
313 


128 
130 
137 
145 
158 
174 


200 
230 
260 
267 
271 
270 


'239 
247 

276 
306 
349 
419 



The results indicate that the heat transmitted per degree of difference of 
temperature in general increases as the temperature of the condensing 
water is increased. 

The amount transmitted is much larger with the steam on the outside of 
the coil than with the steam inside the coil. This may be explained in part by 
the fact that the condensing water when inside the coil flows over the sur- 
face of conduction very rapidly, and is more efficient for cooling than when 
contained in a tank outside of the coil. 

This result is in accordance with that found by Mr. Thomas Craddock, 
which indicated that the rate of cooling by transmission of heat through 
metallic surfaces was almost wholly dependent on the rate of circulation of 
the cooling medium over the surface to be cooled. 

Transmission of Heat in Condenser Tubes. {Eng'g, Dec. 
10, 1875, p. 449.).— In 1874 B. C. Nichol made experiments for determining the 
rate at which heat was transmitted through a condenser tube. The results 
went to show that the amount of heat transmitted through the walls of the 
tube per estimated degree of mean difference of temperature increased 
considerably with this difference. For example: 
Estimated mean difference of Vertical Tube. 

temperature between inside and 

outside of tube,, degrees Fahr. . 
Heat-units transmitted per hour 

per square foot of surface per 

degree of mean diff . of temp .... 422 531 561 

These results seem to throw doubt upon Mr. Isherwood's statement that 
the rate of evaporation per degree of difference of temperature is the same 
for all temperatures. 

Mr. Thomas Craddock found that water was enormously more efficient 
than air for the abstraction of heat through metallic surfaces in the process 
of cooling. He proved that the rate of cooling by transmission of heat 
through metallic surfaces depends upon the rate of circulation of the cool- 
ing medium over the surface to be cooled. A tube filled with hot water, 
moved by rapid rotation at the rate of 59 ft. per second, through air, lost as 
much heat in one minute as it did in still air in 12 minutes. In water, at a 
velocity of 3 ft. per second, as much heat was abstracted in half a minute as 
was abstracted in one minute when it was at rest in the water. Mr. Crad- 
dock concluded, further, that the circulation of the cooling flujd became of 



Horizontal Tube 



128 151.9 J52.9 111.6 146.2 150.4 



610 



737 



474 



greater importance as the difference of temperature on the two sides of the 
plate became less. (Clark, R. T. D.. p. 461.) 

Heat Transmission through Cast-iron Plates Pickled in 
Nitric Acid.— Experiments by R. C. Carpenter (Trans. A. S. M. E., xii 
179) show a marked change in the conducting power of the plates (from 
steam to water), due to prolonged treatment with dilute nitric acid. 

The action of the nitric acid, by dissolving the free iron and not attacking 
the carbon, forms a protecting surface to the iron, which is largely com- 
posed of carbon. The following is a summary of results: 



Character of Plates, each plate 8.4 in 
by 5.4 in., exposed surface 27 sq. ft. 



Increase in 
Tempera- 
ture of 
3.125 lbs. of 
Water 
each 
Minute. 



Proportionate 

Thermal Units 

Transmitted for 

each Degree of 

Difference of 

Temperature per 

Square Foot per 

Hour. 



Rela- 
tive 
Trans- 
mission 

of 
Heat. 



Cast iron— untreated skin on, but 

clean, free from rust 

Cast iron— nitric acid, 1% sol., 9 days . 

" ' 1% sol., 18 days. 

" " \% sol., 40 days. 

" " 5% sol., 9 days. . 

" ' " 5% sol., 40 days. 

Plate of pine wood, same dimensions 

as the plate of cast iron 



13.90 
11.5 
9.7 
9.6 

io!e 



113.2 
97.7 
80.08 
77.8 
87.0 



1.! 



100.0 
86.3 
70.7 



68.5 
1.6 



<?° 



The effect of covering cast-iron surfaces with varnish has been investi- 
gated by P. M. Chamberlain. He subjected the plate to the action of strong 
acid for a few hours, and then applied a non conducting varnish. One sur- 
face only was treated. Some of his results are as follows: 

170. As finished— greasy. 

152. " " washed with benzine and dried. 

169. Oiled with lubricating oil. 

162. After exposure to nitric acid sixteen hours, then oiled (lin- 
j2-s5c*"j seed oil.) 

'3 «-§ I i 166 After exposure to hydrochloric acid twelve hours, then oiled 
^ &2 b (linseed oil.) 

ts « § I After exposure to sulphuric acid 1, water 2, for 48 hours, 

ffi 8 " ** 117 f tben oiled ' varnished, and allowed to dry for 24 hours. 

Transmission of Heat through Solid Plates from Air 
or other Dry Gases to Water. (From Clark on the Steam Engine.) 
—The law of the transmission of heat from hot air or other gases to water, 
through metallic plates, has not been exactly determined by experiment. 
The general results of experiments on the evaporative action of different 
portions of the heating surface of a steam-boiler point to the general law 
that the quantity of heat transmitted per degree difference of temperature 
is practically uniform for various differences of temperature. 

The communication of heat from the gas to the plate surface is much 
accelerated by mechanical impingement of the gaseous products upon the 
surface. 

Clark says that when the surfaces are perfectly clean, the rate of trans- 
mission of heat through plates of metal from air or gas to water is greater 
for copper, next for brass, and next for wrought iron. But when the sur- 
faces are dimmed or coated, the rate is the same for the different metals. 

With respect to the influence of the conductivity of metals and of the 
thickness of the plate on the transmission of heat from burnt gases to 
water, Mr. Napier made experiments with small boilers of iron and copper 
placed over a gas-flame. The vessels were 5 inches in diameter and 2}£ 
inches deep. From three vessels, one of iron, one of copper, and one of iron 
sides and copper bottom, each of them 1/30 inch in thickness, equal quanti- 
ties of water were evaporated to dryness, in the times as follows ; 



CONDUCTION AND CONVECTION OF HEAT. 475 



Water. 


Iron Vessel. 


Copper Vessel. 


iron ana uopp 

Vessel. 


4 ounces 
11 

4 " 


19 minutes 

33 

50 

35.7 " 


18.5 minutes 

30.75 

44 


36.83 minutes 



Two other vessels of iron sides 1/30 inch thick, one having a J4-inch copper 
bottom and the other a J^-inch lead bottom, were tested against the iron 
and copper vessel, 1/30 inch thick. Equal quantities of water were evapo- 
rated in 54, 55, and 53)4 minutes respectively. Taken generally, the results 
of these experiments show that there are practically out slight differences 
between iron, copper, and lead in evaporative activity, and that the activity 
is not affected by the thickness of the bottom. 

Mr. W. B. Johnson formed a like conclusion from the results of his obser- 
vations of two boilers of 160 horse-power each, made exactly alike, ex- 
cept that one had iron flue-tubes and the other copper flue-tubes. No dif- 
ference could be detected between the performances of these boilers. 

Divergencies between the results of different experimenters are attribut- 
able probably to the difference of conditions under which the heat was 
transmitted, as between water or steam and water, and between gaseous 
matter and water. On one point the divergence is extreme: the rate of 
transmission of heat per degree of difference of temperature. Whilst from 
400 to 600 units of heat are transmitted from water to water through iron 
plates, per degree of difference per square foot per hour, the quantity of 
heat transmitted between water and air, or other dry gas, is only about 
from 2 to 5 units, according as the surrounding tar is at rest or in movement. 
In a locomotive boiler, where radiant heat was brought into play, 17 units 
of heat were transmitted through the plates of the fire-box per degree of 
difference of temperature per square foot per hour. 

Transmission of Heat through Plates and. Tubes from 
Steam or Hot Water to Air.— The transfer of heat from steam or 
water through a plate or tube into the surrounding air is a complex opera- 
tion, in which the internal and external conductivity of the metal, the radi- 
ating power of the surface, and the convection of heat in the surrounding 
air are all concerned. Since the quantity of heat radiated from a surface 
varies with the condition of the surface and with the surroundings, according 
to laws not yet determined, and since the heat carried away by convection 
varies with the rate of the flow of the air over the surface, it is evident that 
no general law can be laid down for the total quantity of heat emitted. 

The following is condensed from an article on Loss of Heat from Steam- 
pipes, in The Locomotive, Sept. and Oct., 1892. 

A hot steam pipe is radiating heat constantly off into space, but at the 
same time it is cooling also by convection. Experimental data on which to 
base calculations of the heat radiated and otherwise lost by steam-pipes are 
neither numerous nor satisfactory. 

In Box's Practical Treatise on Heat a number of results are given for the 
amount of heat radiated by different substances when the temperature of 
the air is 1° Fahr. lower than the temperature of the radiating body. A 
portion of this table is given below. It is said to be based on Peclet's ex- 
periments. 

Heat Units Radiated per Hour, per Sjuare Foot op Surface, for 
1° Fahrenheit Excess in Temperature. 

Copper, polished — 0327 j Sheet-iron, ordinary 5662 

Tin, polished 0440 Glass 5948 

Zinc and brass, polished 0491 | Cast iron, new 6480 

Tinned iron, polished 0858 I Common steam-pipe, inferred.. .6400 

Sheet-iron, polished 0920 Cast and sheet iron, rusted 6868 

Sheet lead 1329 ] Wood, building stone, and brick .7358 

When the temperature of the air is about 50° or 60° Fahr., and the radiat- 
ing body is not more than about 30° hotter than the air, we may calculate 
the radiation of a given surface by assuming the amount of heat given off 
by it in a given time to be proportional to the difference in temperature be- 
tween the radiating body and the air. This is " Newton's law of cooling." 
But when the difference in temperature is great, Newton's law does not hold 
good; the radiation is no longer proportional to the difference in tempera- 
ture, but must be calculated by a complex formula established experiment, 
ally by Duloug and Petit. Box has computed a table from this formula, 
which greatly facilitates its application, and which is given below : 



4? 6 SEAT. 

Factors for Reduction to Dulong's Law of Radiation. 



Differences in Tem- 


Temperatu 


re of the Air 


on the Fahrenheit Scale. 


perature between 




Radiating Body 






















and the Air. 


32° 


50° 


59° 


68° 


86° 


104° 


122° 


140° 


158° 


176° 


194° 


212° 


Deg. Fahr. 


























18 


1.00 


1.07 


1.12 


1.16 


1.25 


1.36 


1.47 


1.58 


1.70 


1.85 


1.99 


2.15 


36 


1.03 


1.08 


1.16 


1.21 


1.30 


1.40 


1.52 


1.68 


1.76 


1.912.06 


2.23 


54 


1.07 


1.16 


1.2D 


1.25 


1.35 


1.45 


1.58 


1.70 


1.83 


1.99|2.14 


2.31 


72 


1.12 


1.20 


1.25 


1.30 


1.40 


1.52 


1.64 


1.76 


1.90 


2.07.2.23 


2.40 


90 


1.16 


1.25 


1.31 


1.36 


1.46 


1.58 


1.71 


1.84 


1.98 


2.15 1 2.33 


2.51 


108 


1.21 


1.31 


1.36 


1.42 


1.52 


1.65 


1.78 


1.92 


2.07 


2.28j2.42 
2.34 ! 2.52 


2.62 


126 


1.26 


1.36 


1.42 


1.48 


1.50 


1.72 


1.86 


2.00 


2.16 


2.72 


144 


1.32 


1.42 


1.48 


1.54 


1.65 


1.79 


1.94 


2.08 


2.24 


2. 4412.64 


2.83 


162 


1.37 


1.48J1.54 


1.60 


1.73 


1.86 


2.02 


2.17 


2.34 


2.54'2.74 


2.96 


180 


1.44 


1.55 


1.61 


1.68 


1.81 


1.95 


2.11 


2.27 


2.46 


2.6612.87 


3.10 


198 


1.50 


1.62 


1.69 


1.75 


1.89 


2.04 


2.21 


2.38 




2.7813.00 


3.24 


216 


1.58 


1.69 


1.76 


1.83 


1.97 


2.13 


2.32 


2.48 


2.68 


2.91 3.13 


3.38 


234 


1.64 


1.77 


1.84 


1.90 


2.06 


2.28 


2.43 




2.80 


3. 03(3.28 


3.46 


252 


1.71 


1.85 


1.92 


2.00 


2.15 


2.33 


2.52 


2.71 


2.92 


3.183.43 


3.70 


270 


1.79 


1.93 


2.01 


2.09 


2.22 


2.44 


2. 6 A 


2.84 


3.06 


3.32 3.58 


3.87 


288 


1.89 


■ 


2.12 


2.20 


2.37 


51 


2.78 




3.22 


3.50J3.77 


4.07 


306 


1.98 


2.13 


2.22 


2.31 


2.4!) 


2.69 


2.90 


3.12 


3.37 


3.663.95 


4.26 


324 


2.07 


2.23 


2.33 


2.42 


2.62 


2.81 


3 04 




3.53 


3.844.14 


4.46 


342 


2.17 


2.34 


2.-14 


2.54 


2.73 


2.95 


3.19 


3.44 


3.70 


4.0214.34 


4.68 


360 


2.27 


2.45 


2.56 




2.80 


3.09 


3.35 


3.60 


3.88 


4.224.55 


4.91 


378 


2.39 


2.57 


B8 




3.00 


! 


3.51 




4.08 


4.424.77 


5.15 


396 


2.50 


2.70 2.81 


2.93 


3.15 


3.40 


3.68 


3.97 


4.28 


4.64J5.01 


5.40 


414 


2.63 


2.84'a.95 


3.07 


3.31 


3.51 


3.87 


4.12 


1.48 


4.87|5.26 


5.67 


432 


2.76 


2.98 


3.10 


3.23 


3.47 


3.76 


4.10 


4.32 


4.61 


5.125.33 


6.04 



The loss of heat by convection appears to be independent of the nature of 
the surface, that is, it is the same for iron, stone, wood, and other materials. 
It is different for bodies of different shape, however, and it varies with the 
position of the body. Thus a vertical steam-pipe will not lose so much heat 
by convection as a horizontal one will; for the air heated at the lower part 
of the vertical pipe will rise along the surface of the pipe, protecting it to 
some extent from the chilling action of the surrounding cooler air. For a 
similar reason the shape of a bod}' has an important influence on the result, 
those bodies losing most heat whose forms are such as to allow the cool air 
free access to every part of their surface. The following table from Box 
gives the number of heat units that horizontal cylinders or pipes lose by 
convection per square foot of surface per hour, for one degree difference in 
temperature between the pipe and the air. 

Heat Units Lost by Convection from Horizontal Pipes, per Square 

Foot of Surface per Hour, for a Temperature 

Difference of 1° Fahr. 



External 




External 




External 




Diameter of 


Heat Units 


Diameter 


Heat Units 


Diameter 


Heat Units 


Pipe 


Lost. 


of Pipe 


Lost. 


of Pipe 


Lost. 


in inches. 




in inches. 




in inches. 




2 


0.728 


7 


0.509 


18 


0.455 


3 


0.626 


8 


0.498 


24 


0.447 


4 


0.574 


9 


0.489 


36 


0.438 


5 


0.544 


10 


0.482 


48 


0.434 


6 


0.523 


12 


0.472 







The loss of heat by convection is nearly proportional to the difference in 
temperature between the hot body and the air; but the experiments of 



CONDUCTION" AND CONVECTION OF HEAT. 



477 



Dulcmg and Peclet show that this is not exactly true, and we may here also 
resort to a table of factors for correcting the results obtained by simple 
proportion. 

Factors fob Reduction to Dulong's Law of Convection. 



Difference 




Difference 




Difference 




in Temp. 




in Temp. 




in Temp. 




between Hot 


Factor. 


between Hot 


Factor. 


between 


Factor. 


Body and 




Body and 




Hot Body 




Air. 




Air. 




and Air. 




18° F. 


0.94 


180° F. 


1.62 


342° F. 


1.87 


36° 


1.11 


198° 


1.65 


360° 


1.90 


54° 


1.22 


216° 


1.68 


378° 


1.92 


72° 


1.30 


234° 


1.72 


396° 


1.94 


90° 


1.37 


252° 


1.74 


414° 


1.96 


108° 


1.43 


270° 


1.77 


432° 


1.98 


126° 


1.49 


288° 


1.80 


450° 


2.00 


144° 


1.53 


306° 


1.83 


468° 


2.02 


162° 


1.58 


324° 


1.85 







Example in the Use of the Tables. — Required the total loss of heat by 
both radiation and convection, per foot of length of a steam-pipe 2 11/32 
in. external diameter, steam pressure 60 lbs., temperature of the air in the 
room 68° Fahr. 

Temperature corresponding to 60 lbs. equals 307°; temperature difference 
= 307 - 68 = 239°. 

Area of one foot length of steam-pipe = 2 11/32 x 3.1416 -*- 12 = 0.614 sq. 
! t. 

Heat radiated per hour per square foot per degree of difference, from 
table 064 

Radiation loss per hour by Newton's law = 239° X .614 ft. X .64 = 93.9 
heat units. Same reduced to conform with Dulong's law of radiation: factor 
from table for temperature difference of 239° and temperature of air 68° = 
1.93. 93.9 X 1.93 = 181.2 heat units, total loss by radiation. 

Convection loss per square foot per hour from a 2 11/32-inch pipe: by in- 
terpolation from table, 2" = .728. 3" = .626, 2 11/32" = .693. 

Area, .614 X .693 + 239° = 101.7 heat units. Same reduced to conform with 
Dulong's law of convection: 101.7 X 1.73 (from table) = 175.9 heat units per 
hour. Total loss by radiation and convection = 181.2 + 175.9 = 357.1 heat 
units per hour. Loss per degree of difference of temperature per square 
foot of surface per hour = 357.1 -s- 239 = 1.494 heat units. 

It is not claimed, says The Locomotive, that the results obtained by this 
method of calculation are strictly accurate. The experimental data are not 
sufficient to allow us to compute the heat-loss from steam-pipes with any 
great degree of refinement; yet it is believed that the results obtained as 
indicated above will be sufficiently near the truth for most purposes. Ah 
experiment by Prof. Ordway, in a pipe 2 11/32 in. diam. under the above 
conditions (Trans. A. S. M. E., v. 73), showed a condensation of steam of 181 
grammes per hour, which is equivalent to a loss of heat of 358.7 heat units 
per hour, or within half of one" per cent of that given by the above calcula- 
tion. 

According to different authorities, the quantity of heat given off by steam 
and hot-water radiators in ordinary practice of heating of buildings by 
direct radiation varies from 1.8 to about 3 heat units per hour per square 
foot per degree of difference of temperature. 

The lowest figure is calculated from the following statement by Robert 
Briggs in his paper on "American Practice in Warming Buildings by 
Steam " (Proc. Inst. C. E., 1882, vol. lxxi): " Each 100 sq. ft. of radiating 
surface will give off 3 Fahr. heat units per minute for each degree F. of dif- 
ference in temperature between the radiating surface and the air in which 
it is exposed." 

The figure 2 1/2 heat units is given by the Nason Manufacturing Company 
in their catalogue, and 2 to 2 1/4 are given by many recent writers. 

For the ordinary temperature difference in low-pressure steam- heating, 
say 212° - 70° = 142° F.; 1 lb. steam condensed from 212° to water at the 



478 HEAT. 

same temperature gives up 965.7 heat units. A loss of 2 heat units per sq. 
ft. per hour per degree of difference, under these conditions, is equivalent 
to 2 X 142-4-965 = 0.3 lbs. of steam condensed per hour per sq. ft. of heating 
surface. (See also Heating and Ventilation.) 

Transmission of Heat through Walls, etc., of Buildings 
(Nason Manufacturing Co.). (See also Heating and Ventilation.) — Heat 
has the remarkable property of passing through moderate thicknesses of air 
and gases without appreciable loss, so that air is not warmed by radiant 
heat, but by contact with surfaces that have absorbed the radiation. 

Powers of Different Substances for Transmitting Heat. 

Window-glass 1000 Bricks, rough 200 to 250 

Oak or walnut 66 Bricks, whitewashed 200 

"White pine 80 Granite or slate 250 

Pitch-pine 100 Sheet iron 1030tolll0 

Lath or plaster 75 to 100 

A square foot of glass will cool 1.279 cubic feet of air from the tempera- 
ture inside to that outside per minute, and outside wall surface is generally 
estimated at one fifth of the rate of glass in cooling effect. 

Box, in his " Practical Treatise on Heat," gives a table of the conducting 
powers of materials prepared from the experiments of Peclet. It gives the 
quantity of heat in units transmitted per square foot per hour by a plate 1 
inch in thickness, the two surfaces differing in temperature 1 degree: 

Fine-grained gray marble 28.00 

Coarse-grained white marble 22.4 

Stone, calcareous, fine 16.7 

Stone, calcareous, ordinary 13.68 

Baked clay, brickwork 4.83 

Brick-dust, sifted 1.33 

Hood, in his "Warming and Ventilating of Buildings," p. 249, gives the 
results of M. Depretz, which, placing the conducting power of marble at 1.00, 
give .483 as the value for firebrick. 

THERMODYNAMICS. 

Thermodynamics, the science of heat considered as a form of 
energy, is useful in advanced studies of the theory of steam, gas, and air 
engines, refrigerating machines, compressed air, etc. The method of treat- 
ment adopted by the standard writers is severely mathematical, involving 
constant application of the calculus. The student will find the subject 
thorougly treated in the recent works by Rontgen (Dubois's translation), 
.Wood, and Peabody. 

First Law of Thermodynamics.— Heat and mechanical energy 
are mutually convertible in the ratio of about 778 foot-pounds for the British 
thermal unit. (Wood.) Heat is the living force or vis viva due to certain 
molecular motions of the molecules of bodies, and this living force may be 
stated or measured in units of heat or in foot-pounds, a unit of heat in 
British measures being equivalent to 772 [778] foot-pounds. (Trowbridge, 
Trans. A. S. M. E., vii. 727.) 

Second Law of Thermodynamics.— The second law has by dif- 
ferent writers been stated in a variety of ways, and apparently with ideas 
so diverse as not to cover a common principle. (Wood, Therm., p. 389.) 

It is impossible for a self-acting machine, unaided by any external agency, 
to convert heat from one body to another at a higher temperature. (Clau- 
sius.) 

If all the heat absorbed be at one temperature, and that rejected be at 
one lower temperature, then will the heat which is transmuted into work be 
to the entire heat absorbed in the same ratio as the difference between the 
absolute temperature of the source and refrigerator is to the absolute tem- 
perature of the source. In other words, the second law is an expression for 
the efficiency of the perfect elementary engine. (Wood.) 

The living force, or vis viva, of a body (called heat) is always proportional 
to the absolute temperature of the body. (Trowbridge.) 

Q Q rp _ rp 

The expression v ' V2 — -J-= — - may be called the symbolical or al- 
gebraic enunciation of the second law,— the law which limits the efficiency 
of heat engines, and which does not depend on the nature of the working 
medium employed. (Trowbridge.) Q x and T x — quantity and absolute 



PHYSICAL PROPERTIES OF GASES. 479 

temperature of the heat received, <? 2 aod T 2 = quantity and absolute tem- 
perature of the heat rejected. 
T — T 
The expression 1 — - represents the efficiency of a perfect heat engine 

ivnich receives all its heat at the absolute temperature T x , and rejects heat 
at the temperature T 2 , converting into work the difference between the 
quantity received and rejected. 

Example. — What is the efficiency of a perfect heat engine which receives 
heat at 388° F. (the temperature of steam of 200 lbs. gauge pressure) and 
rejects heat at 100° F. (temperature of a condenser, pressure 1 lb. above 
vacuum). 

388 + 459.2 - 100 + 459.2 *.■ 
- 888+15*2 = W °> "^ 

In the actual engine this efficiency can never be attained, for the difference 
between the quantity of heat received into the cylinder and that rejected 
into the condenser is not all converted into work, much of it being lost by 
radiation, leakage, etc. In the steam engine the phenomenon of cylinder 
condensation also tends to reduce the efficiency. 

PHYSICAL PROPERTIES OF GASES. 

(Additional matter on this subject will be found under Heat, Air, Gas, and 
Steam.) 

When a mass of gas is enclosed in a vessel it exerts a pressure against the 
vails. This pressure is uniform on every square inch of the surface of the 
vessel; also, at any point in the fluid mass the pressure is the same in every 
direction. 

In small vessels containining gases the increase of pressure due to weight 
may be neglected, since all gages are very light; but where liquids are con- 
cerned, the increase in pressure due to their weight must always be taken 
into account. 

Expansion of Gases, Marriotte's I*aw. — The volume of a gas 
diminishes in the same ratio as the pressure upon it is increased. 

This law is by experiment found to be very nearly true for all gases, and 
3 known as Boyle's or Mariotte's law. 

If p = pressure at a volume v, and p x = pressure at a volume v x , p 1 v 1 = 

pv; Pi — — p; pv = a constant. 

The constant, C, varies with the temperature, everything else remaining 
the same. 

Air compressed by a pressure of seventy -five atmospheres has a volume 
about 2% less than that computed from Boyle's law, but this is the greatest 
divergence that is found below 160 atmospheres pressure. 

Law of Charles.— The volume of a perfect gas at a constant pressure 
is proportional to its absolute temperature. If v a be the volume of a gas 
at 32° F., and v x the volume at any other temperature, t t , then 

ft 1 + 459.2\ / :" t x .- 32°\ 

V - = V °V 491.2 )' V - = V + -49lX>»' 
or v x = [1 + 0.002036(^ - 32°)]tv 
If the pressure also change fromp top l5 

„ _ v *o(*i+459.2\ 
1 °pA 49L2~/- 

The Densities of Gases and Vapors are simply proportional to 
their atomic weights. 

Avogadro's Liaw.— Equal volumes of all gases, under the same con- 
ditions of temperature and pressure, contain the same number of mole- 
cules. 

To find the weight of a gas in pounds per cubic foot at 32° F., multiply 
half the molecular weight of the gas by .00559. Thus 1 cu. ft. marsh-gas, CH4, 

12 4- 4 
== -~— X .00559 = .0447 lb, 



480 PHYSICAL PROPERTIES OF GASES. 

When a certain volume of hydrogen combines with one half its volume of 
oxygen, there is produced an amount of water vapor which will occupy the 
same volume as that which was occupied by the hydrogen gas when at the 
same temperature and pressure. 

Saturation-point of Vapors.— A. vapor that is not near the satura- 
tion-point behaves like a gas under changes of temperature and pressure; 
but if it is sufficiently compressed or cooled, it reaches a point where it be- 
gins to condense: it then no longer obeys the same laws as a gas, but its 
pressure cannot be increased by diminishing the size of the vessel containing 
it, but remains constant, except when the temperature is changed. The 
only gas that can prevent a liquid evaporating seems to be its own vapor. 

Dalton's Law of Gaseous Pressures.—- Every portion of a mass 
of gas inclosed in a vessel contributes to tiie pressure against the sides of 
the vessel the same amount that it would have exerted by itself had no 
other gas been present. 

Mixtures of Vapors and Gases.— The pressure exerted against 
the interior of a vessel by a given quantity of a perfect gas enclosed in it 
is the sum of the pressures which any number of parts into which such quan- 
tity might be divided would exert separately, if each were enclosed in a 
vessel of the same bulk alone, at the same temperature. Although this law 
is not exactly true for any actual gas, it is very nearly true for many. Thus 
if 0.080728 lb. of air at 32° F., being enclosed in a vessel of one cubic foot 
capacity, exerts a pressure of one atmosphere or 14.7 pounds, on each square 
inch of the interior of the vessel, then will each additional 0.080728 lb. of air 
which is enclosed, at 32°, in the same vessel, produce very nearly an addi- 
tional atmosphere of pressure. The same law is applicable to mixtures of 
gases of different kinds. For example, 0,12344 lb. of carbonic-acid gas, at 
32°, being enclosed in a vessel of one cubic foot in capacity, exerts a pressure 
of one atmosphere; consequently, if 0.080728 lb. of air and 0.12344 lb. of 
carbonic acid, mixed, be enclosed at the temperature of 32°, in a vessel of 
one cubic foot of capacity, the mixture will exert a pressure of two atmos- 
pheres. As a second example: Let 0.080728 lb. of air, at 212°, be enclosed in 
a vessel of one cubic foot; it will exert a pressure of 

212 + 459.2 . MC , 
oc . , ,. n n — 1 .366 atmospheres. 

32 + 409.2 

Let 0.03797 lb. of steam, at 212°, be enclosed in a vessel of one cubic foot ; it 
will exert a pressure of one atmosphere. Consequently, if 0.080728 lb. of air 
and 0.03797 lb. of steam be mixed and enclosed together, at 212°, in a vessel of 
one cubic foot, the mixture will exert a pressure of 2.366 atmospheres. It is 
a common but erroneous practice, in elementary books on physics, to de- 
scribe this law as constituting a difference between mixed and homogeneous 
gases; whereas it is obvious that for mixed and homogeneous gases the law 
of pressure is exactly the same, viz., that the pressure of the whole of a 
gaseous mass is the sum of the pressures of all its parts This is one of the 
laws of mixture of gases and vapors. 

A second law is that the presence of a foreign gaseous substance in con- 
tact with the surface of a solid or liquid does not affect the density of the 
vapor of that solid or liquid unless there is a tendency to chemical com- 
bination between the two substances, in which case the density of the 
vapor is slightly increased. (Rankine, S. E., p. 239.) 

Flow of Gases.— By the principle of the conservation of energy, it may 
be shown that the velocity with which a gas under pressure will escape into 
a vacuum is inversely proportional to the square root of its density; that is, 
oxygen, which is sixteen times as heavy as hydrogen, would, under exactly 
the same circumstances, escape through an opening only one fourth as fast 
as the latter gas. 

Absorption of Gases foy Liquids.— Many gases are readily ab- 
sorbed by water. Other liquids also possess this power in a greater or less 
degree. Water will for example, absorb its own volume of carbonic-acid 
gas, 430 times its volume of ammonia, 2^g times its volume of chlorine, and 
only about 1/20 of its volume of oxygen. 

The weight of gas that is absorbed by a given volume of liquid is propor- 
tional to the pressure. But as the volume of a mass of gas is less as the 
pressure is greater, the volume which a given amount of liquid can absorb 
at a certain temperature will be constant, whatever the presstire. Water, 
for example, can absorb its own volume of carbonic-acid gas at atmospheric 
pressure; it will also dissolve its own volume if the pressure is twice as 
great, but in that case the gas will be twice as dense, and consequently twice 
the weight of gas is dissolved, 



PRESSURE OF THE ATMOSPHERE. 



481 



AIR. 



Properties of Air.— Air is a mechanical mixture of the gases oxygen 
and nitrogen; 21 parts O and 79 parts N by volume, 23 parts O and 77 parts 
N by weight. 

The weight of pure air at 32° F. and a barometric pressure of 29.92 inches 
of mercury, or 14.6963 lbs. per sq. in., or 2116.3 lbs. per sq. ft., is .080728 lbs. 
The volume of 1 lb. is 12.387 cubic feet. At any other temperature and 

barometric pressure its weight in lbs. per cubic foot is W~ '' , 

where B = height of the barometer, T= temperature Fahr., and 1.3253 = 
weight in lbs. of 459.2 c. ft. of air at 0° F. and one inch barometric pressure. 
Air expands 1/491.2 of its volume for every increase of 1° F., and its volume 
varies inversely as the pressure. 

Volume, Density, and Pressure of Air at Various 
Temperatures. (D. K. Clark.) 





Volume at Atmos. 




Pressure at Constant 




Pressure. 


Density, lbs. 
per Cubic Foot at 
Atmos. Pressure. 


Volume. 


Fahr. 












Cubic Feet 


Compara- 




Lbs. per 
Sq. In. 


Compara- 




in 1 lb. 


tive Vol. 




tive Pres. 





11.583 


.881 


.086331 


12.96 


.881 


32 


12.387 


.943 


.080728 


13.86 


.943 


40 


12.586 


.958 


.079439 


14.08 


.958 


50 


12.840 


.977 


.077884 


14.36 


.977 


62 


13.141 


1.000 


.076097 


14.70 


1.000 


70 


13.342 


1.015 


.074950 


14.92 


1.015 


80 


13.593 


1 .034 


.073565 


15.21 


1.034 


90 


13.845 


1.054 


.072230 


15.49 


1.054 


100 


14.096 


1.073 


.070942 


15.77 


1.073 


110 


14.344 


1.092 


.069721 


16.05 


1.092 


120 


14.592 


1.111 


.06^500 


16.33 


1.111 


130 


14.846 


1.130 


.067361 


16.61 


1.130 


140 


15.100 


1.149 


.066221 


16.89 


1.149 


150 


15.351 


1.168 


.065155 


17.19 


1.168 


160 


15.603 


1.187 


.064088 


17.50 


1.187 


170 


15.854 


1.206 


.063089 


17.76 


1.206 


180 


16.106 


1.226 


.062090 


18.02 


1.226 


200 


16.606 


1.264 


.060210 


18.58 


1.264 


210 


16.860 


1.283 


.059313 


18.86 


1.283 


212 


16.910 


1.287 


.059135 


18.92" 


1.287 



The Air-manometer consists of a long vertical glass tube, closed at 
theuppir end, open at the lower end, containing air, provided with a scale, 
and immersed, along with a thermometer, in a transparent liquid, such as 
water or oil, contained in a strong cylinder of glass, which communicates 
with the vessel in which the pressure is to be ascertained. The scale shows 
the volume occupied by the air in the tube. 

Let v be that volume, at the temperature of 32° Fahrenheit, and mean 
pressure of the atmosphere, p ; let v x be the volume of the air at the tem- 
perature t, and under the absolute pressure to be measured p, ; then 



Pi = " 



_ (t + 459.2° ) Po v 



Pressure of the Atmosphere at Different Altitudes. 

At the sea-level the pressure of the air is 14.7 pounds per square inch; at 
J4 of a mile above the sea-level it is 14.02 pounds; at % mile, 13.33; at $£ 
mile, 12,66; at 1 mile, 12.02; at V& mile, 11.42; at \\i mile, 10.88; and at 2 



482 



miles, 9.80 pounds per square inch. For a rough approximation we may- 
assume that the pressure decreases Y» pound per square inch for every 1000 
feet of ascent. 

It is calculated that at a height of about 3V£ miles above the sea-level the 
weight of a cubic foot of air is only one half what it is at the surface of the 
earth, at seven miles only one fourth, at fourteen miles only one sixteenth, 
at twenty-one miles only on9 sixty-fourth, and at a height of over forty- ' 
five miles it becomes so attenuated as to have no appreciable weight. 

The pressure of the atmosphere increases with the depth of shafts, equal 
to about one inch rise in the barometer for each 900 feet increase in depth: 
this may be taken as a rough-and-ready rule for ascertaining the depth of 
shafts. 

Pressure of tlie Atmosphere per Square Inch and per 
Square Foot at Various Readings of the Barometer. 

Rule.— Barometer in inches x .4908 = pressure per square inch; pressure 
per square inch x 144 = pressure per square foot. 





Pressure 


Pressure 




Pressure 


Pressure 




per Sq. In. 


per Sq. Ft. 




per Sq. In. 


per Sq. Ft. 


in. 


lbs. 


lbs.* 


in. 


lbs. 


lbs.* 


28.00 


13.74 


1978 


29.75 


14.60 


2102 


28.25 


13.86 


1995 


30.00 


14.72 


2119 


28.50 


13.98 


2013 


30.25 


14.84 


2136 


28.75 


14.11 


2031 


30.50 


14.96 


2154 


29.00 


14.23 


2049 


30.75 


15.09 


2172 


29.25 


14.35 


2066 


31.00 


15.21 


2190 


29.50 


14.47 


2083 









* Decimals omitted. 
For lower pressures see table of the Properties of Steam. 
Barometric Readings corresponding with Different 
Altitudes, in French and English Measures. 





Read- 




Reading 




Reading 




Reading 


Alti- 
tude. 


ing of 
Barom- 


Altitude. 


of 
Barom- 


Alti- 
tude. 


of 
Barom- 


Altitude. 


of 
Barom- 




eter. 




eter. 




eter. 




eter. 


meters. 


mm. 


feet. 


inches. 


meters. 


mm. 


feet. 


inches. 





762 


0. 


30. 


1147 


660 


3763.2 


25.98 


21 


760 


68.9 


29.92 


1269 


650 


4163.3 


25.59 


127 


750 


416.7 


29.52 


1393 


640 


4568.3 


25.19 


234 


740 


767.7 


29.13 


1519 


630 


4983.1 


24.80 


342 


730 


1122.1 


28.74 


1647 


620 


5403.2 


24.41 


453 


720 


1486.2 


28.35 


1777 


610 


5830.2 


24.01 


564 


710 


1850.4 


27.95 


1909 


600 


6243. 


23.62 


678 


700 


2224.5 


27.55 


2043 


590 


6702.9 


23.22 


793 


690 


2599.7 


27.16 


2180 


580 


7152.4 


22.83 


909 


680 


2962.1 


26.77 


2318 


570 


7605.1 


22.44 


1027 


670 


3369.5 


26.38 


2460 


560 


8071. 


22.04 



Levelling by the Barometer and hy Boiling Water. 

(Trautwine.) — Many circumstances combine to render the results of this 
kind of levelling unreliable where great accuracy is required. It is difficult 
to read off from an aneroid (the kind of barometer usually employed for 
engineering purposes) to within from two to five or six feet, depending on 
its size. The moisture or dryness of the air affects the results; also winds, 
the vicinity of mountains, and the daily atmospheric tides, which cause 
incessant and irregular fluctuations in the barometer. A barometer hang- 
ing quietly in a room will often vary 1/4 of an inch within a few hours, cor- 
responding to a difference of elevation of nearly 100 feet. No formula can 
possibly be devised that shall embrace these sources of error. 



MOISTURE IN THE ATMOSPHERE. 



483 



To Find the Difference in Altitude of Two Places.— Take 

from the table the altitudes opposite to tue two boiling temperatures, or to 
the two barometer readings. Subtract the one opposite the lower reading 
from that opposite the upper reading. The remainder will be the required 
height, as a rough approximation. To correct this, add together the two 
thermometer readings, and divide the sum by 2, for their mean. From 
table of corrections for temperature, take out the number under this mean. 
Multiply the approximate height just found by this number. 

At 70° F. pure water will boil at 1° less of temperature for an average of 
about 550 feet of elevation above sea-level, up to a height of 1/2 a mile. At 
the height of 1 mile, 1° of boiling temperature will correspond to about 560 
feet of elevation. In the table the mean of the temperatures at the two 
stations is assumed to be 32°F., at which no correction for temperature is 
necessary in using the table. 



^ftd 


£ 


I>1- 


^•s ^ 


a . 






s" 




if^h 


2.2 


$%i$ 




2.9 

eg 


5"Sl« 




03"" 


'**~ (i«M 


« •- 


PQ 


< & 


M 


< & 


pq S 


M 


< OQ 


184° 


16.79 


15,221 


196 


21.71 


8,481 


208 


27.73 


2,063 


185 


17.16 


14,649 


197 


22.17 


7,932 


208.5 


28.00 


1,809 


186 


17.54 


14,075 


198 


22.64 


7,381 


209 


28.29 


1,539 


187 


17.93 


13,498 


199 


23.11 


6,843 


209.5° 


28.56 


1,290 


188 


18.32 


12,934 


200 


23.59 


6,304 


210 


28.85 


1,025 


189 


18.72 


12,367 


201 


24.08 


5,764 


210.5 


29.15 


754 


190 


19.13 


11,799 


202 


24.58 


5,225 


211 


29.42 


512 


191 


19.54 


11,243 


203 


25.08 


4,697 


211.5 


29.71 


255 


192 


19.96 


10,685 


204 


25.59 


4,169 


212 


30.00 


S.L.= 


193 


20.39 


10,127 


205 


26.11 


3,642 


212.5 


30.30 


-261 


194 


20.82 


9,579 


206 


26.64 


3,115 


213 


30.59 


-511 


195 


21.26 


9,031 


207 


27.18 


2,589 









Corrections for Temperature. 



Mean temp. F. in shade. ! 10° I 20° I 
Multiply by .933 |.954|.975|. 



M 40° I 50° 

> 1.016|l.036|l.058|l. 



70° 180° 

.079|1. 100|1. 121 1.143 



I 100«| 



IWoisture in the Atmosphere.— Atmospheric air always contains 
a small quantity of carbonic-acid gas and a varying quantity of aqueous 
vapor. Pure mountain air contains about 3 to 4 parts of carbonic acid in 
10,000. A properly ventilated room should contain not more than six parts 
in 10,000. 

The degree of saturation or relative humidity of the air is determined by 
the use of the dry and wet bulb thermometer. The degree of saturation for 
a number of different readings of the thermometer is given in the following 
table : 

Indications op the Hygrometer (Dry and Wet Bulb), from 
Mr. Glaisher's Observations at Greenwich. 





Difference of Temperature or Degrees of Cold in the Wet- 




bulb Thermometer. 


Temperature 
of the Air, 
Fahrenheit. 






































1 


2 3 


4 


5 6 


7 


8 9 10 


11 


12 


13 


14 


1516 


17 


18 


19 


20 


21 


22 23 24 




Degrees of Humidity, Saturation being 100. 


32° 


87 


75 














































42° 






7K 


72 


m 


6(1 


54 


49 


44 


40 


36 


33 


30 


27 






















52° 


W 


St* 




74 


69 


64 


59 


54 


50 


46 


42 


39 


36 


33 


30 


27 


25 
















62° 


94 


88 


82 


77 


72 


67 


62 


58 


54 


50 


47 


44 


41 


38 


35 


32 


30 


28 


26 


21 










72° 


04 


S9 


84 


79 


74 




65 


61 


57 


54 


51 


48 


45 


42 


39 


36 


34 


32 


30 


28 


26 


24 


23 


22 


82° 




90 


85 






72 




64 


60 


57 


54 


51 


48 


45 


42 


40 


3 y 


35 


33 


31 


29 


27 




25 


92° 


95 


90 


85 


81 


" 


73 


71 


66 


62 


59 


5t; 


53 


50 


47 


45 


43 


41 


38 


36 


34 


32 


30 


28 


26 



484 air. 

Weights of Air, Vapor of Water, and Saturated Mixtures 

of Air and Vapor at Different Temperatures, under 

the Ordinary Atmospheric Pressure of 29.921 

inches of Mercury. 





r 3 H 

III 





Mixtures of Air Saturated with Vapor. 






Weight of Cubic Foot of the 




© 


§3 


Elastic 
Force of 
the Air in 
Mixture 


Mixture of Air and Vapor. 


Weight 
of 


3."S 


cS t« 3 


%% 








Vapor 


~s3 J5 

33 a 

P.? 


©'IS 


£° 


of Airand 
Vapor, 


Weight 

of the 

Air, lbs. 


Weight 
of the 


Total 
W'ghtof 


mixed 
with lib. 

of Air, 
pounds. 


11 


'33 tw <d 




Inches of 
Mercury. 


Vapor, 
pounds. 


Mixture, 
pounds. 


0° 


.0864 


.044 


29.877 


.0863 


.000079 


.086379 


.00092 


12 


.0342 


.074 


29.849 


.0840 


.000130 


.084130 


.00155 


22 


.0824 


.118 


29.803 


.0821 


.000202 


.082302 


.00245 


32 


.0807 


.181 


29.740 


.0802 


.000304 


.080504 


.00379 


42 


.0791 


.267 


29.654 


.0784 


.000440 


.078810 


.00561 


52 


.0776 


.388 


29.533 


.0766 


.000627 


077227 


.00819 


62 


.0761 


.556 


29.365 


.0747 


.000881 


'. 075581 


.01179 


72 


.0747 


.785 


29.136 


.0727 


.001221 


.073921 


.01680 


82 


.0733 


1.092 


28.829 


.0706 


.001667 


.072267 


.02361 


92 


.0720 


1.501 


28.420 


.0684 


.002250 


.070717 


.03289 


102 


.0707 


2.036 


27.885 


.0659 


.002997 


.068897 


.04547 


112 


.0694 


2.731 


27.190 


.0631 


.003946 


.067046 


.06253 


122 


.0682 


3.621 


26.300 


.0599 


.005142 


.065042 


.08584 


132 


.0671 


4.752 


25.169 


.0564 


.006639 


.063039 


.11771 


142 


.0660 


6.165 


23.756 


.0524 


.008473 


.060873 


.16170 


152 


.0649 


7.930 


21.991 


.0477 


.010716 


.058416 


.22465 


162 


.0638 


10.099 


19.822 


.0423 


.013415 


.055715 


.31713 


172 


.0628 


12.758 


17.163 


.0360 


.016682 


.052682 


.46338 


182 


.0618 


15.960 


13.961 


.0288 


.020536 


.049336 


.71300 


192 


.0609 


19.828 


10.093 


.0205 


.025142 


.045642 


1.22613 


202 


.0600 


24.450 


5.471 


.0109 


.030545 


.041445 


2.80230 


212 


.0591 


29 921 


0.000 


.0000 


.036820 


.036820 


Infinite. 



The weight in lbs. of the vapor mixed with 100 lbs. of pure air at any 
given temperature and pressure is given by the formula 

62.3 X E 29.92 
29.92 - E X p ' 

where E = elastic force of the vapor at the given temperature, in inches of 
mercury; p = absolute pressure in inches of mercury, = 29.92 for ordinary 
atmospheric pressure. 
Specific Heat of Air at Constant Volume and at Constant 

Pressure.— Volume of 1 lb. of air at 32° F. and pressure of 14.7 lbs. per sq. 
in. = 12.387 cu. ft. = a column 1 sq. ft. area X 12.387 ft. high. Raising temper- 
1 



Work done = 2116 lbs. per sq. ft. X .02522 = 53.37 foot-pounds, or 53.37 h- 778 
= .0686 heat units. 

The specific heat of air at constant pressure, according to Regnault, is 
0.2375; but this includes the work of expansion, or .0686 heat units; hence 
the specific heat at constant volume = 0.2375 — .0686 = 0.1689. 

Ratio of specific heat at constant pressure to specific heat at constant 
volume = .2375 -*- .1689 = 1.406. (See Specific Heat, p. 458.) 

Flow of Air through Orifices.— The theoretical velocity in feet 
per second of flow of any fluid, liquid, or gas through an orifice is v = 
V%gh = 8.02 VTi, in which h = the " head " or height of the fluid in feet 
required to produce the pressure of the fluid at the level of the orifice. 

h = £-. The quantity of flow in cubic feet per second is equal to the product 



FLOW OF AIR IK PIPES. 485 

of this velocity by the area of the orifice, in square feet, multiplied by a 
"coefficient of flow," which takes into account the contraction of the vein 
or flowing stream, the friction of the orifice, etc. 

For air flowing through an orifice or short tube, from a reservoir of the 
pressure©! into a reservoir of the pressure^, Weisbach gives the follow- 
ing values for the coefficient of flow, obtained from his experiments. 

Flow of Air through an Orifice. 

Coefficient c in formula v = c V2gh. 
Diameter \ Ratio of pressures p x ^p 2 1.05 1.09 1.43 1.65 1.89 2.15 

1 centimetre, f Coefficient 555 .5S9 .692 .724 .754 .788 

Diameter I Ratio of pressures 1.05 1.09 1.36 1.67 2.01 .... 

2.14 centimetres f Coefficient 558 .573 .634 .678 .723.... 

Flow of Air through a Short Tube. 

Diam. 1 cm., (. Ratio of pressures Pi-npa 1.05 1.10 1.30 

Length 3 cm. (Coefficient 730 .771 .830 

Diam. 1.414 cm., | Ratio of pressures 1.41 1.69 

Length 4.242 cm. \ Coefficient 813 .822 

Diam 1 cm j Ratio of pressures 1.24 1.38 1.59 1.85 2.14 .... 

L Se h rounrd> 979 •«» ■*» -™ ™ ' " 

Fliegner's Equations for Flow of Air from a Reservoir through an 
Orifice. (Peabody's Thermodynamics, p. 135.) 

For Pi > 2pa, G = 0.530 F J^ ; 

Vt 1 



Pl >2pa, Q = l.mF^/ pa( ^- pa H 

G = flow of air through the orifice in lbs. per sec, F — area of orifice in 
square inches, p 1 = pressure in reservoir in lbs. per sq. in., pa — pressure of 
atmosphere, T x — absolute temperature, Fahrenheit, of air in reservoir. 

Clark (Rules, Tables, and Data, p. 891) gives, for the velocity of flow of air 
through an orifice due to small differences of pressure, 

,= 0j /fxrr, 2 x( 1 + l^)x^, 

or, simplified, 



!Ci/(l+.00203U- 32^ 



/p 

in which V— velocity in feet per second; 2g — 64.4; h — height of the column 
of water in inches, measuring the difference of pressure; / = the tempera- 
ture Fahr. ; and p = barometric pressure in inches of mercury. 773.2 is the 
volume of air at 32° under a pressure of 29.92 inches of mercury when that of 
an equal weight of water is taken as 1. 



For 62° F., the formula becomes V = 363C A/ -, and if p = 29.92 inchesF = 

66.35C Vh 

The coefficient of efflux C, according to Weisbach, is: 
For conoidal mouthpiece, of form of the contracted vein, 

with pressures of from .23 to 1.1 atmospheres C — .97 to .99 

Circular orifices in thin plates C = .56 to .79 

Short cylindrical mouthpieces C — .81 to .84 

The same rounded at the inner end C = .92 to .93 

Conical converging mouthpieces C — .90 to .99 

Flow of Air in Pipes.— Hawksley (Proc. Inst. C. E.. xxxiii, 55) 

states that his formula for flow of water in pipes v = 48 a/ — =r- may also 

be employed for flow of air. In this case H. — height in feet of a column of 
air required to produce the pressure causing the flow, or the loss of head 



486 



for a given flow; v = velocity in feet per second, D = diameter in feet, L - 
length in feet. 

If the head is expressed in inches of water, h, the air being taken at 
62° F , its weight per cubic foot at atmospheric pressure — .0761 lb. Then 

H = 076 ^" x 12 - 68.37i. If d = diameter in inches, D = — and the formula 



becomes v = 114.5 



- / hd ■ 



»hich h — inclies of. water column, d = dia 



Lv* 



Lv* 



The quantity in cubic feet per second is 

"144 T L ' a Y .39/,' ' l --^w- 



-- .7851 - 



The horse-power required to drive air through a pipe is the volume Q in 
cubic feet per second multiplied by the pressure in pounds per square f.ot 
and divided by 550. Pressure in pounds per square foot = P — inches -f 
water column x 5.196, whence horse-power = 



= QP _ Qh 



Q 3 L 
41.3d 5 " 



If the head or pressure causing the flow is expressed in pounds per square 
inch = p, then h = 27.71p, and the above formulae become 



363,300p 



Q=c 



y £ ' P - 363,300d' 

iTi/ifE. v - Q* L ■ d - //ZM 



QlUp 



= .2&18Qp = .02421^. 



Volume of Air Transmitted in Cubic Feet per Minute in 
Pipes of Various Diameters. 











Formula Q = 


^d>v x 60. 
144 








>>* 


Actual Diameter of Pipe in Inches. 


£ OS 


1 


2 


3 


4 


5 


6 


8 


10 


12 


16 


20 


24 


frft 


























1 


.327 


1.31 


2.95 


5.24 


8.18 


11.78 


20.94 


32.73 


47.12 


83.77 


130.9 


188.5 


2 


.655 


2.62 


5.89 


10.47 


16.36 


23.56 


41 89 


65.45 


94.25 


167.5 


261.8 


377 


3 


.982 


3.93 


8.84 


15.7 


24.5 


35.3 


62.8 


98.2 


141.4 


251.3 


392.7 


565.5 


4 


1.31 


5,24 


11.78 


20.9 


32.7 


47.1 


83.8 


131 


188 


335 


523 


754 


5 


1.64 


6.54 


14.7 


26.2 


41 


59 


104 


163 


235 


419 


654 


942 


6 


1.96 


7.85 


17.7 


31.4 


49.1 


70.7 


125 


196 


283 


502 


785 


1131 


7 


2.29 


9.16 


20.6 


36.6 


57.2 


82.4 


146 


229 


330 


586 


916 


1319 


8 


2.62 


10 5 


23.5 


41.9 


65.4 


94 


167 


262 


377 


670 


1047 


1508 


9 


2.95 


11.78 


26.5 


47 


73 


106 


188 


294 


424 


754 


1178 


1C96 


10 


3.27 


13.1 


29.4 


52 


82 


118 


209 


327 


471 


838 


1309 


1885 


12 


3.93 


15.7 


35.3 


63 


98 


141 


251 


393 


565 


1005 


1571 


2262 


15 


4.91 


19.6 


44.2 


78 


122 


177 


314 


491 


707 


1256 


1963 


2827 


18 


5.89 


23.5 


53 


94 


147 


212 


377 


589 


848 


1508 


2356 


3393 


20 


6.54 


26.2 


59 


105 


164 


235 


419 


654 


942 


1675 


2618 


3770 


24 


7.85 


31.4 


71 


125 


196 


283 


502 


785 


1131 


2010 


3141 


4524 


25 


8.18 


32.7 


73 


131 


204 


294 


523 


818 


1178 


2094 


3272 


4712 


28 


9.16 


36.6 


82 


146 


229 


330 


586 


916 


1319 


2346 


3665 


5278 


30 9.8 


39.3 


88 


157 


245 


353 


628 


982 


1414 


2513 


3927 


5655 



FLOW OF AIR IN" PIPES. 487 

In Hawksley's formula and its derivatives the numerical coefficients are 
constant. It is scarcely possible, however, that they can be accurate except 
within a limited range of conditions. In the case of water it is found that 
the coefficient of friction, on which the loss of head depends, varies with the 
length and diameter of the pipe, and with the velocity, as well as with the 
condition of the interior surface. In the case of air and other gases we 
have, in addition, the decrease in density and consequent increase in volume 
and in velocity due to the progressive loss of head from one end of the pipe 
to the other. 

Clark states that according to the experiments of D'Aubisson and those of 
a Sardinian commission on the resistance of air through long conduits or 
pipes, the diminution of pressure is very nearly directly as the length, and 
as the square of the velocity and inversely as the diameter. The resistance 
is not varied by the density. 

If these statements are correct, then the formulae h = — — and h = ¥-—■ 

cd c'd a 

and their derivatives are correct in form, and they may be used when the 
numerical coefficients c and c' are obtained by experiment. 

If we take the forms of the above formulas as correct, and let C be a vari- 
able coefficient, depending upon the length, diameter, and condition of sur- 
face of the pipe, and possibly also upon the velocity, the temperature and 
the density, to be determined by future experiments, then for h = head in 
inches of water, d = diameter in inches, L = length in feet, v — velocity in 
; feet per second, and Q = quantity in cubic feet per second: 



•-y$ 



Lv* m 
'' CVi ' 



.005454C 



• loss of 



33683(g 2 L 
CVi ' 




For difference or loss of pressure p in pounds per square inch, 

h = 27.71p Vh = 5.264 \'p\ 

/m3Q*L. _ 1213Q2L 



d -\/ 



(For other formulae for flow of air, see Mine Ventilation.) 

Loss of Pressure in Ounces per Square Inch.— B. F. Sturte- 
vant Company uses the following formulae : 



Lv* m _ \ / 250Q0dp 1 . _ Lifl . 
Pl ~ 25000d ' V ~ L '■ 25000?!.' 

in which p t = loss of pressure in ounces per square inch, v = velocity of air 
in feet per second, and L — length of pipe in feet. If p is taken in pounds 
per square inch, these formulae reduce to 

._ nnno .Lv2 .00158 Vdp , .00000251^2 

p = .0000020—— ; v = =-^ — — ; d = ■ . 

^ d L p 

trom tne common rormuia (weisoacns;, p — f 

which /= .0001608. 

The following table is condensed from one given in the catalogue of B. F. 
Sturtevant Company. 

Loss of pressure in pipes 100 feet long, in ounces per square inch. For 
any other length, the loss is proportional to the length, 



488 



AIR. 



'-5.9 

>>53 

'5 -O 


Diameter of Pipe in Inches. 


1 


2 


3 


4 


5 


6 


7 


8 9 


10 


11 


12 


13 v 


Loss of Pressure in Ounces. 


600 


.400 


.200 


.133 


.100 


.080 


.067 


.057 


.050 


.044 


040 


036 


033 


1200 


1.600 


.800 


.533 


.400 


.320 


.267 


.229 


.200 


.178 


.160 


.145 


133 


1800 


3.60C 


1.800 


1.200 


.900 


.720 


.600 


.514 


.450 


400 


360 


327 


.300 


2400 


6.400 


3.200 


2.133 


1.600 


1.280 


1.067 


.914 


.800 


.711 


.640 


582 


.533 


3000 


10. 


5. 


3.333 


2.5 


2. 


1.667 


1 421) 


1.250 


1 111 


1 000 


.909 


833 


3600 


14.4 


7.2 


4.8 


3.6 


2.88 


2.4 


2.057 


1.8 


1 6 


1.44 


1.309 


1 200 


4200 




9.8 


6.553 


4.9 


3.92 


3.267 


2.8 


2.45 


2 178 


1.96 


1 782 


1 633 


4800 




12.8 


8.533 


6.4 


5.12 


4.267 


3.657 


3.2 


2.844 


2.56 


2 327 


2 133 


6000 




20. 


13.333 


10.0 


8.0 


6.667 


5.714 


5.0 


4.444 


4.0 


3.636 






Diameter of Pipe in Inches. 






14 


16 


18 


20 


22 


24 


28 


32 


36 


40 


44 


48 




Loss of Pressure in Ounces. 


600 


.029 


.026 


.022 


.020 


.018 


.017 


.014 


.012 


.011 


.010 


.009 


.008 


1200 


.114 


.100 


.089 


.080 


.073 


.067 


.057 


.050 


.044 


.040 


.036 


.033 


1800 


.257 


.225 


.200 


.180 


.164 


.156 


.129 


.112 


.100 


.090 


082 


.075 


2400 


.457 


.400 


.356 


.320 


.291 


.267 


.239 


.200 


.178 


.160 


145 


.133 


3600 


1.029 


.900 


.800 


.720 


.655 


.600 


.514 


.450 


.400 


.360 


.327 


.300 


4200 


1.400 


1.225 


1.089 


.980 


.891 


.817 


.700 


.612 


.544 


.490 


.445 


408 


4800 


1.829 


1.600 


1.422 


1.280 


1.164 


1.067 


.914 


.800 


.711 


640 


582 


.533 


6000 


2.857 


2.500 


2.222 


2.000 


1.818 


1.667 


1.429 


1.250 


1.111 1.000 


.909 


.833 



Effect of Bends in Pipes. (Norwalk Iron Works Co.) 
Radius of elbow, in diameter of pipe =53 2 1J^ 1J4 1 % M 
Equivalent lgths. of straight pipe, diams 7.85 8.24 9.03 10.36 12.7217.51 35.09 121.2 

Compressed-air Transmission. (Frank Richards, Am. Mach., 
March 8, 1894 )— The volume of free air transmitted may be assumed to be 
directly as the number of atmospheres to which the air is compressed. 
Thus, if the air transmitted be at 75 pounds gauge-pressure, or six atmos- 
pheres, the volunle of free air will be six times the amount given in the 
table (page 486). It is generally considered that for economical transmission 
the velocity in main pipes should not exceed 20 feet per second. In the 
smaller distributing pipes the velocity should be decidedly less than this. 

The loss of power in the transmission of compressed air in general is not 
a serious one, or at all to be compared with the losses of power in the opera- 
tion of compression and in the re-expansion or final application of the air. 

The formulas for loss by friction are all unsatisfactory. The statements 
of observed facts in this line are in a more or less chaotic state, and self- 
evidently unreliable. 

A statement of the friction of air flowing through a pipe involves at least 
all the following factors: Unit of time, volume of air, pressure of air, diam- 
eter of pipe, length of pipe, and the difference of pressure at the ends of 
the pipe or the head required to maintain the flow. Neither of these factors 
can be allowed its independent and absolute value, but is subject to modifi- 
cations in deference to its associates. The flow of air being assumed to be 
uniform at the entrance to the pipe, the volume and flow are not uniform 
after that. The air is constantly losing some of its pressure and its volume 
is constantly increasing. The velocity of flow is therefore also somewhat 
accelerated continually. This also modifies the use of the length of the 
pipe as a constant factor. 

Then, besides the fluctuating values of these factors, there is the condition 
of the pipe itself. The actual diameter of the pipe, especially in the 
smaller sizes, is different from the nominal diameter. The pipe may be 
straight, or it may be crooked and have numerous elbows. Mr. Richards 
considers one elbow as equivalent to a length of pipe, 



FLOW OF COMPRESSED AIR IN" PIPES. 



489 



Head or Additional Pressure in pounds per sq. in. 
required to deliver Air at ¥5 Pounds Gau$;e-pressure 
through Pipes of Various Sizes and Lengths. (Frank 

Richards.) 



Length in feet. 



;- = 



50 
100 
150 
200 



100 
150 
200 
250 
300 



100 
200 



400 
500 



100 

200 



400 
500 



250 
500 



50 100 300 500 1,000 



Loss of pre 

.245 .49 

.981! 1.962 

3.925 7.85 

8.829 17.66 



lbs. p. 



1*4" Pipe. 



1.000 
1,250 
1,500 



Length in feet. 



.64 
1. 
1.44 



of pre 
24 

54 



1.5 

2.16 



400 


1,000 


ssure, 


lbs. p. 


.4 


.8 


.9 


1.8 


1.6 


3.2 


2.5 


5. 


3.6 


7.2 



2,000 

sq. in. 
1.6 
3.6 
6.4 
10. 
14.4 



25 


.056 


.112 


50 


.224 


.449 


100 


.897 


1.79 


150 


2.02 


3.94 


200 


3.59 


7.18 



.336 
1.35 



.017 


.034 


.103 


.171 


.068 


.137 


.411 


.685 


.274 


.548 


1.64 


2.74 


.616 


1.23 


3.69 


6.16 


1.09 


2.19 


6.57 


10.96 



.34 
1.37 
5. 





500 


1,000 


2,000 


4,000 


500 


.11 


.22 


.44 


.88 


1,000 


.44 


.881 


1.76 


3.52 


1,500 


.99 


1.98 


3.96 


7.92 


2,000 


1.76 


3.52 


7.04 


14.08 


2,500 


2.75 


5.5 


11. 





5,000 

1.1 
4.4 
9.9 



.019 


.038 


.114 


.19 


.076 


.152 


.457 


.761 


.171 


.343 


1.03 


1.71 


.304 


.609 


1.83 


3 04 


.476 


.952 


2.86 


4.76 


.685 


1.37 


4.11 


6.85 



6.09 
9.53 
13, 



1,000 
1,500 
2,000 
2,500 
3,000 



1,000 


2,000 


4,000 


5,000 
1.77 


.354 


.708 


1.42 


.799 


1.599 


3.2 


3.99 


1.417 


2.83 


5.67 


7.09 


2.22 


4.44 


8.89 


11.1 


3.18 


6.37 


12.7 


15.9 



10,000 

3.54 
7.99 
14.17 



200 


300 


500 


1,000 


2,000 


.087 


.13 


.217 


.434 


.87 


.347 


.521 


.868 


1.74 


3.47 


.781 


1.17 


1.95 


3.91 


7.81 


1.39 


2.08 


3.47 


6.94 


13.89 


2.17 


3.25 


5.42 


10.85 


21.7 



2,000 
2,500 
3,000 
4.000 
5,000 



2,000 


4,000 


8,000 


10,000 


.598 

.935 

1.25 

2.39 

3.74 


1.19 
1.87 
2.49 
4.79 

7.48 


2.39 
3.74 
4.99 

9.58 
14.97 


2.99 
4.68 
6.24 
11.97 
18.71 



4.48 
7.02 



.0333 


.05 


.0833 


.166 


.133 


.2 


.333 


.666 


.3 


.45 


.75 


1.5 


.533 


.8 


1.33 


2.66 


.833 


1.25 


2.08 


4.16 



.33 
1.33 
3 
5.33 



2,500 


.286 


5,000 


1.14 


7,500 


2.57 


10,000 


4.57 



.57 
2.29 
5.15 
9.14 



1.14 
4.57 
10.29 



1.43 
5.71 

12.86 



2.15 
8.56 



.0832 


.125 


.208 


.416 


.332 


.499 


.832 


1.66 


.748 


1.12 


1.87 


3.75 


1 328 


1.99 


3.33 


6.66 


2.08 


3.12 


5.2 


10.4 



3.32 
7.49 



2,500 
5,000 
7,500 
10,000 



2,000 


4,000 

.22 

.88 
1.98 
3.52 


8,000 


10,000 

.55 

4*95 
8.81 


.11 
.44 
.99 
1.76 


.44 
1.76 
3.96 

7.05 



20,000 

1.101 

4.4 

9.91 

17 6 



Although Mr. Richards does not give any formula with this table, an 
nspection of it shows that for any given diameter the loss of head is 



490 



AIK. 



taken to vary directly as the length and as the square of the quantity 
delivered, but for a given quantity and length the loss of head appears to 
vary inversely as some higher power of the diameter than the fifth, ap- 
proximately the 5.5 power; or, in other words, that the coefficient of fric- 

/pd* 



tion is variable. If we take the formula of the form Q' - 



-V 1 



■ ~~,V~J~v an d solve for c'-- 



■ y d 5 p 



- cubic feet of free air 



per minute, we find values of the coefficient as follows: 



For diameter, inches 
Value of c' — 



4 
552 



10 

664 



12 

676 



The following table is condensed from one given by F. A. Halsey in the 
catalogue of the Rand Drill Co.: 



-2 0) 


Cubic feet of free air compressed to a gauge-pressure of 60 lbs. 




and passing through the pipe each minute. 




50 


100 


200 


400 


600 


800 


1000 


1500 


2000 


3000 


4000 


-)000 


s'g 


Loss of pressure in lbs. per square inch for each 1000 ft. 


^ 


of straight pipe. 


1 


10.40 
























1M 


2.63 
























1W 


1.22 


4.89 






















2 


.35 


1.41 


5.64 




















2U 


.14 


.57 


2.30 


9.20 


















3 




.20 


.78 


3.14 


7.05 
















4 






.20 


.80 
.26 


1.81 
.59 
.23 


3.22 
1.04 
.41 


5.02 
1.63 
.64 


3.66 
1.46 


6.50 

2.59 


5.81 


10.30 




5 








6 










8 












.10 


.16 


.37 


.65 


1.47 


2.61 


4.08 


10 
















.12 


.21 


.47 
.19 


.84 
.34 
.16 


1 30 


19 
















^3 


14 




















.24 

























This table appears to follow more closely than does Richards' table the 

law of the formula p = vf /2 , 5 , but the coefficients differ considerably from 

those of Richards. Solving for C", we obtain— 

For diameter, inches . . 2 4 5 6 8 10 12 14 

Value of C 471 442 443 448 437 436 435 431 

Comparing some of the losses of pressure in the two tables, we find- 
Length, feet 1000 1000 1000 5000 5000 5000 

Quantity, cu. ft 1000 1000 1000 4000 4000 4000 

Diameter, inches 4 5 6 8 10 12 

Loss, Richards 3.2 .881 .354 7.48 2.29 .88 

" Halsey 5.02 1.63 _ .64 13.05 4.20 170 

The two tables are not calculated for the same amount of compression, 
but the difference is not sufficient to account for the difference in the coeffi- 
cients. If we multiply the coefficients derived from Halsey's table by 5/4, 
the ratio of the pressures 75 and 60 lbs., they become for a 2-inch pipe 589, 
and for a 12-inch pipe 531, against Richards's figures of 453 and 676 for the 
same pipes. To compare Richards's figures for loss of pressure with Hal- 
sey's, the former should be multiplied by 25/16. In the absence of experi- 
mental data no opinion can be formed as to which table is the more accurate, 
but either one is probably of sufficient accuracy for practical purposes. 






MEASUREMENT OF VELOCITY OF AIR. 



491 



Mr. Richards, in Am. Mack., Dec. 27, 1894, publishes a new formula, viz.: 



' 10,000d 5 a 



; "=</- 



OOOdWfp. 



10,000d 5 ap 



10,000p ' 



in which V = actual volume of compressed air delivered, in cubic feet per 
minute (not the volume of free air, as in the other formulae), L — length of 
pipe in feet, d = internal diameter of pipe in inches, p = head or additional 
pressure in pounds per square inch required to maintain the flow, and a is 
a coefficient varying with the diameter of the pipe. Its value for different 
nominal diameters of wrought-iron pipe is given by Mr. Richards as follows: 



Diam. in 


Val. of a. 


Diam. in. Val. of a. 


Diam. in 


Val. of a. 


Diam. 


in. Val. of a 


1 


.35 


2% .65 


5 


.93 


12 


1.26 


m 


.5 


3 .73 


6 


1. 


16 


1.34 


IX 


.66 


SH -79 


8 


1.125 


20 


1.4 


2 


.56 


4 .84 


10 


1.2 


24 


1.45 



The values of a for the 1 and 1*4 inch pipes appear inconsistent with the 
values for the other sizes, because the nominal diameters of these two sizes 
are relatively much less than tiieir actual diameters, 1.38 aud 1.61 inches, re- 
spectively. 

The following values of the fifth power of d and of d 5 a are given by Mr. 
Richards to facilitate calculations: 





Fifth Powers of d. 






Value 


of d 5 a. 




1" . 


... 1 


5" 


3,125 


1" ... 


35 


5".... 


.. 2,918.75 


1M".. 


.... 3.05 


6" 


7,776 


134".. 


.... 1.525 


6".... 


7,776 


m».-. 


.. . 7.59 


8" 


32,768 


1li"... 


... 5.03 


8".... 


36,864 




.... 32 


10" 


100.000 


2" ... 


.... 18.08 


10".... 


. . 120,000 


m"~ 


.... 97.65 


12" .... 


248,832 


2%"... 


.. 63.47 


12".... 


.. 313,528 


3" .. 


.... 243 


16" 


1.048,576 


3" . . . 


....177.4 


16"... 


.. 1,405,091 


»**".. 


....525 


20" 


3,200.000 


«&"... 


....413.2 


20"... 


.. 4,480.000 


4" .. 


....1024 


24" 


7.962,624 


4" 


....860.2 


24".... 


..11,545,805 



In order to compare Mr. Richards' new formula for volume of compressed 

/ r)d b 
air transmitted with the formula Q' = c' ju ~-, in which Q is the volume 

of free air, = 5F"if the air is compressed to 5 atmospheres, we have 



Q> z=5V=500Va X 



\/ 1 



and from the values of a given by Mr. Richards we find values of c' as 
follows: . 



For diameter, nominal, inches = 
Value of c' 



2 4 6 8 10 12 
! 374 458 500 530 548 561 



Measurement of the Velocity of Air in Pipes by an Ane- 
mometer.— Tests were made by B. Donkin, Jr. (Inst. Civil Engrs. 1892), 
to compare the velocity of air in pipes from 8 in. to 24 in. diam., as shown by 
an anemometer 2% in. diam. with the true velocity as measured by the time 
of descent of a gas-holder holding 1622 cubic feet. A table of the results 
with discussion is given in Eng'g News, Dec. 22, 1892. In pipes from 8 in. to 20 
in. diam. with air velocities of from 140 to 690 feet per minute the anemome- 
ter showed errors varying from 14.5$ fast to 10$ slow. With a 24-inch pipe 
and a velocity of 73 ft. per minute, the anemometer gave from 44 to 63 feet, 
or from 13.6 to 39.6$ slow. The practical conclusion drawn from these ex- 
periments is that anemometers for the measurement of velocities of air in 
pipes of these diameters should be used with great caution. The percentage 
of error is not constant, and varies considerably with the diameter of the 
pipes and the speeds of air. The use of a baffle, consisting of a perforated 
plate, which tended to equalize the velocity in the centre and at the sides in 
some cases diminished the error. 



492 



The impossibility of measuring the true quantity of air by an anemometer 
held stationary in one position is shown by the following figures, given by 
Wm. Daniel (Proc. Inst. M. E., 1875), of the velocities of air found at different 
points in the cross-sections of two different airways in a mine. 

Differences of Anemometer Readings in Airways. 
8 ft. square. 5 X 8 f t. 



1712 


1795 


1859 


1329 


1622 


1685 


1782 


1091 


1477 
1262 


1344 


1524 


1049 


1356 


1293 


1333 



1170 

948 


1209 


1288 


1104 


1177 


1134 


1049 


1106 



Average 1469. 



Average 1132. 



Equation of Pipes.— It is frequently desired to know what number 
of pipes of a given size are equal in carrying capacity to one pipe of a larger 
size. At the same velocity of flow the volume delivered by ttvo pipes of 
different sizes is proportional to the squares of their diameters; thus, one 
4-inch pipe will deliver the same volume as four 2-inch pipes. With the same 
head, however, the velocity is less in the smaller pipe, and the volume de- 
livered varies about as the square root of the fifth power (i.e., as the 2.5 
power). The following table has been calculated on this basis. The figures 
opposite the intersection of any two sizes is the number of the smaller-sized 
pipes required to equal one of the larger. Thus, one 4-inch pipe is equal to 
5.7 2-inch pipes. 



5 S3 

5'" 


1 


2 


3 


4 


5 


6 


7 


8 


9 


10 


12 


14 


16 


18 


20 


24 


2 
3 


5.7 
15.fi 


1 
2.8 


1 




























4 


32 


5.7 


2.1 


1 


























5 


55,1) 


9.9 


3.6 


1.7 


1 
























6 


SS.2 


15.6 


5 7 


2.8 


1.6 


1 






















7 


130 


22.9 


8 3 


4.1 


2.3 


1.5 


1 




















8 


181 


32 


11. ^ 


5.7 


3.2 


2.1 


1.4 


1 


















9 


243 


43. 


15.6 


7.6 


4.3 


2.8 


1.9 


1.3 


1 
















10 


ma 


55 9 


20.3 


9 9 


5,7 


3.6 


2.4 


1.7 


1 3 


1 














11 


401 


70 9 


25 . 7 


12.5 


7 2 


4.6 


3.1 


2,2 


1.7 


1.3 














12 


m 


KS 2 


32 


ir - 


si 


5.7 


3,8 


2.8 


2 1 


1.6 


1 












13 


609 


108 


39.1 


19 


10.9 


7.1 


4.7 


3.4 


2.5 


1.9 


1 2 












14 


733 


130 


47 


22.'.) 


18.1 


h.;j 


5.7 


4.1 


3.0 


2.3 


1.5 


1 










15 


787 


154 


55. 9 


27.2 


15.6 


9.9 


6.7 


4.8 


3.6 


2 8 


1.7 


1.2 










16 




181 


65.7 


32 


18.: J 


11.7 


7.9 


5.7 


4.2 


3.2 


2.1 


1.4 


1 








17 




•211 


76 4 


37 2 


21 3 


13.5 


9.2 


6 , 6 


4,9 


3,8 


2,4 


1.6 


1 2 








18 




-'43 


88.2 


43 




15.6 


10 6 


7.6 


5.7 


4.3 


2.8 


1.9 


1.3 


1 






19 




27S 


101 


49 1 


28 . 1 


17.8 


12.1 


8.7 


6.5 


5 


3.2 


2.1 


1.5 


1.1 






20 




316 


115 


55 9 


32 


20,3 


13 8 


9.9 


7.4 


5.7 


3.6 


2.4 


1.7 


1.3 


1 




22 




101 


146 


70.!) 


40 . 6 


25.7 


17.5 


12 5 


9.3 


7' 2 


4.6 


3 1 


2.2 


1.7 


1.3 




24 




m 


181 


8S.2 


50.5 


32 


21.8 


15.6 


11.6 


8.9 


5.7 


3 8 


2.8 


2.1 


1.6 


1 


26 




.ion 


221 


108 


61.7 


39.1 


26.6 


19. 


14.2 


10.9 


7.1 


4.7 


3.4 


2.5 


1.9 


1.2 


28 




T33 


266 


130 


7-4.2 


47 


32 


22.9 


17 1 


13.1 


8.3 


5.7 


4,1 


3 


2.3 


1.5 


30 




T87 


316 


154 


88. 2 


55 9 


38 


27.2 


20.3 


15.6 


9 9 


6.7 


4.8 


3.6 


2.8 


1.7 


36 






(9!) 


243 


130 


88 2 


60 


43 


32 


24.6 


15.6 


10. (i 


7.6 


5.7 


4.3 


2.8 


42 






733 


357 


205 


130 


88. -J 


63.2 


47 


36.2 


19 


15,6 


11.2 


8.3 


6.4 


4.1 


48 








499 


286 


181 


123 


88.2 


62.7 


50.5 


32 


21.8 


15.6 


11.6 


8.9 


5.7 


54 








670 


383, 


243 


165 


118 


88.2 


67.8 


43 


29 2 


20 9 


15.6 


12 


7.6 


60 








787 


499 


316 


215 


154 


115 






38 


27.2 




15.6 


9.9 



493 



Loss of Pressure in Compressed Air Pipe-main, at 
St. Gothard Tunnel. 











(E. Stockalper.) 












s 

i 

s 


per second 
air.orequi- 
volume at 

pheric pres- 

ud 32° F. 




o'cH 
'3 3 II 




«"3 

.S c 

O u 


Observed Pressures. 




1 


o 
.S 


6 

ts-S 1 


Loss of 


"Value 
of c' 

in for- 
mula 


•r 


OS 


® S a o =* 


5^5 


-oaS 


« Ch 


£& 




"fl 


Pressure. 


P = 


V 


£ 


~£iiS£ 


e3 Oi> 






5 &CQ, 


0^3 




Q'L 


.i: 


o o >tf w 


"So el 


««H 




£ S 




c' 2 d 6 




> 


> 


§ 


is: 


g 


Oh 


h 
























lbs. 
per 






No 


in. 


cu.ft. 


cu.ft. 


den. 


lbs. 


feet. 


at. 


at. 


sq.in. 


% 




. ( 


r sr 


J- 33.056] 


6.534 


.00650 


2.609 


19.32 


5.60 


5.24 


5.292 


0.4 


610 


l \ 


5,91 


7.063 


.00603 


2 669 


37.14 


5.24 


5.00 


3.528 


4.6 


515 


H 


;,s? 


[■ 28.002 -1 


5.509 


.00514 


1.776 


16.30 


4.35 


4.13 


3.234 


5.1 


519 


5 91 


5.863 


.00482 


1.776 




4.13 










H 


7 ST 


j- 18.364] 


5.262 


.00449 


1.483 


15.58 


3.84 


3.65 


2.793 


5.0 


466 


5.91 


5.580 


.00423 


1.483 


29.34 


3.65 


3.54 


1.617 


3.0 


422 



The length of the pipe 7.87 in diameter was 15,092 ft., and of the smaller 
pipe 1712.6 ft. The mean temperature of the air in the large pipe was 70° F. 
and in the small pipe 80° F. 

WIND. 
Force of tlie Wind.— Smeaton in 1759 published a table of the 
velocity and pressure of wind, as follows: 

Velocity and Force of Wind, in Pounds per Square Inch. 



16 



u . 


55 v. 


33 "2 


2*~ 




p x p, 


fa 




1.47 


0.005 


2.93 


0.020 


4.4 


0.044 


5.87 


0.079 


7.33 


0.123 


8.8 


0.177 


10.25 


0.241 


11.75 


0.315 


13.2 


0.400 


14.67 


0.49-2 


17.6 


0.708 


20.5 


0.964 


22.00 


1.107 


23.45 


1.25 



Common Appelte 

tion of the 

Force of Wind. 



Hardly percepti- 
ble. 
- Just perceptible. 



Gentle pleasant 
wind. 



Pleasant brisk 



i,. 


53-d 


a« -r 3 




Png 






fa 


2 ero 
owp. 
fa 


18 


26.4 


1.55 


20 


29.34 


1.968 


25 


36.67 


3.075 


30 


44.01 


4.429 


35 


51.34 


6.027 


40 


58.68 


7.873 


45 


66.01 


9.963 


50 


73.35 


12.30 


55 


80.7 


14.9 


60 


88.02 


17.71 


66 


95.4 


20.85 


70 


102.5 


24.1 


75 


110. 


27.7 


80 


117.36 


31.49 


100 


146.67 


49.2 



Common Appella- 
tion of the 
Force of Wind. 



>Very brisk. 

y High wind. 

1 

j- Very high storm. 



Hurricane. 
Immense hurri- 



The pressures pnr square foot in the above table correspond to the 
formula P = 0.005F ra , in which V is the velocity in miles per hour. Eng^g 
News. Feb. 9, 1893, says that the formula was never well established, and 
has floated chiefly on Smeaton's name and for lack of a better. It was put 
forward only for surfaces for use in windmill practice. The trend of 
modern evidence is that it is approximately correct only for such surfaces, 
and that for large solid bodies it often gives greatly too large results. 
Observations by others are thus compared with Smeaton's formula: 

Old Smeaton formula P = .005FS 

As determined by Prof. Martin P = .004 V* 

" Whipple and Dines P=.0029F 2 



494 air. 

At 60 miles per hour these formulas give for the pressure per square foot, 
18, 14.4 and 10.44 lbs., respectively, the pressure varying by all of them as 
the square of the velocity. Lieut. Crosby's experiments (Eng'g, June 13, 
1890), claiming to prove that P = fV instead of P = fV 2 , are discredited. 

A. R. Wolff (The Windmill as a Prime Mover, p. 9) gives as the theoretical 

pressure per sq. ft. of surface, P = —£-, in which d = density of air in pounds 
per cu. ft. = ' - — ~^ — - ; p being the barometric pressure per square 

foot at any level, and temperature of 32° F., t any absolute temperature, 
Q = volume of air carried along per square foot in one second, v — velocity 

of the wind in feet per sec, g = 32.16. Since Q = v cu. ft. per sec, P= — . 

Multiplying this by a coefficient 0.93 found by experiment, and substituting 

^ - ii ' ' i e ^ t, 1,4 • -d 0.017431 Xp . 

the above value of d, he obtains P = — - ■, and when p 



- 2116.5 lbs. per sq ft. or average atmospheric pressure at the sea-level, 

36 8929 
> = - — — -^ — , an expression in which the pressure is shown to vary 



v 2 
with the temperature; and he gives a table showing the relation between 
velocity and pressure for temperatures from 0° to 100° F., and velocities 
from 1 to 80 miles per hour. For a temperature of 45° F. the pressures agree 
with those in Smeaton's table, for 0° F. they are about 10 per cent greater, 
and for 100° 10 per cent less. Prof. H. Allen Hazen, Eng^g News, July 5, 
1890, says that experiments with whirling arms, by exposing plates to direct 
wind, and on locomotives with velocities running up to 40 miles per hour, 
have invariably shown the resistance to vary with V 2 . In the formula 
P = .005SF 2 , in which P — pressure in pounds, S — surface in square feet, 
V= velocity in miles per hour, the doubtful question is that regarding 
the accuracy of the first two factors in the second member of this equation. 
The first factor has been variously determined from .003 to .005 [it has been 
determined as low as .0014.— Ed. Eng'g News]. 

The second factor has been found in some experiments with very short 
whirling arms and low velocities to vary with the perimeter of the plate, 
but this entirely disappears with longer arms or straight line motion, and 
the only question now to be determined is the value of the coefficient. Per- 
haps some of the best experiments for determining this value were tried in 
France in 1886 by carrying flat boards on trains. The resulting formula in 
this case was, for 44.5 miles per hour, p = .00535SF 2 . 

Mr. Crosby's whirling experiments were made with an arm 5.5 ft. long. 
It is certain that most serious effects from centrifugal action would be set 
up by using such a short arm, and nothing satisfactory can be learned with 
arms less than 20 or 30 ft. long at velocities above 5 miles per hour. 

Prof. Kernot, of Melbourne {Engineering Record, Feb. 20, 1894), states that 
experiments at the Forth Bridge showed that the average pressure on sur- 
faces as large as railway carriages, houses, or bridges never exceeded two 
thirds of that upon small surfaces of one or two square feet, such as have 
been used at observatories, and also that an inertia effect, which is frequently 
overlooked, may cause some forms of anemometer to give false results 
enormously exceeding the correct indication. Experiments of Mr. O. T. 
Crosby showed that the pressure varied directly as the velocity, whereas all 
the early investigators, from the time of Smeaton onwards, made it vary as 
the square of the veloi ity. Experiments made by Prof . Kernot at speeds 
varying from 2 to 15 miles per hour agreed with the earlier authorities, and 
tended to negative Crosby's results. The pressure upon one. side of a cube, 
or of a block proportioned like an ordinary carriage, was found to be .9 of 
that upon a thin plate of the same area. The same result was obtained for 
a square tower. A square pyramid, whose height was three times its base, 
experienced .8 of the pressure upon a thin plate equal to one of its sides, but 
if an angle was turned to the wind the pressure was iucreased by fully 20%. 
A bridge consisting of two plate-girders connected by a deck at the top was 
found to experience .9 of the pressure on a thin plate equal in size to one 
girder, when the distance between the girders was equal to their depth, and 
this was increased by one fifth when the distance between the girders was 



WINDMILLS. 495 

double the depth. A lattice-work in which the area of the openings was 55% 
of the whole area experienced a pressure of 80% of that upon a plate of the 
same area. The pressure upon cylinders and cones was proved to be equal 
to half that upou the diametral planes, and that upon an octagonal prism to 
be 20^ greater than upon the circumscribing cylinder. A sphere was sub- 
ject to a pressure of .36 of that upon a thin circular plate of equal diameter. 
A hemispherical cup gave the same result as the sphere; when its convexity 
was turned to the wind the pressure was 1.15 of that on a flat plate of equal 
diameter. When a plane surface parallel to the direction of the wind was 
brought nearly into contact with a cylinder or sphere, the pressure on the 
latter bodies was augmented by about 20$, owing to the lateral escape of the 
air being checked. Thus it is possible for the security of a tower or chimney 
to be impaired by the erection of a building nearly 'touching it on one side. 

Pressures of Wind. Registered in Stornis.— Mr. Frizell has 
examined the published records of Greenwich Observatory from 1849 to 1869, 
and reports that the highest pressure of wind he finds recorded is 41 lbs. 
per sq. ft., and there are numerous instances in which it was between 30 and 
40 lbs. per sq. ft. Prof. Henry says that on Mount Washington, N. H., a ve- 
locity of 150 miles per hour has been observed, and at New York City 60 
miles an hour, and that the highest winds observed in 1870 were of 72 and 63 
miles per hour, respectively. 

Lieut. Dun woody, U. S. A., says, in substance, that the New England coast 
is exposed to storms which produce a pressure of 50 lbs. per sq. ft. Engi- 
neering Neivs, Aug. 20, 1880. 

WINDMILLS. 

Power and Efficiency of Windmills.— Rankine, S. E., p. 215. 
gives the following: Let Q = volume of air which acts on the sail, or part 
of a sail, in cubic feet per second, v = velocity of the wind in feet per 
second, s = sectional area of the cylinder, or annular cjdinder of wind, 
through which the sail, or part of the sail, sweeps in one revolution, c = a 
coefficient to be found by experience; then Q = cvs. Rankine, from experi- 
mental data given by Smeaton, and taking c to include an allowance for 
friction, gives for a wheel with four sails, proportioned in the best manner, 
c = 0.75. Let A = weather angle of the sail at any distance from the axis, 
i.e., the angle the portion of the sail considered makes with its plane of 
revolution. This angle gradually diminishes from the inner end of the sail 
to the tip; u = the velocity of the same portion of the sail, and E = the effi- 
ciency. The efficiency is the ratio of the useful work performed to whole 
energy of the stream of wind acting on the surface s of the wheel, which 

Dsv 3 
energy is -= — , D being the weight of a cubic foot of air. Rankine's formula 

for efficiency is 

E =^f = C \^ sia2A ~ g ct-^^+/)-/)h 

in which c = 0.75 and / is a coefficient of friction found from Smeaton's 
data = 0.016. Rankine gives the following from Smeaton's data: 

A = weather-angle =7° 13° 19° 

V -f- v = ratio of speed of greatest effi- 
ciency, for a given weather- 
angle, to that of the wind =2.63 1.86 1.41 

E = efficiency =0.24 0.29 0.31 

Rankine gives the following as the best values for the angle of weather at 
different distances from the axis: 

Distance in sixths of total radius. .. 1 2 3 4 5 6 
Weather angle ... 18° 19° 18° 16° 12^° 7° 

But Wolff (p. 125) shows that Smeaton did not term these the best angles, 
but simply says they " answer as well as any,' 1 '' possibly any that were in ex- 
istence in'his time. Wolff says that they "cannot in the nature of things 
be the most desirable angles." Mathematical considerations, he says, con- 
clusively show that the angle of impulse depends on the relative velocity of 
each point of the sail and the wind, the angle growing larger as the ratio be- 
comes greater. Smeaton's angles do not fulfil this condition. Wolff devel- 



496 



AIR. 



ops a theoretical formula for the best angle of weather, and from it 
calculates a table for different relative velocities of the blades (at a distance 
of one seventh of the total length from the centre of the shaft) and the wind, 
from which the following is condensed: 



Ratio of the 

Speed of Blade 

at 1/7 of Radius 

to Velocity of 

Wind. 



Distance from the axis of the wheel in sevenths of radius. 



Best angles of weather. 



0.10 
0.15 
0.20 
0.25 
0.30 
0.35 
0.40 
0.45 
0.50 



42° 9' 


39° 21' 


36° 39' 


34° 6' 


31° 43' 


29° 31' 


40 44 


36 39 


32 53 


29 31 


26 34 


24 


39 21 


34 6 


29 31 


25 40 


22 30 


19 54 


37 59 


36 43 


26 34 


22 30 


19 20 


16 51 


36 39 


29 31 


24 


19 54 


16 51 


14 32 


35 21 


27 30 


21 48 


17 46 


14 52 


12 44 


34 6 


25 40 


19 54 


16 


13 17 


11 19 


32 53 


24 


18 16 


14 32 


11 59 


10 10 


31 43 


22 30. 


16 51 


13 17 


10 54 


9 13 



27° 30' 
21 48 
17 46 
14 52 
12 44 
11 6 



58 



The effective power of a windmill, as Smeaton ascertained by experiment, 
varies as s, the sectional area of the acting stream of wind; that is, for simi- 
lar wheels, as the squares of the radii. 

The value 0.75, assigned to the multiplier c in the formula Q = cvs, is 
founded on the fact, ascertained by Smeaton, that the effective power of a 
windmill with sails of the best form, and about 15^ ft. radius, with a breeze 
of 13 ft. per second, is about 1 horse-power. In the computations founded 
on that fact, the mean angle of weather is made — 13°. The efficiency of 
this wheel, according to the formula and table given, is 0.29, at its best 
speed, when the tips of the sails move at a velocity of 2.6 times that of the 
wind. 

Merivale (Notes and Formulas for Mining Students), using Smeaton's co- 
efficient of efficiency, 0.29, gives the following: 
U = units of work in foot lbs. per sec. ; 

W = weight, in pounds, of the cylinder of wind passing the sails each 
second, the diameter of the cylinder being equal to the diameter 
of the sails ; 
V = velocity of wind in feet per second; 
H.P. = effective horse-power; 

TT-KE1. °- 29 wv * 

64 ' ~ 64 X 550' 

A. R. Wolff, in an article in the American Engineer, gives the following 
(see also his treatise on Windmills): 
Let c = velocity of wind in feet per second; 

n = number of revolutions of the windmill per minute; 

6<>i &n b 2 , b x be the breadth of the sail or blade at distances l , l x , 7 2 , 

Z 3 , and I, respectively, from the axis of the shaft; 
l — distance from axis of shaft to beginning of sail or blade proper; 
I — distance from axis of shaft to extremity of sail proper; 
v oi v ii v 2* v 3> v x — tne ve l°city of the sail in feet per second at dis- 
tances l . l l% l 2 , Z, respectively, from the axis of the shaft; 
a , «i, a 2 , a 3 , a x = the angles of impulse for maximum effect at dis- 
tances l , l t , l 2 . l s , I respectively from the axis of the shaft; 
a = the angle of impulse when the sails or blocks are plane surfaces, 

so that there is but one angle to be considered; 
N = number of sails or blades of windmill; 
K = .93. 

d — density of wind (weight of a cubic foot of air at average tempera- 
ture and barometric pressure where mill is erected); 
W '= weight of wind-wheel in pounds; 
/ = coefficient of friction of shaft and bearings; 
D = diameter of bearing of windmill in feet. 



WINDMILLS. 



4£>? 



The effective horse-power of a windmill with plane sails will equal 

(l-l )Kc*dN J . . v 

£— X mean of ^ (sin a cos a)b cos a 

v x .. \ f\V X .05236?il> 
v x (sin a cos a) b x cos a J— - r^c . 

The effective horse-power of a windmill of shape of sail for maximum 
effect equals 

N(l - l )Kdc 3 „ .12 sin 2 a -l. 2 sin* a x - 1 , 

n " X mean off . . ° 6 , r-^ b x . . . 

22Q0g \ sin 2 a °' sin 2 a x 1 



2 sin 2 a x - 1 

■ : b. 

sin 2 a x 



fW X .05236w£> 
550 



The mean value of quantities in brackets is to be found according to 
Simpson's rule. Dividing I into 7 parts, finding the angles and breadths 
corresponding to these divisions by substituting them in quantities within 
brackets will be found satisfactory. Comparison of these formulae with the 
only fairly reliable experiments in windmills (Coulomb's) showed a close 
agreement of results. 

Approximate formulae of simpler form for windmills of present construc- 
tion can be based upon the above, substituting actual average values for a, 
c, d, aud e. but since improvement in the present angles is possible, it is 
better to give the formulae in their general and accurate form. 

Wolff gives the following table based on the practice of an American 
manufacturer. Since its preparation, he says, over 1500 windmills have been 
sold oh its guaranty (1885), and in all cases the results obtained did not vary 
sufficiently from those presented to cause any complaint. The actual re- 
sults obtained are in close agreement with those obtained by theoretical 
analysis of the impulse of wind upon windmill blades. 

Capacity of the Windmill. 







a> 
















§ oi> 




a 






P^ 


§5° 


3 


ii 


£^ 


Gallons of Water raised per Minute to 


"3 <D 


ffi£5 


o 
s 
o 


I! 

+i CD 

£ ft 


an Elevation of— 


S * 

III 

> hM o 


O 3-u 
&^3 


I 

he 


25 


50 


75 


100 


150 


200 


|o|| 




> 


feet. 


feet. 


feet. 


feet. 


feet. 


feet. 


3 3 £ 


< 


wheel 






















8^ ft. 


16 


70 to 75 


6.162 


3.016 










0.04 


8 


10 " 


16 


60 to 65 


19.179 


9.563 


6.638 


4.750 






0.12 


8 


12 " 


16 


55 to 60 


33.941 


17.952 


11.851 


8.485 


5.680 




0.21 


8 


14 " 


16 


50 to 55 


45.139 


22.569 


15.304 


11.246 


7.807 


4 998 


0.28 


8 


16 " 


16 


45 to 50 


64 600 


31.654 


19.542 


16.150 


9.771 


8.075 


0.41 


8 


18 " 


16 


40 to 45 


97.682 


52.165 


32.513 


24.421 


17.485 


12.211 


0.61 


8 


20 " 


16 


35 to 40 


124.950 


63.750 


40.800 


31.248 


19.284 


15.938 


0.78 


8 


25 " 


16 


30 to 35 


212.381 


106.964 


71.604 


49.725 


37.349 


26.741 


1.34 


8 



These windmills are made in regular sizes, as high as sixty feet diameter of 
wheel; but the experience with the larger class of mills is too limited to 
enable the presentation of precise data as to their performance. 

If the wind can be relied upon in exceptional localities to average a higher 
velocity for eight hours a day than that stated in the above table, tbe per- 
formance or horse-power of the mill will be increased, and can be obtained 
by multiplying the figures in the table by the ratio of the cube of the higher 
average velocity of wind to the cube of the velocity above recorded. 

He also gives the following table showing the economy of the windmill. 
All the items of expense, including both interest and repairs, are reduced to 
the hour by dividing the costs per annum by 365 x 8 = 2920; the interest, 



493 



AIR. 



etc., for the twenty-four hours being charged to the eight hours of actual 
work. By multiplying the figures in the 5th column by 584, the first cost of 
the windmill, in dollars, is obtained. 







Economy of the Windmill. 












-3 




to 


Expense of Actual Useful Power 








I=>f 


o =•£ 


Developed, in cents, per hour. 


i a 












Designation 


D O 




umbei 
* Day 
s Qua 
ised. 


or Interest on 
First Cost (Firs 
Cost, including 
Cost of Wind- 
mill, Pump, an 
Tower. 5% per 
annum). 


T3 — & 

5 C+J 


5 






ffi cd 


of Mill. 


o ^ 

o w 
3 


|| 

_> CO 

'3 o 


verage N 
Hours pe 
which thi 
will be ra 


or Repair 
Deprecia 
of First C 
annum). 


a 
< 


o 


«i 


aO O 




O 


< 


fc 


fe 


Ik 


h 


H 


s 


8J^ ft. wheel 


370 


0.04 


8 


0.25 


0.25 


0.06 


0.04 


0.60 


15.0 


10 " " 


1151 


0.12 


8 


0.30 


0.30 


0.06 


0.04 


0.70 


5.8 


12 " 


2036 


0.21 


8 


0.36 


0.36 


06 


0.04 




3.9 


14 " 


2708 


0.28 


8 


0.75 


0.75 


0.06 


0.07 


1.63 


5.8 


16 " 


3876 


0.41 


8 


1.15 


1.15 


06 


0.07 


2.43 


5.9 


18 " 


5861 


061 


8 


1.35 


1.35 


(1 Of, 


l) 07 




4.6 


20 " 


7497 


0.79 


8 


1.70 


1.70 


0.06 






4.5 


25 " 


12743 


1.34 


8 


2.05 


2.05 




i Id 


4.26 


3.2 



Lieut. I. N. Lewis (Eng'g Mag., Dec. 1894) gives a table of results of ex- 
periments with wooden wheels, from which the following is taken : 



Diameter 
of wheel, 

Feet. 



Velocity of Wind, miles per hour. 



Actual Useful Horse-power developed. 



16 
20 

25 



V* 



m 
w 

3 



IV2 



4y* 



1 
214 



5^ 



The wheels were tested by driving a differentially wound dynamo. The 
" useful horse-power " was measured by a voltmeter and ammeter, allow- 
ing 500 watts per horse-power. .Details of the experiments, including the 
means used for obtaining the velocity of the wind, are not given. The re- 
sults are so far in excess of the capacity claimed by responsible manufactu- 
rers that they should not be given credence until established by further 
experiments. ' 

A recent article on windmills in the Iron Age contains the following: Ac- 
cording to observations of the United States Signal Service, the average 
velocity of the wind within the range of its record is 9 miles per hour for 
the year along the North Atlantic border and Northwestern States, 10 miles 
on the plains of the West, and 6 miles in the Gulf States. 

The horse-powers of windmills of the best construction are proportional 
to the squares of their diameters and inversely as their velocities; for ex- 
ample, a 10-ft. mill in a 16-mile breeze will develop 0.15 horse-power at 65 
revolutions per minute; and with the same breeze 

A 20-ft. mill, 40 revolutions, 1 horse-power. 

A 25-ft. mill, 35 revolutions, \% horse-power. 

A 30-ft. mill, 28 revolutions, 3^| horse-power. 

A 40-ft. mill, 22 revolutions, 7J4 horse-power. 

A 50-ft. mill, 18 revolutions, 12 horse-power. 

The increase in power from increase in velocity of the wind is equal to the 

square of its proportional velocity; as for example, the 25-ft. mill rated 



COMPRESSED AIR. 499 

above for a 16-niile wind will, with a 32-mile wind, have its horse-power in- 
creased to 4 X 1% = 7 horse-power, a 40-ft. mill in a 32-mile wind will run 
up to 30 horse-power, and a 50-ft. mill to 48 horse-power, with a small de 
d notion for increased friction of air on the wheel and the machinery. 

The modern mill of medium and large size will run and produce work in a 
4-mile breeze, becoming very efficient in an 8 to 16-mile breeze, and increase 
its power with safety to the running-gear up to a gale of 45 miles per hour. 

Prof. Thurston, in an article on modern uses of the windmill, Engineer- 
hut Magazine, Feb. 1893, says : The best mills cost from about $600 for the 
10-ft. wheel of % horse-power to $1200 for the 25-ft. wheel of l^j horse-power 
or less. In the estimates a working-day of 8 hours is assumed ; but the ma- 
chine, when used for pumping, its most common application, may actually 
do its work 24 hours a day for clays, weeks, and even months together, 
whenever the wind is "stiff" enough to turn it. It costs, for work done in 
situations in which its irregularity of action is no objection, only one half or 
one third as much as steam, hot-air, and gas engines of similar power. At 
Faversbam, it is said, a 15-horse-power mill raises 2,000,000 gallons a month 
from a depth of 100 ft., saving 10 tons of coal a month, which would other- 
wise be expended in doing the work by steam. 

Electric storage and lighting from the power of a windmill has been tested 
on a large scale for several years bj r Charles F. Brush, at Cleveland, Ohio. 
In 1887 he erected on the grounds of his-dwelling a windmill 56 ft. in diam- 
eter, that operates with ordinary wind a dynamo at 500 revolutions per 
minute, with an output of 12,000 amperes— 16 electric horse-power— charging 
a storage system that gives a constant lighting capacity of 100 16 to 20 
candle-power lamps. The current from the dynamo is automatically regu- 
lated to commence charging at 330 revolutions and 70 volts, and cutting the 
circuit at 75 volts. Thus, by its 21 hours' work, the storage system of 408 
cells in 12 parallel series, each cell having a capacity of 100 ampere hours, is 
kept in constaut readiness for all the requirements of the establishment, it 
being fitted up with 350 incandescent lamps, about 100 being in use each 
evening. The plant runs at a mere nominal expense for oil. repairs, and at- 
tention. (For a fuller description of this plant, and of a more recent one at 
Marblehead Neck, Mass., see Lieut. Lewis's paper in Engineering Magazine, 
Dec. 1894, p. 475.) 

COMPRESSED AIR. 

Heating of Air by Compression.— Kimball, in his treatise on Physi- 
cal Properties of Gases, says: When air is compressed, all the work which is 
done in the compression Is converted into heat, and shows itself in the rise in 
temperature of the compressed gas. As the gas becomes hotter it is com- 
pressed with more difficulty; so in practice many devices are employed to 
carry off the heat as fast as it is developed, and keep the temperature down. 
But it is not possible in any way to totally remove this difficulty. But, it may 
be objected, if all the work done in compression is converted into heat, and 
if this heat is got rid of as soon as possible, then the work may be virtually 
thrown away, and the compressed air can have no more energy than it had 
before compression. It s true that the compressed gas has no more energy 
than the gas had before compression, if its temperature is no higher, but 
the advantage of the compression lies in bringing its energy into more avail- 
able form. 

The total energy of the compressed and uncompressed gas is the same at 
the same temperature, but the available energy is much greater in the former. 

The rise in temperature due to compression is so great that if a mass of 
air at 32° F. is compressed to one fourth its original volume, its temperature 
will be raised 376° F., if no heat is allowed to escape. 

When the compressed air is used in driving a rock-drill, or any other piece 
of machinery, it gives tip energy equal in amount to the work it does, and 
its temperature is accordingly greatly reduced. 

Causes of IiOss of Energy in Use of Compressed Air. 
(Zahner, on Transmission of Power by Compressed Air.)— 1. The compression 
of air always develops heat, and as the compressed air always cools down to 
the temperature of the surrounding atmosphere before it is used, the me- 
chanical equivalent of this dissipated heat is work lost. 

2. The heat of compression increases the volume of the air, and hence it 
is necessary to carry the air to a higher pressure in the compressor in order 
that we may finally have a given volume of air at a given pressure, and at 
the temperature of the surrounding atmosphere. The work spent in effect- 
ing this excess of pressure is work lost. 



500 



3. The great cold which results when air expands against a resistance 
forbids expansive working, which is equivalent to saying, forbids the reali- 
zation of a high degree of efficiency in the use of compressed air. 

4. Friction of the air in the pipes', leakage, dead spaces, the resistance of- 
fered by the valves, insufficiency of valve-area, inferior workmanship, and 
slovenly attendance, are all more or less serious causes of loss of power. 

The first cause of loss of work, namely, the heat developed by compres- 
sion, is entirely unavoidable. The whole of the mechanical energy which 
the compressor-piston spends upon the air is converted into heat. This heat 
is dissipated by conduction and radiation, and its mechanical equivalent is 
work lost. The compressed air, having again reached thermal equilibrium 
with the surrounding atmosphere, expands and does work in virtue of its 
intrinsic energy. 

The intrinsic energy of a fluid is the energy which it is capable of exert- 
ing against a piston in changing from a given state as to temperature and 
volume, to a total privation of heat and indefinite expansion. 

Volumes, Mean Pressures per Stroke, Temperatures, etc., 
in the Operation of Air-compression from 1 Atmosphere 
and 60° Fahr. (F. Richards, Am. Alack., March 30, 1893.) 



5 
■r. 

© 


1 
o 

a 




< 
£3 © 

> o 

o 

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p. 




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S ° 
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< 

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^ O 

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ftfl 

~ ■- - 

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Pi 

© .. O 
■- © O 
&.* O 

B O 




a 

«H O 
O O 

P. 


5 




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goo 


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K 




04J 


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O 


< 

2 


>* 


> 


a 


H 


1 


< 


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3 


H 


1 


3 


4 


5 


6 


2 


3 


4 


5 


6 


7 





1 


1 


1 








60° 


80! 6. 442|. 1552 


.267 


27.38 


36.64 


432 


1 


1.068 


.9363 


.95 


.96 


.975 


71 


851 6.782 .1474 


2566 


28.16 


37.94 


447 


2 


1.136 


.8803 


.91 


1.87 


1.91 


80.4 


90; 7.1221.1404 


.248 


28.89 


39.18 


459 


3 


1.204 


.8305 


.876 


2.72 


2.8 


88.9 


95! 7.462!. 134 
100 1 7.802 .1281 


.24 


29.57 


40.4 


472 


4 


1.272 


.7861 


.84 


3.53 


3.67 


98 


.232 


30.21 


41.6 


485 


5 


1.34 


.7462 


.81 


4.3 


4 5 


106 


105! 8.142J.1228 


.2254 


30.81 


42.78 


496 


10 


1.68 


.5952 


.69 


7.62 


8.27 


145 


110! 8. 483!. 1178 


.2189 


31.39 


43.91 


507 


15 


2.02 


.495 


.606 


10.33 


11.51 


178 


lisl 8 823' 1133 


.2129 


31.98 


44.98 


518 


20 


2.36 


.4237 


.543 


12.62 


14.4 


207 


120! 9.163 .1091 


.2073 


32.54 


46.04 


529 


25 


2.7 


.3703 


.494 


14.59 


17.01 


234 


125 1 9.503 .1052 


.2020 


33.07 


47.06 


540 


3ii 


3.04 


3289 


.4538 


16.34 


19.4 


252 


130 9.843 .1015 


.1969 


33.57 


48.1 


550 


35 


3.381 


.2957 


.42 


17.92 


21.6 


281 


135 10.183 


.0981 


.1922 


34.05 


49.1 


560 


40 


3.721 


.2687 


.393 


19.32 


23.66 


302 


140 30.523 


.095 


.1878 


34.57 


50.02 


570 


45 


4.061 


.2462 


37 


20.57 


25.59 


321 


145 10.864 


.0921 


.1837 


35.09 


51. 


580 


no 


4.401 


.2272 


.35 


21.69 


27.39 


339 


150 11.204 


.0892 


.1796 


35.48 


51.89 


589 


55 


4.741 


.2109 


.331 


22.76 


29.11 


357 


160 11.88 


.0841 


.1722 


36.29 


53.65 


607 


CO 


5.081 


.1968 


.3141 


23.78 


30.75 


375 


170 12.56 


.0796 


.1657 


37.2 


55.39 


624 


65 


5.423 


.1844 


.301 


24.75 


32.32 


389 


180 13.24 


.0755 


.1595 


37.96 


57.01 


640 


70 


5.762 


.1735 


.288 


25.67 


33.83 


405 


190 13.92 


.0718 


.154 


38.68 


58.57 


657 


75 


6.102 


.1639 


.276 


26.55 


35.27 


420 


200 14.6 


.0685 


.149 


39.42 


60.14 


672 



Column 3 gives the volume of air after compression to the given pressure 
and after it is cooled to its initial temperature. After compression air loses 
its heat very rapidly, and this column may be taken to represent the volume 
of air after compression available for the purpose for which the air has 
been compressed. 

Column 4 gives the volume of air more nearly as the compressor has to 
deal with it. In any compressor the air will lose some of its heat during 
compression. The slower the compressor runs the cooler the air and the 
smaller the volume. 

Column 5 gives the mean effective resistance to be overcome by the air- 
cylinder piston in the stroke of compression, supposing the air to remain 
constantly at its initial temperature. Of course it will not so remain, but 
this column is the ideal to be kept in view in economical air-compression. 



COMPRESSED AIR. 



501 



Column 6 gives the mean effective resistance to be overcome by the pis- 
ton, supposing that there is no cooling of the air. The actual mean effec- 
tive pressure will be somewhat less than as given in this column; but for 
computing the actual power required for operating air-compressor cylinders 
the figures in this column may be taken and a certain percentage added — 
say 10 per cent— and the resuk will represent very closely the power required 
by the compressor. 

The mean pressures given being for compression from one atmosphere 
upward, they will not be correct for computations in compound compression 
or for any other initial pressure. 

Loss Due to Excess of Pressure caused by Heating In 
the Compression-cylinder.— If the air during compression weie 
kept at a constant temperature, the compression-curve of an indicator-dia- 
gram taken from the cylinder would be an isothermal curve, and would fol- 
low the law of Boyle and Marriotte, pv = a constant, or pjV, = PoV Q , or 

Pi =Po — i Po ana " v o being the pressure and volume at the beginning of 

compression, a,ndp 1 v 1 the pressure and volume at the end, or at any inter- 
mediate point. But as the air is heated during compression the pressure 
increases faster than the volume decreases, causing the work required for 
any given pressure to be increased. If none of the heat were abstracted 
by radiation or by injection of water, the curve of the diagram would be an 

adiabatic curve, with the equation p x = p (^— ) ' Cooling the air dur- 
ing compression, or compressing it in two cylinders, called compounding, 
and cooling- the air as it passes from one cylinder to the other, reduces the 
exponent, of this equation, and reduces the quantity of work necessary to 
effect a given compression. F. T. Gause (Am. Much., Oct. 20, 1892), describ- 
ing the operations of thePopp air-compressors in Paris, says : The greatest 
saving realized in compressing in a single cylinder was 33 per cent of that 
theoretically possible. In cards taken from the 2000 H.P. compound com- 
pressor at Quai De La Gare. Paris, the saving realized is 85 per cent of the 
theoretical amount. Of this amount only 8 per cent is due to cooling dur- 
ing compression, so that the increase of economy in the compound com- 
pressor is mainly due to cooling the air between the two stages of compres- 
sion. A compression-curve with exponent 1.25 is the best result that was 
obtained for compression in a single cylinder and cooling with a very fine 
spray. The curve with exponent 1.15 is that which must be realized in a 
single cylinder to equal the present economy of the compound compressor 
at Quai De La Gare. 



Horse-power required to 
compress one cubic foot of 
Free Air per minute to a 
given Pressure with no cooling 
of the air during the compression; 
also the horse-power required, sup- 
posing the air to be maintained at 
constant temperature during the 
compression. (Richards.) 

Gauge- Air not Air constant 



70 
60 
50 
40 



cooled. 

.22183 

.20896 

.19521 

.17989 

.164 

.14607 

.12433 

.10346 

.076808 

.044108 

.024007 



Temperature, 
.14578 
.13954 
.13251 
.12606 
.11558 
.10565 
.093667 
.079219 
.061188 



Horse - power required to 
deliver one cubic foot of 
Air per minute at a given 
Pressure with no cooling of the 
air during the compiessiou; also the 
horse-power required, supposing the 
air to be maintained at constaut 
temperature during the compres- 
sion. (Richards.) 



Gauge- 
pressure. 
100 
90 
80 
70 
60 
50 
40 
30 



Air not 
cooled. 
1.7317 
1.4883 
1.25779 
1.03683 
.83344 
.64291 
.46271 
.31456 
.181279 
.074106 
.032172 



In computing the above table an allowance of 10 per cent 
for friction of the compressor. 



Air constant 
Temperature. 
1.13801 

.99387 

.8538 

.72651 

.58729 

.465 

.34859 

.24086 

.14441 

.06069 

.027938 

has been made 



502 



TaMe for Adiabatlc Compression or Expansion of Air. 

(Proc. Inst. M.E., Jan. 1881, p. 123.) 



Absolute 


Pressure. 


Absolute Temperature. 


"Volume. 


Ratio of 


Ratio of 


Ratio of 


Ratio of 


Ratio of 


Ratio of 


Greater 


Less to 


Greater 


Less to 


Greater 


Less to 


to Less. 


Greater. 


to Less. 


Greater. 


to Less. 


Greater. 


(Expan- 


(Compres- 


(Expan- 


(Compres- 


(Compres- 


(Expan- 


sion.) 


sion.) 


sion.) 


sion.) 


sion.) 


sion.) 


1.8 


.833 


1.054 


.948 


1.138 


.879 


1.4 


.714 


1.102 


.907 


1.270 


.788 


1.6 


.625 


1.146 


.873 


1.396 


.716 


1.8 


.556 


1.186 


.843 


1.518 


.659 


2.0 


.500 


1.222 


.818 


1.636 


.611 


2.2 


.454 


1.257 


.796 ■ 


1.750 


.571 


2.4 


.417 


1.289 


.776 


1.862 


.537 


2.6 


.385 


1.319 


.758 


1.971 


.507 


2.8 


.357 


1.348 


.742 


2.077 


.481 


3.0 


.333 


1.375 


.727 


2.182 


.458 


3.2 


.312 


1.401 


.714 


2.284 


.438 


3.4 


.294 


1.426 


.701 


2.384 


.419 


3.6 


.278 


1.450 


.690 


2.483 


.403 


3.8 


.263 


1.473 


.679 


2.580 


.388 


4.0 


.250 


1.495 


.669 


2.676 


.374 


4.2 


.238 


1.516 


.660 


2.770 


.361 


4.4 


.227 


1.537 


.651 


2.863 


.349 


4.6 


.217 


1.557 


.642 


2.955 


.338' 


4.8 


.208 


1.576 


.635 


3.046 


.328 


5,0 


.200 


1.595 


.627 


3.135 


.319 


6.0 


.167 


1.681 


.595 


3.569 


.280 


7.0 


.143 


1.758 


.569 


3.981 


.251 


8.0 


.125 


1.828 


.547 


4.377 


.228 


9.0 


.111 


1.891 


.529 


4.759 


.210 


10.0 


.100 


1.950 


.513 


5.129 


.195 



Mean Effective Pressures for tlie Compression Part only 
of the Stroke when compressing and delivering Air 
from one Atmosphere to given Gauge-pressure in a Sin- 
gle Cylinder. (b\ Richards, Am. Much., Dec. 14, 1893.) 



Gauge- 


Adiabatic 


Isothermal 


Gauge- 


Adiabatic 


Isothermal 


pressure. 


Compression 


Compression . 


pressure. 


Compression. 


Compression. 


1 


.44 


.43 


45 


13.95 


12.62 


2 


.96 


.95 


50 


15.05 


13.48 


3 


1.41 


1.4 


55 


15.98 


14.3 


4 


1.86 


1.84 


60 


16.89 


15.05 


5 


2.26 


2.22 


65 


17.88 


15.76 


10 


4.26 


4.14 


70 


18.74 


16.43 


15 


5.99 


5.77 


75 


19.54 


17.09 


20 


7.58 


7.2 


80 


20.5 


17.7 


25 


9.05 


8.49 


85 


31.22 


18.3 


30 


10.39 


9.66 


90 


22. 


18.87 


35 


11.59 


10.72 


95 


22.77 


19.4 


40 


12.8 


11.7 


100 


23.43 


19.92 



The mean effective pressure for compression only is always lower than 
the mean effective pressure for the whole work 



COMPRESSED AIR. 



503 



Mean and Terminal Pressures of Compressed Air used 
Expansively for Gauge-pressures from 60 to 100 lbs. 

(Frank Richards, Am. Much., April 13, 1893.) 



Initial 
















Pres- 


60. 


70. 


80. 


90. 


100. 


sure. 
















ojb 


,2 


13 CD 


a , £ 


% © 


<6 | 


^ £ 

.2 i. 8 


a , £ 


"3 g 


6 


3 £ 


4J o 


a3 u S 










as L 3 


•9^3 


§ - P 




o 3 


©'=3 '{I 
ft 


H ft 


ft 


$ ft 




£3f 

H ft 


ft 
39 04 


£ ft 


ft 


H ft 


.25 


23.6 


JO. 65 


28.74 


12.07 


33.89 13.49 


U.91 


44.19 


1.33 


.30 


28.9 


13.77 


34.75 


.6 


40.61 1 


2.44 


46.46 


4.27 


53.32 


6.11 


Vb 


32.13 


.96 


38.41 


3.09 


44.69 


5.22 


50.98 


7.35 


57.26 


9.48 


.35 


33.66 


2.33 


40.15 


4.38 


46.64, 


6.66 


53.13 


8.95 


59.62 


11.23 


y% 


35.85 


3.85 


42.63 


6.36 


49.411 


7.88 


56.2 


11.39 


62.98 


13.89 


.40 


37.93 


5.64 


44.99 


8.39 


52.05 


11.14 


59.11 


13.88 


66.16 


16.64 


.45 


41.75 


10.71 


49.31 


12.61 


56.9 1 


15.86 


64 45 


19.11 


72.02 


22.36 


.50 


45.14 


13.26 


53.16 


17. 


61.18 


20.81 


69.19 


24.56 


77.21 


2S.33 


.60 


50.75 


21.53 


59.51 


26.4 


68.28 


31.27 


77.05 


36.14 


85.82 


41.01 


y& 


51.92 


23.69 


60.84 


28.85 


69.76 


34.01 


78.69 


39.16 


87.61 


44.32 


Vb 


53.67 


27.94 


62.83 


33.03 


71.99 


38.68 


81.14 


44.33 


90.32 


49.97 


.70 


54.93 


30.39 


64 25 


36.44 


73.57| 


42.49 


82.9 


48.54 


92.22 


54.59 


.75 


56.52 


35.01 


66.05 


41.68 


75.59 1 


48.35 


85.12 


55.02 


94.66 


61.69 


.80 


57.79 


39.78 


67.5 


47.08 


77.2 


54.38 


86.91 


61.69 


96.61 


68.99 


Vs 


59.15 


47.14 


69.03 


55.43 


78.92, 


63.81 


88.81 


72. 


98.7 


80.28 


.90 


59.46 


49.65 


69.38 


58.27 


79 31 1 


66.89 


89.24 


75.52 


99.17 


87.82 



The pressures in the table are all gauge-pressures except those in italics, 
which are absolute pressures (above a vacuum). 



Straight-line 


Air-compressors, Ingersoll-Sergeant 






Rock-drill Co. 






Diameter 
Steam- 
cylinder, 
inches. 


Diameter 

of Air- 
cylinder, 
inches. 


Length 

of 
Stroke, 
inches. 


No. of 
Revolu- 
tions 
per 
minute. 


Piston 
Speed 
in feet 

per 
minute. 


Cubic Feet 
Free Air 

per minute 
(Theo- 
retical). 


Horse- 
power 

of 

Boiler 

required. 


4 


m 


10 


175 


291 


28 


6 


5 


b}4 


10 


175 


291 


42 


8 


6 


12 


160 


320 


66 


10 


7 


m 


12 


160 


320 


91 


12 


8 


&A 


12 


160 


320 


117 


15 


9 


9H 


12 


160 


320 


148 


20 


10 


IO14 


14 


155 


361 


207 


30 


12 


12^4 

14J4 


14 


155 


361 


295 


40 


14 


18 


120 


360 


398 


55 


16 


16J4 


18 


120 


360 


518 


70 


18 


1814 


24 


94 


376 


683 


100 


20 


20^ 


24 


94 


376 


840 


130 


22 


22J4 


30 


75 


375 


1011 


155 


24 


24^4 


30 


75 


375 


1202 


t.'00 



The same sizes are made to be driven by belt or gearing. 
Compressors at High Altitudes.— Cubic feet of compressed air 
delivered by air-compressors at high altitudes, expressed as a percentage of 
the air delivered at the sea-level. 



Altitude above Sea- | 
level, feet. j 





1000 


2000 


3000 


4000 


5000 


60C0 


7000 


8000 


9000 


10000 


Air delivered, per cent. . 


100 


97 


94 


91 


89 


86 


84 


81 


78 


70 


74 



504 



AIR. 



Standard Air-compressors driven by Steam. 

(Norwalk Iron Works Co.) 
In the following list the large air-cylinder gives the capacity of the ma- 
chine. For actual capacity, allowance of 10 per cent may be made for 
contingencies. The small piston only encounters the pressure of the final 
compression. 



3 . js 
lis 


teg 


u . fee's 

sssZ 

HOt«C 




c *> a 
oQ£c 


oretical 
pacity, 
bic feet 
r minute, 
ee Air. 


& 
"ft 

E 


00 

2 3 
Kft 


ft 

ft 


63 

ft 
'ft 


<L £ 


3*fc 


©cfi 


|oao 


3*& 




^Oofth 


<v 


i 


c6 


£ ft 


Q 


i-3 


Q 


Q 


ti 


H 


CO 


H- 


<5 


tc 


8 


10 


5 


8 


200 


116 


2 


2U 


2 


3^ 


15 


10 


12 


6M 


10 


190 


207 


2V, 


3 


2^ 


3 4 


28 


14 


16 


9^ 


14 


150 


427 


3 


4 


4 


l 


55 


20 


24 


13^ 


20 


110 


960 


5 


6 


5 


1M 


125 


26 


30 


1% 


24 


90 


1659 


fi 


8 


6 


iM 


215 


32 


36 


21^ 


30 


80 


2686 


7 


10 


8 


m 


350 



Double-compound Compressors, 

(Norwalk Iron Works Co ) 



Diameter 


Length 


Air- 


of 


cylinder. 


Stroke. 


10 


12 


12 


12 


14 


16 


16 


16 


20 


20 


20 


24 


22 


24 


26 


30 


28 


30 


32 


36 



Com- 
pressing 
cylinder. 



9^ 



18£ 



gljj 



High- 
pressure 

Steam- 
cylinder. 



7^ 

10 
10 
14 
14 
14 
18 
18 



Low- 
pressure 
Steam- 
cylinder. 



12 
12 
16 
16 
22 
22 
22 



Steam 
pipe. 



Revolu- 
tions 
per 

minute. 



190 
190 
150 
150 
120 
110 
110 



Capac'y 

cubic 

feet 

Free 

Air per 

minute. 



427 

558 
872 
960 
1160 
1659 



Mountain or High-altitude Compressors. 

(Norwalk Iron Works Co.] 







6C 






At Sea- 


At 2000 


At 6000 


At 10,000 


<.- 


©3 

n 


O GO j_- 

G O >-. 
.SCO 


$§.£ 

.2co o 


E S 

o c 

Ha 

£3 
£ ft 


level. 


feet. 


feet. 


feet. 


.2 « 




i © 

Sh o 
O ft 


'5 

S3 

ft 


•L J O 
O ft 


'3 

eS 

ft 


, 3 

V >. 
t c 
- ft 


>> 
'3 

03 

ft 


. 3 

o a 


ft 


J 


Q 


A 


ti 


o 


w 


O 


w 


O 


w 


o 


W 


12 


12 


7 


10 


190 


298 


35 


2N0 


34 


24-1 


32 


214 


30 


16 


16 


9U 


14 


150 


558 


70 


524 


68 


462 


64 


405 


60 


20 


20 


13W 


18 


120 


872 


110 


819 


107 


72-,' 


100 


634 


94 


22 


24 


13U 


20 


110 


1160 


145 


lO'.IO 


140 


9(H) 


132 


843 


124 


26 


30 


17^ 


24 


90 


1659 


215 


1500 


207 




195 


1200 


184 



The delivery and power of the compressors decrease as the height in- 
creases. As the capacity decreases in a greater ratio than the power 
necessary to compress, it follows that operations at a high altitude are more 
expensive than at sea-level. At 10,000 feet this extra expense amounts to 
over 20 per cent. 



COMPRESSED AIR. 505 

Rand Drill Co.'s Air-coiiipressors. 



■ ) 



and ■{ 

B 

i 

A f 

and I 

B 

Geared 



C 1 





u 


Theo 




S3 S 




of Air- 




eyliuders 


•£ 2 




in 


5,'- 




inches. 


O S3 

> 

pel 


Free. 


10x16 |d*.: 


100 
100 


145.44 

290.88 


14x22] I'- 


85 
85 


333.20 
666.40 


16^x30 1|; 


75 
75 


556.83 
1113.66 


18x30-1 D "" 


75 
75 


662.68 
1325.36 


20x48] »;;; 


50 
50 


872.66 
1745.32 


28x4s]S;;; 


40 


1368.34 


40 


- 


32x48] 1;;; 


40 
40 


1787.22 
3574.44 


32 x 60 ] |- • • 


35 
35 


1954.77 
3909.55 


36 x 60 ] |- • 


30 
30 


2120.61 
4241.22 


8x12 


120 


83.78 


10x14 


110 


139.95 


12x16 


100 


209.44 


14x22 


95 


372.40 


16x24 


90 


502.66 


17^x24 .... 


90 


601.29 


20x30... . 


80 


872.67 



Theoretical Volume of Air delivered in cubic 
feet per minute, at Sea-level. 



Compressed to a Gauge-pressure of- 



10 


20 


40 


60 


80 


100 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


86.56 


61.61 


39.08 


28.62 


22.57 


18.64 


173.12 


123.23 


78.17 


57.24 


45.15 


37.28 


198.31 


141.10 


89.51 


65.54 


51.93 


42.67 


396.61 


282.20 


179.01 


131.07 


103.86 


85.31 


331.39 


235.89 


149.64 


109.57 


86.43 


71.36 


662.79 


471.79 


299.28 


219.15 


172.86 


142.72 


394.39 


280.73 


178.08 


130.40 


102.86 


84.92 


788.78 


561.46 


356.17 


260.81 


205.72 


169.84 


519.36 


369.69 


234.51 


171.72 


135.46 


111.84 


1038.72 


739.38 


469.03 




270.92 


223.68 


814.36 


579.67 


367.72 




212.40 


175.36 


1628.71 


1159.34 


735.45 


538.54 


424.80 


350.73 


1063.65 


757.12 


480.29 


351.70 


277.42 


229.05 


2127.30 


1514.24 


960.58 


703.40 


554.85 


458.10 


1163.37 


828.10 


525.32 


384.67 


303.43 


250.52 


: 


1656.20 


1050.63 


769.34 


606.86 


501.05 


1262.07 


898.35 


572.07 


417.72 


329.16 


272.82 


2524.14 


1796.70 


1144.14 


835.44 


658.32 


545.64 


49.86 


35.49 


22; 51 


16.49 


13.00 


10.74 


83.27 


59 29 


37.62 


27.50 


21.72 


17.94 


124.65 


88.73 


56.28 


41.22 


32.51 


26.66 


221.64 


157.70 


100.04 


73.25 


58.04 


47.69 


299.15 


212.94 


135.08 


98.92 


78.03 


64.42 


357.85 


254.95 


161.60 


118.33 


93.33 


77.06 


519.36 


369.69 


234.52 


171.73 


135.46 


111.84 



*S, Single; D, Duplex. 

Practical Results with Compressed Air.— Compressed-air 
System at the Chapiu Mines, Iron Mountain, Mich. — These mines are three 
miles from the falls which supply the power. There are four turbines at 
the falls, one of 1000 horse-power and three of 900 horse-power each. The 
pressure is 60 pounds at 60° Fahr. Each turbine runs a pair of compressors. 
The pipe to the mines is 24 inches in diameter. The power is applied at the 
mines to Corliss engines, running pumps, hoists, etc., and direct to rock- 
drills. 

A test made in 1888 gave 1430.27 horse-power at the compressors, and 390.17 
horse-power as the sum of the horse-power of the engines at the mines. 
Therefore, only 27$ of the power generated was recovered at the mines. 
This includes the loss due to leakage and the loss of energy in heat, but not 
the friction in the engines or compressors. (F. A. Pocock, Trans. A. I. M. E., 
1890.) 

W. L. Saunders (Jour. F. I. 1892) says: " There is not a properly designed 
compressed-air installation in operation to-day that loses over h% by trans- 
mission alone. The question is altogether one of the size of pipe; and if 
the pipe is large enough, the friction loss is a small item. The largest com- 
pressed-air power plant in America is that at the Chapin Mines in Michigan, 
where power is generated at Quinnesec Falls, and transmitted three miles. 
This is is not an economical plant, but the loss of pressure as shown by the 
gauge is only 2 lbs., and this is the loss which may be laid strictly to trans- 
mission. 

" The loss of power in common practice, where compressed air is used to 
drive machinery in mines and tunnels, is about 70$. I refer to cases where 
common American air-compressors are used, and where the air is trans- 
mitted far enough to lose its heat .of compression and is exhausted without 



506 AIR. 

reheating. In the best practice, with the best air-compressors, and without 
reheating, the loss is about 60*. 

" These losses may be reduced to a point as low as 20% by combining the 
best systems of reheating with the best air-compressors. 11 

Prof. Kennedy says compressed air transmission system is now being 
carried on, on a "large commercial scale, in such a fashion that a small motor 
four miles away from the central station can indicate in round numbers 10 
horse-power, for 20 horse power at the station itself, allowing for the value 
of the coke used in heating the air. 

The limit to successful reheating lies in the fact that air-engines cannot 
work to advantage at temperatures over 350°. 

The efficiency of the common system of reheating is shown by the re- 
sults obtained with the Popp system in Paris. Air is admitted to the re- 
heater at about 83°, and passes to the engine at about 315°, thus being in- 
creased in volume about 42*. The air used in Paris is about 11 cubic feet of 
free air per minute per horse-power. The ordinary practice in America 
with cold air is from 15 to 25 cubic feet per minute per horse-power. When 
the Paris engines were worked without reheating the air consumption was 
increased to about 15 cubic feet per horse-power per minute. The amount 
of fuel consumed during reheating is trifling. 

Efficiency of Compressed-air Engines.— The efficiency of an 
air-engine, that is, the percentage which the power given out by the air-en- 
gine bears to that required to compress the air in the compressor, depends 
on the loss by friction in the pipes, valves, etc., as well as in the engine itself. 
This question is treated at length in the catalogue of the Norwalk Iron Works 
Co., from which the following is condensed. As the friction increases the 
most economical pressure increases. In fact, for any given friction in a 
pipe, the pressure at the compressor must not be carried below a certain 
limit. The following table gives the lowest pressures which should be used 
at the compressor with varying amounts of friction in the pipe: 

Friction, lbs 2.9 5.8 8.8 11.7 14.7 17.6 20.5 23.5 26.4 29.4 

Lbs. at Compressor... 20.5 29.4 38.2 47. 52.8 61.7 70.5 76.4 82.3 88.2 
Efficiency* 70.9 64.5 60.6 57.9 55.7 54.0 52.5 51.3 50.2 49.2 

An increase of pressure will decrease the bulk of air passing the pipe and 
its velocity. This will decrease the loss by friction, but we subject ourselves 
to a new loss, i.e. the diminishing efficiencies of increasing pressures. Yet as 
each cubic foot of air is at a higher pressure and therefore carries more 
power, we will not need as many cubic feet as before, for the same work. 
With so many sources of gain or loss, the question of selecting the proper 
pressure is not to be decided hastily. 

The losses are, first, friction of the compressor. This will amount ordinarily 
to 15 or 20 per cent, and cannot probably be reduced below 10 per cent. 
Second, the loss occasioned by pumping the air of the engine-room, rather 
than the air drawn from a cooler place. This loss varies with the season and 
amounts from 3 to 10 per cent. This can all be saved. The third loss, or series 
of losses, arises in the compressing cylinder, viz., insufficient supply, difficult 
discharge, defective cooling arrangements, poor lubrication, etc. The fourth 
loss is found in the pipe. This loss varies with the situation, and is subject 
to somewhat complex influences. The fifth loss is chargeable to fall of 
temperature in the cylinder of the air-engine. Losses arising from leaks 
are often serious. 

Air should be drawn from outside the engine-room, and from as cool a 
place as possible. The gain amounts to one per cent for every five degrees 
that the air is taken in lower than the temperature of the engine-room. 
The inlet conduit should have an area at least 50* of the area of the air- 
piston, and should be made of wood, brick, or other non-conductor of heat. 

Discharge of a compressor having an intake capacity of 1000 cubic feet 
per minute, and volumes of the discharge reduced to cubic feet at atmos- 
pheric pressure and at temperature of 62 degrees Fahrenheit: 

Temperature of Intake, F 0° 32° 62° 75 c 80° 90° 100° 1 10° 

Relative volume discharged, cubic ft.. . 1135 1060 1000 975 966 949 932 916 

Requirements of Rock-drills Driven by Compressed 
Air. (Norwalk Iron Works Co.)— The speed of the drill, the pressure of 
air, and the nature of the rock affect the consumption of power of rock- 
drills. 

A three-inch drill using air at 30 lbs. pressure made 300 blows per minute 
and consumed the equivalent of 64 cubic, feet of free air per minute, The 



COMPRESSED AIR. 



507 



same drill, with air of 58 lbs. pressure, made 450 blows per minute and con- 
sumed 160 cubic f> j et of free air per minute. At Hell Gate different machines 
doing the same work used from 80 to 150 cubic feet free air per minute. 

An average consumption may be taken generally from 80 to 100 cubic feet 
per minute, according to the nature of the work. 

The Popp Compressed-air System in Paris.— A most exten- 
sive system of distribution of power by means of compressed air is that of 
M. Popp, in Paris. One of the central stations is laid out for 24,000 horse- 
power. For a very complete description of the system, see Engineering, 
Feb. 15, June 7, 21. and 28, 1889, and March 13 and 29, April 10, and May 1, 
1891. Also Proc. Inst. M. E., July, 1889. A condensed description will be 
found in Modern Mechanism, p. 12. 

Utilization of Compressed Air in Small Motors.— In the 
earliest stages of the Popp system in Paris it was recognized that no good 
results could be obtained if the air were allowed to expand direct into the 
motor; not only did the formation of ice due to the expansion of the air 
rapidly accumulate and choke the exhaust, but the percentage of useful 
work obtained, compared with that put into the air at the central station, 
was so small as to render commercial results hopeless. 

After a number of experiments M. Popp adopted a simple form of cast- 
iron stove lined with fire-clay, heated either by a gas jet or by a small coke 
fire. This apparatus answered the desired purpose until some better ar- 
rangement was perfected, and the type was accordingly adopted through- 
out the whole system. The economy resulting from the use of an improved 
form was very marked, as will be seen from the following table. 





Efficiency of Air-heating Stoves. 








© 

o 

w 
be 

a 

3 


P. 

is 3 
c3 O 

< 


Temperature 
of Air in Oven. 


Value of Heat Absorbed 
per Hour. 


Nature of 
Stove. 


.2 -a 




O 

Eh 


Per Square 
Foot of 
Heat i n g 
Surface. 


£0 


Cast-iron box j 
stoves j 

Wrought-i r o n 
coiled tubes.. 


sq. ft. 
14 
14 

46.3 


cub.ft. 
20,342 
11,054 

38,428 


45 

45 

41 


215 
361 

347 


cal. 
17,900 
17,200 

39,200 


cal. 

1278 
1228 

830 


cal. 

2032 
2058 

2545 



The results given in this table were obtained from a large number of 
trials. From these trials it was found that more than 70# of the total num- 
ber of calories in the fuel employed was absorbed by the air and trans- 
formed into useful work. Whether gas or coal be employed as the fuel, the 
amount required is so small as to be scarcely worth consideration; accord- 
ing to the experiments carried out it does not exceed 0.2 lb. per 
horse-power per hour, but it is scarcely to be expected that in regular prac- 
tice this quantity is not largely exceeded. The efficiency of fuel consumed 
in this way is at least six times greater than when utilized in a boiler and 
steam-engine. 

According to Prof. Riedler, from 15$ to 20$ above the power at the central 
station can be obtained by means at the disposal of the power users, and it 
has been shown by experiment that by heating the air to 480° F. an in- 
creased efficiency of 30% can be obtained. 

A large number of motors in use among the subscribers to the Compressed 
Air Company of Paris are rotary engines developing 1 horse-power and 
less, and these in the early times of the industry were very extravagant in 
their consumption. Small rotary engines, working cold air without expan- 
sion, used as high as 2330 cu. ft. of air per brake horse-power per 
hour, and with heated air 16$4 cu. ft. Working expansively, a 1 horse- 
power rotary engine used 1469 cu. ft. of cold air, or 960 cu. ft. of heated air, 
and a 2- horse-power rotary engine 1059 cu. ft. of cold air. or 847 cu. ft. of air, 
heated to about 50° C. 

The efficiency of this type of rotary motors, with air heated to 50° C , may 
now be assumed at 43$. With such an efficiency the use of small motors in 



II. 


III. 


IV. 


69.7 


85 


71 


356 


388 


46 


68 




77 


350 


310 


243 


477 


376 


316 


791 


900 


1148 



508 air. 

many industries becomes possible, while in cases where it is necessary to 
have a constant supply of cold air economy ceases to be a matter of the 
first importance. 

The following table shows the results of tests of a small rotary engine used 
for driving sewing-machines, and indicating about a tenth of a horse-power: 

Trials of a Small Rotary Riedinger Engine. 

Numbers of trials I. II. 

Initial air-pressure, lbs. per sq. in 86 71.8 

Initial temperature, deg. Fahr 54° 338° 

Ft.-lbs. per sec, measured on the brake 51.63 34.07 

Revolutions per minute 384 300 

Consumption of air per 1 horse-power per hour 1377 988 

The following table shows the results obtained with a one-half horse- 
power variable expansive Riedinger rotary engine. These trials represent 
the best practice that has been obtained up to the present time (1890). The 
volumes of air were in all cases taken at atmospheric pressure: 

Trials of a 5-Horse-power Riedinger Rotary Engine. 

Numbers of trials I. 

Initial pressure of air, lbs. per sq. in 54 

" temperature of air, deg. Fahr 338 

Final " " " " " .. 77 

Revolutions per minute 335 

Ft.-lbs. per second, measured on brake . . 271 
Consumption of air per horse-power per 

hour 883 

Trials made with an old single-cylinder 80-horse-power Farcot steam-en- 
gine, indicating 72 horse-power, gave a consumption of air per brake horse- 
power as low as 465 cu. ft. per hour. The temperature of admission was 
320° F., and of exhaust 95° F. 

Prof. Elliott gives the following as typical results of efficiency for various 
systems of compressors and air-motors : 

Simple compressor and simple motor, efficiency S9A% 

Compound compressor and simple motor, " 44.9 

" " " compound motor, efficiency 50.7 

Triple compressor and triple motor, " 55.3 

The efficiency is the ratio of the indicated horse-power in the motor cylin- 
ders to the indicated horse-power in the steam-cylinders of the compressor. 
The pressure assumed is 6 atmospheres absolute, and the losses are equal 
to those found in Paris over a distance of 4 miles. 

Summary of Efficiencies of Compressed-air Transmission 
at Paris, between the Central Station at St. Fargeau and 
a 10-horse-power Motor Working with Pressure Re- 
duced to 4>o Atmospheres. 

(The figures below correspond to mean results of two experiments cold and 

two heated.) 
1 indicated horse-power at central station gives 0.845 indicated horse-power 
in compressors, and corresponds to the compression of 348 cubic feet of air 
per hour from atmospheric pressure to 6 atmospheres absolute. (The weight 
of this air is about 25 pound s.} 

0.845 indicated horse-power in compressors delivers as much air as will do 
0.52 indicated horse-power in adiabatic expansion after it has fallen in tem- 
perature to the normal temperature of the mains. 

The fall of pressure in mains between central station and Paris (say 5 kilo- 
metres) reduces the possibility of work from 0.52 to 0.51 indicated horse- 
power. 

The further fall of pressure through the reducing valve to 4J^ atmospheres 
(absolute) reduces the possibility of work from 0.51 to 0.50. 

Incomplete expansion, wire-drawing, and other such causes reduce the 
actual indicated horse-power of the motor from 0.50 to 0.39. 

By heating the air before it enters the motor to about 320° F., the actual 
indicated horse-power at the motor is. however, increased to 0.54. The ratio 

of gain by heating the air is, therefore, r 1 ^ = 1.38. 



COMPRESSED AIR. 500 

In this process additional heat is supplied by the combustion of about 0.39 
pounds of coke per indicated horse-power per hour, and if this be taken into 
account, the real indicated efficiency of the whole process becomes 0.47 
instead of 0.54. 

Working with cold air the work spent in driving the motor itself reduces 
the available horse-power from 0.39 to 0.26. 

Working with heated air the work spent in driving the motor itself reduces 
the available horse-power from 0.54 to 0.44. 

A summary of the efficiencies is as follows : 

Efficiency of main engines 0.845. 

Efficiency of compressors 0.52 -^ 0.845 = 0.61. 

Efficiency of transmission through mains 0.51 -v- 0.52 = 0.98. 

Efficiency of reducing valve 0.50-i- 0.51 = 0.98. 

The combined efficiency of the mains and reducing valve between 5 and 
4)4 atmospheres is thus 0.98 X 0.9S = 0.96. If the reduction had been to 4, 
3J^, or 3 atmospheres, the corresponding efficiencies would have been 0.93, 
0.89, and 0.85 respectively. 

Indicated efficiency of motor 0.39 -h- 0.50 = 0.78. 

Indicated efficiency of whole process with cold air 0.39. Apparent indi- 
cated efficiency of whole process with heated air 0.54. 

Real indicated efficiency of whole process with heated air 0.47. 

Mechanical efficiency of motor, cold, 0.67. 

Mechanical efficiency of motor, hot, 0.81. 

Most of the compressed air in Paris is used for driving motors, but the 
work done by these is of the most varied kind. A list of motors driven from 
St. Fargeau station shows 225 installations, nearly all motors working at 
from % horse-power to 50 horse-power, and the great majority of them more 
than two miles away from the station. The new station at Quai de la Gare 
is much larger than the one at St. Fargeau. Experiments on the Riedler 
air-compressors at Paris, made in December, 1891, to determine the ratio 
between the indicated work done by the air-pistons and the indicated work 
in the steam-cylinders, showed a ratio of 0.8997. The compressors are driven 
by four triple-expansion Corliss engines of 2000 horse-power each. 

Shops Operated by Compressed Air.— The Iron Age, March 2, 
1893, describes the shops of the Wuerpel Switch and Signal Co. .East St. Louis, 
the machine tools of which are operated by compressed air, each of the 
larger tools having its own air engine, and the smaller tools being belted 
from shafting driven by an air engine. Power is supplied by a compound 
compressor rated at 55 horse-power. The air engines are of the Kriebel 
make, rated from 2 to 8 horse-power. 

Pneumatic Postal Transmission.— A paper by A Falkenau, 
Eng'rs Club of Philadelphia, April 1894, entitled the "First United States 
Pneumatic Postal System, 1 ' gives a description of the system used in London 
and Paris, and that recently introduced in Philadelphia between the main 
post-office and a substation. In London the tubes are 214 and 3 inch lead 
pipes laid in cast-iron pipes for protection. The carriers used in 2^-inch 
tubes are but 1*4 inches diameter, the remaining space being taken up by 
packing. Carriers are despatched singly. First, vacuum alone was used; 
later, vacuum and compressed air. The tubes used in the Continental cities 
in Europe are wrought iron, the Paris tubes being 2^ inches diameter. 
There the carriers are despatched in trains of six to ten, propelled by a' 
piston. In Philadelphia the size of tube adopted is 6J-6 inches, the tubes 
being of cast iron bored to size. The lengths of the outgoing and return 
tubes are 2928 feet each. The pressure at the main station is 7 lbs., at the 
substation 4 lbs., and at the end of the return pipe atmospheric pressure. 
The compressor has two air-cylinders 18 X 24 in. Each carrier holds about 
200 letters, but 100 to 150 are taken as an average. Eight carriers may be 
despatched in a minute-, giving a delivery of 48,000 to 72,000 letters per hour. 
The time required in transmission is about 57 seconds. 

The Mekarski Compressed-air Tramway at Berne, 
Switzerland. (Eng'g Neivs, April 20, 1893.)— -The Mekarski system has 
been introduced in Berne, witzerland, on a line about two miles long, with 
grades of 0.25$? to 3.7$ and 5.2$. A special feature of the Mekarski system is 
the heating of the air, to maintain it at a constant temperature, by passing 
it through superheated water at 330° F. The air thus becomes saturated 
with steam, which subsequently partly condenses, its latent heat being 
absorbed by the expanding air. The pressure in the car reservoirs is 440 
lbs. per sq. in. 

The engine is constructed like an ordinary steam tramway locomotive. 



510 AIR. 

and drives two coupled axles, the wheel-base being 5.2 ft. It has a pair of 
outside horizontal cylinders, 5.1 x 8.6 in.; four coupled wheels, 27.5 in. 
diameter. The total weight of the ear including compressed air is 7.25 tons, 
and with 30 passengers, including the driver and conductor, about 9.5 tons. 

The authorized speed is about 7 miles per hour. Taking the resistance 
due to the grooved rails and to curves under unfavorable conditions at 30 
lbs. per ton of car weight, the engine has to overcome on the steepest grade, 
b%, a total resistance of about 0.63 ton, and has to develop 25 H.P. At the 
maximum authorized working pressure in cylinders of 176 lbs. persq. in. the 
motors can develop a tractive force of 0.64 ton. This maximum is, there- 
fore, just sufficient to take the car up the h.2% grade, while on the flatter 
sections of the line the working pressure does not exceed 73 to 147 lbs. per 
sq. in. Sand has to be frequently used to increase the adhesion on the 2% to 
5% grades. 

Between the two car frames are suspended ten horizontal compressed-air 
storage-cylinders, varying in length according to the available space, but of 
uniform inside diameter of 17.7 in., composed of riveted 0.27-in. sheet iron, 
and tested up to 588 lbs. per sq. in. These cylinders have a collective 
capacity of 64.25 cu. ft., which, according to Mr. Mekarski's estimate, 
should have been sufficient for a double trip, 3% miles. The trial trips, 
however, showed this estimate to be inadequate, and two further small 
storage-cylinders had therefore to be added of 5.3 cu. ft. capacity each, 
bringing the total cubic contents of the 12 storage-cylinders per car up to 
75 cu. ft., divided into two groups, the working and the reserve battery, the 
former of 49 cu. ft. the latter of 26 cu. ft. capacity. 

From the results of six official trips, the pressure and the mean consump- 
tion of air during a double journey per motor car are as follows: 

Working, Reserve, 
Storage-cylinders. ^ P er ^fjf 

Pressure of air on starting 440 440 

Pressure of air at end of up journey 176 260 

Pressure of air at end of down journey 103 176 

Lbs. 

Consumption of air at end of up journey 92 

Consumption of air during down journey 31 

This has been fully confirmed by the working experience of 1891, when 
the consumption of air per motor car and double journey was as follows: 

Minimum, 103 lbs „. . 28 lbs. per car-mile. 

Maximum, 154 lbs 42 " " " 

Mean, 123 lbs 35 " 

The principal advantages of the compressed-air system for urban and 
suburban tramway traffic as worked at Berne consist in the smooth 
and. noiseless motion ; in the absence of smoke, steam, or heat, of overhead 
or underground conductors, of the more or less grinding motion of most 
electric cars, and of the jerky motion to which underground cable traction 
is subject. On all these grounds the system has vindicated its claims as 
being preferable to any other so far known system of mechanical traction 
for street tramways. Its disadvantages, on i he other hand, consist in the 
extremely delicate adjustment of the different parts of the system, in the 
comparatively small supply of air carried by one motor car, which necessi- 
tates the car returning to the depot for refilling after a run of only four 
miles or 40 minutes, although on the Nogent and Paris lines the cars, 
which are, moreover, larger, and carry outside passengers on the top, 
run seven miles, and the loading pressure is 517 lbs. per sq. in. as against 
oidy 440 lbs. at Berne. 

Longer distances in the same direction would involve either more power- 
ful motors, a larger number of storage-cylinders, and consequently heavier 
cars, or loading stations every four or seven miles; and in this respect the 
system is manifestly inferior to electric traction, which easily admits of a 
line of 10 to 15 miles in length being continuously fed from one central 
station without the loss of time and expense caused by reloading. 

The cost of -working the Berne line is compared in the annexed table 
with some other tramways worked under similar conditions by horse and 
mechanical traction for the year 1891. As is seen, both in the case of com- 
pressed air and of electric traction, the cost of working is considerably 



FANS AND BLOWERS. 511 

increased where steam at a high cost of fuel has to be used instead of 
hydraulic power. Given the latter, the cost of working by air is about the 
same as that by steam-locomotives or steam-cars; but over both of these 
last-named, compressed-air offers, at equal cost and for such short lines 
with constant traffic, certain advantages: 

Constr. Opera- 



1891. Length of Line, 


Motive Power, and equip't 


tion, 


miles. 


per mile. 


p. car mi 


Geneva, city 8.68 


Horse $60,800 


19.4 cts 


Zurich, city 5.58 


Horse 39,700 


11.6 


Geneva, suburban 40.30 


Steam locomotive. 32,000 


13.2 


Mulhouse, city 18.00 


Steam locomotive.22,400 


17.8 


Montreux, suburban 6.82 


Hydro-electric . . 20,800 


10.4 


Florence, suburban 4 . 96 


Steam -electric .... 32,000 


20.0 


Tours, suburban 6.20 


Steam cars 19,200 


17.2 


Nogent (Paris), suburban 7.44 


Steam-compr. air.46,100 


25.6 


Berne, city . . 1.86 


Hydro-compr. air.48,950 


17.8 



For description of the Mekarski system as used at Nantes, France, see 
paper by Prof. D. S. Jacobus, Trans. A. I. M. E. xix. 553. 

Compressed Air for Working Underground Pumps in 
Mines.— Eng'g Record, May 19, 1894, describes an installation of com- 
pressors for working a number of pumps in the Nottingham No. 15 Mine, 
Plymouth, Pa., which is claimed to be the largest in America. The com- 
pressors develop above 2300 H.P., and the piping, horizontal and vertical, is 
6000 feet in length. About 25,000 gallons of water per hour are raised. 

FANS AND BLOWERS. 

Centrifugal Fans.— The ordinary centrifugal fan consists of a num- 
ber of blades fixed to arms, revolving on a shaft at high speed. The width 
of the blade is parallel to the axis of the shaft. Most engineers 1 reference 
books quote the experiments of VV. Buckle, Proc. Inst. M.E., 1847, as still 
standard. Mr. Buckle's conclusions are given below, together with data of 
more recent experiments. 

Experiments were made as to the proper size of the inlet openings and on 
the proper proportions to be given to the vane. The inlet openings in the 
sides of the fan-chest were contracted from 17^ in., the original diameter, 
to 12 and 6 in. diam., when the following results were obtained: 

First, that the power expended with the opening contracted to 12 in. diam. 
was as 2)4, to 1 compared with the opening of 17J^ in. diam. ; the velocity of 
the fan being nearly the same, as also the quantity and density of air 
delivered. 

Second, that the power expended with the opening contracted to 6 in. 
diam. was as 2% to 1 compared with the opening of 17J^ in. diam.; the 
velocity of the fan being nearly the same, and also the area of the efflux 
pipe, but the density of the air decreased one fourth. 

These experiments show that the inlet openings must be made of sufficient 
size, that the air may have a free and uninterrupted action in its passage to 
the blades of the fan; for if we impede this action we do so at the expense 
of power. 

With a vane 14 in. long, the tips of which revolve at the rate of 236.8 ft. 
per second, air is condensed to 9.4 ounces per square inch above the pres- 
sure of the atmosphere, with a power of 9.6 H. P. ; but a vane 8 inches long, 
the diameter at the tips being the same, and having, therefore, the same 
velocity, condenses air to 6 ounces per square inch only, and takes 12 H. P. 

Thus the density of the latter is little better than six tenths of the former, 
while the power absorbed is nearly 1.25 to 1. Although the velocity of the 
tips of the vanes is the same in each case, the velocities of the heels of the 
respective blades are very different, for, while the tips of the blades in each 
case move at the same rate, the velocity of the heel ofthe 14-inch is in the 
ratio of 1 to 1.67 to the velocity of the heel of the 8-inch blade. The 
longer blades approaching nearer the centre, strikes the air with less velo- 
city, and allows it to enter on the blade with greater freedom, and with 
considerably less force than the shorter one. The inference is, that the 
short blade must take more power at the same time that it accumulates a 
less quanta of air. These experiments lead to the conclusion that the 
length of the vane demands as great a consideration as the proper 
diameter of the inlet opening. If there were no other object in view, it 



512 



AIR. 



would be useless to make the vanes of the fan of a greater width than the 
inlet opening can freely supply. On the proportion of the length and width 
of the vane and the diameter of the inlet opening rest the three most im- 
portant points, viz., quantity and density of air, and expenditure of power. 

In the 11-inch blade the tip has a velocity 2.6 times greater than the 
heel; and, by the laws of centrifugal force, the air will have a density 2.6 
times greater at the tip of the blade than that at the heel. The air cannot 
enter on the heel with a density higher than that of the atmosphere; but in 
its passage along the vane it becomes compressed in proportion to its 
centrifugal force. The greater the length of the vane, the greater will be 
the difference of the centrifugal force between the heel and the tip of the 
blade; consequently the greater the density of the air. 

Reasoning from these experiments, Mr. Buckle recommends for easy ref- 
erence the following proportions for the construction of the fan: 

1. Let. the width of the vanes be one fourth of the diameter; 2. Let the 
diameter of the inlet openings in the sides of the fan-chest be one half the 
diameter of the fan; 3. Let the length of the vanes be one fourth of the 
diameter of the fan. 

In adopting this mode of construction, the area of the inlet openings in 
the sides of the fan-chest will be the same as the circumference of the heel 
of the blade, multiplied by its width; or the same area as the space 
described by the heel of the blade. 

Best Proportions of Fans. (Buckle.) 

Pressure from 3 ounces to 6 ounces per square inch; or 5.2 inches 
to 10.1 inches of Water. 



Diameter 


Vanes. 


Diameter 
of Inlet 
Open- 
ings. 


Diameter 
of Fan. 


Vanes. 


Diameter 
of Inlet 
Open- 
ings. 




Width. 


Length. 


Width. 


Length. 


ft. ins. 

3 

3 6 

4 


ft. ins. 
9 

10 J* 

1 


ft. ins. 

9 

10% 

1 


ft. ins. 

1 6 

1 9 

2 


ft. ins. 

4 6 

5 

6 


ft. ins. 
/ 6 


ft. ins. 

i m 

1 3 
1 6 


ft. ins. 

2 3 

2 6 

3 



Pressure from 6 ounces to 9 ounces per square inch, and upwards, 
or 10.4 inches to 15.6 inches of Water. 



3 





7 


1 


1 


4 


6 


10% 


1 4% 


1 9 


3 


6 


8)4 


1 Wz 


1 3 


5 





1 o 


1 6 


2 


4 





o 9% 


i sy 2 


1 6 


6 





1 2 


1 10 


2 4 



The dimensions of the above tables are not laid down as prescribed limits, 
but as approximations obtained from the best results in practice. 

Experiments were also made with reference to the admission of air into 
the transit or outlet pipe. By a slide the width of the opening into this pipe 
was varied from 12 to 4 inches. The object of this was to proportion the 
opening to the quantity of air required, and thereby to lessen the power 
necessary to drive the fan. It was found that the less this opening is made, 
provided we produce sufficient blast, the less noise will proceed from the 
fan; and by making the tops of this opening level with the tips of the vane, 
the column of air has little or no reaction on the vanes. 

The number of blades may be 4 or 6. The case is made of the form of 
an arithmetical spiral, widening the space between the case and the revolv- 
ing blades, circumferentially, from the origin to the opening for discharge. 

The following rules deduced from experiments are given in Spretson's 
treatise on Casting and Founding: 

The fan-case should be an arithmetical spiral to the extent of the depth 
of the blade at least. 

The diameter of the tips of the blades should be about double the diameter 
of the hole in the centre; the width to be about two thirds of the radius of 
the tips of the blades, The velocity of the tips of the blades should be rather 



FANS AND BLOWERS. 513 

more than the velocity due to the air at the pressure required, say one 
eighth more velocity. 

In some cases, two fans mounted on one shaft would be more useful than 
one wide one, as in such an arrangement twice the aiva of inlet opening is 
obtained as compared with a single wide fan. Such an arrangement may 
be adopted where occasionally half the full quantity of air is required, as 
one of them may be put out of gear, thus saving power. 

Pressure due to Velocity of the Fan-blades.— " By increas- 
ing the number of revolutions of the fan the head or pressure is increased, 
the law being that the total head produced is equal (in centrifugal fans) to 
twice the height due to the velocity of the extremities of the blades, or 

v i 
H = — approximatelyin practice" (W. P. Trowbridge, Trans. A. S. M. E., 

vii. 536.) This law is analogous to that of the pressure of a jet striking a 
plane surface. T. Hawksley, Proc. Inst. M. E., 1882, vol. lxix.. says: "The 
pressure of a fluid striking a plane surface perpendicularly and then escap- 
ing at right angles to its original path is that due to twice the height h due 
the velocity." 

(For discussion of this question, showing that it is an error to take the 
pressure as equal to a column of air of the height h = -u 2 -j- 2g, see Wolff on 
Windmills, p. 17.) 

Buckle says: " From the experiments it further appears that the velocity 
of the tips of the fan is equal to nine tenths of the velocity a body would 
acquire in falling the height of a homogeneous column of air equivalent to 
the density." D. K. Clark (R. T. & D., p. 924), paraphrasing Buckle, appar 
ently, says: " It further appears that the pressure generated at the circum 
ference is one ninth greater than that which is due to the actual circumfer- 
ential velocity of the fan." The two statements, however, are not in 

harmony, for if v = 0.9 V2gH, H = n a ? _ = 1.234 ~ and not 14 ~. 
0.81 x 2(? 2g 9 2g 

If we take the pressure as that equal to a head or column of air of twice 
the height due the velocity, as is correctly stated by Trowbridge, the para- 
doxical statements of Buckle and Clark— which would indicate that the 
actual pressure is greater than the theoretical— are explained, and the 

v 2 , — , 

formula becomes H- .617— and v - 1.273 VgH = 0.9 \ 2gH, in which H 

is the head of a column producing the pressure, which is equal to twice the 

theoretical head due the velocity of a falling body (orh = —\, multiplied 

by the coefficient .617. The difference between 1 and this coefficient ex- 
presses the loss of pressure due to friction, to the fact that the inner por- 
tions of the blade have a smaller velocity than the outer edge, and probably 
to other causes. The coefficient 1.273 means that the tip of the blade must 
be given a velocity 1.273 times that theoretically required to produce the 
head H. 

To convert the head H expressed in feet to pressure in lbs. per sq. in. 
multiply it by the weight of a cubic foot of air at the pressure and tempera- 
ture of the air expelled from the fan (about .08 lb. usually) and divide by 
144. Multiply this by 16 to obtain pressure in ounces per sq. in. or by 2.035 
to obtain inches of mercury, or by 27.71 to obtain pressure in inches of 
water column. Taking .08 as the weight of a cubic foot of air, 

p lbs. per sq. in. = ,00001066u 2 ; v = 310 \ |T nearly; 

Pi ounces per sq. in. = .0001706-u 2 ; v = 80 \/px " 

p 2 inches of mercury = .00002169u 2 ; v — 220 \ p~ 2 " 

p 3 inches of water = .0002954i> 2 ; v= 60 4-^ " 

in which v = velocity of tips of blades in feet per second. 

Testing the above formula by the experiment of Buckle with the vane 
14 inches long, quoted above, we have p = .00001066w 2 = 9.56 oz. The ex- 
periment grave 9.4 oz. 

Testing it by the experiment of H. I. Snell, given below, in which the 
circumferential speed was about 150 ft. per second, we obtain 3.85 ounces, 
while the experiment gave from 2.38 to 3.50 ounces, according to the amount 
of opening for discharge. The numerical coefficients of the above formulae 
are all based on Buckle's statement that the velocity of the tips of the fan 
js equal to nine tenths of the velocity a body would acquire in falling the 



514 



height of a homogeneous column of air equivalent to the pressure. Should 
other experiments show a different law, the coefficients can be corrected 
accordingly. It is probable that they will vary to some extent with differ- 
ent proportions of fans and different speeds. 

Taking the formula y — 80 Vpi, we have for different pressures in ounces 
per square inch the following velocities of the tips of the blades in feet per 
second: 



p x = ounces per square inch.. 
v = feet per second 



2 3 4 5 
113 139 160 179 1 



10 12 



I 212 226 253 



14 
299 

A rule in App. Cyc. Mech, article " Blowers," gives the following velocities 
of circumference for different densities of blast in ounces: 3, 170; 4, 180; 5, 
195; 6, 205; 7, 215. 

The same article gives the following tables, the first of which shows that 
the density of blast is not constant for a given velocity, but depends on the 
ratio of area of nozzle to area of blades: 



Velocity of circumference, feet per second. 

Area of nozzle -=- area of blades 

Density of blast, oz. per square inch 



150 150 150 170 200 200 220 

2 i y 2 ya % V6 Vs 

12 3 4 4 6 6 



Quantity of Air of a Given Density Delivered by a Fan. 
Total area of nozzles in square feet X velocity in feet per minute corre- 
sponding to density (see table) = air delivered in cubic feet per minute. 



S Velocity, feet 
perTq Ce m. ^ minute ' 

1 5000 

2 7000 

3 8600 

4 10,000 



JESTS' Velocity, feet 

.er?qin. Per m ' n - 

5 11,000 

6 12,250 

7 13,200 

8 14,150 



Density, 
ounces 
per sq. in. 
9 
10 
11 
12 



Velocity, feet 
per minute. 

15,000 
15,800 
16,500 
17,300 

Experiments with Blowers. (Henry I. Snell, Trans. A. S. M. E. 
ix. 51.)— The following tables give velocities of air discharging through an 
aperture of any size under the given pressures into the atmosphere. The 
volume discharged can be obtained by multiplying the area of discharge 
opening by the velocity, and this product by the coefficient of contraction: 
.65 for a thin plate and .93 when the orifice is a conical tube with a conver- 
gence of about 3.5 degrees, as determined by the experiments of Weisbach. 

The tables are calculated for a barometrical pressure of 14.69 lbs. (= 
235 oz.), and for a temperature of 50° Fahr., from the formula V "== \/2gh. 

Allowances have been made for the effect of the compression of the air, 
but none for the heating effect due to the compression. 

At a temperature of 50 degrees, a cubic foot of air weighs .078 lbs., and 
calling g — 32.1602, the above formula may be reduced to 



V x = 60 1/31.5812 X (235 - P) X P, 

where Vi = velocity in feet per minute. 

P — pressure above atmosphere, or the pressure shown by gauge, in oz. 
per square inch. 



Pressure 

per sq. in. 

in inches of 

water. 


Corre- 
sponding 
Pressure in 
oz. per sq. 
inch. 


Velocity 
due the 
Pressure in 
feet per 
minute. 


Pressure 

per sq. in. 

in inches of 

water. 


Corre- 
sponding 
Pressure in 
oz. per sq. 
inch. 


Velocity due 

the Pressure 

in feet per 

minute. 


1/32 


.01817 


696.78 


% 


.36340 


3118.38 


1/16 


.03634 


987.66 


Va 


.43608 


3416.64 


H 


.07268 


1393.75 


Vs 


.50870 


3690.62 


3/16 


.10902 


1707.00 


1 


.58140 


3946.17 


Va 


.14536 


1971.30 


m 


.7267 


4362.62 


5/16 


.18170 


2204.16 


Wz 


.8721 


4836.06 


% 


.21804 


2414.70 


Wa 


1.0174 


5224.98 


Xt 


.29072 


2788.74 


2 


1.1628 


5587.58 



FANS AND BLOWERS. 



515 



Press- 


Velocity 


Press- 


Velocity 


Press- 


Velocity 




Velocity 


ure 


due the 


ure 


due the 


ure 


due the 


Pressure 


due the 


in oz. . 


Pressure 


in oz. 


Pressure 


in oz. 


Pressure 


in oz. 


Pressure 


per sq. 


in ft. per 


persq. 


in ft. per 


per sq. 


in ft. per 


persq. in. 


in ft. per 


inch. 


minute. 


inch. 


minute. 


inch. 


minute. 




minute. 


.25 


2,582 


2.25 


7,787 


5.50 


12,259 


11.00 


17,534 


.50 


3,658 


2.50 


8,213 


6.00 


12,817 


12.00 


18,350 


.75 


4,482 


2.75 


8,618 


6.50 


13,354 


13.00 


19,138 


1.00 


5,178 


3.00 


9,006 


7.00 


13.873 


14.00 


19,901 


1.25 


5,792 


3.50 


9,739 


7.50 


14,374 


15.00 


20.641 


1.50 


6,349 


4.00 


10,421 


8.00 


14,861 


16.00 


21,360 


1.75 


6,861 


4.50 


11,065 


9.00 


15,795 






2.00 


7,338 


5.00 


11,676 


10.00 


16,684 







Pressure in ounces Velocity in feet 
per square inch. per minute. 



.04 
.05 



516.90 
722.64 
895.26 
1033.86 
1155.90 



Pressure in ounces Velocity in feet per 
per square inch. minute. 



12C6.24 
1367.76 



1550.70 
1635.00 



Experiments on a Fan with. Varying Discharge-opening 
Revolutions nearly constant. 



1485 
1465 
1468 
1500 
1426 



OB'S 

•go 


<E 


.2 o 








*g 


H a« 






Ill 


o 
a 

i 


■a e "§ 


eS w 




u 


Sp> 




-O 






"S u ^ 


< 


O 


> 


w 


<J 





3.50 





.80 




6 


3.50 


406 


1.15 


353 


10 


3.50 


676 


1.30 


520 


20 


3.50 


1353 


1.95 


694 


28 


3.50 


1894 


2.55 


742 


36 


3.40 


2400 


3.10 


774 


40 


3.25 


2605 


3.30 


790 


44 


3.00 


2752 


' 3.55 


775 


48 


3.00 


3002 


3.80 


790 


89.5 


2.38 


3972 


4.80 


827 



>,S % «a 3 


Sa 


heore 
min. 
disch 
IH.P 

Press 


CD cS c 

SE » o 


EH 


H 


1048 




1048 


.337 


1048 


.496 


1048 


.66 


1048 


.709 


1078 


.718 


1126 


.70 


1222 


.635 


1222 


.646 


1544 


.536 



The fan wheel was 23 inches in diameter, %% inches wide at its periphery, 
and had an inlet of 12)^ inches in diameter on either side, which was 
partially obstructed by the pulleys, which were 5 9/16 inches in diameter. It 
had eight blades, each of an area of 45.49 square inches. 

The discharge of air was through a conical tin tube with sides tapered at 
an angle of 3^ degrees. The actual area of opening was 7% greater than 
given in the tables, to compensate for the vena contracta. 

In the last experiment, 89.5 sq. in. represents the actual area of the mouth 
of the blower less a deduction for a narrow strip of wood placed across it for 
the purpose of holding the pressure-gauge. In calculating the volume of air 
discharged in the last experiment the value of vena contractu is taken at .80. 



516 



AIR. 



Experiments were undertaken for the purpose of showing the results ob- 
tained by running the same fan at different speeds with the discharge-open- 
ing the same throughout the series. 

The discharge-pipe was a conical tube 8% inches inside diameter at the 
end, having an area of 56.74, which is 7% larger than 53 sq. inches ; therefore 
53 square inches, equal to .368 square feet, is called the area of discharge, as 
that is the practical area by which the volume of air is computed. 

Experiments on a Fan with Constant Discharge-open- 
ing and Varying Speed.— The first four columns are given by Mr. 
Snell, the others are calculated by the author. 



a 

i 

> 


o 
a 

o 
a 

w 

1 


o 

S CO 

2! 

o u 


CO 

o 
ft 

o 


EH ft 

Is 


& ol.e, 

fcl II 
Is 5 


^1 
! Ill 

111 


lit 

it. 
ill 


o 
o 

CO 


s 

a 

Z 
ft 

c 
'5 

5F 


ti 


Ph 


> 


W 


> 


> 


O 


> 


Eh 


H 


600 


.50 


1336 


.25 


60.2 


56.6 


85.1 


3,630 


.182 


73 


800 


.88 


1787 


.70 


80.3 


75.0 


85.6 


4,856 


.429 


61 


1000 


1.38 


2245 


1.35 


100.4 


94. 


85.4 


6,100 


.845 


6?, 


1200 


2.00 


2712 


2.20 


120.4 


113. 


85.1 


7,370 


1.479 


67 


1400 


2.75 


3177 


3.45 


140.5 


133. 


84.8 


8,633 


2.283 


66 


1600 


3.80 


3670 


5.10 


160.6 


156. 


82.4 


9,973 


3.803 


74 


1800 


4.80 


4172 


8.00 


180.6 


175. 


82.4 


11,337 


5.462 


68 


2000 


5.95 


4674 


11.40 


200.7 


195. 


85.6 


12,701 


7.586 


67 



Mr. Snell has not found any practical difference between the efficiencies 
of blowers with curved blades and those with straight radial ones. 

From these experiments, says Mr. Snell, it appears that we may expect to 
receive back 65$ to 75$ of the power expended, and no more. 

The great amount of power often used to run a fan is not due to the fan 
itself, but to the method of selecting, erecting, and piping it. 

(For opinions on the relative merits of fans and positive rotary blowers, 
see discussion of Mr. Snell 's paper, Trans. A. S. M. E, ix. 66, etc.) 

Comparative Efficiency of Fans and Positive Blowers.— 
(H. M. Howe, Trans. A. I. M. E., x. 482.)— Experiments with fans and positive 
(Baker; blowers working at moderately low pressures, under 20 ounces, show 
that they work more efficiently at a given pressure when delivering large 
volumes {i.e., when working nearly up to their maximum capacity) than 
when delivering comparatively small volumes. Therefore, when great vari- 
ations in the quantity and pressure of blast required are liable to arise, the 
highest efficiency would be obtained by having a number of blowers, always 
driving them up to their full capacity, and regulating the amount of blast 
by altering the number of blowers at work, instead of having one or two 
very large blowers and regulating the amount of blast by the speed of the 
blowers. 

There appears to be little difference between the efficiency of fans and of 
Baker blowers when each works under favorable conditions as regards 
quantity of work, and when each is in good order. 

For a given speed of fan, any diminution in the size of the blast-orifice de- 
creases the consumption of power and at the same time raises the pressure 
of the blast ; but it increases the consumption of power per unit of orifice 
for a given pressure of blast. When the orifice has been reduced to the 
normal size for any given fan, further diminishing it causes but 
slight elevation of the blast pressure: and, when the orifice becomes com- 
paratively small, further diminishing it causes no sensible elevation of the 
blast pressure, which remains practically constant, even when the orifice is 
entirely closed. 

Many of the failures of fans have been due to too low speed, to too small 
pulleys, to improper fastening of belts, or to the belts being too nearly ver- 
tical; in brief, to bad mechanical arrangement, rather than to inherent de- 
fects in the principles of the machine. 



FANS AND BLOWERS. 



51? 



If several fans are used, it is probably essential to high efficiency to pro- 
vide a separate blast pipe for each (at least if the fans are of different size 
or speed), while any number of positive blowers may deliver into the same 
pipe without lowering their efficiency. 

Capacity of Fans and Blowers. 

The following tables show the guaranteed air-supply and air-removal of 
leading forms of blowers and exhaust fans. The figures given are often 
exceeded in practice, especially when the blowers and fans are driven at 
higher speeds than stated. The ratings, particularly of the blowers, are 
below those generally given in catalogues, but it was the desire to present 
only conservative and assured practice. (A. R. Wolff on Ventilation.) 



Quantity of Air 


SUPPLIED 


to Buildings by 


Blowers 


op Various Sizes. 








Capacity 
cu. ft. 








Capacity 


Diam- 


Ordinary 


Horse- 


Diam- 


Ordinary 


Horse- 


cu. ft. 


eter of 


Number 


power 


against a 
Pressure 
of 1 ounce 


eter of 


Number 


power 


against a 
Pressure 
of 1 ounce 


Wheel 


of Revs. 


to Drive 


Wheel 


of Revs. 


to Drive 


in feet. 


per mm. 


Blower. 


in feet. 


per mm. 


Blower. 








per sq. in 








per sq. in. 


4 


350 


6. 


10,635 


9 


175 


29 


56,800 


5 


325 


9.4 


17,000 


10 


160 


35.5 


70,340 


6 


275 


13.5 


29,618 


12 


130 


49.5 


102,000 


7 


230 


18.4 


42,700 


11 


110 


66 


139,000 


8 


200 


24 


46,000 


15 


100 


77 


160,000 



If the resistance exceeds the pressure of one ounce per square inch, of 
above table, the capacity of the blower will be correspondingly decreased, 
or power increased, and allowance for this must be made when the distrib- 
uting ducts are small, of excessive length, and contain many contractions 
and bends. 

Quantity of Air moved by an Approved Form of Exhaust Fan, the 
fan discharging directly from room into the atmosphere. 



Diam- 
eter of 
Wheel 
in feet. 


Ordinary 
Number 
of Revs, 
per min. 


Horse- 
power 
to Drive 
Fan. 


Capacity 
in cu. ft. 
per min. 


Diam- 
eter of 
Wheel 
in feet. 


Ordinary 
Number 
of Revs, 
per min. 


Horse- 
power 
to Drive 
Fan. 


Capacity 
in cu. ft. 
per min. 


2.0 
2.5 
3.0 
3.5 


600 
550 
500 

500 


0.50 
0.75 

1.00 
2.50 


5,000 
8,000 
12,000 
20,000 


4.0 
5.0 
6.0 

7.0 


475 
350 
300 
250 


3.50 
4.50 
7.00 
9.00 


28,000 
35,000 
50,000 
80,000 



The capacity of exhaust fans here stated, and the horse-power to drive 
them, are for free exhaust from room into atmosphere. The capacity de- 
creases and the horse-power increases materially as the resistance, resulting 
from lengths, smallness and bends of ducts, enters as a factor. The differ- 
ence in pressures in the two tables is the main cause of variation in the re- 
spective records. The fan referred to in the second table could not be used 
with as high a resistance as one ounce per square inch, the rated resistance 
of the blowers. 



518 



CENTRIFUGAL FANS. 



Pressures, Velocities, Vol nine 
Required, etc. (B. F. 



of Air, Horse-Power 

Sturtevant Co.) 



^ _ ~ >> 

.5 --3 « 

C bjC 

. 3 s a 

COO'" 

ill. 


elocity in feet per minute of Air 
(at 50 °F.) escaping into open air 
through any shaped hole from 
any pipe or reservoir in which 
the Air is compressed. 


ubic feet of Air per minute (at 
50° F.), which may be discharged 
through a proper shaped mouth- 
piece, the diameter of which 
must be 1.362 inches, the area 
being 1 07 square inches. 


&& o 

P.-I.8 

111 I 


!ubic feet of Air per minute that 
may be discharged with one H. 
P., no allowance heing made for 
friction in the blast-machine 
(whatever power that friction 
amounts to must be added). It 
makes no difference how the Air 
is discharged, provided the pres- 
sure is steady, the same as given 
in the first column. 


Number of mouth-pieces de- 
scribed in column 3, required to 
discharge one H. P. of wind, no 
allowance being made for fric- 
tion in the blast-machine. 


Ph 


t> 


O 


< 


# 


+- 


1 


2 


3 


4 


5 


6 


Ya 


2584.80 


17.944 


0.001224 


14662.76 


817.00 


g 


3657.60 


25.400 


0.003463 


7333.70 


288.70 


4482.00 


31.124 


0.005659 


4889.11 


157.08 


t 


5175.00 


35.93 


0.0098 


3666.62 


102.05 


2 


7338.24 


50.96 


0.0278 


1833.00 


35.970 


3 


9006 42 


62.54 


0.0512 


1222.30 


19.540 


4 


10421.58 


72.37 


0.0789 


916.27 


12.660 


5 


11676.00 


81.08 


0.1106 


733.39 


9.015 


6 


12817.08 


89.01 


0.1456 


611.10 


6.867 


7 


13872.72 


96.34 


0.1839 


523.81 


5.440 


8 


14861.16 


103.20 


0.2251 


458.43 


4.440 


9 


15795.06 


109.69 


0.2692 


407.42 


3.715 


10 


16683.51 


115.86 


0.3160 


366.69 


3.165 


11 


17533.50 


121.76 


0.3652 


333.40 


2.738 


12 


18350 34 


127.43 


0.4170 


305.56 


2.398 


13 


19138.26 


132.90 


0.4712 


282.05 


2.136 


. 14 


19900.68 


138.20 


0.5277 


261.91 


1.895 


*15 


20640.48 


143.34 


0.5864 


244.44 


1.705 


16 


21360.00 


148.33 


0.6473 


229.17 


1.545 


17 


22060.80 


153.26 


0.7103 


215.77 


1.408 


18 


22745.40 


157.96 


0.7754 


203.71 


1.290 


19 


23415.00 


162.60 


0.8426 


192.98 


1.187 


20 


24070.80 


167.16 


0.9118 


183.33 


1.097 



* Always give the wind a good wide opening into the furnace or forge ; 
see by this table how much more wind can be discharged with one H. P. at 
low pressure than at high. 

This table shows the great advantage of large tuyeres, large pipes, large 
blower, and slow speed when the nature of the w r ork will admit. 

t Number of forges driven with 1.2 H. P. with Sturtevant blower. 



CENTRIFUGAL FANS. 



519 



Engines, Fans, and Steam-coils combined for the 
Blower System of Heating. (Buffalo Forge Co.) 



a 

H 

Is 


P* 


H > 
o . 

■*8 


° 13 1 

i?© © $ 
'1 fl S & 


O C v 


©^ 

? eg.S o 




8-2 ^ 

•si® 


3*3 


8'5b 
m 


© 




fgls 




5oB 


03 




©■'Wco 


4x3 


52 


450 


8,740 


1,200 


49 x 38 


3.1 


1,000 


12 


4x4 


60 


425 


11,000 


1,525 


51 x 45 


4 


1,200 


15 


5x4 


70 


390 


15,280 


1,700 


52 x 50 


4.5 


1,600 


20 


5x5 


80 


360 


19,900 


2,200 


52 x 56 


6 


2,000 


25 


6x5 


90 


330 


25,900 


2,450 


59 x 74 


7.2 


2,500 


30 


6x6 


100 


290 


32,500 


2,700 


62 x 84 


9.1 


3,000 


35 


7x6 


110 


260 


39,300 


3,200 


69 x 94 


11 


3,500 


42 


7x7 


120 


235 


49,161 


3,900 


79 x 104 


13.5 


4,000 


48 


8x7 


130 


210 


57,720 


4,500 


83 x 111 


15 


4,500 


54 


8x8 


150 


180 


81,120 


5.300 


87 x 133 


20 


5,000 


62 


10 x 9 


170 


165 


101,250 


6,000 


92 x 148 


22 


6,000 


72 



The Sturtevant Steel Pressure- blower, applied to 
Cupola Furnaces. 









.2* 


«H . 




id 


u 


Power Saved by Reducing 


H ® 


— © o 


0) E 


d 


2 ©*! 

© eeq 


5~' 


the Speed and Pressure of 
Blast. 


o o 


c "2 2* 
S =o 


&c© 


o2 








d 

© 

a 

XII 


.« 


Ph 


d 


• ml 0h 
N © 

Oh W 


£ 


s 


s° 


& 


3 © 
O ft 




5~ 


5* 


a 


w 


a 


1 


22 


1200 


*4 


324 


4135 


0.5 














2 


26 


1900 


5.7 


507 


3756 


6 


1 


3445 


5 


o.s 


8100 


4 


0.6 


3 


30 


2880 


8 


768 


3250 


7 


1.8 


3000 


6 


1.5 




5 


1.1 


4 


35 


4130 


10.7 


1102 


3100 


8 


3 


2900 


7 


2 5 




6 


2. 


5 


40 


6178 


14.2 


1646 


2900 


10 


5.5 


2560 


8 


4 




7 


3.3 


6 


46 


8900 


18.7 


2375 


2820 


12 


9.7 


2550 


10 


7 4 




8 


5.3 


7 


53 


12500 


24.3 


3353 


2600 


14 


16. 


23S0 


12 


12.7 


2 inn 


10 


9.4 


8 


60 


1656C 


32 


4416 


2270 


14 


22. 


2100 


12 


16.7 




10 


12.7 


9 


72 


2380C 


43 


6364 


2100 


16 


35. 


1960 


14 128.4 




12 


22.4 


10 


84 


33300 


60 


8880 


1815 


16 


48. 


1700 


14 39.6 




12 


31.7 



*One square inch of blast is sufficient for one forge-fire, or 90 square 
inches area of cupola furnaces. 

The speed given is regulated so as to give the pressure of blast stated in 
ounces per square inch. 

The term " square inches of blast " refers to the area of a proper shaped 
mouth-piece discharging blast into the open air. 

The melting capacity per hour in pounds of iron is made up from an 
average of tests on a few of the best cupolas found, and is reliable in cases 
where the cupolas are well constructed and driven with the greatest force 
of blast given in the table. 

For tables of the steel pressure -blower as applied to forge-fires, and for 
sizes, etc., of other patterns of blowers and exhausters, see catalogue of 
B. F. Sturtevant Co. 

(For other data concerning Cupolas, see Foundry Practice.) 

Diameter of Blast-pipes for Pressure-blowers for Cupola 
Furnaces and Forges. (B. F. Sturtevant Co.) 

The following table has been constructed on this basis, namely : Allowing 
a loss of pressure of y% oz. in the process of transmission through any length 
of pipe of any size as a standard, the increased friction due to lengthening 
the pipe has been compensated for by an enlargement of the pipe sufficient 



520 



AIR. 



to keep the loss still at % oz. The quantities of air in the left-hand column 
of each division indicate the capacity of the given blower when working 
under pressures of 4, 8, 12, and 16 ozs. Thus a No. 6 Blower will force 2678 
cubic ft. of air, at 8 oz. pressure, through 50 ft. of 12J4-in. pipe, with a loss 
of % oz pressure. If it is desired to force the air 300 ft. without an increased 
loss by friction, the pipe must be enlarged to 17*4 hi. diameter. 



Blower No. 1. 



Blower No. 



515 
635 
740 



Lengths of Blast-pipe in Feet. 



50 100 150 200 300 



S15 



Diameter in inches. 



734 vm 

S%. I 9 
9 9% 



1872 
2678 



Lengths of Blast-pipe in Feet. 



50 100 150 200 300 



Diameter in inches. 



10% 


12% 


13M 


13% 


15 


12V, 


14 


15*6 


16 


17M 


1314 


15% 


16% 


1% 


18% 


14% 


16*6 


17y * 


18% 


20*6 



Blower No. 2. 



Blower No. 7. 



7*4 I 8*4 



734 m m 

9 I 9% 1 10*4 

9% 1 10% 1 11 

10% 11 UH 



2592 


12 


1334 


15 15% 

17*4 18*4 


3708 


13% 


15% 


45V2 


1^/8 


17% 
18% 


18% 19% 


5328 


16 


20 21*4 



17% 
1934 



Blower No. 3. 



Blower No. 8. 



720 


7*4 


8*4 


9 


9% 


1030 


834 


9i/o 


1U46 


11 


•1270 


9*6 


1034 


11*4 


11% 


1480 


9% 


11 


12 


M% 



1014 
1134 

13*6 



3312 


13*4 


15% 


16% 


17% 


4738 


1514 


Vi% 


19% 


20% 


5842 


16% 


19% 


2034 


22 


6808 


l'<% 


20*4 


22% 


«% 



18% 



Blower No. 4. 



Blower No. 9. 



1008 


8*4 


9% 

10% 


10*4 


10% 


1442 


W», 


11% 


12i/ 3 


1778 


103/ 8 


11% 


12% 


13% 


2072 


11 


12% 


1334 


14% 



4320 


14% 


17 


183/6 


19% 


6180 


17 


19Vo 


21 V A 


22% 


7620 


18% 


21% 


23% 


243^ 


8880 


19% 


^% 


24% 


26 



21% 
24% 
26% 



Blower No. 5. 



Blower No. 10. 



1440 
2060 
2540 



9% 


10% 


11% 
1334 


12% 


11 


12% 


14*6 


11% 


13% 


14% 


15% 


1234 


14% 


15% 


16% 



15% 



5760 


16% 


19 


20% 


21% 


8240 


18% 


2134 


2334 

25% 


25% 


10160 


20% 


2514 


273/ 8 


11840 


22% 


27% 


29% 



27*4 

29% 
31% 



CENTRIFUGAL FANS. 521 

Centrifugal Ventilators for Mines.— Of different appliances for 
ventilating mines various forms of centrifugal machines having proved their 
efficiency have now i lmost completely replaced all others. Most if not all 
of the machines in use in this country are of this class, being either open- 
periphery fans, or closed, with chimney and spiral casing, of a more or less 
modified Guibal type. The theory of such machines has been demonstrated 
by M.v. Daniel Murgue in " Theories and Practices of Centrifugal Ventilating 
Machines, " translated by A. L Stevenson, and is discussed in a paper by R. 
Van A. Norris, Trans. A. I. M. E. xx. 637. From this paper the following for- 
mulae are taken: 

Let a = area in sq. ft. of an orifice in a thin plate, of such area that its re- 
sistance to the passage of a given quantity of air equals the 
resistance of the mine; 
o = orifice in a thin plate of such area that its resistance to the pas- 
sage of a given quantity of air equals that of the machine; 
Q — quantity of air passing in cubic feet per minute ; 
V= velocity of air passing through a. in feet per second; 
F = velocity of air passing through o in feet per second; 
h = head in feet air -column to produce velocity V; 
h = head in feet air-column to produce velocity V . 

; V2gh; Q = 0.65a Y2gh; 

a = v = equivalent orifice of mine; 

0.65 Y2gh 

or, reducing to water-gauge in inches and quantity in thousands of feet per 
minute, 




Q = 0.65oF ; V = Y2gh - Q = 0.65o V2gh ; 
equivalent orifice of machine. 

The theoretical depression which can be produced by any centrifugal ven- 
tilator is double that due to its tangential speed. The»formula 



in which Tis the tangential speed, Fthe velocity of exit of the air from the 
space between the blades, and H the depression measured in feet of air- 
column, is an expression for the theoretical depression which can be pro- 
duced by an uncovered ventilator; this reaches a maximum when the air 
leaves the blades without speed, that is, V= 0, and H = T 2 -t- 2g. 

Hence the theoretical depression which can be produced by any uncovered 
ventilator is equal to the height due to its tangential speed, and one half- 
that which can be produced by a covered ventilator with expanding 
chimney. 

So long as the condition of the mine remains constant: 

The volume produced by any ventilator varies directly as the speed of 
rotation. 

The depression produced by any ventilator varies as the square of the 
speed of rotation. 

For the same tangential speed with decreased resistance the quantity of 
air increases and the depression diminishes. 

The following table shows a few results, selected from Mr. Norris's paper, 
giving the range of efficiency which may be expected under different cir 
cumstances. Details of these and other fans, with diagrams of the results 
are given in the paper. 



522 



ALU. 



Experiments on Mine-ventilating Fans. 





.2 - 

OS 


"3 . 

& ft 

£,05 


<1 V 

2 3 


2^ 


.3 fa 


Ms 

© 0)« 

*.£& 


0) s 


a 

U 

% 

o . 
ft.il 

o 


S o 


0) 

a 
'5b . 

a I 

>>fa 
§■« 
.2 c 


Out 

> <D 3 


5 




'*"[=! 


s ft 
o 


!S i 


5 o 


r0 » 
SO 


fc 


^O 


l" 


O^M 


fe 


s 




o a 


O 


O 1-1 




K 


Sft 


fa 


fa 




84 


5517 


236,684 


2818 


3040 


4290 


1.80 


67.13 


88.40175.9 


1 £ 


A 


100 


6282 


336,862 


3369 


'3040 


5393 


2.50 


132.70 


155.43 85.4 


1 


111 


6973 


347,396 


3130 


3040 


5002 


3.20 


175.17 


209.64 83.6 


ff 




123 


7727 


394,100 


3204 


3040 


5100 


3.60 


223.56 


295.2175.7 


Bl 


100 


6282 


188,888 


1889 


1520 


3007 


1.40 


41.67 


97.9942.5 


J^ 


130 


8167 


274,876 


2114 


1520 


3366 


2.00 


86.63 


194.95 44.6 


22 


0-j 


59 


3702 


59,587 


1010 


1520 


1610 


1.20 


11.27 


16.76 67.83 




83 


5208 


82,969 


1000 


1520 


1593 


2.15 


27.86 


48.5457.38 




»■] 


40 


3140 


49,611 


1240 


3096 


1580 


0.87 


6.80 


13.8249.2 


32 


70 


5495 


137,760 


1825 


3096 


2507 


2.55 


55.35 


67.44 82.07 




I 


50 


2749 


147,232 


2944 


1522 


5356 


0.50 


11.60 


28.55 


40.63 




E 1 


69 


3793 


205,761 


2982 


1522 


5451 


1.00 


32.42 


45.98 


70.50 


83 


1 


96 


5278 


299,600 


3121 


1522 


5676 


2.15 


101.50 


120.64 


84.10 




H 


200 


7540 


133,198 


666 


746 


1767 


3.35 


70.30 


102.79 68.40 


26.9 


200 


7540 


180,809 


904 


746 


2398 


3 05 


86.89 


129.07,67.30 


38.3 


1 


200 


7540 


209,150 


1046 


746 


2774 


2.80 


92.50 


150.08 61.70 


46.3 


r 


10 


785 


28,896 


2890 


3022 


3680 


0.10 


0.45 


1.30 


35. 






20 


1570 


57,120 


2856 


3022 


3637 


0.20 


1.80 


3.70 


49. 




1 


25 


1962 


66,640 


2665 


3022 


3399 


0.29 


2.90 


6.10 


48. 






30 


2355 


73,080 


2436 


3022 


3103 


0.40 


4.60 


9.70 


47. 


52 


°l 


35 


2747 


94,080 


2688 


3022 


3425 


0.50 


7.40 


15.00 


48. 




40 


3140 


112,000 


2800 


3022 


3567 


0.70 


12.30 


24.90 


49. 






50 


3925 


132,700 


2654 


3022 


3381 


0.90 


18.80 


38.80 


48. 




i 


60 


4710 


173,600 


2893 


3022 


3686 


1.35 


36.90 


66.40 


55. 






70 


5495 


203,280 


2904 


3022 


3718 


1.80 


57.70 


107.10 


54. 




i 


80 


6280 


222,320 


2779 


3022 


3540 


2.25 


78.80 


152.60 


52. 





Type of Fan. Diam. Width. 

A. Guibal, double 20 ft. 6 ft. 

B. Same, only left hand running. 20 6 

C. Guibal 20 6 

D. Guibal 25 8 

E. Guibal, double...., 17^ 4 

F. Capell 12 10 

G. Guibal 25 8 



No. Inlets. 
4 



Diam. Inlets. 

8 ft. 10 in. 

8 10 

8 10 
11 6 

8 

7 
12 



An examination of the detailed results of each test in Mr. Norris's table 
shows a mass of contradictions from which it is exceedingly difficult to draw 
any satisfactory conclusions. The following, he states, appear to be more 
or less warranted by some of the figures : 

1. Influence of the Condition of the Airways on the Fan.— Mines with 
varying equivalent orifices give air per 100 feet periphery-motion of fan, 
within limits as follows, the quantity depending on the resistance of the 
mine : 



Equivalent Cu Ft. Air per 
Orifice. 100 ft. Periphery- 
speed. 



Under 20 sq. ft. 
20 to 30 
30 to 40 
40 to 50 
50 to 60 



1100 to 1700 
1300 to 1800 
1500 to 2500 
2300 to 3500 
2700 to 4800 



Aver- 
age. 

1300 
1600 
2100 
2700 
3500 



Equivalent Cu. Ft. Air per Aver 

Orifice. 100 ft. Periphery- age. 

speed . 

3300 to 5100 4000 

4000 to 4700 4400 

3000 to 5600 4800 



60 to 70 
70 to 80 
80 to 90 
90 to 100 



100 to 1 1 4 5200 to 6200 5700 

The influence of the mine on the efficiency of the fan does not seem to be 
very clear. Eight fans, with equivalent orifices over 50 square feet, give 



CENTRIFUGAL FANS. 523 

pfficieiiv.ies over 70# ; four, with smaller equivalent mine-orifices, give about 
the same figures ; while, on the contrary, six fans, with equivalent orifices of 
over 50 square feet, give lower efficiencies, as do ten fans, all drawing from 
mines with small equivalent orifices. 

It would seem that, on the whole, large airways tend to assist somewhat 
in attaining large efficiency. 

2. Influence of the Diameter of the Fan.— This seems to be practically nil, 
the only advantage of large fans being in their greater width and the lower 
speed required of the engines 

3. Influence of the Width of a Fan— This .appears to be small as regards 
the efficiency of the machine ; but the wider fans are, as a rule, exhausting 
more air. 

4. Influence of Shape of Blades.— This appears, within reasonable limits, 
to be practically nil. Thus, six fans with tips of blades curved forward, 
three fans with flat blades, and one with' blades curved back to a tangent 
with the circumference, all give very high efficiencies- over 70$. 

5. Influence of the Shape of the Spiral Casing.— This appears to be con- 
siderable The shapes of spiral casing in use fall into two classes, the first 
presenting a large spiral, beginning at or near the point of cut-off, and the 
second a circular casing reaching around three quarters of the circumference 
of the fan, with a short spiral reaching to the evasee chimney. 

Fans having the first form of casing appear to give in almost every case 
large efficiencies. 

Fans that have a spiral belonging to the first class, but very much con- 
tracted, give only medium efficiencies. It seems probable that the proper 
shape of spiral casing would be one of such form tnat the air between each 
pair of blades could constantly and freely discharge into the space between 
the fan and casing, the whole being swept along to the evasee chimney. This 
would require a spiral beginning near the point of cut-off, enlarging by 
gradually increasing increments to allow for the slowing of the air caused by 
its friction against the casing, and reaching the chimney with an area such 
that the air could make its exit with its then existing speed— somewhat less 
than the periphery-speed of the fan. 

6. Influence of the Shutter. —This certainly appears to be an advantage, as 
by it the exit area can be regulated to suit the varying quantity of air given 
by the fan, and in this way re-entries can be prevented. It is not uncommon 
to find shutterless fans into the chimneys of which bits of paper may be 
dropped, which are drawn into the fan, make the circuit, and are again 
thrown out. This peculiarity has not been noticed with fans provided with 
shutters. 

7. Influence of the Speed at which a Fan is Run.— It is noticeable that 
most of the fans giving high efficiency were running at a rather high 
periphery velocity. The best speed seems to be between 5000 and 6000 feet 
per minute. 

The fans appear to reach a maximum efficiency at somewhere about the 
speed given, and to decrease rapidly in efficiency when this maximum point 
is passed. 

In discussion of Mr. Norris's paper, Mr. A. H. Storrs says: From the "cu- 
bic feet per revolution " and '' cubical contents of fan-blades," as given in the 
table, we find that the enclosed fans empty themselves from one half to 
twice per revolution, while the open fans are emptied from one and three- 
quarter to nearly three times. This for fans of both types, on mines cover- 
ing the same range of equivalent orifices. One open fan, on a very large 
orifice, was emptied nearly four times, while a closed fan, on a still larger 
orifice, only shows one and one-half times. For the open fans the "cubic 
feet per 100 ft. motion " is greater, in proportion to the fan width and equiv- 
alent orifice, than for the enclosed type. Notwithstanding this apparently 
free discharge of the open fans, they show very low efficiencies. 

As illustrating the very large capacity of centrifugal fans to pass air, i± 
the conditions of the mine are made favorable, a 16-ft. cliam. fan, 4 ft. 6 in. 
wide, at 130 revolutions, passed 360,000 cu. ft. per min., and another, of same 
diameter, but slightly wider and with larger intake circles, passed 500,000 cu. 
ft , the water-gauge in both instances being- about J4 in. 

T. D, Jones says : The efficiency reported in some cases by Mr. Norris is 
larger than I have ever been able to determine by experiment. My own ex- 
periments, recorded in the Pennsylvania Mine Inspectors' Reports from 1875 
to 1881, did not show more than 60$ to 65$, 



524 



DISK FANS. 

Experiments made with a Blackmail Disk Fan, 4 ft. 

diam., by Geo. A. Suter, to determine the volumes of air delivered under 
various conditions, and the power required; with calculations of efficiency 
and ratio of increase of poAver to increase of velocity, by G. H. Babcock. 
(Trans. A. S. M. E., vii. 547) : 



a 

u 
<v 

ft 
> 


3-d ft 
Q 


53 

f 

o 
W 


c3 he"- 
to 


1-1 O • 
° $ v 

•2 8? ft 




O s3 S 

P5 U 


5? . 
§ ^ 

a PL 


a k 

a 8 


>> 

§ as 




25,797 
32,575 
41,929 
47,756 
For 


0.65 
2.29 
4.42 
7.41 

series 














1.682 


440 
534 
612 




1.257 
1.186 
1.146 

1.749 


1.262 
1.287 
1.139 
1.851 


3.523 
1.843 
1.677 
11.140 


5.4 
2.4 
3.97 
4. 




.9553 

1.062 

.9358 








340 


20,372 
26,660 
31,649 
36,543 
For 


0.76 
1.99 
3.86 
6.47 
series 














.7110 


453 
536 
627 




1.332 
1.183 
1.167 
1.761 


1.308 
1.187 
1.155 
1.794 


2.618 
1.940 
1.676 
8.513 


3.55 
3.86 
3.59 
3.63 




.6063 
.5205 
.4802 








340 
430 
534 
570 


9,983 
13,017 
17,018 
18,649 
For 


1.12 
3.17 
6.07 
8.46 
series 


0.28 
0.47 
0.75 

0.87 


'i!265' 
1.242 

1.068 
1.676 


'l'.304 

1.307 
1.096 
1.704 


"2". 837 
1.915 
1.394 
7.554 


'3!93' 
2.25 
3.63 
3.24 


i!65' 

1.74 

1.60 
1.81 


.3939 
.3046 
.3319 
.3027 


330 


8,399 

10,071 

11,157 

For 


1.31 
3.27 
6.00 
series 


0.26 
0.45 
0.75 








'6*31 

3.66 
5.35 


siofr 

4.96 
3.72 


.2631 


437 
516 


1.324 
1.181 
1.563 


1.199 

1.108 
1.329 


3.142 

1.457 
4.580 


.2188 
.2202 



Nature of the Experiments.— First Series: Drawing air through 30 ft. of 
48-in. diam. pipe on inlet side of the fan. 

Second Series: Forcing air through 30 ft. of 48-in. diam. pipe on outlet side 
of the fan. 

Third Series: Drawing air through 30 ft. of 48-in. pipe on inlet side of the 
fan— the pipe being obstructed by a diaphragm of cheese-cloth. 

Fourth Series: Forcing air through 30 ft. of 48-in. pipe on outlet side of fan 
—the pipe being obstructed by a diaphragm of cheese cloth. 

Mr. Babcock says concerning these experiments : The first four experi- 
ments are evidently the subject of some error, because the efficiency is such 
as to prove on an average that the fan was a source of power sufficient to 
overcome all losses and help drive the engine besides. The second series is 
less questionable, but still the efficiency in the first two experiments is larger 
than might be expected. In the third and fourth series the resistance of the 
cheese-cloth in the pipe reduces the efficiency largely, as would be expected. 
In this case the value has been calculated from the height equivalent to tne 
water-pressure, rather than the actual velocity of the air. 

This record of experiments made with the disk fan shows that this kind of 
fan is not adapted for use where there is any material resistance to the flow 
of the air. In the centrifugal fan the power used is nearly proportioned to 
the amount of air moved under a given head, while in this fan the power re- 
quired for the same number of revolutions of the fan increases very mate- 
rially with the resistance, notwithstanding the quantity of air moved is at the 
same time considerably reduced. In fact, from the inspection of the third 
and fourth series of tests, it would appear that the power required is very 
nearly the same for a given pressure, whether more or less air be in motion. 
It would seem that the main advantage, if any, of the disk fan over the cen- 
trifugal fan for slight resistances consists in the fact that the delivery is the 
full area of the disk, while with centrifugal fans intended to move the same 
quantity of air the opening is much smaller, 



DISK FAHS. 



525 



It will be seen by columns 8 and 9 of the table that the power used in- 
creased much more rapidly than the cube of the velocity, as in centrifugal 
fans. The different experiments do not agree with each other, but a general 
average may be assumed as about the cube root of the eleventh power. 

Cubic Feet of Air removed by Exhaust Disk-wheel per 

minute, (Buffalo Forge Co.) 



Number 
of Revo- 
lutions of 
Wheel 

per 
minute, 



Diameter of Wheel. 



!4 Inch. 30 Inch. 36 Inch. 42 Inch. 48 Inch. 54 Inch. ]60 Inch. 72 Inch. 



Amount of Air in cubic feet per minute. 



100.. 
150.. 
200.. 
250.. 
300.. 
350.. 
400.. 
450. . 
500 . 
550. 
600. 
650. . 
700. 



1,307 
1,684 
2,014 
2,375 
2,770 
3,197 
3,656 
4,148 
4.671 
5.221 



3,338 
1,042 



3.594 
1,541 
3,550 

3.621 
7.755 
B,950 



23,420 



4,245 


6,059 


6,405 


9,154 


8,686 


12,410 


11,098 


15,822 


13,641 


19,408 


16,315 


23,147 


19,119 


27,048 


22,055 


31,112 


25,127 


35,338 


28,325 


39,727 


31,518 


44,277 


34,310 


48,992 


36,940 


53,-858 



: 8,387 
12,822 
17,457 
22.292 
^7.:«7 
32.565 
37,997 
43,632 
49,467 
55,152 
60,401 



14,936 
22,926 
31,267 
39.956 
48,996 
58,386 
67,985 
76,900 



Efficiency ©t Disk Fans. — Prof. A. B. W. Kennedy {Industries, Jan. 
17, 1890) made a series of tests on two disk fans, 2 and 3 ft. diameter, known 
as the Verity Silent Air-propeller. The principal results and conclusions 
are condensed below. 

In each case the efficiency of the fan, that is, the quantity of air delivered 
per effective horse-power, increases very rapidly as the speed diminishes, 
so that lower speeds are much more economical than higher ones. On the 
other hand, as the quantity of air delivered per revolution is very nearly 
constant, the actual useful work done by the fan increases almost directly 
with its speed. Comparing the large and small fans with about the same 
air delivery, the former (running at a much lower speed, of course) is much 
the more economical. Comparing the two fans running at the same speed, 
however, the smaller fan is very much the more economical. The delivery 
of air per revolution of fan is very nearly directly proportional to the area 
of the fan's diameter. 

The air delivered per minute by the 3-ft. fan is nearly 12.522 cubic feet 
(R being the number of revolutions made by the fan per minute). For the 
2-ft. fan the quantity is 5.7R cubic feet. For either of these or any other 
similar fans of which the area is A square feet, the delivery will be about 
1.8AR cubic feet. Of course any change in the pitch of the blades might 
entirely change these figures. 

The net H.P. taken up is not far from proportional to the square of the 
number of revolutions above 100 per minute. Thus for the 3-ft. fan the net 
(R - 100)2 n m n ti n ti „ , „„ . (R— 100)2 



H.P. 



, while for the 2-ft. fan the net H.P. is 



200,000 ' "*""" l "' """ ~ *"■ """ """ """ "■■"■ " 10 1,000,000 ' 

The denominators of these two fractions are very nearly proportional in- 
versely to the square of the fan areas or the fourth power of the fan diam- 
eters. The net H.P. required to drive a fan of diameter D feet or area A 
square feet, at a speed of R revolutions per minute, will therefore be ap- 

. . . DKR - 100)2 A i( B _ 10 0)2. 

proximately -^^^ or 'j^^. 

The 2-ft. fan was noiseless at all speeds. The 3-ft. fan was also noiseless 
up to over 450 revolutions per minute. 



526 



Speed of fan, revolutions per minute. 

Net H.P. to drive fan and belt. 

Cubic feet of air per minute 

Mean velocity of air in 3-ft. flue, feet 

per minute 

Mean velocity of air in flue, same 

diameter as fan 

Cu.f t.of air per min.per effective H.P. 
Motion given to air per rev. of fan, ft. 
Pubic feet of a ir per rev, of fan 



Propeller, 
2 ft. diam. 



750 
0.42 



1.7' 
5.58 



676 
0.32 

3,830 

543 

1,220 

11,970 

1.81 

5.66 



577 
0.227 
3,410 



15.000 
1.88 
5.90 



Propeller, 
3 ft. diam. 



576 
1.02 
7,400 

1,046 



7,250 
l."~ 

12. 



459 
0.575 



10,070 
1.7f 
12.6 



373 
0.324 
4,470 



13,800 
1.70 
12.0 



POSITIVE ROTARY BLOWEKS. (P. H. & F. M. Roots.) 



Size number 

Cubic feet per revolution 

Revolutions per minute, 
Smith fires " 

Furnishes blast for Smith 
fires 

Revolutions per minute for J ' 
cupola, melting iron ) 



Size of cupola, inches, 
side lining 



5 
200 



350 300 275 
2 6 10 

to to to 

4 8 14 

275 



225 
24 



30 
200 



4 
13 
150 



■:!: 



Will melt iron per hour, tons- 



375 325 300 275 
18 24 30 36 
to to to to 
24 30 36 42 

1^ 2V 2 3 4% 
to 



175 
47 
to 
67 

170 
to 

250 
42 



42 
100 

to 
150 

70 

to 
100 
150 

to 
200 

50 



135 
137 
to 
175 

72 



to 


to 


to 


2 


3 


4^ 


m 


5^ 


8 



60 2-55's 
12}4 17% 



to 
7 12 16% 22% 

Horse-power required 1 2 3^ 5^ 8 llj^ 17% 27 40 

The amount of iron melted is based on 30.000 cubic feet of air per ton of 
iron. The horse-power is for maximum speed and a pressure of % pound, 
ordinary cupola pressure. (See also Foundry Practice.) 

BLOWING-ENGINES. 



Blast-furnace 


Blowing-engines of the Variable Puppet- 


valve Cut-off Type. (Philada. Engineering Wo 


ks.) 


Diameter 


Diameter 




Shop 


Revolu- 


Displace- 


Maximum 


of 


of 




Weights. 


tions, 




Blast-pres- 


Steam - 


Blowing- 




approxi- 


ordinary 




sure for Reg- 


cylinder. 


cylinder. 




mate. 


speed. 


ordinary 
speed. 


ular Work. 


in. 


in. 


in. 


pounds. 




cubic feet. 


lbs. persq.in. 


28 


66 


36 


80,000 


60 


8,550 


10 


28 


66 


48 


90,000 


50 


9,500 


10 


32 


72 


48 


106.000 


50 


11,308 


12 


36 


72 


48 


130.000 


50 


11,308 


15 


36 


84 


48 


140,000 


50 


15,392 


11 


36 


84 


60 


165,000 


40 


15,392 


11 


42 


84 


48 


165,000 


50 


15,392 


15 


42 


84 


60 


190,000 


40 


15,392 


15 


42 


90 


48 


170,000 


50 


17,700 


13 


42 


90 


60 


195,000 


40 


17,700 


13 


48 


96 


48 


220,000 


50 


20.000 


15 


48 


96 


60 


280.000 


40 


20,000 


15 



The blowing-engines of the country are usually very wasteful of steam. 
by reason of wire-drawing valve-gear, and especially of slow piston-speed. 
The latter is perhaps the greatest and the least recognized of all steam- 
engine defects. Almost any expense to increase the economy of blowing- 
engines is warranted. (A. L. Holley, Trans. A. I. M. E., vol. iv. p. 81.) 



STEAM-JET BLOWER, AND EXHAUSTER. 



527 



The calculations of power, capacity, etc., of blowing-engines are the same 
as those for air-compressors. They are built without any provision for 
cooling the air during compression. About 400 feet per minute is the usual 
piston-speed for recent forms of engines, but with positive air-valves, which 
have been introduced to some extent, .this speed may be increased. The 
efficiency of the engine, that is, the ratio of the I.H.P. of the air cylinder to 
that of the steam cylinder, is usually taken at 90 per cent, the losses by 
friction, leakage, etc., being taken at 10 per cent. 

STEAM-JET BLOWER AND EXHAUSTER. 

A blower and exhauster is made by L. Schutte & Co., Philadelphia, on 
the principle of the steam-jet ejector. The following is a table of capacities: 



Size 
No. 


Quantity of 
Air per hour 

in 
cubic feet. 


Diameter of 
Pipes in inches. 


Size 

No. 


Quantity of 
Air per hour 

in 
cubic feet. 


Diameter of 
Pipes in inches. 


Steam. 


Air. 


Steam. 


Air. 


000 

00 



1 

2 

3 

4 


1,000 
2,000 
4,000 
6,000 

12.000 

18,000 
24,000 


M 
M 

2 


V 

2 V2 
4 


5 
6 

8 
9 
10 


30,000 
36,000 
42,000 
48,000 
54,000 
60,000 


2^ 
ft 

3 

3^ 


5 
6 
6 

7 
S 



The admissible vacuum and counter pressure, for which the apparatus is 
constructed, is up to a rarefaction of 20 inches of mercury, and a counter- 
pressure up to one sixth of the steam-pressure. 

The table of capacities is based on a steam- pressure of about 60 lbs., and 
a counter-pressure of about 8 lbs. With an increase of steam-pressure or 
decrease of counter-pressure the capacity will largely increase. 

Another steam-jet blower is used for boiler-firing, ventilation, and similar 
purposes where a low counter-pressure or rarefaction meets the require- 
ments. 

The volumes as given in the following table of capacities are under the 
supposition of a steam-pressure of 45 lbs. and a counter-pressure of, say, 
2 inches of water : 





Cubic 


Diameter 


Diameter in 




Cubic 


Diam. 


Diameter in 


Size 

No. 


feet of 


of 


inches of— 


Size 
No. 


feet of 


of 


inches of— 


Aii- 
delivered 


Steam- 
pipe in 




Air de- 
livered 


Steam - 
pipe in 














per hour. 


inches. 


Inlet 


Disch. 




per hour 


inches. 


Tnlet. 


Disch. 


00 


6,000 


% 


4 


3 


4 


250,000 


1 


17 


14 





12,000 


Mi 


5 


4 


6 


500,000 


iy s 


24 


20 


1 


30,000 


y* 


8 


6 


8 


1,000,000 


32 


27 


2 


60,000 


M 


11 


8 


10 


2,000,000 


2 


42 


36 


3 


125,000 


l 


14 


10 













The Steam-jet as a Means for Ventilation.— Between 1810 
and 1850 the steam-jet was employed to a considerable extent for veutilat- 
ing English collieries, and in 1852 a committee of the House of Commons 
reported that it was the most powerful and at the same time the cheapest 
meihod for the ventilation of mines ; but experiments made shortly after- 
wards proved that this opinion was erroneous, and that furnace ventilation 
was less than half as expensive, and in consequence the jet was soon aban- 
doned as a permanent method of ventilation. 

For an account of these experiments see Colliery Engineer, Feb. 1890. 
The jet, however, is sometimes advantageously used as a substitute, for 
instance, in the case of a fan standing for repairs, or after an explosion, 
when the furnace may not be kept going, or in the case of the fan having 
been rendered useless. 



528 HEAT1H0 AND VtfNTILATtOH. 



HEATING AND VENTILATION. 

Ventilation. (A. R. Wolff, Stevens Indicator, April, 1890.)— The pop- 
ular impression that the impure air falls to the bottom of a crowded room 
is erroneous. There is a constant mingling of the fresh air admitted with 
the impure air due to the law of diffusion of gases, to difference of temper- 
ature, etc. The process of ventilation is one of dilution of the impure ?ir 
by the fresh, and a room is properly ventilated in the opinion of the hygien- 
ists when the dilution is such that the carbonic acid in the air does not ex- 
ceed from 6 to 8 parts by volume in 10,000. Pure country air contains about 
4 parts C0 2 in 10,000. and badly-ventilated quarters as high as 80 parts. 

An ordinary man exhales 0.6 of a cubic foot of C0 2 per hour. New York 
gas gives out 0.75 of a cubic foot of C0 2 for each cubic foot of gas burnt. 
An ordinary lamp gives out 1 cu. ft. of C0 2 per hour. An ordinary candle 
gives out 0.3 cu. ft. per hour. One ordinary gaslight equals in Vitiating 
effect about 5J^ men, an ordinary lamp 1% men, and an ordinary candle ^ 
man. 

To determine the quantity of air to be supplied to the inmates of an un- 
lighted room, to dilute the air to a desired standard of purity, we can estab- 
lish equations as follows: 
Let v = cubic feet of fresh air to be supplied per hour; 

r = cubic feet of C0 2 in each 10,000 cu. ft. of the entering air: 

R = cubic feet of C0 2 which each 10,000 cu. ft. of the air in the room 

may contain for proper health conditions; 
n = number of persons in the room; 
.6 = cubic feet of C0 2 exhaled by one man per hour. 

Then ■ -J- .6?i equals cubic feet of C0 2 communicated to the room dur- 
ing one hour. 

This value divided by v and multiplied by 10,000 gives the proportion of 
C0 2 in 10,000 parts of the air in the room, and this should equal B, the stan- 
dard of purity desired. Therefore 



or the quantity of air to be supplied per person is 3000 cubic feet per hour. 

If the original air in the room is of the purity of external air, and the cubic 
contents of the room is equal to 100 cu. ft. per inmate, only 3000 - 100 = 2900 
cu. ft. of fresh air from without will have to be supplied the first hour to 
keep the air within the standard purity of 6 parts of C0 2 in 10,000. If the 
cubic contents of the room equals 200 cu. ft. per inmate, only 3000 — 200 = 2800 
cu. ft. will have to be supplied the first hour to keep the air within the 
standard purity, and so on. 

Again, if we only desire to maintain a standard of purity of 8 parts of 
carbonic acid in 10,000, equation (1) gives as the required air-supply per hour 

v = -£ — -n = 1500n, or 1500 cu. ft. of fresh air per inmate per hour. 

Cubic feet of air containing 4 parts of carbonic acid in 10,000 necessary per 
person per hour to keep the air in room at the composition of 

6 7 8 9 10 15 20 j parte of rarbonic acid in 

3000 2000 1500 1200 1000 545 375 cubic feet. 

If the original air in the room is of purity of external atmosphere (4 parts 
of carbonic acid in 10,000), the amount of air to be supplied the first hour, 
for given cubic spaces per inmate, to have given standards of purity not 
exceeded at the end of the hour is obtained from the following table : 



10 ' 000 Llo^oo- + -H ] ftPfl= S9222L. . 


. . . (1) 
. . . (2) 


v E — r 

and B at 6, v - „ r n - 3000?i, 


b — 4 





VENTILATION. 



529 



Cubic Feet 

of 

Space 

in Room 


Proportion of Carbonic Acid in 10,000 Parts of the Air 
be Exceeded at End of Hour. 


not to 


6 


7 


8 


9 


10 


15 


20 


Individual. 


Cubic Fe 


3t of Air, of Composition 4 Parts of Carbonic Acid in 
10,000, to be Supplied the First Hour. 


100 
200 
300 
400 
500 
600 
700 


2900 

2800 
2700 
2600 
2500 
2400 
2300 
2200 
2100 
2000 
1500 
1000 
500 


1900 
1800 
1700 
1600 
1500 
1400 
1300 
1200 
1100 
1000 
500 
None 


1400 
1300 
1200 
1100 
1000 
900 
800 
700 
600 
500 
None 


1100 
1000 
900 
800 
700 
600 
500 
400 
300 
200 
None 


900 
800 
700 
600 
500 
400 
300 
200 
100 
None 


445 
345 
245 
145 
45 
None 


275 

175 

75 

None 


800 






900 






1000 
1500 






2000 








2500 

























It is exceptional that systematic ventilation supplies the 3000 cubic feet 
per inmate per hour, which adequate health considerations demand. Large 
auditoriums in which the cubic space per individual is great, and in which 
the atmosphere is thoroughly fresh before the rooms are occupied, and the 
occupancy is of two or three hours' duration, the systematic air-supply may 
be reduced, and 2000 to 2500 cubic feet per inmate per hour is a satisfactory 
allowance. 

Hospitals where, on account of unhealthy excretions of various kinds, the 
air-dilution must be largest, an air-supply of from 4000 to 6000 cubic feet per 
inmate per hour should be provided, and this is actually secured in some 
hospitals. A report dated March 15, 1882, by a commission appointed to 
examine the public schools of the District of Columbia, says : 

" In each class-room not less than 15 square feet of floor-space should be 
allotted to each pupil. In each class-room the window-space should not be 
less than one fourth the floor-space, and the distance of desk most remote 
from the window should not be more than one and a half times the height of 
the top of the window from the floor. The height of the class-room should 
never exceed 14 feet. The provisions for ventilation should be such as to 
provide for each person in a class-room not less than 30 cubic feet of fresh 
air per minute (1800 per hour), which amount must be introduced and 
thoroughly distributed without creatiug unpleasant draughts, or causing any 
two parts of the room to differ in temperature more than 2° Falir., or the 
maximum temperature to exceed 70° Fahr." 

When the air enters at or near the floor, it is desirable that the velocity of 
inlet should not exceed 2 feet per second, which means larger sizes of 
register openings and flues than are usually obtainable, and much higher 
velocities of inlet than two feet per second are the rule in practice. The 
velocity of current into vent-flues can safely be as high as 6 or even 10 feet 
per second, without being disagreeably perceptible. 

The entrance of fresh air into a room is co-incident with, or dependent on, 
the removal of an equal amount of air from the room. The ordinary means 
of removal is the vertical vent-duct, rising to the top of the buildingr. Some- 
times reliance for the production of the current in this vent-duct is placed 
solely on the difference of temperature of the air in the room and that of 
the external atmosphere: sometimes a steam coil is placed within the flue 
near its bottom to heat the air within the duct; sometimes steam pipes 
(risers and returns) run up the duct performing the same functions; or steam 
jets within the flue, or exhaust fans, driven by steam or electric power, act 
directly as exhausters; sometimes the heating of the air in the flue is ac- 
complished by gas-jets. 

The draft of such a duct is caused by the difference of weight of ths 



530 



HEATING AND VENTILATION. 



heated air in the duct, and a column of equal height and cross-sectional area 
of weight of the external air. 

Let d — density, or weight in pounds, of a cubic foot of the external air. 

Let d 1 = density, or weight in pounds, of a cubic foot of the heated air 
within the duct. 

Let h = vertical height, in feet, of the vent-duct. 

h(d — d,) = the pressure, in pounds per square foot, with which the air is 
forced into and out of the vent-duct. 

This pressure can be expressed in height of a column of the air of density 

within the vent-duct, and evidently the height of such column of equal 

... ,, h(d — d t ) /Q , 

presssure would be_, — - (o) 

Or, if t = absolute temperature of external air, and t x = absolute temper- 
ature of the air in vent-duct in the form, then the pressure equals 

h(t 1 - t), 



(4) 

The theoretical velocity, in feet per second, with which the air would 
travels through the vent-duct under this pressure is 



.y^pi'^-V'^ <»> 

The actual velocity will be considerably less than this, on account of loss 
due to friction. This friction will vary with the form and cross-sectional 
area of the vent duct and its connections, and with the degree of smooth- 
ness of its interior surface. On this account, as well as to prevent leakage 
of air through crevices in the wall, tin lining of vent-flues is desirable. 

The loss by friction may be estimated at approximately 50$, and so we find 
for the actual velocity of the air as it flows through the vent-duct : 



v = -a/ 2gh- 



(*i - t) 



or, approximately, 



v= A a/ h- 



[t t - t) 



(6) 



If V= velocity of air in vent-duct, in feet per minute, and the external air 
be at 32° Fahr., since the absolute temperature on Fahrenheit scale equals 
thermometric temperature plus 459.4, 



= 240^-^! 



(7) 



from which has been computed the following table : 

Quantity of Air, in Cubic Feet, Discharged per Minute 
through a Ventilating J>uct, of which the Cross-sec- 
tional Area is One Square Foot (the External Tempera- 
ture of Air being 32° Fahr.). 





Excess of Temperature of Air in Vent-duct above that of 


Height of 








External Air. 






















feet. 






















5° 


10° 


15° 


20° 


25° 


30° 


50° 


100° 


150° 


10 


77 


108 


133 


153 


171 


188 


242 


342 


419 


15 


94 


133 


162 


188 


210 


230 


297 


419 


514 


20 


108 


153 


188 


217 


242 


265 


342 


484 


593 


25 


121 


171 


210 


242 


271 


297 


383 


541 


663 


30 


133 


188 


230 


265 


297 


325 


419 


593 


726 


35 


143 


203 


248 


286 


320 


351 


453 


640 


784 


40 


153 


217 


265 


306 


342 


375 


484 


656 


838 


45 


162 


230 


282 


325 


363 


398 


514 


476 


889 


50 


1 171 


242 


297 


342 


383 


419 


541 


278 


937 



Multiplying the figures in above table by 60 gives the cubic feet of air dis- 
charged per hour per square foot of cross-section of vent-duct. Knowing 



MINE-VENTILATION". 531 

the cross-sectional area of vent-ducts we can find'the total discharge; or 
for a desired air-removal, we can proportion the cross-sectional area of 
vent-ducts required. 

Artificial Cooling of Air lor Ventilation. (Engineering 
News, July 7, 1892.) — A pound of coal used to make steam for a fairly effi- 
cient refrigerating-machine can produce an .actual cooling effect equal to 
that produced by the melting of 16 to 46 lbs. of ice, the amount varying 
with the conditions of working. Or, 855 heat-units per lb. of coal converted 
into work in the refrigerating plant (at the rate of 3 lbs. coal per horse- 
power hour) will abstract 2275 to 6545 heat-units of heat from the refriger- 
ated body. If we allow 2000 cu. ft. of fresh air per hour per person as suffi- 
cient for fair ventilation, with the air at an initial temperature of 80° F., its 
weight per cubic foot will be .0736 lb.; hence the hourly supply per person 
will weigh 2000 x .0736 lb. = 147.2 lbs. To cool this 10°, the specific heat of 
air being 0.238. will require the abstraction of 147.2 X 0.238 X 10 - 350 heat- 
units per person per hour. 

Taking the figures given for the refrigerating effect per pound of coal as 
above stated, and the required abstraction of 350 heat-units per person per 
hour to have a satisfactory cooling effect, the refrigeration obtained from a 
pound of coal will produce this cooling effect for 2275 -h- 350 = 6J^ hours with 
the least efficient working, or 6545 -=- 350 = 18.7 hours with the most efficient 
working. With ice at $5 per ton, Mr. Wolff computes the cost of cooling with 
ice at about $5 per hour per thousand persons, and concludes that this is too 
expensive for any generaluse. With mechanical refrigeration, however, if 
we assume 10 hours' cooling per person per pound of coal as a fair practical 
service in regular w : ork, we have an expense of only 15 cts. per thousand 
persons per hour, coal being estimated at S3 per short ton. This is for fuel 
alone, and the various items of oil. attendance, interest, and depreciation on 
the plant, etc., must be considered in making up the actual total cost of 
mechanical refrigeration. 

Mine-ventilation— Friction of Air in Underground Pas- 
sages.— In veuiilaiin^ a mine or other underground passage the resistance 
to be overcome is, according to most writers on the subject, proportional to 
the extent of the f rictional surface exposed ; that is, to the product lo of the 
length of the gangway by its perimeter, to the density of the air in circula- 
tion, to the square of its average speed, v, and lastly to a coefficient k, whose 
numerical value varies according to the nature of the sides of the gangway 
and the irregularities of its course. 

The formula for the loss of head, neglecting the variation in density as 

unimportant, is p = , in which p — loss of pressure in pounds per square 

foot, s = square feet of»rubbing-surface exposed to the air, v the velocity of 
the air in feet per minute, a the area of the passage in square feet, and k the 
coefficient of friction. W. Fairley, in Colliery Engineer, Oct. and Nov. 
1893, gives the following formulas for all the quantities involved, using the 
same notation as the above, with these additions : /*, = horse-power of ven- 
tilation; I = length of air-channel; o = perimeter of air-channel; q = quan- 
tity of air circulating in cubic feet per minute; u — units of work, in foot- 
pounds, applied to circulate the air: w — water-gauge in inches. Then, 



_ lesv 2 _ ksv^q _ ksv 3 



v — pv 
5 2qiv 



> 2 sv 3 sv 2 ■+■ a sv 2 -*- a 

pa 
~ kv 2 o ' 
_ pa 
~'kvH' 

v* _ u _ 5 n w _ I / 2l\ ks — fcs« 3 _ _M_ 
T ~q " \f ks) « 3 a>v 



532 



HEATING AND VENTILATION. 



7. pa = ksv* = [ i/~\ ks = !?; pa3 = fcsg*. 

a u ksv 3 /pa / u 

p p Y ks y ks 

9 s = P?L = _ — _ 3P_ _ 2^a _ Zo 
fcv 2 ~~ kv 3 kv 3 kv 3 

10. w = qp = vpa = SV q = fcsv' = o.2qw = 33,0007i. 
pa a y fcs y fcs y fcs 

fcs fcs fcs 
. . p fcsv 3 

14 ' W = -572 = ^a 

To find the quantity of air with a given horse-power and efficiency (e) of 
engine: 

h X 33,000 X e 

q = i — • 

The value of fc, the coefficient of friction, as stated, varies according to 
the nature of the sides of the gangway. Widely divergent values have been 
given by different authorities (see Colliery Engineer, Nov. 1893), the most 
generally accepted one until recently being probably that of J. J. Atkinson, 
.0000000217, which is the pressure per square foot in decimals of a pound for 
each square foot of rubbing-surface and a velocity of one foot per minute. 
Mr. Fairley, in his " Theory and Practice of Ventilating Coal-mines, ,, gives a 
value less than half of Atkinson's, or .00000001 ; and recent experiments by D. 
Murgue show that even this value is high under most conditions. Murgue's 
results are given in his paper on Experimental Investigations in the Loss of 
Head of Air-currents in Underground Workings, Tnans. A. I. M. E., 1893. 
vol. xxiii. 63. His coefficients are given in the following table, as determined 
in twelve experiments: 

Coefficient of Loss of 

Head by Friction. 
French. British. 

f Straight, normal section 00092 .000,000,00486 

Rock. J Straight, normal section 00094 .000,000,00497 

gangways. | Straight, large section 00104 .000,000,00549 

[Straight, normal section 00122 .000,000,00645 

f Straight, normal section 00030 .000,000,00158 

Brick-lined | Straight, normal section .00036 . 000,000,00190 

arched -( Continuous curve, normal section 00062 .000,000,00328 

gangways. | Sinuous, intermediate section 00051 .000,000,00269 

(.Sinuous, small section 00055 .000,000,00291 

rr- k *a ( Straight, normal section 00168 .000,000,00888 

nmoerea J straight, normal section 00144 .000,000.00761 

gangways. ( slightly sinuous, small section 00238 .000,000,01257 

The French coefficients which are given by Murgue represent the height 
of water-gauge in millimetres for each square metre of rubbing-surface and 
a velocity of one metre per second. To convert them to the British measure 
of pounds per square foot for each square foot of rubbing-surface and a 
velocity of one foot per minute they have been multiplied by the factor of 
conversion, .000005283. For a velocity of 1000 feet per minute, since the loss 
of head varies as v a , move the decimal point in the coefficients six places to 
the right. 



FANS AND HEATED CHIMNEYS FOR VENTILATION. 533 

Equivalent Orifice.— The head absorbed by the working-chambers 
of a mine cannot be computed a priori, because the openings, cross-pas- 
sages, irregular-shaped gob-piles, and daily changes in the size and shape of 
the chambers present much too complicated a network for accurate 
analysis. In order to overcome this difficulty Murgue proposed in 1872 the 
method of equivalent orifice. This method consists in substituting for the 
mine to be considered the equivalent thin-lipped orifice, requiring' the same 
height of head for the discharge of an equal volume of air. The area of 
this orifice is obtained when the head and the discharge are known, by 
means of the following formulae, as given by Fairley: 
Let Q = quantity of air in thousands of cubic feet per minute; 
to — inches of water-gauge; 
A — area in square feet of equivalent orifice. 
Then 

A = ±KQ = __L ; * _ AX_VK. _ / Qy 

Vio 2.7 Via y_ 0.37 ' w ~ °" 3C9 X \a ' ' 

Motive Column or the Head of Air Due to Differences 
of Temperature, etc. (Fairley.) 
Letii = motive column in feet; 

T = temperature of upcast; 

/ = weight of one cubic foot of the flowing air; 

t = temperature of downcast; 

D = depth of downcast. 

Then 

To find diameter of a round airway to pass the same amount of air as a 
square airway the length and power remaining the same: 

Let D = diameter of round airway, A = area of s quare airway; O = peri- 
b / ~A 3 X 3.1416 
meter of square airway. Then D 3 — 4/ .7S54 3 x O ' 

If two fans are employed to ventilate a mine, each of which when worked 
separately produces a certain quantity, which may be indicated by A and B 
then the quantity of air that will pass wLien the two fans are worked together 
will be a/A 3 -f B 3 . (For mine-ventilating fans, see page 521.) 

Relative Efficiency of Fans and Heated Chimneys for 
Ventilation.— W. P. Trowbridge, Trans. A. S. M. E. vii. 531, gives a theo- 
retical solution of the relative amounts of heat expended to remove a given 
volume of impure air by a fan and by a chimney. Assuming the total effi- 
ciency of a fan to be only 1/25, which is made up of an efficiency of 1/10 for 
the engine, 5/10 for the fan itself, and 8/10 for efficiency as regards friction, 
the fan requires an expenditure of heat to drive it of only 1/38 of the amount 
that would be required to produce the same ventilation by a chimney 100 ft. 
high. For a chimney 500 ft. high the fan will be 7.6 times more efficient. 

In all cases of moderate ventilation of rooms or buildings where the air 
is heated before it enters the rooms, and spontaneous ventilation is pro- 
duced by the passage of this heated air upwards through vertical flues, 
no special heat is required for ventilation; and if such ventilation be suffi- 
cient, the process is faultless as far as cost is concerned. This is a condition 
of things which may be realized in most dwelling-houses, and in many halls, 
schoolrooms, and public buildings, provided inlet and outlet fines of ample 
cross-section be provided, and the heated air be properly distributed. 

If a more active ventilation be demanded, but such as requires the small- 
est amount of power, the cost of this power may outweigh the advantages 
of the fan. There are many cases in which steam-pipes in the base of a 
chimney, requiring no care or attention, may be preferable to mechanical 
ventilation, on the ground of cost, and trouble of attendance, repairs, etc. 

* Murgue gives A — _ -, and Norris A = — — rdr. See page 521 , ante. 



534 HEATING AND VENTILATION. 

The following' figures are given by Atkinson (Coll. Engr., 1889), showing 
the minimum depth at which a furnace would be equal to a ventilating- 
machine, assuming that the sources of loss are the same in each case, i.e., 
thar, the loss of fuel in a furnace from the cooling in the upcast is equivalent 
to the power expended in overcoming the friction in the machine, and also 
assuming that the ventilating-machine utilizes 60% of the engine-power. The 
coal consumption of the engine per I.H.P. is taken at 8 lbs. per hour: 

Average temperature in upcast 100° F. 150° F. 200° F: 

Minimum depth for equal economy... 960 yards. 1040 yards. 1130 yards. 

Heating and Ventilating of Large Buildings. (A. R. 

Wolff, Jour. Frank. Inst., 1893.)— The transmission of heat from the interior 
to the exterior of a room or building, through the walls, ceilings, windows, 
etc., is calculated as follows : 

S = amount of transmitting surface in square feet; 
t = temperature F. inside, / = temperature outside; 
K = a coefficient representing, for various materials composing buildings, 
the loss by transmission per square foot of surface in British ther- 
mal units per hour, for each degree of difference of temperature 
on the two sides of the material ; 
Q — total heat transmission — SK (t - t ). 

This quantity of heat is also the amount that must be conveyed to the 
room in order to make good the loss by transmission, but it does not cover 
the addiiional heat to be conveyed on account of the change of air for pur- 
poses of ventilation. The coefficients K given below are those prescribed by 
law by the German Government in the design of the heating plants of its 
public buildings, and generally used in Germany for all buildings. They 
have been converted into American units by Mr. Wolff, and he finds that 
they agree well with good American practice: 

Value of K for Each Square Foot of Brick Wall. 

Th brick e wall f [ 4 " 8 " 12 " 16 " 20 " 24 " 28 " 32 " 36 " 40 " 
'£=0.68 0.46 0.32 0.26 0.23 0.20 0.174 0.15 0.129 0.115 

1 sq. ft., wooden-beam construction, ) as flooring, K — 0.083 

planked over or ceiled, j as ceiling-, £"=0.104 

1 sq. ft., fireproof construction, i as flooring, K = 0.1 -J4 

floored over, j as ceiling, K— 0.145 

1 sq. ft., single window K = 0.776 

1 sq. ft., single skylight £=1.118 

1 sq. ft. , double window K = 0.518 

1 sq. ft., double skylight K = 0.621 

1 sq. ft., door K = 0.414 

These coefficients are to be increased respectively as follows: 10% when the 
exposure is a northerly one, and winds are to be counted on as important 
factors; 10% when the building is heated during the daytime only, and the 
location of the building is not an exposed one; 30% when the building is 
heated during the daytime only, and the location of the building is exposed; 
50% when the building is heated during the winter months intermittently, 
with long intervals (say days or weeks) of non-heating. 

The value of the radiating-surface is about as follows: Ordinary bronzed 
cast-iron radiating-surfaces, in American radiators (of Bundy or similar 
type), located in rooms, give out about 250 heat-units per hour for each 
square foot of surface, with ordinary steam-pressure, say 3 to S lbs. per sq. 
in., and about 0.6 this amount with ordinary hot-water heating. 

Non-painted radiating-surfaces, of the ordinary "indirect" type (Climax 
or pin surfaces), give out about 400 heat-units per hour for each square foot 
of heating-surface, with ordinary steam-pressure, say 3 to 5 lbs. per sq. in.; 
and about 0.6 this amount with ordinary hot-water heating. 

A person gives out about 400 heat-units per hour; an ordinary gas-burner, 
about 4800 heat-units per hour; an incandescent electric (16 candle-power) 
lij-rht, about 1600 heat-units per hour. 

The following example is given by Mr. Wolff to show the application of 
the formula and coefficients: 

Lecture-room 40 X 60 ft., 20 ft. high, 48,000 cubic feet, to be heated to 
69° F.; exposures as follows: North wall, 60 x 20 ft., with four windows, 
each 14 x 4 feet, outside temperature 0° F. Room beyond west wall and 



HEATING AND VENTILATING OF LARGE BUILDINGS. 535 



room overhead heated to 69°, except a double skylight in ceiling, 14 X 24 ft., 
exposed to the outside temperature of 0°. Store-room beyond east wall at 
30°. Door X 12 ft. in wall. Corridor beyond south wall heated to 59°. 
Two doors, 6 X 12, in wall. Cellar below, temperature 36°. 
The following table shows the calculation of heat transmission: 



fa © 

1^ 


Kind of Transmitting 
Surface. 


o5.S • 
Ho 


Calculation 

of Area of 

Transmitting 

Surface. 


Is 

w ° 


7 
k 


"3 . 
SB 

Eh 


69° 
69 
33 
33 


Outside wall ... 

Four windows (single) 

Inside wall (store-room) 


36" 
36" 
24" 
36" 


63 X 22 - 448 

4x 8X 14 
42X22- 72 

6X12 
45X22- 72 

6X12 
17X22- 72 

6X 12 

32 X 42 - 336 
14X24 
62 X 42 


938 
448 
852 

72 
918 

72 
302 

72 

1,008 

336 

2,604 


9 

72 
4 

19 
2 
5 
1 
5 

10 

43 
4 


8,442 

32,256 

3,408 

1,368 

1,836 

360 

302 

360 

10,080 

14,448 

10,416 


10 
10 
10 
10 
69 
69 


Inside wall (corridor) 

Door 

Inside wall (corridor) 

Door 

Roof 


33 






Supplementary allowance, r 
" " i 

Exposed location and intern 
Total thermal units 












orth outside wall, 1C 
lorth outside windov 

littent day or night 


% 


83,276 
844 




ps, 10% 

jse, 30$ . . . . 


3,226 
87,346 
26.204 












113.550 



If we assume that the lecture- room must be heated to 69 degrees Fahr. in 
the daytime when unoccupied, so as to be at this temperature when first 
persons arrive, there will be required, ventilation not being considered, and 
bronzed direct low-pressure steam-radiators being the heating media, about 
113,550 -h 250 = 455 sq. ft. of radiating-surface. (This gives a ratio of about 
105 cu. ft. of contents of room for each sq. ft. of heating-surface.) 

If we assume that there are 160 persons in the lecture-room, and we pro- 
vide 2500 cubic feet of fresh air per person per hour, we will supply 160 X 

2500 = 400,000 cubic feet of air per hour (i.e., ' - over eight changes of 

contents of room per hour). 

To heat this air from 0° Fahr. to 69° Fahr. will require 400,000 X 0.0189 X 
69 = 521,640 thermal units per hour (0.0189 being the product of a weight of 
a cubic foot by the specific heat of air). Accordingly there must be provided 
521,640 h-400 = 1304 sq. ft. of indirect surface, to heat the air required for 
ventilation, in zero weather. If the room were to be warmed entirely indi- 
rectly, that is, by the air supplied to room (including the heat to be'conveyed 
to cover loss by transmission through walls, etc.), there would have to be 
conveyed to the fresh-air supply 521,640 + 113,550 = 635,190 heat-units. This 
would imply the provision of an amount of indirect heating-surface of the 
" Climax " type of 635,190 -*- 400 = 1589 sq. ft., and the fresh air entering the 
room would have to be at a temperature of about 84° Fahr., viz., 69° = 

The above calculations do not, however, take into account that 160 per- 
sons in the lecture-room give out 160 X 400 = 64.000 thermal units per hour; 
and that, say, 50 electric lights give out 50 x 1600 — 80,000 thermal units per 
hour; or, say, 50 gaslights. 50 X 4800 = 240,000 thermal units per hour. The 
presence of 160 people and the gas-lighting would diminish considerably the 
amount of heat required. Practically, it appears that the heat generated 
by the presence of 160 people, 64,000 heat-units, and by 50 electric lights, 
80,000 heat-units, a total of 144,000 heat-units, more than covers the amount 
of heat transmitted through walls, etc. Moreover, that if the 50 gaslights 
give out 240,000 thermal units per hour, the air supplied for ventilation must 
enter considerably below 69° Fahr., or the room will be heated to an 
unbearably high temperature. If 400,000 cubic feet of fresh air per hour 



536 



HEATIHG AHt) VEHTILATIOK. 



are supplied, and 240,000 thermal units per hour generated by the gas must 
be abstracted, it means that the air must, under these conditions, enter 

400 000 V 0189 = about 3 ~° less tnan 84°, or at about 52° Fahr. Further- 
more, the additional vitiation due to gaslighting would necessitate a rmich 
larger supply of fresh air than when the vitiation of the atmosphere by the 
people alone is considered, one gaslight vitiating the air as much as five 
men. 

Various Rules for Computing Radiating-surface.— The 
following rules are compiled from various sources. They are more in the 
nature of "rule-of -thumb " rules than those given by Mr. Wolff, quoted 
above, but they may be useful for comparison. 

Divide the cubic feet of space of the room to be heated, the square feet 
of wall surface, and the square feet of the glass surface by the figures 
given under these headings in the following table, and add the quotients 
together; the result will be the square feet of radiating-surface required. 
(F. Schumann.) 

Space, Wall and Glass Surface which One Square Foot of Radiating- 
surface will Heat. 





2 

It 
k§ 

as a 

m 

1 
3 
5 


o 

'2 

3 
O 

.2 

O D 

ft* - 
m 

190 
210 

225 


Exposure of Rooms. 


0) 


All Sides. 


Northwest. : 


Southeast. 


c3 

.q 
O 

< 


Wall 

Surface, 

sq. ft. 


Glass 

Surface, 

sq. ft. 


Wall 

Surface, 
sq. ft. 


Glass 

Surface, 

sq. ft. 


Wall 

Surface, 
sq. ft. 


Glass 
Surface, 
sq. ft. 


Once 
per 
hour. 


13.8 
15.0 
16.5 


7 

7.7 

8.5 


15.87 
17.25 
18.97 


8.05 
8.85 
9.77 


16.56 
18.00 
19.80 


8.4 
9.24 
10.20 


Twice 
per 
hour. 


1 
3 

5 


75 
82 
90 


11.1 
12.1 
13.0 


5.7 
6.2 
6.7 


12.76 
13.91 
14.52. 


6.55 
7.13 
7.60 


13.22 
14.52 
15.60 


6.84 
7.44 

8.04 



Emission of Heat-units per square foot per Hour from Cast-iron Pipes 
or Radiators. Temp, of Air in Room, 70° F. (F. Schumann.) 



Mean Temperature of 


By Contact. 


By Radi- 
ation. 


By Radiation 
and Contact. 


Heated Pipe, Radia- 
tor, etc. 


Air quiet. 


Air 
moving. 


Air quiet. 


Air 
moving. 


Hot water 140° 


55.51 
65.45 
75.68 
86.18 
96.93 
107.90 
119.13 
130.49 
142.20 
153.95 
165.90 
178.00 
189.90 
202.70 
215.30 
228.55 
240.85 


92.52 
109.18 
126.13 
143.30 
161.55 
179.83 
198.55 
217.48 
237.00 
256.58 
279.83 
296.63 
31.6.50 
337.83 
358.85 
380.91 
401.41 


59.63 
69.69 
80.19 
91.12 
102.15 
114.45 
127.00 
139.96 
155.27 
169.56 
184.58 
200.18 
214.36 
233.42 
251.21 
267.73 
279.12 


115.14 
135.11 
155 87 
177.30 
199.43 
222.35 
246.13 
270.49 
297.47 
323.51 
350.48 
378.18 
404.26 
436.12 
466.51 
496.28 
519.97 


152 15 


" 150° 

44 160° 

" 170° 


178.87 
206.32 
234.42 


180° 

" 190° 

44 200° 

14 44 or steam.. 210° 

Steam 220° 

230° 


264.05 
294.28 
325.55 
357.48 
392.27 
426.14 


•' 240° 

250° 


464.41 
496.81 


" 260° 

270° 

44 ...280° 


530.86 
571.25 
610.06 


290° 

300° 


648.64 
680.53 



INDIRECT HEATING-SURFACE. 537 

Radiating-surface required for Different Kinds of Buildings. (From 
practice of the Dubuque Steam Supply Co., External Air 0° F. Chas. A. 
Smith.) 



Cubic ft. of Room heated 
by 1 sq. ft. of Surface. 
Direct Indirect 
System. System. 

Dwellings 50 40 

Stores, wholesale 125 100 

retail 100 80 



Cubic ft. of Room heated 

by 1 sq. ft. of Surface. 

Direct Indirect 

System. System. 

Banks, offices, drug-stores 70 60 

Large hotels 125 100 

Churches 200 150 



The Nason Mfg. Co.'s catalogue gives the following: One square foot of 
surface will heat from 40 to 100 cu. ft. of space to 75° in — 10° latitudes. 
This range is intended to meet conditions of exposed or corner rooms of 
buildings, and those less so, as intermediate ones of a block. As a general 
rule, 1 sq. ft. of surface will heat 70 cu. ft. of air in outer or front rooms and 
100 cu. ft. in inner rooms. In large stores in cities with buildings on each 
side, 1 to 100 is ample. 

Approximate Proportions of Radiating-surfaces. 
One square foot radiating-surface will heat: 

Indwellings, In hall, stores, In churches, large 





schoolrooms, 


lofts, factories, 


auditoriums, 




offices, etc. 


etc. 


etc. 


By direct radiation. . . 


60 to 80 ft. 


75 to 100 ft. 


150 to 200 ft. 


By indirect radiation . 


40 to 50 " 


50 to 70 •« 


100 to 140 " 



Isolated buildings exposed to prevailing north or west winds should have 
a generous addition made to the heating-surface on their exposed sides. 

The following rule is given in the catalogue of the Babcock & Wilcox Co., 
and is also recommended by the Nason Mfg. Co.: 

Radiating surface may be calculated by the rule: Add together the square 
feet of glass in the windows, the number of cubic feet of air required to be 
changed per minute, and one twentieth the surface of external wall and 
roof; multiply this sum by the difference between the required temperature 
of the room and that of the external air at its lowest point, and divide the 
product by the difference in temperature between the steam in the pipes 
and the required temperature of the room. The quotient is the required 
rad latin g-surf ace in square feet. 

Overhead Steam-pipes. (A. R. Wolff, Stevens Indicator, 1887.)— 
When the overhead system of steam-heating is employed, in which system 
direct radiating-pipes, usually 1*4 in. in diam., are placed in rows overhead, 
suspended upon horizontal racks, the pipes running horizontally, and side 
by side, around the whole interior of the building, from 2 to 3 ft. from the 
walls, and from 2 to 4 ft. from the ceiling, the amount of 1J4 in. pipe re- 
quired, according to Mr. C. J. H. Woodbury, for heating mills (for which 
use this system is deservedly much in vogue), is about 1 ft. in length for 
every 90 cu. ft. of space. Of course a great range of difference exists, due 
to the special character of the operating machinery in the mill, both in re- 
spect to the amount of air circulated by the machinery, and also the aid to 
warming the room by the friction of the journals. 

Indirect Heating-surface.— J. H. Kinealy, in Heating and Ven- 
tilation, May 15, 1894, gives the following formula, deduced from results of 
experiments by C. B. Richards, W. J. Baldwin, J. H. Mills, and others, upon 
indirect heaters of various kinds, supplied with varying amounts of air per 
hour per square foot of surface: 

N= cubic feet of air, reduced to 70° F., supplied to the heater per square 

foot of heating-surface per hour; 
T = temperature of the steam or water in the heater; 
Tx = temperature of the air when it enters the heater; 
T 2 = temperature of the air when it leaves the heater. 

As the formula is based upon an average of experiments made upon all 
sorts of indirect heaters, the results obtained by the use of the equation 
may in some cages be slightly too small and in others slightly too large, 



538 HEATING AND VENTILATION. 

although the error will in no case be great. No single formula ought to be 
expected to apply equally well to all dispositions of heating-surface in in- 
direct heaters, as the efficiency of such heater can be varied between such 
wide limits by the construction and arrangement of the surface. 

In indirect heating, the efficiency of the radiating-surface will increase, 
and the temperature of the air will diminish, when the quantity of the air 
caused to pass through the coil increases. Thus 1 sq. ft. radiating-surface, 
with steam at 212°, has been found to heat 100 cu. ft. of air per hour from 
zero to 150°, or 30J cu. ft. from zero to 100° in the same time. The best re- 
sults are attained by using indirect radiation to supply the necessary venti- 
lation, and direct radiation for the balance of the heat. (Steam.) 
■■ In indirect steam-heating the least flue area should be 1 to 1*4 sq- in. 
to every square foot of heating-surface, provided there are no long horizon- 
tal reaches in the duct, with little rise. The register should have twice the 
area of the duct to allow for the fretwork. For hot water heating from 25% 
to '30% more heating-surface and flue area should be given than for low- 
pressure steam. (Engineering Record, May 26, 1894.) 

Boiler Heating-surface Required. (A. R. Wolff, Stevens Indi- 
cator, 1887.) — When the direct system is used to heat buildings in which the 
street floor is a store, and the upper floors are devoted to sales and stock- 
rooms and to light manufacturing, and in which the fronts are of stone or 
iron, and the sides and the rear of building of brick— a safe rule to follow is to 
supply 1 sq. ft. of boiler heating-surface for each 700 cu. ft., and 1 sq. ft. of 
radiating-surface for each 100 cu. ft. of contents of building. 

For heating mills, shops, and factories, 1 sq. ft. of boiler heating-surface 
should be supplied for each 475 cu. ft. of conteuts of building; and the same 
allowance should also be made for heating exposed wooden dwellings. For 
heating foundries and wooden shops, 1 sq. ft. of boiler heating-surface 
should be provided for each 400 cu. ft. of contents; and for structures in 
which glass enters very largely in the construction— such as conservatories, 
exhibition buildings, and the like— 1 sq. ft. of boiler heating-surface should 
be provided for each 275 cu. ft. of contents of building. 

When the indirect system is employed, the radiator-surface and the boiler 
capacity to be provided will each have to be, on an average, about 25$ more 
than where direct radiation is used. This percentage also marks approxi- 
mately the increased fuel consumption in the indirect system. 

Steam (Babcock& Wilcox Co.) has the following: 1 sq. ft. of boiler-surface 
will supply from 7 to 10 sq. ft. of radiating-surface, depending upon the size 
of boiler and the efficiency of its surface, as well as that of the radiating- 
surface. Small boilers for house use should be much larger proportionately 
than large plants. Each horse-power of boiler will supply from 240 to 360 
ft. of 1-in. steam pipe, or 80 to 120 sq. ft. of radiating surface. Cubic feet 
of space has little to do with amount of steam or surface required, but is a 
convenient factor for rough calculations. Under ordinary conditions 1 
horse-power will heat, approximately, in — 

Brick dwellings, in blocks, as in cities 15,000 to 20,000 cu. ft. 

" stores " " 10,000 " 15,000 " 

" dwellings, exposed all round 10,000 " 15,000 " 

mills, shops, factories, etc 7,000 " 10,000 " 

Wooden dwellings, exposed 7,000 " 10,000 " 

Foundries and wooden shops 6,000 " 10,000 " 

Exhibition buildings, largely glass, etc 4,000 " 15,000 " 

Proportion of Grate-surface to Radiator-surface. 

(J. R. Willett, Heating and Ventilation, Feb. 1894.) 

R sq lf f t° r ~ SUI f ' } 10 ° 200 400 600 80 ° 100 ° 1200 140 ° 160 ° 1800 2000 

G 'sq te hi UrfaCe ' f 120 208 362 501 630 754 8T2 986 110 ° 1210 131 ° 

Steam-consumption in Car-heating. 

C, M. & St. Paul Railway Tests. (Engineering, June 27, 1890, p. 764.) 

Water of Condensation 

Outside Temperature. Inside Temperature. per Car per Hour. 

40 70 70 lbs. 

30 70 85 

JO 70 100 



REGISTERS AND COLD-AIR DUCTS. 



539 



Internal Diameters of Steam Supply-mains, with Total 
Resistance equal to 2 inches of Water-column.* 

Steam, Pressure 10 lbs. per square inch above atm., Temperature 239° F. 

Formula, d ^ 0.5374 a/ ^-; 



where d = internal diameter in inches; 



g = s 


.2 cub 


c feet of steam pei 


minute per 


100 sq. 


ft, of 


radiat 


ng-su 


face ; 


l — length of ma 


ns in feet; h 


= 159.3 feet head of steam to produce flow. 




Inter 


ial Diameters in inches for Lengths of Mains from 1 ft. to 600 ft. 


1ft. 


10 ft 


20 ft. 


40 ft. 


60 ft. 


80 ft. 


100 ft. 


200 ft. 


300 ft. 


400 ft. 


600 ft. 


sq.ft. 


inch. 


inch. 


inch. 


inch. 


inch. 


inch. 


inch. 


inch. 


inch. 


inch. 


inch. 


1 


0.075 


119 


0.136 


0.157 


0.170 


0.180 


0.189 


0.216 


0.234 


0.248 


0.270 


10 


0.19 


0.30 


0.34 


0.39 


0.43 


0.45 


0.47 


0.54 


0.59 


0.62 


0.68 


20 


0.25 


0.39 


0.45 


0.52 


0.56 


60 


0.62 


0.72 


0.78 


0.82 


0.89 


40 


0.33 


0.52 


0.60 


0.69 


0.74 


0.79 


0.82 


0.95 


1.03 


1.09 


1.18 


60 


0.39 


0.61 


0.71 


0.81 


0.87 


0.93 


0.97 


1.11 


1.21 


1.28 


1.39 


HO 


0.43 


0.68 


0.79 


0.90 


0.98 


1.04 


1.09 


1.25 


1.35 


1.43 


1.55 


100 


0.47 


0.75 


0.86 


0.99 


1.07 


1.14 


1.19 


1.36 


1.48 


1.57 


1.70 


200 


62 


0.99 


1.14 


1.30 


1.41 


1.50 


1.57 


1.80 


1.95 


2.07 


2.24 


300 


0.73 


1.16 


1.34 


1.53 


1.66 


1.76 


1.84 


2 12 


2.30 


2.43 


2.64 


400 


0.82 


1.30 


1.50 


1.72 


1.86 


1.98 


2.07 


2.37 


2.57 


2.73 


2.96 


500 


0.90 


1.43 


1.64 


1.88 


2.04 


2.16 


2.26 


2.60 


2.81 


2.98 


3.23 


600 


0.97 


1.53 


1.76 


2.03 


2.20 


2.33 


2.43 


2.79 


3.03 


3.21 


3.48 


800 


1.09 


1.72 


1.98 


2.27 


2 46 


2.61 


2.73 


3.13 


3.40 


3.60 


3.90 


1,000 


1.19 


1.88 


2.16 


2.48 


2.69 


2.85 


2.98 


3.43 


3.71 


3.94 


4.27 


1,200 


1.28 


2.04 


2.33 


2.67 


2.90 


3.07 


3.21 


3.68 


4.00 


4.23 


4.59 


1,400 


1.36 


2.15 


2.47 


2.84 


3.08 


3.26 


3.41 


3.92 


4.25 


4.50 


4.83 


1,600 


1.43 


2.27 


2.61 


3.00 


3.25 


3.44 


3.60 


4.13 


4.49 


4.75 


5.15 


1,800 


1.50 


2.38 


2.74 


3.14 


3.41 


3.61 


3.78 


4.34 


4.70 


4.98 


5.40 


2,000 


1.57 


2.48 


2.85 


3.28 


3.55 


3.76 


3.93 


4.52 


4.90 


5.19 


5.63 


3,000 


1.84 


2.92 


3.36 


3.85 


4.18 


4.43 


4 63 


5.32 


5.77 


6.11 


6.63 


4,000 


2.07 


3.28 


3.76 


4.32 


4.69 


4.96 


5.19 


5.96 


6.47 


6.85 


7.44 



* From Robert Briggs's paper on American Practice of Warming Buildings 
by Steam (Proc. Inst. C. E., 1882, vol. lxxi). 

For other resistances and pressures above atmosphere multiply by the 
respective factors below : 

Water col . C in. 12 in. 24 in. I Press, ab. atm. lbs. 3 lbs. 30 lbs. 60 lbs. 
Multiply by 0.8027 0.6988 0.6084 | Multiply by 1.023 1.015 0.973 0.948 

Registers and Cold-air Ducts for Indirect Steam Heating. 
—The Locomotive gives the following table of openings for registers and 
cold-air ducts, which has been found to give satisfactory results. The cold- 
air boxes should have \y% sq. in. area for each square foot of radiator suface, 
and never less than % the sectional area of the hot air ducts. The hot air 
ducts should have 2 sq. in. of sectional area to each square foot of radiator 
surface on the first floor, and from 1^ to 2 inches on the second floor. 











Heating Surface 
in Stacks. 


Cold-air Supply, First Floor. 


Size 
Register. 


Supply, 
2d Floor. 




inches 


inches 


inches 


30 square feet 


45 square inches = 5 by 9 


9 by 12 


4 by 10 


40 •' 


60 " = 6 by 10 


10 by 14 


4 by 14 


50 " 


75 ,l " = 8 by 10 


10 by 14 


5 by 15 


60 " 


90 " " = 9 by 10 


12 by 15 


6 by 15 


70 


108 " " = 9 by 12 


12 by 19 


6 by 18 


80 " 


120 " " = 10 by 12 


12 by 22 


8 by 15 


90 " 


135 " " = 11 by 12 


14 by 24 


9 by 15 


100 " 


150 ■■ " =12 by 12 


16 by 20 


12 by 12 



The sizes in the table approximate to the rules given, and it will be found 
that they will allow an easy flow of air and a full distribution throughout the 
room to be heated. 



540 



HEATING AND VENTILATION. 



Physical Properties of Steam and Condensed Water, 
under Conditions of Ordinary Practice in Warming by 
Steam. (Brig^s.) 



A 


( Steam-pressure j above atm. . . 
} per square inch | total 


lbs. 

lbs. 



14.7 


3 

17.7 


10 
24.7 


30 
44.7 


60 
74.7 


F, 




Fahr. 
Fahr. 
Fahr. 

>■ units 

Fahr. 


212° 
60° 
152° 

456 

965° 


222° 
60 
162° 

486 

958° 


239° 
60° 
179° 

537 

946° 


274° 
60° 
214° 

642 

921° 


307° 


€ 
D 

E 

F 


Temperature of air 

Difference = B — C 

1 Heat given out per minute per 
■I 100 sq. ft. of radiatiug-sur- 

I face = D X 3 
Latent heat of steam 


60* 

247° 

741 

898° 


G 
H 

J 


Volume of 1 lb. weight of steam 
Weight of 1 cubic foot of steam 
( Volume Q of steam per minute 
< to give out E units 

( =EXG-F. 


cu. ft. 
lb. 

tcu.ft. 


26.4 
0.0380 

12.48 


22.1 
0.0452 

11.21 


16.2 
0.0618 

9.20 


9.24 
0.1082 

6.44 


5.70 
0.1752 

4.70 


K 
L 
M 


( Weight of 1 cubic foot of con- 

< densed water at tempera- 
[ ture B, 

I Volume of condensed water to 

< return to boiler per minute 
j =JXH-K, 

i Head of steam equivalent to 
■< 12 inches water-column 
I =K-4-H. 


t lbs. 
tcu.ft. 
V feet 


59.64 
0.0079 
1569 


59.51 

0.0085 
1317 


59.05 

0.0096 
955.5 


58.07 
0.0120 
536.7 


57.03 
0.0144 
325.5 


N 

P 
B 

S 


Steam-supply Mains. 

fHead h of steam, equivalent 
J to assumed 2 inches water- 
1 column for producing steam 
L flow Q, = M -f- 6, 
j Internal diameter d of tube* 
\ for flow Q when I = 1 foot, 
Do. do. when I = 100 feet, 
Ratios of values of d. 


}■ feet 

J 

>■ inch 
inch 
ratio 


261.5 

0.484 

1.217 
1.023 


219.5 

0.481 

1.207 
1.015 


159.3 

0.474 
1.190 
1.000 


89.45 

0.461 
1.158 
0.973 


54.25 

0.449 

1.128 
0.948 


T 

U 

V 
W 


Water-Return Mains. 

( Head h assumed at J^-inch 
< water-column for producing 

[ full-bore water-flow Q, 

j Internal diameter d of tube* 

j for flow Q when I = 1 foot, 
Do. do. when I = 100 feet, 
Ratios of values of d 


V foot 

y inch 

inch 
ratio 


0.0417 

0.147 
0.368 
0.926 


0.0417 

151 
0.379 
0.952 


0.0417 

0.158 

0.398 
1.000 


0.0417 

0.173 
0.434 
1.092 


0.0417 

0.186 
0.468 
1.176 



* P, P, U, V are each determined from the formula d = 0.5374 



m 



Size of Steam Pipes for Steam Heating. (See also Flow of 
Steam in Pipes.)— Sizes of vertical main pipes. Direct radiation. (J. R. 
Willett, Heating and Ventilation, Feb., 1894.) 

Diameter of pipe, inches. 1 1J4 1}4 2 2^ 3 3^ 4 5 6 
Sq. ft. of radiator surface 40 70 110 220 360 560 810 1110 2000 3000 
A horizontal branch pipe for a given extent of radiator surface should be 
one size larger than a vertical pipe for the same surface. No return from a 
main should be more than two sizes smaller than the feed at its commence- 
ment (or than its largest dimension). 

A. R. Wolff (Stevens Indicator, 1887) says: For determining the cross- 
sectional area of pipes (in square inches) for steam mains and returns it 
will be ample to allow a constant of .375 sq. in. for each hundred square 



HEATING A GREENHOUSE BY STEAM. 541 

feet of heating-surface in coils and radiators, when exhaust steam is used, 
.19 sq. in. when live steam is used, and .09 sq. in. for the return. If the cross- 
sectional areas thus obtained are each mulitplied by 1.273, and the square root 
extracted from each product, the respective figures obtained will represent 
the proper diameters in inches of the several steam-pipes referred to. 

Steam, by the Babcock & Wilcox Co., says : Where the condensed water 
is returned" to the boiler, or where low pressure of steam is used, the diame- 
ter of mains leading from the boiler to the radiating-surface should be 
equal in inches to one tenth the square root of the radiating-surface, mains 
included, in square feet. Thus a 1-inch pipe will supply 100 square feet of 
surface, itself included. Return-pipes should be at least % inch in diame- 
ter, and never less than one half the diameter of the main— longer returns 
requiring larger pipe. A thorough drainage of steam-pipes will effectually 
prevent all cracking and pounding noises therein. 

The Nason Mfg. Co. gives the following : 
Radiating-surface in square Size of Steam- Size of Return- 

feet to be supplied. pipes. pipes. 

125 1M 1 

125to200 \\i V/ A 

200 to 500 2 \% 

500 to 1000 2)4 2 

1000tol500 * 3 2}4 

1500to2500 33^ 3 

When mains and surfaces are very much above the boiler the pipes need 
not be as large as given above; under very favorable circumstances and 
conditions a 4-inch pipe may supply from 2000 to 2500 sq. ft. of surface, a 6- 
inch pipe for 5000 sq. ft., and a 10-inch pipe for 15,000 to 20,000 sq. ft., if the 
distance of run from boiler is not too great. Less than lj^-inch pipe should 
not be used horizontally in a main unless for a single radiator connection. 
The return sizes named are large enough in ordinary pipe-work, though 
when horizontal pipes with many fittings are used they should be of the 
same diameter as the steam-pipes. 

Generally, when condensation is returned to the boiler by gravity, the 
diameter of mains in inches should equal one tenth of the square root of the 
radiating-surfaces in square feet; thus a 1-inch pipe will supply 100 sq. ft. of 
surface, or with 900 sq. ft. the supply-pipe should be V900 = 30 -s- 10 = 3" 
diameter. 

Heating a Greenhouse by Steam.— Wm. J. Baldwin answers a 
question in the American Macliinist as below: With five pounds steam- 
pressure, how many square feet or inches of heating-surface is necessary to 
heat 100 square feet of glass on the roof, ends, and sides of a greenhouse 
in order to maintain a night heat of 55° to 65°, while the thermometer out- 
side ranges at from 15° to 20° below zero ; also, what boiler-surface is neces- 
sary ? What is the best way to set pipes in a greenhouse — hang them or lay 
them down ? Which is the best for the purpose to use— 2" pipe or 134" pipe? 

Ans.— Reliable authorities agree that 1.25 to 1.50 cubic feet of air in an 
enclosed space will be cooled per minute per sq. ft. of glass as many degrees 
as the internal temperature of the house exceeds that of the air outside. 
Between -f- 65° and — 20° there will be a difference of 85°, or, say, one cubic 
foot of air cooled 127.5° F. for each sq ft. of glass for the most extreme 
condition mentioned. Multiply this by the number of square feet of 
glass and by 60, and we have the number of cubic feet of air cooled 1° per 
hour within the building or house. Divide the number thus found by 48, and 
it gives the units of heat required, approximately. Divide again by 953, 
and it will give the number of pounds of steam that must be condensed from 
a pressure and temperature of five pounds above atmosphere to water at 
the same temperature in an hour to maintain the heat. Each square foot 
of surface of pipe will condense from 34 to nearly y% lb. of steam per hour, 
according as the coils are exposed or well or poorly arranged, for which 
an average of \£ lb. may be taken. According to this, it will require 3 sq. ft. 
of pipe surface per lb. of steam to be condensed. Proportion the heating- 
surface of the boiler to have about one fifth the actual radiating-surface, if 
you wish to keep steam over night, and proportion the grate to burn not 
more than six pounds of coal per sq. ft. of grate per hour. With very slow 
combustion, such as takes place in base-burning boilers, the grate might be 
proportioned for four to five pounds of coal per hour. It is cheaper to make 
coils of 1J4" pipe than of 2", and there is nothing to be gained by using 2" 
pipe unless the coii^ are very long. The pipes in a greenhouse should be 



542 HEATING AND VENTILATION. 

under or in front of the benches, with every chance for a good circulation 
of air. " Header" coils are better than "return-bend 11 coils for this purpose. 

Mr. Baldwin's rule may be given the following form : Let H = heat-units 
transferred per hour, T ~ temperature inside the greenhouse, t — tempera- 
ture outside, 8= sq. ft. of glass surface; then H = 1.5S(!T- t) X 60 -=- 48 
= 1.8755(7' - t). Mr. Wolff's coefficient K for single skylights would give 
H= 1.1l8S(r- t). 

Heating a Greenhouse by Hot Water.— W. M. Mackay, of the 
Richardson & Boynton Co., in a lecture before the Master Plumbers' Asso- 
ciation, N. Y., 1889, says : I find that while greenhouses were formerly 
heated by 4-inch and 3-inch cast-iron pipe, on account of the large body of 
water which they contained, and the supposition that they gave better satis- 
faction and a more even temperature, florists of long experience who 
have tried 4-inch and 3-inch cast-iron pipe, and also 2 inch wrought-iron 
pipe for a number of years in heating their greenhouses by hot water, 
and who have also tried steam-heat, tell me that they get better satisfaction, 
greater economy, and are able to maintain a more even temperature with 2- 
inch wrought-iron pipe and hot water than by any other system they have 
used. They attribute this result principally to the fact that this size pipe 
contains less water and on this account tUe beat can be raised and lowered 
quicker than by any other arrangement of pipes, and a more uniform tem- 
perature maintained than by steam or any other system. 

HOT- WATER HEATING. 

(Nason Mfg. Co.) 

There are two distinct forms or modifications of hot- water apparatus, de- 
pending upon the temperature of the water. 

In the first or open-tank system the water is never above 212° tempera- 
ture, and rarely above 200°. This method always gives satisfaction where 
the surface is sufficiently liberal, but in making it so its cost is considerably 
greater than that for a steam-heating apparatus. 

In the second method, sometimes called (erroneously) high-pressure hot- 
water heating, or the closed-system apparatus, the tank is closed. If it is 
provided with a safety-valve set at 10 lbs. it is practically as safe as the open- 
tank system. 

Law of Velocity of Flow.— The motive power of the circulation 
in a hot -water apparatus is the difference between the specific gravities of 
the ascending and the descending pipes. This effective pressure is very 
small, and is equal to about one grain for each foot in height for each de- 
gree difference between the pipes; thus, with a height of 12" in " up " pipe, 
and a difference between the temperatures of the up and down pipes of 8°, 
the difference in their specific gravities is equal to 8.16 grains on each square 
inch of the section of return-pipe, and the velocity of the circulation is pro- 
portioned to these differences in temperature and height. 

To Calculate Velocity of Flow.— Thus, with a height of ascend- 
ing pipe equal to 10' and a difference in temperatures of the flow and return 
pipes of 8°, the difference in their specific gravities will equal 81.6 grains, or 
-h 7000 = .01166 lbs., or X 2.31 (feet of water in one pound ) = . 0269 ft., and by 
the law of falling bodies the velocity will be equal to 8 V.0-J69 = 1.312 ft. per 
second, or X 60 = 78.7 ft. per minute. In this calculation the effect of fric- 
tion is entirely omitted. Considerable deduction must be made on this 
account. Even in apparatus where length of pipe is not great, and with 
pipes of larger areas and with few bends or angles, a large deduction for 
friction must be made from the theoretical velocity, while in large and 
complex apparatus with small head, the velocity is so much reduced by 
friction that sometimes as much as from 50$ to 90$ must be deducted to ob- 
tain the true rate of circulation^ 

Main flow-pipes from the heater, from which branches may be taken, are 
to be preferred to the practice of taking off nearly as many pipes from the 
heater as there are radiators to supply. 

It is not necessary that the main flow and return pipes should equal in 
capacity that of all their branches. The hottest water will seek the highest 
level, while gravity will cause an even distribution of the heated water if the 
surface is properly proportioned. 

It is good practice to reduce the size of the vertical mains as they ascend, 
say at the rate of one size for each floor. 

As with steam, so with hot water, the ninoa ™yst De unconfined to allow 



HOT- WATER HEATING. 



543 



for expansion of the pipes consequent on having their temperatures in- 
creased. 

An expansion tank is required to keep the apparatus filled with water, 
which latter expands 1/24 of its bulk on being heated from 40° to 212°, and 
the cistern must have capacity to hold certainly this increased bulk. It is 
recommended that the supply cistern be placed on level with or above the 
highest pipes of the apparatus, in order to receive the air which collects in 
the mains and radiators, and capable of holding at least 1/20 of the water 
in the entire apparatus. 

Approximate Proportions of Radiatiiig-surfaces to 
Cubic Capacities of Space to be Heated. 



One Square Foot of Ra- 


In Dwellings, 


In Halls, Stores, 


In Churches, 


diating-surface will 


School-rooms, 


Lofts, Facto - 


Large Audito- 


heat with— 


Offices, etc. 


, ries, etc. 


riums, etc. 


High temperature di- ) 








rect hot- water radi- V 


50 to 70 cu. ft. 


65 to 90 cu. ft. 


130 to 180 cu. ft. 


ation ) 








Low temperature di- ) 








rect hot-water radi- > 


30 to 50 " " 


35 to 65 " " 


70 to 130 " " 










High temperature in- 1 




direct hot- water ra- V 


30 to 60 " " 


35 to 75 " " 


70 to 150 " " 


diation ) 








Low temperature in- ) 








direct hot-water ra- >■ 


20 to 40 " " 


25 to 50 " " 


50 to 100 " " 


diation J 









Diameter of Main and Branch Pipes and square feet of coil 
surface they will supply, in a low-pressure hot-water apparatus (212°) for 
direct or indirect radiation, when coils are at different altitudes for direct 
radiation or in the lower story for indirect radiation: 



04 oa 


lS a 
o.2 




Ej£> 


.a * 


Direct Radiation. Height of Coil above Bottom of Boiler, 




■§■■2 




. a 


MA 


























Q 





10 


20 


30 40 
sq. ft. 'sq.ft. 


50 


60 


70 


80 


90 


100 




sq. ft. 


sq. ft. 


sq. ft. 


sq. ft. 


sq. ft. 


sq. ft. 


sq.ft. 


sq. ft. 


sq. ft. 


% 


49 


50 


52 


53 


55 


57 


59 


61 


63 


65 


68 


1 


87 


89 


92 


95 


98 


101 


103 


108 


112 


116 


121 


4\i 


136 


140 


144 


149 


153 


158 


161 


169 


175 


182 


189 


w<& 


196 


202 


209 


214 


222 


228 


235 


243 


252 


261 


271 




349 


359 


370 


380 


393 


405 


413 


433 


449 


465 


483 


m 


546 


561 


577 


595 


613 


633 


643 


678 


701 


727 


755 


3 


785 


807 


835 


856 


888 


912 


941 


974 


1009 


1046 


1086 


3^ 


1069 


1099 


1132 


1166 


1202 


1241 


1283 


1327 


1374 


1425 


1480 


4 


1395 


1436 


1478 


1520 


1571 


1621 


1654 


1733 


1795 


1861 


1933 


*H 


1767 


1817 


1871 


1927 


1988 


2052 


2120 


2193 


2272 


2356 


2445 


5 


2185 


2244 


2309 


2376 


2454 


2531 


2574 


2713 


2805 


2907 


3019 


6 


3140 


3228 


3341 


3424 


3552 


3648 


3763 


3897 


4036 


4184 


4344 


7 


4276 


4396 


4528 


4664 


4808 


4964 


5132 


5308 


5496 


5700 


5920 


8 


5580 


5744 


5912 


60S0 


6284 


6484 


6616 


6932 


7180 


7444 


7735 


9 


7068 


7268 


7484 


7708 


7952 


8208 


8482 


8774 


9088 


9424 


9780 


10 


8740 


8976 


9236 


9516 


9816 


10124 


10296 


10852 


11220 


11628 


12076 


11 


10559 


10860 


11180 


11519 


11879 


12262 


12666 


13108 


13576 


14078 


14620 


12 


12560 


12912 


13364 


13696 14208 


14592 


15052 


15588 


16144 


16736 


17376 


13 


14748 


15169 


15615 


16090 |16591 


17126 


17697 


18307 


18961 


19633 


20420 


14 


17104 


17584 


18109 


18656 ! 19232 


19856 


20528 


21232 


21984 


22800 


23680 


15 


19634 


20195 


20789 


21419 22089 


22801 


23561 


24373 


25244 


26179 


27168 


16 


22320 


22978 


23643 


24320 25136 


25936 


2G464 


27728 


28720 


29776 


30928 



544 HEATING AND VENTILATION. 

The best forms of hot- water- heating boilers are proportioned about as 
follows: 

1 sq. ft. of grate-surface to about 40 sq. ft. of boiler-surface. 
1 " " boiler- " " 5 " " radiating-surface. 

1 " " grate- " " 200 " " 

Rules for Hot-water Heating.— J. L. Saunders (Heating and 
Ventilation, Dec. 15, 1894) gives the following : Allow 1 sq. ft. of radiating 
surface for every 3 ft. of glass surface, and 1 sq. ft. for every 30 sq. ft. of 
wall surface, also 1 sq. ft. for the following numbers of cubic feet of space 
in the several cases mentioned. 

In dwelling-houses: Libraries and dining-rooms, first floor. . 35 to 40 cu. ft. 

Reception halls, first floor 40 to 50 " " 

Stairhalls, " " 40 to 55 '• " 

Chambers above, " " 50 to 65 " " 

Libraries, sewing-rooms, nurseries, etc., 

above first floor 45 to 55 " " 

Bath-rooms 30 to 40 " " 

Public-schoolrooms 60 to 85 " " 

Offices 50 to 65 " " 

Factories and stores 65 to 90 " " 

Assembly halls and churches 90 to 150 " " 

To find the necessary amount of indirect radiation required to heat a room: 
Find the required amount of direct radiation according to the foregoing 
method and add 50$. This if wrought-iron pipe coil surface is used ; if cast- 
iron pin indirect -stack surface is used it is advisable to add from 70$ to 80$. 

Sizes of hot-air flues, coLt-air ducts, and registers for indirect work. — 
Hot-air flues, first floor: Make the net internal area of the flue equal to 
% sq. in. to every square foot of radiating surface in the indirect stack. Hot- 
air flues, second floor: Make the net internal area of the flue equal to % sq. in. 
to every square foot of radiating surface in the indirect stack. 

Cold-air ducts, first floor : Make the net internal area of the duct equal 
to % sq. in. to every square foot of radiating surface in the indirect stack. 
Cold air ducts, second floor : Make the net internal area of the duct equal 
to H> sq. in. to every square foot of radiating surface in the indirect stack. 

Hot-air registers should have their net area equal in full to the area of the 
hot-air flues. Multiply the length by the w r idth of the register in inches ; % 
of the product is the net area of register. 

Arrangement of Mains for Hot-water Heating. (W. M. 
Mackay, Lecture before Master Plumbers' Assoc, N. Y., 1889 )— There are 
two different systems of mains in general use, either of which, if properly 
placed, will give good satisfaction. One is the taking of a single large-flow 
main from the heater to supply all the radiators on the several floors, with a 
corresponding return main of the same size. The other is the taking of a 
number of 2-inch wrought-iron mains from the heater, with the same num- 
ber of return mains of the same size, branching off to the several radiators 
or coils with l^-inch or 1-inch pipe, according to the size of the radiator or 
coil. A 2-inch main will supply three 1 14-inch or four 1-inch branches, and 
these branches should be taken from the top of the horizontal main with a 
nipple and elbow, except in special cases where it is found necessary to retard 
the flow of water to the near radiator, for the purpose of assisting the circu- 
lation in the far radiator ; in this case the branch is taken from the side of 
the horizontal main. The flow and return mains are usually run side by side, 
suspended from the basement ceiling, and should have a gradual ascent from 
the heater to the radiators of at least 1 inch in 10 feet. It is customary, and 
an advantage where 2-inch mains are used, to reduce the size of the main at 
every point where a branch is taken off. 

The single or large main system is best adapted for large buildings ; but 
there is a limit as to size of main which it is not wise to go beyond— gener- 
ally 6- inch, except in special cases. 

The proper area of cold- air pipe necessary for 100 square feet of indirect 
radiation in hot-water heating is 75 square inches, while the hot-air pipe 
should have at least 100 square inches of area. There should be a damper in 
the cold-air pipe for the purpose of controlling the amount of air admitted to 
the radiator, depending on the severity of the weather. 



BLOWER SYSTEM OF HEATING AND VENTILATING. 545 

THE BLOWER SYSTEM OF HEATING AND 
VENTILATING. 

The sj^stem provides for the use of a fan or blower which takes its supply 
of fresh air from the outside of the building to be heated, forces it over 
steam coils, located either centrally or divided up into a number of indepen- 
dent groups, and then into the several ducts or flues leading to the various 
rooms. The movement of the warmed air is positive, and "the delivery of 
the air to the various points of supply is certain and entirely independent 
of atmospheric conditions. For engines, fans, and steam-coils used with the 
blower system, see page 519. 

Experiments with Radiators of 60 sq. ft. of Surface. 
(Mech. News, Dec, 1893.) — After having determined the volume and tem- 
perature of the warm air passing through the flues and radiators from 
natural causes, a fan was applied to each flue, forcing in air, and new sets of 
measurements were made. The results showed that more than t\\ o and one- 
third times as much air was warmed with the fans in use, and the falling off 
in the temperature of this greatly increased air-volume was only about 12. 6$. 
The condensation of steam in the radiators with the forced-air circulation 
also was only 66%$ greater than with natural air draught. One of the 
several sets of test figures obtained is as follows : 

^ ■ Natural Forced- 

Draught air 
in Flue. Circulation. 

Cubic feet of air per minute 457.5 1227 

Condensation of steam per minute in ounces 11.7 19.6 

Steam pressure in radiator, pounds 9 9 

Temperature of air after leaving radiator 142° 124° 

" " " before passing through radiator. 61° 61° 

Amount of radiating surface in square feet 60 60 

Size of flue in both cases 12 x 18 inches. 

There was probably an error in the determination of the volume of air in 
these tests, as appears from the following calculation. (W. K.) Assume 
that 1 lb. of steam in condensing from 9 lbs. pressure and cooling to the tem- 
perature at which the water may have been discharged from the radiator 
gave up 1000 heat-units, or 62.5 h. u. per ounce; that the air weighed .076 lb. 
per cubic foot, and that its specific heat is .238. We have 

Natural Forced 
Draught. Draught. 

Heat given up by steam, ounces x 62 5 — 731 1225 H.U. 

Heat received by air, cu. ft. x. 076 xdiff. of tern. x. 238= 673 1399 '.' 

Or, in the case of forced draught the air received \\% more heat than the 
steam gave out, which is impossible. Taking the heat given up by the steam 
as the correct measure of the work done by the radiator, the temperature 
of the steam at 237°, and the average temperature of the air in the case of 
natural draught at 102° and in the other case at 93°, we have for the tem- 
perature difference in the two cases 135° and 144° respectively; dividing 
these into the heat- units we find that each square foot of radiating surface 
transmitted 5.4 heat-units per hour per degree of difference of temperature, 
in the case of natural draught, and 8.5 heat-units in the case of forced 
draught. 

In the Women's Homoeopathic Hospital in Philadelphia, 2000 feet of 
one-inch pipe heats 250,000 cubic feet of space, ventilating as well; this 
equals one square foot of pipe surface for about 350 cubic feet of space, or 
less than 3 square feet for 1000 cubic feet. The fan is located in a sepa- 
rate building about 100 feet from the hospital, and the air, after being heated 
to about 135°, is conveyed through an underground brick duct with a loss of 
only five or six degrees in cold weather. (H. I. Snell, Trans. A. S. M. E ,ix. 106. 

Heating a Building to 7O F. Inside when the Outside 
Temperature is Zero.— It is customary in some contracts for heating 
to guarantee that the apparatus will heat the interior of the building to 70° 
in zero weather. As it may not be practicable to obtain zero weather for 
the purpose of a test, it may be difficult to prove the performance of the 
guarantee. E. E. Macgovern, in Engineering Record, Feb. 3, 1894, gives a 
calculation tending to show that a test may be made in weather of a higher 
temperature than zero, if the heat of the interior is raised above 70°. The 
higher the temperature of the rooms the lower is the efficiency of the radi- 
ating-surface, since the efficiency depends upon the difference between the 



546 HEATING AND VENTILATION. 

temperature inside of the radiator and the temperature of the room. He 
concludes that a heating apparatus sufficient to heat a given building to 70° 
in zero weather with a given pressure of steam will be found to heat the 
same building, steam-pressure constant, to 110° at 60°, 95° at 50°, 82° at 40°, 
and 74° at 32°, outside temperature. The accuracy of these figures, however 
has not been tested by experiment. 

The following solution of the question is proposed by the author. It gives 
results quite different from those of Mr. Macgovern, but, like them, lacks ex- 
perimental confirmation. 
Let S = sq. ft. of surface of the steam or hot-water radiator; 
W = sq. ft. of surface of exposed walls, windows, etc.; 
Ts = temp, of the steam or hot water, T x = temp, of inside of building 

or room, T = temp, of outside of building or room; 
a — heat-units transmitted per sq. ft. of surface of radiator per hour 

per degree of difference of temperature; 
b — average heat-units transmitted per sq. ft. of walls per hour, per 
degree of difference of temperature, including allowance for 
ventilation. 
It is assumed that within the range of temperatures considered Newton's 
law of cooling holds good, viz., that it is proportional to the difference of 
temperature between the two sides of the radiating-surface. 

bW 
Then aS(Ts - T x ) = bW{T x - T )._ Let - - = C ; then 



T x = CKTj. - T ) ; T x = 



aS 
Ts + CT . Ts - T x 



1 _j_ C ' ~ Ti 

If T t = 70, and T = 0, C = Ts ~ 7 ° . 

LetTs = 140°, 213.5°, 308°; 

Then C = 1, 2.05, 3.4. 

From these we derive the following: 
Temperature of Outside Temperatures, T . 

Steam or Hot - 20° - 10° 0° 10° 20° 

Water, Ts. Inside Temperatures, T,. 



140° 
213.5 



54.5 62.3 



70 


75 


80 


85 


90 


70 


76.7 


83.4 


90.2 


96.9 


70 


77.7 


85.5 


93.2 


100.9 



If eating by Electricity.— If the electric currents are generated oy 
a dynamo driven by a steam-engine, electric heating will prove very expen- 
sive, since the steam-engine wastes in the exhaust-steam and by radiation 
about 90$ of the heat-units supplied to it. In direct steam-heating, with a 
good boiler and properly covered supply-pipes, we can utilize about 60$ of 
the total heat value of the fuel. One pound of coal, with a heating value of 
13,000 heat-units, would supply to the radiators about 13,000 X -CO = 7800 
heat-units. In electric heating, suppose we have a first-class condensing- 
engine developing 1 H.P. for every 2 lbs. of coal burned per hour. 
This would be equivalent to 1,980,000 ft.-lbs. -h 778 = 2545 heat-units, or 1272 
heat-units for 1 lb. of coal. The friction of the engine and of the dynamo and 
the loss by electric leakage, and by heat radiation from the conducting 
wires, might reduce the heat-units delivered as electric current to the elec- 
tric radiator, and these converted into heat to 50% of this, or only 636 heat- 
units, or less than one twelfth of that delivered to the steam -radiators in 
direct steam -heating. Electric heating, therefore, will prove uneconomical 
unless the electric current is derived from water or wind power, which would 
otherwise be wasted. (See Electrical Engineering.) 



WEIGHT OF WATEB. 



547 



WATER. 



Expansion of Water.— The following table gives the relative vol- 
umes of water at different temperatures, compared with its volume at 4° C. 
according to Kopp, as corrected by Porter. 



Cent. 


Fahr. 


Volume . 


Cent. 


Fahr. 


Volume. 


Cent, 


Fahr. 


Volume. 


40 


39.1° 


1.00000 


35° 


95° 


1.00586 


70° 


158° 


1.02241 


5 


41 


1.00001 


40 


104 


1.00767 


75 


167 


1.02548 


10 


50 


1.00025 


45 


113 


1.00967 


80 


176 


1.02872 


15 


59 


1.00083 


50 


122 


1.01186 


85 


185 


1.03213 


20 


68 


1 .00171 


55 


131 


1.01423 


90 


194 


1.03570 


25 


77 


1.00286 


60 


140 


1.01678 


95 


203 


1.03943 


30 


86 


1.00425 


65 


149 


1.01951 


100 


212 


1.04332 



Weight of 1 cu. ft. at c 
ft. at 212° F. 



U° F. = 62.4245 lb. -- 1.04332 = E 



S3, weight of 1 cu. 



"Weight of Water at Different Temperatures.— The weight 
of water at maximum density, 39.1°, is generally taken at the figure given 
by Rankine, 62.425 lbs. per cubic foot. Some authorities give as low as 
62.379. The figure 62.5 commonly given is approximate. The highest 
authoritative figure is 62.425. At 62° F. the figures range from 62.291 to 62.360. 
The figure 62.355 is generally accepted as the most accurate. 

At 32° F. figures given by different writers range from 62.379 to 62.418. 
Clark gives the latter figure, and Hamilton Smith, Jr., (from Rosetti,) gives 
62.416. 

Weight of Water at Temperatures above 212° F.— Porter 
(Richards' "Steam-engine Indicator,' 1 p. 52) says that nothing is known 
about the expansion of water above 212°. Applying formulae derived from 
experiments made at temperatures below 212°, however, the weight and 
volume above 212° may be calculated, but in the absence of experimental 
data we are not certain that the formulae hold good at higher temperatures. 

Thurston, in his " Engine and Boiler Trials," gives a table from which we 
take the following (neglecting the third decimal place given by him) : 





£■§ 




S-S 




J§.2 








•S.2 






2 fe 

KtbD 


K,Q 


eg ,• 
e-Fb'r 




o3 _• 




o.gfbi 




^-3 


sh 


43 3 

to- 


^r 3 


43 3 

4»£s-o 


§5-o 


■s &s 


*£% 


'53 a<2 


e=^ 


£ S.S 


£5.3 


•s&S 


S S-S 


'33 £ 


H 


£ 


H 


£ 


H 


F 


H 


£ 


H 


212 


59.71 


280 


57.90 


350 


55.52 


420 


52.86 


490 


50.03 


220 


59.64 


290 


57.59 


360 


55.16 


430 


52.47 


500 


49.61 


230 


59.37 


300 


57.26 


370 


54.79 


440 


52.07 


510 


49.20 


240 


59.10 


310 


56.93 


380 


54.41 


450 


51.66 


520 


48.78 


250 


58.81 


320 


56.58 


390 


54.03 


460 


51.26 


530 


48.36 


260 


58 52 


330 


56.24 


400 


53.64 


470 


50.85 


540 


47.94 


270 


58.21 


310 


55.88 


410 


53.26 


480 


50.44 


550 


47.52 



Box on Heat gives the following : 



Temperature F 

Lbs. per cubic foot. . 



212° 
59.82 



250° 

58.85 



00° 350° 

r.42 55.94 



400° 450° 500° 600° 
54.34 52.70 51.02 47.64 



te At 212° figures given by different writers (see Trans. A. S. M. E., xiii. 409) 
h'ange from 59.56 to 59.845, averaging about 59.77. 



548 



WATER. 



Weight of Water per Cubic Foot, from 32° to 212° F., and heal- 
units per pouud, reckoned above 32° F.: The following table, made by in- 
terpolating the table given by Clark as calculated from Rankine's formula, 
with corrections for apparent errors, was published by the author in 1884, 
Trans. A. S. M. E., vi. 90. (For heat units above 212° see Steam Tables.) 





If 


& 


ftg'ti 


If 


cc 


«S • 


II 


S 




« 


■J* 

P&JD 


bo 5s o 


'3 




3 


2 & 
§•£ tub 


bjc£ o 


'3 




'3 


1-8 


■-Do 


«3 


JJ23 


33p.fi 


<D 


£+3 -O 


® &<2 


0) 


gssg&s 


CD 


H 


& 


K 




£ 


a 


EH 


£ 


K 


EH ,£ 


W 


32 


62.42 


0. 


78 


62.25 


46.03 


123 


61.68 


91.16 


168 


60.81 


136.44 


33 


62.42 


1. 


79 


62.24 


47.03 


124 


61.67 


92.17 


169 


60.79 


137.45 


34 


62.42 


2. 


80 


62.23 


48.04 


125 


61.65 


93.17 


170 


60.77 


138.45 


35 


62.42 


3. 


81 


02.22 


49.04 


126 


61.63 


94.17 


171 


60.75 


139.46 


36 


62.42 


4. 


82 


62.21 


50.04 


127 


61.61 


95.18 


172 


60.73 


140.47 


3? 


62.42 


5. 


83 


62.20 


51.04 


128 


61.60 


96.18 


173 


60.70 


141.48 


38 


62.42 


6. 


84 


62.19 


52.04 


129 


61.58 


97.19 


174 


60.68 


142.49 


39 


62.42 


7 


85 


62.18 


53.05 


130 


61.56 


98.19 


175 


60.66 


143. CO 


40 


62.42 


8'. 


86 


62.17 


54.05 


131 


61.54 


99.20 


176 


60.64 


144.51 


41 


62.42 


9. 


87 


62.16 


55.05 


132 


61.52 


100.20 


177 


60.62 


145.52 


42 


62.42 


10. 


88 


62.15 


56.05 


133 


61.51 


101.21 


178 


60.59 


146.52 


43 


62.42 


11. 


89 


62.14 


57.05 


134 


61.49 


102.21 


179 


60.57 


147.53 


44 


62.42 


12. 


90 


62.13 


58.06 


135 


61.47 


103.22 


180 


60.55 


148.54 


45 


62.42 


13. 


91 


62.12 


59.06 


136 


61.45 


104.22 


181 


60.53 


149.55 


46 


62.42 


14. 


92 


62.11 


60.06 


137 


61.43 


105.23 


182 


60.50 


150.56 


47 


62.42 


15. 


93 


62.10 


61.06 


138 


61.41 


106.23 


183 


60.48 


151.57 


48 


62.41 


16. 


94 


62.09 


62.06 


139 


61.39 


107.24 


184. 


69.46 


152.58 


49 


62.41 


17. 


95 


62.08 


63.07 


140 


61.37 


108.25 


185 


60.44 


153.59 


50 


62.41 


18. 


96 


62.07 


64.07 


141 


61.36 


109.25 


186 


60.41 


154.60 


51 


62.41 


19. 


97 


62.06 


65.07 


142 


61.34 


110.26 


187 


60.39 


155.61 


52 


62.40 


20. 


98 


62.05 


66.07 


143 


61 32 


111.26 


188 


60.37 


156.62 


53 


62.40 


21.01 


99 


62.03 


67.08 


144 
145 


61.30 




189 


60.34 


157.63 


54 


62.40 


22.01 


100 


62.02 


68.08 


61.28 


113.28 


190 


60.32 


158.64 


55 


62.39 


23.01 


101 


62.01 


69.08 


146 


61.26 


114.28 


191 


60.29 


159.65 


56 


62.39 


24.01 


102 


62.00 


70.09 


147 


61.24 


115.29 


192 


60.27 


160.67 


57 


62.39 


25.01 


103 


61.99 


71.09 


148 


61.22 


116.29 


193 


60.25 


161.68 


58 


62.38 


26.01 


104 


61.97 


72.09 


149 


61.20 


117.30 


194 


60.22 


162 69 


59 


62.38 


27.01 


105 


61.96 


73.10 


150 


61.18 


118.31 


195 


60.20 


163.70 


60 


62.37 


28.01 


106 


61.95 


74.10 


151 


61.16 


119.31 


196 


60.17 


164.71 


61 


62.37 


29.01 


107 


61.93 


75.10 


152 


61.14 


120.32 


197 


60.15 


165.72 


62 


62.36 


30.01 


108 


61.92 


76.10 


153 


61.12 


121.33 


198 


60.12 


166.78 


63 


62.36 


31.01 


109 


61.91 


77.11 


154 


61.10 




199 


60.10 


167.74 


64 


62.35 


32.01 


110 


61.89 


78.11 


155 


61.08 


123.34 


200 


60.07 


168.75 


65 


62.34 


33.01 


111 


61.88 


79.11 


156 


61.06 


124.35 


201 


60.05 


169.77 


66 


62.34 


34.02 


112 


61.86 


80.12 


157 


61.04 


125.35 


202 


60.02 


170.78 


67 


62.33 


35.02 


113 


61.85 


81.12 


158 


61.02 


126.36 


203 


60.00 


171.79 


68 


62.33 


36.02 


114 


61.83 


82.13 


159 


61.00 




204 


59.97 


172.80 


69 


62.32 


37.02 


115 


61.82 


83.13 


160 


60.98 


128.37 


205 


59.95 


173.81 


70 


62.31 


38.02 


116 


61.80 


84.13 


161 


60.96 


129.38 


206 


59.92 


174.83 


71 


62.31 


39.02 


117 


61.78 


85.14 


162 


60.94 


130.39 


207 


59.89 


175.84 


72 


62.30 


40.02 


118 


61.77 


86.14 


163 


60.92 


131.40 


208 


59.87 


176.85 


73 


62.29 


41.02 


119 


61.75 


87.15 


164 


60.90 


132.41 


209 


59.84 


177.86 


74 


62.28 


42.03 


120 


61.74 


88.15 


165 


60.87 


133.41 


210 


59.82 


178.87 


75 




43.03 


121 


61.72 


89.15 


166 


60.85 


134.4-4 


211 


59.79 


179.89 


76 


62.27 


44.03 


122 


61.70 


90.16 


167 


60.83 


135.43 


212 


59.76 


180.90 


77 


62.26 


45.03 





















Comparison of Heads of Water in Feet with Pressures in 
Various Units. 



One foot of water at 39°. 1 Fahr, 



62.425 lbs. on the square foot; 
0.4335 lbs. on the square inch; 
= 0.0295 atmosphere; 
= 0.88.26 inch of mercury at 32° ; 
_ r/r-o o , ( feet of air at 32° and 

1 atmospheric pressure* 



PRESSURE OF WATER. 



549 



One lb. on the square foot, at 39°. 1 Fahr = 0.01602 foot of water; 

One lb. on the square inch " = 5.307 feet of water; 

One atmosphere of 29.922 inches of mercury =33.9 " " " 

One inch of mercury at 32°. 1 = 1.133 " " " 

One foot of air at 32 cleg., and one atmosphere.. = 0.001293 " " " 

One foot of average sea water = 1.026 foot of pure water; 

One foot of water at 62° F = 62.355 lbs. per sq. foot ; 

" " " " "62°F = 0.43302 lbs. per sq. inch; 

One inch of water at 62° F = 0.036085" " " " 

One pound of water on the square inch at 62° F. = 2.3094 feet of water. 

Pressure in Pounds per Square Inch for Different Heads 
of Water. 



At 62° F. 1 foot head = 
per cubic foot. 



0.433 lb. per square inch, .433 X 144 : 



Head, feet. 





1 


2 


3 


4 

1.732 


5 


6 


7 


8 


9 







0.433 


0.866 


1.299 


2.165 


2.598 


3.031 


3.464 


3.897 


10 


4.330 


4.763 


5.196 


5.629 


6.062 


6.495 


6.928 


7.361 


7.794 


8.227 


20 


8.660 


9.093 


9.526 


9.959 


10 392 


10.825 


11.258 


11.691 


12.124 


12.557 


30 


12.990 


13.423 


13.856 


14.289 


14.722 


15.155 


15.588 


16.021 


16.454 


16.8S7 


40 


1 7.820 


17.753 


18.186 


18.619 


19.052 


19.485 


19.918 


20.351 


20.784 


21.217 


50 


21.650 


22.083 


22.516 


22.949 


23.382 


23.815 


24.248 


24.681 


25.114 


25.547 


60 


25.980 


26.413 


26.816 


27.279 


27.712 


28.145 


28.578 


29.011 


29.444 


29.877 


70 


30.310 


30.743 


31.176 


31.609 


32.042 


32.475 


32.908 


33.341 


33.774 


34.207 


80 


34.640 


85.073 


35.506 


35.939 




36.805 


- 


37.671 


38.104 


38.537 


90 


38.970 


39.403 


39.836 


40.269 


40.702 


41.135 


41.568 


42.001 


42.436 


42.867 



Head in Feet of "Water, Corresponding to Pressures in 
Pounds per Square Inch. 



1 lb. per square inch = 2.30947 feet head, 1 atmosphere = 
inch = 33.94 ft. head. 



14.7 lbs. per sq. 



Pressure. 





1 
2.309 


2 


3 


4 


5 


6 


7 


8 


9 







4.619 


6.928 9.238 


11.547 


13.857 


16.166 


18.476 


20.785 


10 


23.0947 25.404 27.714 


30.023 32.333 


34.642 


36.952 


39.261 


41.570 


43.880 


20 


46.1894 48.499 50.808 


53.118 55.427 


57.737 


60.046 


62.356 


64.665 


66.975 


30 


69.2841 71.594 73.903 


76.213 78.522 


80.831 


83.141 


85.450 


87.760 


90.069 


40 


92.3788 94.688 96.998 


99.307 101.62 


103.93 


106.24 


108.55 


110.85 


113.16 


50 


115.4735117.78:120.09 


122.40 124.71 


126.02 


129.33 


131.64 


133.95 


136.26 


60 


138.5682 140.88 143.19 


145.50 147.81 


150.12 


152.42 


154.73 


157.04 


159.35 


70 


161.6629 163. 97jl66. 28 


168.59 170.90 


173 21 


175.52 


177.83 


180.14 


182.45 


80 


184. 7576,187. 07ilS9. 38 


191.69 194 00 


196.31 


198.61 


200.92 


203.23 


205.54 


90 


207.8523 210.16 212.47 


214.78 217.09 

1 


219.40 


221.71 


224.02 


226.33 


228.64 



Pressure of Water due to its Weight.— The pressure of still 
water in pounds per square inch against the sides of any pipe, channel, or 
vessel of any shape whatever is due solely to the " head," or heigrht of the 
level surface of the water above the point at which the pressure is con- 
sidered, and is equal to .43302 lb. per square inch for every foot of head, 
or 62.355 lbs. per square foot for every foot of head (at 62° F.). 

The pressure per square inch is equal in all directions, downwards, up- 
wards, or sideways, and is independent of the shape or size of the containing 
vessel. 

The pressure against a vertical surface, as a retaining-wall, at any point 
is in direct ratio to the head above that point, increasing from at the level 
surface to a maximum at the bottom. The total pressure against a vertical 
Strip of a unit's breadth increases as the area of a right-angled triangle 



550 WATER. 

whose perpendicular represents the height of the strip and whose base 
represents the pressure on a unit of surface at the bottom; that is, it in- 
creases as the square of the depth. The sum of all the horizontal pressures 
is represented by the area of the triangle, and the resultant of this sum is 
equal to this sum exerted at a point one third of the height from the bottom. 
(The centre of gravity of the area ol a triangle is one third of its height.) 

The horizontal pressure is the same it' the surface is inclined instead of 
vertical. 

(For an elaboration of these principles see Trautwine's Pocket-Book, or 
the chapter on Hydrostatics in any work on Physics. For dams, retaining- 
walls, etc., see Trautwine.) 

The amount of pressure on the interior walls of a pipe has no appreciable 
effect upon the amount of flow. 

Buoyancy.— When a body is'immersed in a liquid, whether it float or 
sink, it is buoyed up by a force equal to the weight of the bulk of the liquid 
displaced by the body. The weight of a floating body is equal to the weight 
of the bulk of the liquid that it displaces. The upward pressure or buoy- 
ancy of the liquid may be regarded as exerted at the centre of gravity of 
the displaced water, which is called the centre of pressure or of buoyancy. 
A vertical line drawn through it is called the axis of buoyancy or of flota- 
tion. In a floating body at rest a line joining the centre of gravity and the 
centre of buoyancy is vertical, and is called the axis of equilibrium. When 
an external force causes the axis of equilibrium to lean, if a vertical line be 
drawn upward from the centre of buoyancy to this axis, the point where it 
cuts the axis is called the meiacentre. If the metacentre is above the centre 
of gravity the distance between them is called the metacentric height, and 
the body is then said to be in stable equilibrium, tending to return to its 
original position when the external force is removed. 

Boiling-point. — Water boils at '212° F. (100° C.) at mean atmospheric 
pressure at the sea-level, 14.696 lbs. per square inch. The temperature at 
which water boils at any given pressure is the same as the temperature of 
saturated steam at the same pressure. For boiling-point of water at other 
pressure than 14.696 lbs. per square inch, see table of the Properties of 
Saturated Steam. 

The Boiling-point of "Water may be Raised.— When water 
is entirely freed of air., wmich may be accomplished by freezing or boiling, 
the cohesion of its atoms is greatly increased, so that its temperature may 
be raised over 50° above the ordinary boiling-point before ebullition takes 
place. It was found by Faraday that when such air-freed water did boil, 
the rupture of the liquid was like an explosion. When water is surrounded 
by a film of oil, its boiling temperature may be raised considerably above 
its normal standard. This has been applied as a theoretical explanation in 
the instance of boiler-explosions. 

The freezing-point also may be lowered, if the water is perfectly quiet, to 
- 10° C, or 18° Fahrenheit below the normal freezing-point. (Hamilton 
Smith, Jr., on Hydraulics, p. 13.) The density of water at 14° F. is .99814, its 
density at 39°. 1 being 1, and at 32°, .99987. 

Freezing-point.— Water freezes at 32° F. at the ordinary atmospheric 
pressure, and ice melts at the same temperature. In the melting of 1 pound 
of ice into water at 32° F. about 142 heat-units are absorbed, or become 
latent: and in freezing 1 lb. of water into ice a like quantity of heat is given 
out to the surrounding medium. 

Sea-water freezes at 27° F. The ice is fresh. (Trautwine.) 

Ice and Snow. (From Clark.)— 1 cubic foot of ice at 32° F. weighs 
57.50 lbs. ; 1 pound of ice at 32° F. has a volume of .0174 cu. ft. = 30.067 cu. in. 

Relative volume of ice to water at 32° F., 1.0855, the expansion in passing 
into the solid state being 8.55$. Specific gravity of ice = 0.922, water at 
62°. F. being 1. 

At high pressures the melting-point of ice is lower than 32° F., being at 
the rate of .0133° F. for each additional atmosphere of pressure 

The specific heat of ice is .504, that of water being 1. 

1 cubic foot of fresh snow, according to humidity of atmosphere: 5 lbs. to 
12 lbs. 1 cubic foot of snow moistened and compacted by rain: 15 lbs. to 
50 lbs. (Trautwine). 

Specific Heat of "Water. (From Clark's Steam-engine.) — Calcu- 
lated by means of Regnault's formula, c = 1 -j- 0.00004i -f 0.0000009* 2 , in 
which c is the specific heat of water at any temperature t in centigrade de- 
grees, the specific heat at the freezing-point being 1. 



THE IMPURITIES OF WATER. 



551 



Tempera- 
tures. 


SB'* 3 * 


5** 


Mean Specific 
Heat between 
32° F. and the 
given Temp. 


Tempera- 
tures. 




+3 9 

o*>55 


Mean Specific 
Heat between 
32° F. and the 
given Temp. 


Cent. 


Fahr. 


Cent. 


Fahr. 


0° 


32" 


0.000 


1.0000 




120° 


218° 


217.449 


1.0177 


1.0067 


10 


50 


18.004 


1.0005 


1.0002 


130 


266 


235.791 


1.0204 


1.0076 


20 


68 


36.018 


1.0012 


1.0005 


140 


284 


254.187 


1.0232 


1.0087 


30 


86 


54.047 


1.0020 


1.0009 


150 


302 


272.628 


1.0262 


1.0097 


40 


104 


72.090 


1.0030 


1.0013 


160 


320 


291.132 


1.0294 


1.0109 


50 


122 


90.157 


1.0042 


1.0017 


170 


3~" 


309.690 


1.0328 


1.0121 


60 


140 


108.247 


1.0056 


1.0023 


180 


35d 


328.320 


1.0364 


1.0133 


70 


158 


126.378 


1.0072 


1.0030 


190 


374 


347.004 


1.0401 


1.0146 


SO 


176 


144.508 


1.0089 


1.0035 


200 


392 


365.760 


1.0440 


1.0160 


90 


194 


162.686 


1.0109 


1.0042 


210 


410 


384.588 


1.0481 


1.0174 


100 


212 


180.900 


1.0130 


1.0050 


220 


428 


403.-18!- 


1.0524 


1.0189 


110 


230 


199.152 


1.0153. 


1.0058 


230 


446 


422.47S- 


1.0568 


1.0204 



Compressibility of Water.— Water is very slightly compressible. 
Its compressibility is from .000040 to .000051 for one atmosphere, decreasing 
with increase of temperature. For each foot of pressure distilled water will 
be diminished in volume .0000015 to .0000013. Water is so incompressible 
that even at a depth of a mile a cubic foot of water will weigh only about 
half a pound more than at the surface. 



THE IMPURITIES OF WATER. 

(A. E. Hunt and G. H. Clapp, Trans. A. I. M. E. xvii. : 



3.) 



Commercial analyses are made to determine concerning a given water: 
(1) its applicability for making "3am; (2) its hardness, or the facility with 
which it will " form a lather" necessary for washing; or (3) its adaptation 
to other manufacturing purposes. 

At the Buffalo meeting of the Chemical Section of the A. A. A. S. it was de- 
cided to report all water analyses in parts per thousand, hundred-thousand, 
and million. 

To convert grains per imperial (British) gallons into parts per 100,000, di- 
vide by 0.7. To convert parts per 100,000 into grains per U. S. gallon, mul- 
tiply by 7/12 or .583. 

The most common commercial analysis of water is made to determine its 
fitness for making steam. Water containing more than 5 parts per 100,000 
of free sulphuric or nitric acid is liable to cause serious corrosion, not only 
of the metal of the boiler itself, but of the pipes, cylinders, pistons, and 
valves with which the steam comes in contact. 

The total residue in water used for making steam causes the interior lin- 
ings of boilers to become coated, and often produces a dangerous hard 
scale, which prevents the cooling action of the water from protecting the 
metal against burning. 

Lime and magnesia bicarbonates in water lose their excess of carbonic 
acid on boiling, and often, especially when the water contains sulphuric 
acid, produce, with the other solid residues constantly bein^ formed by the 
evaporation, a very hard and insoluble scale. A larger amount than 100 
parts per 100,000 of total solid residue will ordinarily cause troublesome 
scale, and should condemn the water for use in steam-boilers, unless a 
better supply can be obtained. 

The following is a tabulated form of the causes of trouble with water for 
steam purposes, and the proposed remedies, given by Prof. L. M. Norton. 

Causes of Incrustation. 

1. Deposition of suspended matter. 

2. Deposition of deposed salts from concentration. 

3. Deposition of carbonates of lime and magnesia by boiling off carbonic 
acid, which holds them in solution, 



552 



4. Deposition of sulphates of lime, because sulphate of lime is but slightly 
soluble in cold water, less soluble in hot water, insoluble above 270° F. 

5. Deposition of magnesia, because magnesium salts decompose at high 
temperature. 

6 Deposition of lime soap, iron soap, etc., formed by saponification of 
grease. 

Means for Preventing Incrustation. 

1. Filtration. 

2. Blowing off. 

3. Use of internal collecting apparatus or devices for directing the cir- 
culation. 

4. Heating feed-water. 

5. Chemical or other treatment of water in boiler. 

6. Introduction of zinc into boiler. 

7. Chemical treatment of water outside of boiler. 



Troublesome Substance. 
Sediment, mud, clay, etc. 
Readily soluble salts. 

Bicarbonates of lime, magnesia, } 
iron. j 

Sulphate of lime. 
Chloride and sulphate of magne- ) 
sium. f 

Carbonate of soda in large ) 
amounts. f 

Acid (in mine waters). 

Dissolved carbonic acid and | 
oxygen. j 

Grease (from condensed water). 

Organic matter (sewage). 
Organic matter. 



Tabular View. 

Trouble. 
Incrustation. 



Priming. 
Corrosion. 



Priming. 
Corrosion. 



Remedy or Palliation. 
Filtration ; blowing off. 
Blowing off. 

Heating feed. Addition of 
caustic soda, lime, or 
magnesia, etc. 
j Addition of carb. soda, 
I barium chloride, etc. 
j Addition of carbonate of 
} soda, etc. 

j Addition of barium chlo- 
j ride, etc. 
Alkali. 

I Heating feed. Addition of 
•< caustic soda, slacked 
( lime, etc. 
Slacked lime and filtering. 
Carbonate of soda. 
Substitute mineral oil. 
Precipitate with alum or 
ferric chloride and filter. 
Ditto. 



The mineral matters causing the most troublesome boiler-scales are bicar- 
bonates and sulphates of lime and magnesia, oxides of iron and alumina, 
and silica. The analyses of some of the most common and troublesome 
boiler-scales are given in the following table : 





Analyses 


of Boiler-scale. (Chandler.) 






Sul- 
phate 

of 
Lime. 


Mag- 
nesia. 


Silica. 


Per- 
oxide 

of 
Iron. 


Water. 


Car- 
bonate 

of 
Lime. 


N.Y.C 


&H.R.Ry.,No. 1 

No. 2 
No. 3 
No. 4 
No. 5 
No. 6 

" " No. 7 
" No. 8 

" " No. 9 
No. 10 


74.07 
71.37 
62.86' 
53.05 
46.83 
30.80 
4.95 
0.88 
4.81 
30.07 


9.19 
"18.95' 

"3ili7 

2.61 

2.84 


0.65 
1.76 
2.60 
4.79 
5.32 
7.75 
2.07 
0.65 
2.92 
8.24 


0.08 


1.14 


14.78 


" 


0.92 


1.28 


12.62 


<t 






26! 93* 

86.25 
93.19 


» 


i.08 
1.03 
0.36 


2.44 
0.63 
0.15 



THE IMPURITIES OF WATER. 



553 



Analyses in Parts per 100,000 of Water giving 
Results in Steam-boilers. (A. E. Hunt.) 







, Sjc 






















it- 










































hJ = 


s~ 






















|| 




a 








a> 




G 




§S 


= o 

- i 


J 


^ 


o 






o 


2 


© 
73 
























o — 


5.5 


S3 


o 


o. 


o 

3 


o 


?* 


5 


O 

3 




S-c 


- i. 




fcH 


cc 


U 


1-1 


O 


< 


o 




no 

151 

130 
80 
32 

30 


25 

38 
89 
•21 
70 
82 
50 


119 

1.90 

95 

161 
94 
01 
41 


39 
48 
120 
33 
81 
1.04 
68 


890 
3150 
310 
•210 
219 
28 
890 


590 
990 
21 
38 
210 
1.90 
42 


7S0 
38 
75 
70 
90 
38 
23 


30 
•21 

10 


640 
30 
80 






13.10 


Spring 1 


36 














a u 








Allegheny R., near Oil-works 







Many substances have been added with the idea of causing chemical 
action which will prevent boiler-scale. As a general rule, these do more 
harm than good, for a boiler is one of the worst possible places in which to 
carry on chemical reaction, where it nearly always causes more or less 
corrosion of the metal, and is liable to cause dangerous explosions. 

In cases where water containing large amounts of total solid residue is 
necessarily used, a heavy petroleum oil, free from tar or wax, which is not 
acted upon by acids or alkalies, not having sufficient wax in it to cause 
saponification, and which has a vaporizing-point at nearly 600° F., will give 
the best results in preventing boiler-scale. Its action is to form a thin 
greasy film over the boiler linings, protecting them largely from the action 
of acids in the water and greasing the sediment which is formed, thus pre- 
venting the formation of scale and keeping the solid residue from the 
evaporation of the water in such a plastic suspended condition that it can 
be easily ejected from the boiler by the process of " blowing off. 1 ' If the 
water is not blown off sufficiently often, this sediment forms into a " putty" 
that will necessitate cleaning the boilers. Any boiler using bad water should 
be blown off every twelve hours. 

Hardness of Water.— The hardness of water, or its opposite quality, 
indicated by the ease with which it will form a lather with soap, depends 
almost altogether upon the presence of compounds of lime and magnesia. 
Almost all soaps consist, chemically, of oleate, stearate, and palmitate, of 
an alkaline base, usually soda and potash. The more lime and magnesia in a 
sample of water, the more soap a given volume of the water will decompose, 
so as to give insoluble oleate, palmitate, and stearate of lime and magnesia, 
and consequently the more soap must be added to a gallon of water in order 
that the necessary quantity of soap may remain in solution to form the lather. 
The relative hardness of samples of water is generally expressed in terms 
of the number of standard soap-measures consumed by a gallon of water in 
yielding a permanent lather. 

The standard soap-measure is the quantity required to precipitate one 
grain of carbonate of lime. 

It is commonly reckoned that one gallon of pure distilled water takes one 
soap-measure to produce a lather. Therefore one is deducted from the 
total number of soap-measures found to be necessary to use to produce a 
lather in a gallon of water, in reporting the number of soap-measures, or 
" degrees " of hardness of the water sample. In actually making tests for 
hardness, the " miniature gallon," or seventy cubic centimetres, is used 
rather than the inconvenient larger amount. The standard measure is made 
by completely dissolving ten grammes of pure castile soap (containing 60 per- 
cent olive-oil) in a litre of weak alcohol (of about 35 per cent alcohol). This 
yields a solution containing exactly sufficient soap in one cubic centimeter 
of the solution to precipitate one milligramme of carbonate of lime, or, in 
other words, the standard soap solution is reduced to terms of the " minia- 
ture gallon" of water taken. 

If a water charged with a bicarbonate of lime, magnesia, or iron is boiled, 



554 



WATER. 



it will, on the excess of the carbonic acid being expelled, deposit a, consid- 
erable quantity of the lime, magnesia, or iron, and consequently the water 
will be softer. The hardness of the water after this deposit of lime, after 
long boiling, is called the permanent hardness and the difference between it 
and the total hardness is called temporary hardness. 

Lime salts in water react immediately on soap-solutions, precipitating the 
oleate, palmitate, or stearate of lime at once. Magnesia salts, on the con- 
trary, require some considerable time for reaction. They are, however, 
more powerful hardeners ; one equivalent of magnesia salts consuming as 
much soap as one and one-half equivalents of lime. 

The presence of soda and potash salts softens rather than hardens water. 
Each grain of carbonate of lime per gallon of water causes an increased 
expenditure for soap of about 2 ounces per 100 gallons of water. {Eng^g. 
Neivs, Jan. 31, 1885.) 

Purifying Feed-water for Steam-boilers.— To effect the 
purification of water before and after being fed into a boiler, a device man- 
ufactured by the Albany Steam Trap Company, Albany, N. Y. removes 
the impurities by the process of a continuous circulation of the water from 
the boiler, through the filter and back into the boiler, The scale forming 
impurities that are held in suspension are thus brought in contact with 
and "arrested" by the filtering agent contained in the filter while under 
pressure, and at a temperature limited only by that contained in the boiler. 

It is sometimes desirable, in the removal of the sulphates and carbonates 
from the feed-water, to heat the water up to nearly the same temperature 
as it is in the boiler, and then to filter the same before feeding it into the 
boiler. The operation in a general way is : The water is first forced into the 
usual exhaust -heater by the feed-pump, and there it is heated by the ex- 
haust from the engine, say to 200°, and at this temperature it enters the re- 
heater. The reheater consists of a vertical, cylindrical shell containing a 
series of water pans or shelves, and so arranged that as the water enters it 
it delivered into the top pan, and then overflows into the second, and so on 
down the series to the bottom, and during its transit deposits the scale- 
forming material. The circulating-pump takes the water from the bottom of 
the reheater and forces it through the filter on its way into the boiler. 

Mr. W. B. Coggswell, of the Solvay Process Co.'s Soda Works in Syracuse, 
N. Y., thus describes the system of purification of boiler feed-water in use 
at these works (Trans. A. S. M. E., xiii. 255): 

For purifying, we use a weak soda liquor, containing about 12 to 15 grams 
Na 2 Co 3 per litre. Say 1^ to 2 M 3 (or 397 to 530 gals.) of this liquor is run 
into the precipitating tank. Hot water about 60° C. is then turned in, and 
the reaction of the precipitation goes on while the tank is filling, which re- 
quires about 15 minutes. When the tank is full the water is filtered through 
the Hyatt (4), 5 feet diameter, and the Jewell (1), 10 feet diameter, filters in 
30 minutes. Forty tanks treated per 24 hours. 

Charge of water purified at once 35 M 3 , 9,275 gallons. 

Soda in purifying reagent 15 kgs. Na 2 C0 3 . 

Soda used per 1,000 gallons 3.5 lbs. 

A sample is taken from each boiler every other day and tested for deg. 
Baume, soda and salt. If the deg. B is more than 2, that boiler is blown to 
reduce it below 2 deg. B. 

The following are some analyses given by Mr. Coggswell : 





Lake 

Water, 

grams per 

litre. 


Mud from 
Hyatt 
Filter. 


Scale from 
Boiler- 
tube. 


Scale 
found 

in 
Pump. 




.261 
.186 
.091 
.015 
.087 
.63 


3.70 


51.24 


10.9 






Calcium carbonate 

Magnesium carbonate 


63.37 
1.11 


19.76 
25.21 


87. 




Salt, NaCl 




.14 
2.29 
1.10 




Silica 


15.17 
3.75 


.8 


Iron and aluminum oxide. . . 




1.2 


Total 


1.270 


87.10 


99.74 


99.9 



FLOW OF WATER. 555 

Softening Hard Water for Locomotive Use.— A water-soft- 
ening plant in operation at Fossil, in Western Wyoming, on the Union Pa- 
cific Railway, is described in Encfg News, June 9. 1892. It is the invention 
of Arthur Pennell, of Kansas City. The general plan adopted is to first dis- 
solve the chemicals in aclo-ed tank, and then connect this to the supply main 
so that its contents will be forced into the main tank, the supply-pipe being 
so arranged that thorough mixture ot the solution with the water is ob- 
tained. A waste-pipe from the bottom of the tank is opened from time to 
time to draw off the precipitate. The pipe leading to the tender is arranged 
to draw the water from near the surface. 

A water-tank 24 feet in diameter and 16 feet high will contain about 46,600 
gallons of water. About three hours should be allowed for this amount of 
water to pass through the tank to insure thorough precipitation, giving a 
permissible consumption of about 15,000 gallons per hour. Should more 
than this be required, auxiliary settling-tanks should be provided. 

The chemicals added to precipitate the scale-forming impurities are so- 
dium carbonate and quicklime, varying in proportions according to the rela- 
tive proportions of sulphates and carbonates in the water to be treated. 
Sufficient sodium carbonate is added to produce just enough sodium sulphate 
to combine with the remaining lime and magnesia sulphate and produce 
glauberite or its corresponding magnesia salt, thereby to get rid of the 
sodium sulphate, which produces foaming, if allowed to accumulate. 



HYDRAULICS-FLOW OP WATER. 

Formulae for Discharge of Water though Orifices and 

Weirs.— For rectangular or circular orifices, with the head measured from 
centre of the orifice to the surface of the still water in the feeding reservoir. 

Q= CVtoJHX a (1) 

For weirs with no allowance for increased head due to velocity of approach: 

Q=C%+2gHxLH . . . (2) 

For rectangular and circular or other shaped vertical or inclined orifices; 
formula based on the proposition that each successive horizontal layer of 
water passing through the orifice has a velocity due to its respective head: 

Q = cL% \2y X ( \Hb 3 - VHt*) (3) 

For rectangular vertical weirs: 

Q = c%\2gHxLh (4) 

Q — quantity of water discharged in cubic feet per second; C = approxi- 
mate coefficient for formulas (1) and (2) ; c — correct coefficient for (3) 
and (4). 

Values of the coefficients c and Care given below. 

g = 32.16; V2g = 8.02; H — head in feet measured from centre of orifice 
to level of still water; Hb = head measured from bottom of orifice; Ht = 
head measured from top of orifice; h = H, corrected for velocity of ap- 

4 Va? 
proach, Va, — H-\-- - — ; a = area in square feet; L = length in feet. 

Flow of Water from Orifices.— The theoretical velocity of water 
flowing from an orifice is the same as the velocity of a falling body which 
has fallen from a height equal to the head of water, — \'2gH. The actual 
velocity at the smaller section of the vena contractu is substantially the 
same as the theoretical, but the velocity at the plane of the orifice is 
C \'2gH, in which the coefficient C has the nearly constant value of .62. ^The 
smallest diameter of the vena contractu is therefore about .79 of that of the 
orifice. If C be the approximate coefficient = .62, and c the correct coeffi- 



556 



HYDBATTLICS. 



cient, the ratio - varies with different ratios of the head to the diameter 
c 

H 



10. 
1. 



For vertical rectangular orifices of ratio of head to width W : 

.6 .8 .1 1.5 2. 3. 4. 5. 8. 

9657 .9823 .9890 .9953 .9974 .9988 .9993 .9996 .999* 
c 

For H -¥- D or H -h W over 8, C — c, practically. 



o 



Weisbacb gives the following values of c for circular orifices in a thin wall. 
H — measured head from centre of orifice. 



D ft 


H ft. 




.066 


.33 


.82 


2.0 


3.0 


45. 


340. 


.033 
.066 
.10 
.13 


.711 


.665 


.637 
.629 
.622 
.614 


.628 
.621 
.614 
.607 


.641 


.632 


.600 



For an orifice of D = . 
effective head in feet, 



5 ft. and a well-rounded mouthpiece, H being the 



H = . 



: .959 



l.C 



11.5 



56 
.994 



338 
.994 



Hamilton Smith, Jr., found that for great heads, 312 ft. to 336 ft., with con- 
verging mouthpieces, c has a value of about one. and for small circular 
orifices in thin plates, with full contraction, c — about .60. Some of Mr. 
Smith's experimental values of c for orifices in thin plates discharging into 
air are as follows. All dimensions in feet. 



Circular, in steel, D — 


.«*>. f^Z 


.739 

.6495 


2.43 
.6298 


3.19 
.6264 






Circular, in brass, D = 


.050,{ ^ = 


.185 
.6525 


.536 

.6265 


1.74 
.6113 


2.73 
.6070 


3.57 4.63 
.6060 .6051 


Circular, in brass, D = 


■m\ H c z 


.129 

.6337 


.457 
.6155 


.900 
.6096 


1.73 

.6042 


2.05 3.18 
.6038 .6025 


Circular, in iron, D = 


•™'\ H c = 


.80 
.6061 


1.81 
.6041 


2.81 
.6033 


4.68 
.6026 




Square, in brass, .05 x 


^m 


.313 

.6410 


.877 
.6238 


1.79 

.6157 


2.81 
.6127 


3.70 4.63 
.6113 .6097 


Square, in brass, .10 X 


•io, K = 


.181 
.6292 


.939 
.6139 


1.71 
..6084 


2.75 
.6076 


3.74 4.59 
.6060 .6065 


Rectangular, in brass, 


j H = 


.261 


.917 


1.82 


2.83 


3.75 4.70 


Z= .300, W = .050.. 


. ...1 c = 


.6476 


.6280 


.6203 


.6180 


.6176 .6168 



For the rectangular orifice, L, the length, is horizontal. 

Mr. Smith, as the result of the collation of much experimental data of 
others as well as his own, gives tables of the value of c for vertical orifices, 
with full contraction, with a free discharge into the air, with the inner face 
of the plate, in which the orifice is pierced, plane, and with sharp inner 
corners, so that the escaping vein only touches these inner edges. These 
tables are abridged below. The coefficient c is to be used in the formulas (3) 
and (4) above. For formulas (1) and (2) use the coefficient found from the 

C 
values of the ratios — above. 



HYDRAULIC FORMULAE. 



557 



Values of Coefficient e for Vertical Orifices with Sharp 
Edges, Full Contraction, and Free Discharge into 
Air. (.Hamilton Smith, Jr.) 



^6 



1.0 
3.0 
6.0 

10. 
20. 
100. (?) 



Square Orifices. Length of the Side of the Square, in feet. 



.03 


.04 
.643 


.05 
.637 


.07 
.628 


.10 
.621 


.12 
.616 


.15 


.20 


.40 


.60 


.80 




,611 






.645 


.636 


.630 


.623 


.617 


.613 


.610 


.605 


.601 


.598 


.596 


.636 


.628 


.622 


.618 


.613 


.610 


.608 


.605 


603 


.601 


.600 


.622 


.616 


.612 


.609 


.607 


.606 


.606 


, 605 


.605 


.604 


,603 


.616 


.612 


.609 


.607 


.605 


.605 


.605 


.604 


.604 


.603 


.602 


.611 


608 


.606 


.605 


.604 


.604 


.603 


.603 


.603 


.602 


602 


.605 


.604 


.603 


.602 


.602 


.602 


602 


602 


601 


601 


.601 


.598 


.598 


.598 


.598 


.598 


.598 


.598 


.598 


598 


.598 


.598 



Circular Orifices. Diameters, in feet. 



10. 
20. 
50.(?) 
100.(?) 



.637 
.624 
.617 
.610 
.605! 
.604' 
.601 
.598; 
.5951 
.592 



07 


.10 


.12 


.15 


.20 


.40 


.60 


.80 


62^ 


.618 


.612 


.606 








CIS 


.613 


.609 


.605 


.601 


.596 


.593 


.590 


612 


.608 


.605 


.603 


.600 


.598 


.595 


.593 


ho; 


.604 


.601 


.600 


.599 


.599 


.597 


.596 


603 


.602 


.600 


.599 


.599 


.598 


.597 


.597 




.60(1 


.599 


.599 


.598 


.598 


.597 


.596 


599 


.598 


.598 


.597 


.597 


.597 


.596 


.596 


597 


.596 


.596 


.596 


596 


.596 


.596 


.595 


594 


.594 


.594 


.594 


.594 


.594 


.594 


.593 


592 


■ 502 


.592 


.592 


.592 


.592 


.592 


.592 



.591 
.595 
.596 
.596 
.595 
.594 
.593 
.592 



HTDRAFLIC FORMULJE.-FLOW OF WATER IN 
OPEN AND CLOSED CHANNELS. 

Flow of Water in Pipes.— The quantity of water discharged 
through a pipe depends on the •'head;' 1 that is, the vertical distance be- 
tween the level surface of still water in the chamber at the entrance end of 
the pipe and the level of the centre of the discharge end of the pipe ; 
also upon the length of the pipe, upon the character of its interior surface 
as to smoothness, and upon the number and sharpness of the bends: but 
it is independent of the position of the pipe, as horizontal, or inclined 
upwards or downwards. 

The head, instead of being an actual distance between levels, may be 
caused by pressure, as by a pump, in which case the head is calculated as a 
vertical distance corresponding to the pressure 1 lb. per sq. in. = 2.309 ft. 
head, or 1 ft. head = .433 lb. per sq. in. 

The total head operating to cause flow is divided into three parts: 1. The 
velocity-head, which is the height through which a body must fall in vacuo 
to acquire the velocity with which the water flows into the pipe = v 2 h- 2g, in 
which v is the velocity in ft. per sec. and 2g = 64.32; 2. the entry-head, that 
required to overcome the resistance to entrance to the pipe. With sharp- 
edged entrance the entry-head = about V^ the velocity-head; with smooth 
rounded entrance the entry-head is inappreciable; 3. the friction-head, due 
to the frictional resistance to fl >w within the pipe. 

In ordinary cases of pipes of considerable length the sum of the entry and 
velocity heads required scarcely exceeds 1 foot. In the case of Ions: pipes 
with low heads the sum of the velocity and entry heads is generally so small 
that it may be neglected. 

General Formula for Flow of Water in Pipes or Co nduits. 

Mean velocity in ft. per sec. = c \ n ieau hydraulic ra dius X slope 

„ „ . „ ,, /diameter 

Do. for pipes running full = ca/ x slope, 

in which c is a coefficient determined by experiment. (See pages 559-564.) 



558 



HYDRAULICS. 



m , ■' ' :.. ,. £.rea of wet cross-section 

The mean hydraulic radius = : . 

wet perimeter. 

In pipes running full, or exactly half full, and in semicircular open chan- 
nels running full it is equal to 34 diameter. 

The slope = the head (or pressure expressed as a head, in feet) 

-=- length of pipe measured in a straight line from end to end. 

In open channels the slope is the actual slope of the surface, or its fall per 
unit of length, or the sine of the angle of the slope with the horizon. 

If r = mean hydraulic radius, s = slope = head -h lengthy = velocity in 
feet per second (all dimensions in feet), v = c \'r \/s = c )/rs. 

Quantity of Water Discharged. -If Q = discharge in cubic feet 
per second and a — area of channel, Q = av — uc Vrs< 

a V'r is approximately proportional to the discharge. It is a maximum at 
308°, corresponding to 19/20 of the diameter, and the flow of a conduit 19/20 
full is about 5 per cent greater than that of one completely filled. 

Table giving Fall in Feet per Mile, the Distance on Slope 
corresponding to a Fall of 1 Ft., and also_the Values 

of s and V* for Use in the Formula v — c \ rs. 

s = H-^-L— sine of angle of slope = fall of water-surface (H), in any dis- 
tance (L), divided by that distance. 



Fall in 


Slope, 


Sine of 




Fall in 


Slope, 


Sine of 




Feet 


1 Foot 


Slope, 


Vs. 


Feet 


1 Foot 


Slope, 


Vs. 


per Mi. 


in 


s. 




per Mi. 


in 


s. 




0.25 


21120 


.0000473 


.006881 


17 


310.6 


.0032197 


.056742 


.30 


17600 


.0000568 


.007538 


18 


293.3 


.0034091 


.058S88 


.40 


13200 


.0000758 


.008704 


19 


277.9 


.0035985 


.059988 


.50 


10560 


.0000947 


.009731 


20 


264 


.0037879 


.061546 


.60 


8800 


.0001136 


.010660 


22 


240 


.0041667 


.064549 


.702 


7520 


.0001330 


.011532 


24 


220 


.0045455 


.067419 


.805 


6560 


.0001524 


.012347 


26 


203.1 


.0049242 


.070173 


.904 


5840 


.0001712 


.013085 


28 


188.6 


.0053030 


.072822 


1. 


5280 


.0001894 


.013762 


30 


176 


.0056818 


.075378 


1.25 


4224 


.0002367 


.015386 


35.20 


150 


.0066667 


.081650 


1.5 


3520 


.0002841 


.016854 


40 


132 


.0075758 


.087039 


1.75 


3017 


.0003314 


.018205 


44 


120 


.0083333 


.091287 


2. 


2640 


.0003788 


.019463 


48 


110 


.0090909 


.095346 


2.25 


2347 


.0004261 


.020641 


52.8 


100 


.010 


.1 


2.5 


2112 


.0004735 


.021760 


60 


88 


.0113636 


.1066 


2.75 


1920 


.0005208 


.022822 


66 


80 


.0125 


.111803 


3. 


1760 


.0005682 


.023837 


70.4 


75 


.0133333 


.115470 


3.25 


1625 


.0006154 


.024807 


80 


66 


.0151515 


.123091 


3.5 


1508 


.0006631 


.025751 


88 


60 


.0166667 


.1291 


3.75 


1408 


.0007102 


.026650 


96 


55 


.0181818 


.134839 


4 


1320 


.0007576 


.027524 


105.6 


50 


.02 


.141421 


5 


1056 


.0009470 


.030773 


120 


44 


.0227273 


.150756 


6 


880 


.0011364 


.03371 


132 


40 


.025 


.158114 


7 


754.3 


.0013257 


.036416 


160 


33 


.0303030 


.174077 


8 


660 


.0015152 


.038925 


220 


24 


.0416667 


.204124 


9 


586.6 


.0017044 


.041286 


264 


20 


.05 


.223607 


10 


528 


.0018939 


.043519 


330 


16 


.0625 


.25 


11 


443.6 


.0020833 


.045643 


440 


12 


.0833333 


.288675 


12 


440 


.0022727 


.047673 


528 


10 


.1 


.316228 


13 


406.1 


.0024621 


.04962 


660 


8 


.125 


.353553 


14 


377.1 


.0026515 


.051493 


880 


6 


.1666667 


.408248 


15 


352 


.0028409 


.0533 


1056 


5 


.2 


.447214 


16 


330 


.0030303 


.055048 


1320 


4 


!25 


.5 



HYDRAULIC FORMULAE. 



559 



Values of yr for Circular Pipes, Sewers, and Conduits of 
different Diameters. 

r = mean hydraulic depth = : — = 14 diam. for circular pipes run- 

■ 7 * perimeter * * 

ning full or exactly half full. 



Diam. . 


\ r Y 


Di 


Mil., 


\'r 


Diam., 


Yr 


Diam., 


in Feet. 


ft. in. 


in Feet. 


i't. 


in. 


in Feet. 


ft. 


in. 


in Feet. 


ft, 


111. 


¥s 


.088 


2 




.707 


4 


6 


1.061 


9 




1.500 


y* 


.102 


2 


1 


.722 


4 


7 


1.070 


9 


3 


1.521 


% 


.125 


2 


2 


.736 


4 


8 


1.0S0 


9 


6 


1.541 




.144 


2 


3 


.750 


4 


9 


1.089 


9 


9 


1.561 


m 


.161 


2 


4 


.764 


4 


10 


1.099 


10 




1.581 


V4 


.177 


2 


5 


.777 


4 


11 


1.109 


10 


3 


1.601 


m 


•191 


2 


6 


.790 


5 




1.118 


10 


6 


1.620 


2 


.204 


2 


7 


.804 


5 


1 


1.127 


10 


9 


1.639 


%\& 


.228 


2 


8 


.817 


5 


2 


1.137 


11 




1.658 


3 


.251 


2 


9 


.829 


5 


3 


1.146 


11 


3 


1.677 


4 


.290 


2 


10 


.842 


5 


4 


1.155 


11 


6 


1.696 


5 


.323 


2 


11 


.854 


5 


5 


1.164 


11 


9 


1.714 


6 


.354 


3 




.866 


5 


6 


1.173 


12 




1.732 


7 


.382 


3 


1 


.878 


5 


7 


1.181 


12 


3 


1 .750 


8 


.408 


3 


2 


.890 


5 


8 


1.190 


12 


6 


1.768 


9 


.433 


3 


3 


.901 


5 


9 


1.199 


12 


9 


1.785 


10 


.456 


3 


4 


.913 


5 


10 


1.208 


13 




1.083 


11 


.479 


3 


5 


.924 


5 


11 


1.216 


13 


3 


1.820 


1 


.500 


3 


6 


.935 


6 




1.225 


13 


6 


1.837 


1 1 


.520 


3 


7 


.946 


6 


3 


1.250 


14 




1.871 


1 2 


.540 


3 


8 


.957 


6 


6 


1.275 


14 


6 


1.904 


1 3 


.559 


3 


9 


.968 


6 


9 


1.299 


15 




1.936 


1 4 


.577 


3 


10 


.979 


7 




1.323 


15 


6 


1.968 


1 5 


.595 


3 


11 


.990 


7 


3 


1.346 


16 




2. 


1 6 


.612 


4 




1. 


7 


6 


1.369 


16 


6 


2.031 


1 7 


.629 


4 


1 


1.010 


7 


9 


1.392 


17 




2.061 


1 8 


.646 


4 


2 


1.021 


S 




1.414 


17 


6 


2.091 


1 9 


.661 


4 


3 


1.031 


8 


3 


1.436 


18 




2.121 


1 10 


.677 


4 


4 


1.041 


8 


6 


1.458 


19 




2.180 


1 11 


.692 


4 


5 


1.051 


8 


9 


1.479 


20 




2.236 



Values of the Coefficient c. (Chiefly condensed from P. J. Flynn 
on Flow of Water.)— Almost all the old hydraulic formulae for finding 'the 
mean velocity in open and closed channels have constant coefficients, and are 
therefore correct for only a small range of channels. They have often been 
found to give incorrect results with disastrous effects. Ganguillet and Kut- 
ter thoroughly investigated the American, French, and other experiments, 
and they gave as the result of their labors the formula now generally known 
as Kutter's formula. There are so many varying conditions affecting the 
flow of water, that all hydraulic formulas are only approximations to the 
correct result. 

When the surface-slope measurement is good, Kutter's formula will give 
results seldom exceeding 7^4% error, provided the rugosity coefficient of the 
formula is known for the site. For small open channels D'Arcy's and 
Bazin's formulas, and for cast-iron pipes D'Arcy's formulas, are generally 
accepted as being approximately correct. 

Kutter's Formula for measures in feet is 



f 

I 



1.811 



+ 4U 



.00281 



i'+O 



X Vrs, 



s y " Yr i 
which v — mean velocity in feet per second ; r 



• hydraulic mean 



560 HYDRAULICS. 

depth in feet = area of cross-section in square feet divided by wetted perim- 
eter in lineal feet ; s = fall of water-surface (h) in any distance (I) divided 

by that distance, = -, == sine of slope ; n — the coefficient of rugosity, de- 
pending on the nature of the lining or surface of the channel. If we let the 
first term of the right-hand_side of the equation equal c, we have Chezy's 
formula, v = c Yrs = c X Vr X Vs- 

Values ofn in Kutter's Formula.— The accuracy of Kutter's for- 
mula depends, in a great measure, on the proper selection of the coefficient 
of roughness n. Experience is required in order to give the right value to 
this coefficient, and to this end great assistance can be obtained, in making 
this selection, by consulting and comparing the results obtained from ex- 
periments on the flow of water already made in different channels. 

In some cases it would be well to provide for the contingency of future 
deterioration of channel, by selecting a high value of n, as, for instance, 
where a dense growth of weeds is likely to occur in small channels, and also 
where channels are likely not to be kept in a state of good repair. 

The following table, giving the value of n for different materials, is com- 
piled from Kutter, Jackson, and Hering, and this value of n applies also in 
each instance, to the surfaces of other materials equally rough. 

Value of n in Kutter's Formula for Different Channels. 

n — .009, well-planed timber, in perfect order and alignment ; otherwise, 
perhaps .01 would be suitable. 

n = .010, plaster in pure cement ; planed timber ; glazed, coated, or en- 
amelled stoneware and iron pipes ; glazed surfaces of every sort in perfect 
order. 

n = .011, plaster in cement with one third sand, in good condition ; also for 
iron, cement, and terra-cotta pipes, well joined, and in best order. 

n = .012, unplaned timber, when perfectly continuous on the inside ; 
flumes. 

n — .013, ashlar and well-laid brickwork ; ordinary metal ; earthen and 
stoneware pipe in good condition, but not new ; cement and terra-cotta pipe 
not well jointed nor in perfect order , plaster and planed wood in imperfect 
or inferior condition ; and, generally, the materials mentioned with n — .010, 
when in imperfect or inferior condition. 

n — .015, second class or rough-faced brickwork ; well-dressed stonework ; 
foul and slightly tuberculated iron ; cement and terra-cotta pipes, with im- 
perfect joints and in bad order ; and canvas lining on wooden frames. 

n = .017, brickwork, ashlar, and stoneware in an inferior condition ; tu- 
berculated iron pipes ; rubble in cement or plaster in good order ; fine gravel, 
well rammed, y% to % inch diameter ; and, generally, the materials men- 
tioned with n = .013 when in bad order and condition. 

n — .020, rubble in cement in an inferior condition ; coarse rubble, rough 
set in a normal condition ; coarse rubble set dry ; ruined brickwork and 
masonry ; coarse gravel well rammed, from 1 to l£g inch diameter ; canals 
with beds and banks of very firm, regular gravel, carefully trimmed and 
rammed in defective places ; rough rubble with bed partially covered with 
silt and mud ; rectangular wooden troughs, with battens on the inside two 
inches apart ; trimmed earth in perfect order. 

it = .0225, canals in earth above the average in order and regimen. 

n — .025. canals and rivers in earth of tolerably uniform cross-section ; 
slope and direction, in moderately good order and regimen, and free from 
stones and weeds. 

n — .0^75, canals and rivers in earth below the average in order and regi- 
men. 

n = .030, canals and rivers in earth in rather bad order and regimen, hav- 
ing stones and weeds occasionally, and obstructed by detritus. 

n = .035, suitable for rivers and canals with earthen beds in bad order and 
regimen, and having stones and weeds in great quantities. 

n — .05, torrents encumbered with detritus. 

Kutter's formula has the advantage of being easily adapted to a change 
in the surface of the pipe exposed to the flow of water, by a change in the 
value of n. For cast-iron pipes it is usual to use n = .013 to provide for the 
future deterioration of the surface. 

Reducing Kutter's formula to the form v = cX Vr X Vs, and taking n, the 
coefficient of roughness in the formula = .011, .012, and .013, and s = .001, we 
have the following values of the coefficient c for different diameters cf 
conduit. 



HYDRAULIC FORMULAE. 



561. 



Values of c in Formula v - c x \ r x \ s for Metal Pipes and 
Moderately Smooth Conduits Generally. 

By Kutter's Formula, (s = .001 or greater.) 



Diameter. 


n = .011 


n = .012 


ft = .013 


Diameter. 


n = .011 


ft = .012 


ft = .013 


ft. 


in. 


c = 


c = 


c = 


ft. 


c = 


c = 


c = 





1 
2 

4 


47.1 
61.5 

77.4 






7 
8 
9 


152.7 
155.4 
157.7 


139.2 
141.9 
144.1 


127.9 








130.4 








132.7 




6 


87.4 


77.5 


69.5 


10 


159.7 


146 


134.5 


1 




105.7 


94.6 


85.3 


11 


161.5 


147.8 


136.2 


1 


6 


116.1 


104.3 


94.4 


12 


163 


149.3 


137.7 


2 




123.6 


111.3 


101.1 


14 


165.8 


152 


140.4 


3 




133.6 


120.8 


110.1 


16 


168 


154.2 


142.1 


4 




140.4 


127.4 


116.5 


18 


169.9 


156.1 


144.4 


5 




145.4 


132.3 


121.1 


20 


171.6 


157.7 


146 


6 




149.4 


136.1 


124.8 











For circular pipes the hydraulic mean depth r equals *4 of tlie diameter. 

According to Kutter's formula the value of c, the coefficient of discharge, 
is the same for all slopes greater than 1 in 1000; that is, within these limits 
c is constant. We further find that up to a slope of 1 in 2640 the value of c 
is, for all practical purposes, constant, and even up to a slope of 1 in 5000 
the difference in the value of c is very little. This is exemplified in the 
following : 

Value of c for Different Values of \'r and s in Kutter's 

Formula, with nj= .013. 

-- c V r X Ys 





Slopes. 


Vr 


1 in 1000 


1 in 2500 


1 in 3333.3 


1 in 5000 


1 in 10,000 


.6 

1 
2 


93.6 
116.5 
142.6 


91.5 
115.2 
142.8 


90.4 
114.4 
143.0 


88.4 
113.2 
143.1 


83.3 
109.7 
143.8 



The reliability of the values of the coefficient of Kutter's formula for 
pipes of less than 6 in. diameter is considered doubtful. (See note under 
table on page 564.) 

Values of c for Earthen Channels, by Kutter's Formula, 
for Use in Formula v — c Vrs. 





Coefficient of Roughness, 


Coefficient of Roughness, 






ft = .0225. 






ft = .035. 






Vr in feet. 


Yr in feet. 




0.4 


1.0 


1.8 


2.5 


4.0 


0.4 


1.0 


1.8 


2.5 


4.0 


Slope, 1 in 


c 


c 


c 


c 


c 


c 


c 


c 


c 


c 


1000 


35.7 


62.5 


80.3 


89.2 


99.9 


19.7 


37.6 


51.6 


59.3 


69.2 


1250 


35 5 


62.3 


80.3 


89.3 


100.2 


19.6 


37.6 


51.6 


59.4 


69.4 


1667 


35.2 


62.1 


80.3 


89.5 


100 6 


19.4 


37.4 


51.6 


59.5 


£9.8 


2500 


34.6 


61.7 


80.3 


89.8 


101.4 


19.1 


37.1 


51.6 


59.7 


70.4 


3333 


34. 


61.2 


80.3 


90.1 


102.2 


18.8 


36.9 


51.6 


59.9 


71.0 


5000 


33. 


60.5 


80.3 


90.7 


103.7 


18.3 


36.4 


51.6 


60.4 


72.2 


7500 


31.6 


59.4 


80.3 


91.5 


106.0 


17.6 


35.8 


51.6 


60.9 


73.9 


10000 


30.5 


58.5 


80.3 


92.3 


107.9 


17.1 


35.3 


51.6 


60.5 


75.4 


15840 


28.5 


56.7 


80.2 


93.9 


112.2 


16.2 


34.3 


51.6 


62.5 


78.6 


20000 


27.4 


55.7 


80.2 J 


94.8 


115.0 


15.6 


33.8 


51.5 


63.1 


80.6 



562 



HYDRAULICS. 



Mr. Molesworth, in the 22d edition of his " Pocket-book of Engineering 
Formulae," gives a modification of Kutter's formula as follows : For flow in 
cast-iron pipes, v = c )/rs, in which 



'l + ^+^l 



in which d — diameter of the pipe in feet. 
(This formula was given incorrectly in Molesworth's 21st edition.) 
Molesworth's Formula.- v = Vkrs, in which the values of k are 

as follows : 



Nature of Channel. 


Values of k for Velocities. 


Less than 
4 ft. per sec. 


More than 
4 ft. per sec. 




8800 
7200 
6400 
5300 


8500 


Earth 


6800 




5900 


Rough, with bowlders 


4700 



In very large channels, rivers, etc., the description of the channel affects 
the result so slightly that it may be practically neglected, and k assumed = 
from 8500 to 9000. 

Flynn's Formula.— Mr. Flynn obtains the following expression of 
the value of Kutter's coefficient for a slope of .001 and a value of n = .013 : 

183.72 



- (44.41 5 



The following table shows the close agreement of the values of c obtained 
from Kutter's, Molesworth's, and Flynn's formulas : 



Diameter. 


Slope. 


Kutter. 


Molesworth. 


Flynn. 


6 inches 


lin 40 


71.50 


71.48 


69.5 


6 inches 


1 in 1000 


69.50 


69.79 


69.5 


4 feet 


1 in 400 


117. 


117. 


116.5 


4 feet 


1 in 1000 


116.5 


116.55 


116.5 


8 feet 


1 in 700 


130.5 


130.68 


130.5 


8 feet 


1 in 2600 


129.8 


129.93 


130.5 



Mr. Flynn gives another simplified form of Kutter's formula for use with 
different values of n as follows : 



A 1+ H x i-))" 



In the following table the value of Kis given for the several values of n : 


n 


K 


n 


K 


n 


K 


n 


K 


n 


K 


.009 
.010 
.011 


245.63 
225.51 

209.05 


.012 
.013 
.014 


195.33 

183.72 
137.77 


.015 
.016 
.017 


165.14 
157.6 
150.94 


.018 
.019 
.020 


145.03 
139.73 
184.96 


.021 
.022 

.0225 


130.65 
126.73 
124.9 



If in the application of Mr. Flynn's formula given above within the limits 
of n as given in the table, we substitute for n, K, and Vr their values, we 
have a simplified form of Kutter's formula. 



HYDRAULIC FORMULA. 563 



For instance, when n = .011, and d = 3 feet, we have 
>9.05 



Vrs. 



/ -01l\ ' 



Bazin's Formulae : 

For very even surfaces, fine plastered sides and bed, planed planks, etc., 



v =|/l -*- .0000045(l0.16 -f~) X Vn 



For even surfaces such as cut-stone, brickwork, unplaned planking, mortar, 
etc. : 



v = i/l -*- .000013(4.354 + i) x Vrs. 
For slightly uneven surfaces, such as rubble masonry : 

v = i/l -*- .00006(l.219 + i) x VrsI 
For uneven surfaces, such as earth : 

v = i/l -4- .00035(0.2438 -f -) X Vrs. 
A modification of Bazin's formula, known as D'Arcy's Bazin's : 

- • / 1 000"s 
V ~ ? \ .08534r -f 0.35* 

For small channels of less than 20 feet bed Bazin's formula for earthen 
channels in good order gives very fair results, but Kutter's formula is super- 
seding it in almost all countries where its accuracy has been investigated. 

The last table on p. 561 shows the value of c, in Kutter's formula, for a wide 
range of channels in earth, that will cover anything likely to occur in the 
ordinary practice of an engineer. 

D'Arcy's Formula for clean iron pipes under pressure is 



_ ( rs ) 

V -! ™™,.,™ , .00000162 1 
j .00007726 -\ f 

Flynn's modification of D'Arcy's formula is 
/ 155256 \^ .j— 

in which d = diameter in feet. 

D'Arcy's formula, as given by J. B. Francis, C.E., for old cast-iron pipe 
lined with deposit and under pressure, is 

v = ( U4d* s \y 2 

V.0082(12d + 1 V . . • 
Flynn's modification of D'Arcy's formula for old cast-iron pipe is 

/ 70243.9 \ A/ — 



564 



HYDRAULICS. 



For Pipes Less _than 5 inches in Diameter, coefficients (c) 
in the formula v — c Vrs, from the formula of D'Arcy, Kutter, and Fanning. 



Diam. 

in 
inches. 


D'Arcy, 

for Clean 

Pipes. 


Kutter, 

for 
n = .011 
s = .001 


Fanning, 

for Clean 

Iron 

Pipes 


Diam. 

in 
inches 


D'Arcy, 

for Clean 

Pipes. 


Kutter, 

for 
n = .011 
s = .001 


Fanning, 

for Clean 

Iron 

Pipes. 


% 

H 

m 


59.4 
65.? 

74.5 
80.4 

84.8 
88.1 


32. 

36.1 

42.6 

47.4 

51.9 

55.4 


80.4 
88. 


¥ 

4 

5 


90.7 
92.9 
96.1 
98.5 
101.7 
103.8 


58.8 

61.5 

66. 

70.1 

77.4 

82.9 


92.5 
94.8 

96.6 
103.4 



Mr. Flynn, in giving the above table, says that the facts show that the co- 
efficients diminish from a diameter of 5 inches to smaller diameters, and it 
is a safer plan to adopt coefficients varying with the diameter than a con- 
stant coefficient. No opinion is advanced as to what coefficients should be 
used with Kutter's formula for small diameters. The facts are simply 
stated, giving the results of well-known authors. 

Older Formulae,— The following are a few of the many formulae for 
flow of water in pipes given by earlier writers. As they have constant coef- 
ficients, they are not considered as reliable as the newer formulas. 

Prony, v = 97 Vrs - .08; 

Eytelwein, v = 50 a/- .^ or v = 108 Vrs - 0.13 ; 
y I -\- 50a 

— 3 /— 

Neville, v = 140 Vrs - 11 yrs. 



Hawksley, 



Wd 



54cT 



: head of water in feet; I = 
r — mean hydraulic depth, 



In these formulae d = diameter in feet; h = 
length of pipe in feet; s = sine of slope = y; 

== area -v- wet perimeter = — for circular pipe. 

Mr. Santo Crimp (Eug'g, August 4, 1893) states that observations on flow 
in brick sewers show that the actual discharge is 33% greater than that cal- 
culated by Eytelwein's formula. He thinks Kutter's formula not superior 
to D'Arcy's for brick sewers, the usual coefficient of roughness in the 
former, viz., .013, being too low for large sewers and far too small in the case 
of small sewers. 

D'Arcy's formula for brickwork is 

v = ^rs ; m = a(\ + — ) ; a = .0037285; B = .229663. 
m \ r ' 

VELOCITY OF WATER IN OPEN CHANNELS. 

Irrigation Canals.— The minimum mean velocity required to prevent 
the deposit of silt or the growth of aquatic plants is in Northern India 
taken at \y» feet per second. It is stated that in America a higher velocity 
is required for this purpose, and it varies from 2 to 3J^ feet per second. The 
maximum allowable velocity will vary with the nature of the soil of the 
bed. A sandy bed will be disturbed if the velocity exceeds 3 feet per 
second. Good loam with not too much sand will bear a velocity of 4 feet 
per second. The Cavour Canal in Italy, over a gravel bed, has a velocity of 
about 5 per second. ( Flynn 's "Irrigation Canals.") 

Mean Surface and Bottom Velocities.— According to the for- 
mula of Bazin, 



= Umax — 25.4 Vt'i 



-- vb + 10.87 Vrs. 



VELOCITY OF WATER IN OPEN CHANNELS. 565 



.-•. vb ~ v — 10.87 \ ,r rs, in which v = mean velocity in feet per second, 
vmak - maximum surface velocity in feet per second, vb = bottom velocity 
in feet per second, r = hydraulic mean depth in feet = area of cross-section 
in square feet divided by wetted perimeter in feet, s = sine of slope. 

The least velocity, or that of the particles in contact with the bed, is 
almost as much less than the mean velocity as the greatest velocity is 
greater than the mean. 

Rankine states that in ordinary cases the velocities may be taken as bear- 
ing to each other nearly the proportions of 3, 4, and 5. In very slow cur- 
rents they are nearly as 2, 3, and 4. 

Safe Bottom and Mean Velocities.— Ganguillet & Kutter give 
the following table of safe bottom and mean velocity in channels, calculated 
from the formula v = vb -\- 10.87 \'rs' 



Material of Channel. 



Soft brown earth 

Soft loam 

Sand 

Gravel 

Pebbles 

Broken stone, flint 

Conglomerate, soft slate. 

Stratified rock 

Hard rock 



Safe Bottom Veloc 


Mean Velocity v, 


ity vb, in feet 


in feet per 
second. 


per second. 


0.249 


0.328 


0.499 


0.656 


1.000 


1.312 


1.998 


2.625 


2.999 


3.938 


4.003 


5.579 


4.988 


6.564 


6.006 


8.204 


10.009 


13.127 



Ganguillet & Kutter state that they are unable for want of observations 
to judge how far these figures are trustworthy. They consider them to be 
rather disproportionately small than too large, and therefore recommend 
them more confidently. 

Water flowing at a high velocity and carrying large quanties of silt is very 
destructive to channels, even when constructed of the best masonry. 

Resistance of Soils to Erosion by Water.— W. A. Burr, Eng'g 
Nexus, Feb. 8, 1894, gives a diagram showing the resistance of various soils to 
erosion by flowing water. 

Experiments show that a velocity greater than 1.1 feet per. second will 
erode sand, while pure clay will stand a velocity of 7.35 feet per second. 
The greater the proportion of clay carried by any soil, the higher the per- 
missible velocity. Mr. Burr states that experiments have shown that the line 
describing the power of soils to resist erosion is parabolic. From his dia- 
gram the following figures are selected representing different classes of 
soils: 

Pure sand resists erosion by flow of 1.1 feet per second. 

Sandy soil, 15$ clay 1.2 " " 

Sandy loam, 40$ clay 1.8 " " 

Loamy soil, 65$ clay 3.0 " " 

Clay loam, 85$ clay 4.8 " " 

Agricultural clay, 95$ clay 6.2 " " 

Clay 7.35 " 

Abrading and Transporting Power of Water.— Prof. J. 
LeConte, in his "Elements of Geology," states : 

The erosive power of water, or its power of overcoming cohesion, varies as 
the square of the velocity of the current. 

The transporting power of a current varies as the sixth power of the ve- 
locity. * * * If the velocity therefore be increased ten times, the transport- 
ing power is increased 1,000,000 times. A current running three feet per 
second, or about two miles per hour, will bear fragments of stone of the 
size of a hen's egg, or about three ounces weight. A current of ten miles an 
hour will bear fragments of one and a half tons, and a torrent of twenty 
miles an hour will carry fragments of 100 tons. 

The transporting power of water must not be confounded with its erosive 
power. The resistance to be overcome in the one case is weight, in the 
other, cohesion ; the latter varies as the square : the former as the sixth 
power of the velocity. 

In many cases of removal of slightly cohering material, the resistance is a 



566 HYDRAULICS. 

mixture of these two resistances, and the power of removing material will 
vary at some rate between i> 2 and v*. 

Baldwin Latham has found that in order to prevent deposits of sewage silt 
in small sewers or drains, such as those from 6 inches to 9 inches diameter, 
a mean velocity of not less than 3 feet per second should be produced. 
Sewers from 12 to 24. inches diameter should have a velocity of not less than 
2% feet per second, and in sewers of larger dimensions in no case should the 
velocity be less than 2 feet per second. 

The specific gravity of the materials has a marked effect upon the mean 
velocities necessary to move them. T. E. Blackwell found that coal of a 
sp. gr. of 1.26 was moved by a current of from 1.25 to 1.50 ft. per second, 
while stones of a sp. gr. of 2 32 to 3.00 required a velocity of 2.5 to 2.75 ft. per 
second. 

Chailly gives the following formula for finding the velocity required to 
move rounded stones or shingle : 

v = 5.67 Vag, 

in which v = velocity of water in feet per second, a = average diameter in 
feet of the body to be moved, g = its specific gravity. 

Geo. Y. Wisner, Eng'g News, Jan 10, 1895, doubts the general accuracy of 
statements made by many authorities concerning the rate of flow of a cur- 
rent and the size of particles which different velocities will move. He says: 

The scouring action of any river, for any given rate of current, must be an 
inverse function of the depth. The fact that some engineer has found that 
a given velocity of current on some stream of unknown depth will move 
sand or gravel has no bearing whatever on what may be expected of cur- 
rents of the same velocity in streams of greater depths. In channels 3 to 5 
ft. deep a mean velocity of 3 to 5 ft. per second may produce rapid scouring, 
while in depths of 18 ft. and upwards current velocities of 6 to 8 ft. per 
second often have no effect whatever on the channel bed. 

Grade of Sewers.— The following empirical formula is given in Bau- 
meister's " Cleaning and Sewerage of Cities," for the minimum grade for a 
sewer of clear diameter equal to d inches, and either circular or oval in 
section : 

Minimum grade, in per cent, = , . 

As the lowest limit of grades which can be flushed, 0.1 to 0.2 per cent may 
be assumed for sewers which are sometimes dry, while 0.3 per cent is allow- 
able for the trunk sewers in large cities. The sewers should run dry as 
rarely as possible. 

Relation of Diameter of Pipe to Quantity Discharged.— 
In many cases which arise iu practice the information sought is the diame- 
ter necessary to supply a given quautity of water under a given head. The 
diameter is commonly taken to vary as the two-fifth power of the dis- 
charge. This is almost certainly too large. Hagen's formula, with Prof. 

f O \ 387 
Unwin's coefficients, give d = el j^ V , where c = .239 when d and Q 

\Q)> 

are in feet and cubic feet per second. 

Mr. Thrupp has proposed a formula which makes d vary as the .383 power 
of the discharge, and the formula of M. Vallot, a French engineer, makes d 
vary as the .375 power of the discharge. (Engineering.) 

FliOW OF WATER-EXPERIMENTS AND TABLES. 

The Flow of Water through New Cast-iron Pipe was 

recently measured by S. Bent Russell, of the St. Louis, Mo., Water-works. 
The pipe was 12 inches in diameter, 1631 feet long, and laid on a uniform 
grade from end to end. Under an average total head of 3.36 feet the flow 
was 43,200 cubic feet in seven hours; under an average head of 3.37 feet the 
flow was the same; under an average total head of 3.41 feet the flow was 
46,700 cubic feet in 8 hours and 35 minutes. Making allowance for loss 
of head due to entrance and to curves, it was found that the value of c in 
the formula v = c \/rs was from 88 to 93 (Eng'g Record, April 14, 1894. 

Flow of Water in a 20-inch Pipe 75,000 Feet Long.— A 
comparison of experimental data with calculations by different formulae is 



FLOW OF WATER — EXPERIMENTS AttD TABLES. 567 

given by Chas. B. Brush, Trans. A. S. C. E., 1888. The pipe experimented 
with was that supplying the city of Hoboken, N. J. 

Results Obtained by the Hackensack Water Company, from 1882-1887, 
in Pumping Through a 20-in. Cast-iron Main 75,000 Feet Long. 

Pressure in lbs. per sq. in. at pumping-station: 

95 100 105 110 115 120 125 130 

Total effective head in feet : 

55 66 77 89 100 112 123 135 

Discharge in U. S. gallons in 24 hours, 1 = 1000 : 

2,848 3,165 3,354 3,566 3,804 I 
Actuai velocity in main in feet per second : 

2.00 2.24 2.36 2.52 2.68 



4,116 4,255 



2 92 



3.00 



Cost of coal consumed in delivering each million gals, at given velocities : 
$8.40 $8.15 $8.00 $8.10 $8.30 $8.60 $9.00 $9.60 

Theoretical discharge by D'Arcy's formula : 

2,743 3,004 3,244 3,488 3,699 3,915 4,102 4,297 



Velocities in Smooth Cast-iron Water-pipes from 1 Foot 
to 9 Feet in Diameter, on Hydraulic Grades of 0.5 
Foot to 8 Feet^per Mile; with Corresponding Values 

of c in V- c \rs. (D. M. Greene, in Eng'g News, Feb. 24, 1894.) 



0) S 


!«■- 




hydraulic Grade; Feet per Mile = h. 




sr„ 


■sil 












h = 0.5 


1.0 


1.5 


2.0 


3.0 


4.0 


D. 


r. 


s = 0.0000947 


0.0001894 


0.0002841 


0.0003788 


0.0005682 


0.0007576 


1. 


0.25] 


V= 0.4542 


0.6673 


0.8356 


0.9803 


1.2277 


1.4402 


c= 92.7 


97.0 


99.1 


100.7 


103.0 


104.7 


2. 


0.5 -j 


V= 0.7359 


1.0793 


1.3516 


1.5856 


1.9857 


2.3294 


c= 106.6 


110.9 


113.4 


115.2 


117.9 


119.7 


3. 


0.75J 


V= 0.9733 


1.4298 


1.7906 


2.1017 


2.6306 


3.0860 


c= 115.5 


119.9 


122.6 


124.4 


127.5 


129.5 


4. 


1.0 


F= 1.1883 


1.7456 


2.1861 


2.5645 


3.2116 


3.7676 


c = 122.1 


126.8 


129.7 


131.8 


134.7 


136.9 


5. 


1.25 -j 


V= 1.3872 


2.0379 


2.5521 


2.9939 


3.7493 


4.39S3 


c = 127.5 


132.4 


135.5 


137.6 


140.7 


142.9 


6. 


'•M 


V= 1.5742 


2.3126 


2.8961 


3.3975 


4.2548 


4.9913 


c = 132.1 


137.8 


140.3 


142.6 


145.8 


148.1 


7. 


1.75^ 


V= 1.7518 


2.5736 


3.2230 


3.7809 


4.7350 


5.5546 


c = 135.9 


141.4 


146.0 


146,8 


150.2 


152.5 


8. 


2.0 -j 


V= 1.9218 


2.8234 


3.5358 


4.1479 


5.1945 


6.0936 


c = 139.7 


145.1 


148.4 


150.7 


154.1 


156.5 


9. 


2.25J 


F = 2.0854 


3.0638 


3.8368 


4.5010 


5.6368 


6.6125 


c = 142.9 


148.4 


151.7 


154.2 


157.6 


160.1 



The velocities in this table have been calculated by Mr. Greene's modifi- 
cation of the Chezy formula, which modification is found to give results 
which differ by from 1.29 to — 2.65 per cent (average 0.9 per centj from very 
carefully measured flows in pipes from 16 to 48 inches in diameter, on grades 
from 1.68 feet to 10.296 feet per mile, and in which the velocities ranged 
from 1.577 to 6.195 feet per second. The only assumption made is that the 
modified formula for V gives correct results in conduits from 4 feet to 9 
feet in diameter, as it is known to do in conduits less than 4 feet in diameter. 

Other articles on Flow of Water in long- tubes are to be found in Eng'g 
News as follows : G. B. Pearsons, Sept, 23, 1876; E. Sherman Gould, Feb. 16, 
23, March 9, 16, and 23, 1889; J. L. Fitzgerald, Sept. 6 and 13, 1890; Jas. Duane, 
Jan. 2, 1892; J. T. Fanning, July 14, 1892; A. N. Talbot, Aug. 11, 1892. 



568 



HYDRAULICS. 



Flow of Water in Circular Pipes, Sewers, etc., Flowing 
Full. Based on Kutter's Formula, with n = .013. 

Discbarge in cubic feet per second. 







Slope, or Head Divided by Length of Pipe. 




Diam- 


















eter. 




















lin40 


Iin70 


1 in 100 


1 in 200 


1 in 300 


1 in 400 


1 in 500 


1 in 600 


5 in. 


.456 


.344 


.288 


.204 


.166 


.144 


.137 


.118 


6 " 


.762 


.576 


.482 


.341 


.278 


.241 


.230 


.197 


7 " 


1.17 


.889 


.744 


.526 


.430 


.372 


.355 


.304 


8 " 


1.70 


1.29 


1.08 


.765 


.624 


.54 


.516 


.441 


9 " 


2.37 


1.79 


1.50 


1.06 


.868 


.75 


.717 


.613 


Slope .... 


1 in 60 


1 in 80 


1 in 100 


1 in 200 


1 in 300 


1 in 400 


1 in 500 


1 in 600 


10 in. 


2.59 


2.24 


2.01 


1.42 


1.16 


1.00 


.90 


.82 


11 " 


3.39 


2 94 


2.63 


1.86 


1.52 


1.31 


1.17 


1.07 


12 " 


4.32 


3.74 


3.35 


2.37 


1.93 


1.67 


1.5 


1.37 


13 " 


5.38 


4.66 


4.16 


2.95 


2.40 


2.08 


1.86 


1.70 


14 " 


6.60 


5.72 


5.15 


3.62 


2.95 


2.57 


2.29 


2.09 


Slope... 


1 in 100 


1 in 200 


1 in 300 


1 in 400 


1 in 500 


1 in 600 


1 in 700 


1 in 800 


15 in. 


6.18 


4 37 


3.57 


3.09 


2.77 


2.52 


2.34 


2.19 


16 " 


7.38 


5.22 


4.26 


3.69 


3.30 


3.01 


2.79 


2.61 


18 " 


10.21 


7.22 


5.89 


5.10 


4.56 


4.17 


3.86 


3.61 


20 " 


13.65 


9.65 


7.88 


6.82 


6.10 


5.57 


5.16 


4.83 


22 " 


17.71 


12.52 


10.22 


8.85 


7.92 


7.23 


6.69 


6.26 


Slope 


1 in 200 


1 in 400 


1 in 600 


1 in 800 


1 in 1000 


1 in 1250 


1 in 1500 


1 in 1800 


2 ft. 


15.88 


11.23 


9.17 


7.94 


7.10 


6.35 


5 80 


5.29 


2fr.2iu. 


19.73 


13.96 


11.39 


9.87 


8.82 


7.89 


7.20 


6.58 


2 " 4 " 


24.15 


17.07 


13 94 


12.07 


10.80 


9 66 


8.82 


8.05 


2 " 6 " 


29.08 


20.56 


16.79 


14.54 


13.00 


11.63 


10.62 


9.69 


2 " 8 " 


34.71 


24.54 


20.04 


17 .35 


15.52 


13.88 


12.67 


11.57 


Slope ... 


I in 500 


1 in 750 


1 in 1000 


1 in 1250 


1 in 1500 


1 in 1750 


1 in 2000 


1 in 2500 


2 ft. 10 in. 


25.84 


21.10 


18.27 


16.34 


14.92 


13.81 


12.92 


11.55 


3 " 


30.14 


24.61 


21.31 


19.06 


17.40 


16.11 


15.07 


13.48 


3 " 2 in. 


34.90 


28.50 


24.68 


22.07 


20.15 


18.66 


17.45 


15.61 


3 " 4 " 


40.08 


32 72 


28 34 


25.35 


23.14 


21.42 


20.04 


17.93 


3 " 6 " 


45.66 


37.28 


32.28 


28.87 


26.36 


24.40 


22.83 


20.41 


Slope 


1 in 500 


1 in 750 


1 in 1000 


1 in 1250 


1 in 1500 


1 in 1750 


1 in 2000 


1 in 2500 


3 ft. 8 in. 


51.74 


42.52 


36.59 


32.72 


29.87 


27.66 


25.87 


23.14 


3 " 10 " 


58.36 


47.65 


41.27 


36 91 


33.69 


31.20 


29.18 


26.10 


4 " 


65.47 


53.46 


46.30 


41.41 


37.80 


34.50 


32.74 


29.28 


4 " 6 in. 


89.75 


73.28 


63.47 


56.76 


51.82 


47.97 


44.88 


40.14 


5 " 


118.9 


97.09 


84.08 


75.21 


68.65 


63.56 


59.46 


53.18 


Slope . . 


1 in 750 


1 in 1000 


1 in 1500 


1 in 2000 


1 in 2500 


1 in 3000 


1 in 3500 


1 in 4000 


5fr.6in. 


125.2 


108.4 


88.54 


76.67 


68.58 


62.60 


57.96 


54.21 


6 " 


157.8 


136.7 


111 6 


96.66 


86.45 


78.92 


73.07 


68 35 


6 " 6 " 


195.0 


168.8 


137.9 


119.4 


106.8 


97.49 


90.26 


84.43 


7 " 


237.7 


205.9 


168.1 


145.6 


130.2 


118.8 


110.00 


102.9 




285.3 


247.1 


201.7 


174.7 


156.3 


142.6 


132.1 


123.5 


Slope.... 


1 in 1500 


1 in 2000 


1 in 2500 


1 in 3000 


1 in 3500 


1 in 4000 


1 in 4500 


1 in 5000 


8 ft. 


239.4 


207.3 


195.4 


169.3 


156.7 


146.6 


138.2 


131.1 


8 " 6 in. 


281.1 


243.5 


217.8 


198.8 


184.0 


172.2 


162.3 


154.0 


9 " 


327.0 


283.1 


253.3 


231.2 


214.0 


200.2 


188.7 


179.1 


9 " 6 " 


376.9 


326.4 


291.9 


266.5 


246.7 


230.8 


217.6 


206.4 


10 " 


431.4 


373.6 


334.1 


305.0 


282.4 


264.2 


249.1 


236.3 



For U. S. gallons multiply the figures in the table by 7.4805. 
For a given diameter the quantity of flow varies as the square root of the 
sine of the slope. From this principle the flow for other slopes than those 



FLOW OF WATER IK CTRCULAK PIPES, ETC. 569 



{riven in the table may be found. Thus, what is the flow for a pipe 8 feet 
diameter, slope 1 in 125 ? From the table take Q = 207.3 for slope 1 in 2000. 
The given slope 1 in 125 is to 1 in 2000 as 16 to 1, and the square root of this 
ratio is 4 to 1. Therefore the flow required is 207.3 x 4 = 829.2 cu. ft. 

Circular Pipes, Conduits, etc., Flowing Full. 

"Values of the factor ac \ r in tbe formula Q = ac \'r X \'s correspond- 
ing to different values of the coefficient of roughness, n. (Based on Kutter's 
formula.) 



§ 






Value of 


acVr. 






5 

ft. in. 


n = .010. 


n = .011. 


n = .012. 


n = .013. 


n = .015. 


n = .017. 


6 


6.906 


6.0627 


5.3800 


4.8216 


3.9604 


3 329 


9 


21.25 


18.742 


16.708 


15.029 


12.421 


10.50 


1 


46.93 


41.487 


37.149 


33.497 


27.803 


23 60 


1 3 


86.05 


76.347 


68.44 


61.867 


51.600 


43.93 


1 6 


141.2 


125.60 


112.79 


102.14 


85.496 


72.99 


1 9 


214.1 


190.79 


171.66 


155.68 


130.58 


111.8 


2 


307.6 


274.50 


247.33 


224.63 


188.77 


164 


2 3 


421.9 


377.07 


340.10 


309.23 


260.47 


223.9 


2 6 


559.6 


500.78 


452.07 


411.27 


347.28 


299.3 


2 9 


722.4 


647.18 


584.90 


532.76 


451.23 


388.8 


3 


911.8 


817.50 


739.59 


674.09 


570.90 


493.3 


3 3 


1128.9 


1013.1 


917.41 


836.69 


709.56 


613.9 


3 6 


1374.7 


1234.4 


1118.6 


1021.1 


866.91 


750.8 


3 9 


1652.1 


1484.2 


1345.9 


1229.7 


1045 


906 


4 


1962.8 


1764.3 


1600.9 


1463.9 


1245.3 


1080.7 


4 6 


2682.1 


2413.3 


2193 


2007 


1711.4 


1487.3 


5 


3543 


3191.8 


2903.6 


2659 


2272.7 


1977 


5 6 


4557.8 


4111.9 


3742.7 


3429 


2934.8 


2557.2 


6 


5731.5 


5176.3 


4713.9 


4322 


3702.3 


3232.5 


6 6 


7075.2 


6394.9 


5825.9 


5339 


4588.3 


4010 


7 


8595.1 


7774 3 


7087 


6510 


5591.6 


4893 


7 6 


10296 


9318! 8 


8501.8 


7814 


6717 


5884.2 


8 


12196 


11044 


1008C 


9272 


7978.3 


6995.3 


8 6 


14298 


12954 


11832 


10889 


9377.9 


8226.3 


9 


16604 


15049 


13751 


12663 


10917 


9580.7 


9 6 


19118 


17338 


15847 


14597 


12594 


11061 


10 


21858 


19834 


18134 


16709 


14426 


12678 


10 6 


24823 


22534 


20612 


18996 


16412 


14434 


11 


28020 


25444 


23285 


21464 


18555 


16333 


11 6 


31482 


28593 


26179 


24139 


20879 


18395 


12 


35156 


31937 


29254 


26981 


23352 


20584 


12 6 


39104 


35529 


32558 


30041 


26012 


22938 


13 


43307 


39358 


36077 


33301 


28850 


25451 


13 6 


47751 


43412 


39802 


36752 


31860 


28117 


14 


52491 


47739 


43773 


40432 


35073 


30965 


14 6 


57496 


52308 


47969 


44322 


38454 


33975 


15 


62748 


57103 


52382 


48413 


42040 


37147 


16 


74191 


67557 


62008 


57343 


49823 


44073 


17 


86769 


79050 


72594 


67140 


58387 


51669 


18 


100617 


91711 


84247 


77932 


67839 


60067 


19 


115769 


105570 


96991 


89759 


78201 


69301 


20 


132133 


120570 


110905 


102559 


89423 


79259 



Flow of Water in Circular Pipes, Conduits, etc., Flowing 
under Pressure. 

Based on D'Arcy's formulae for the flow of water through cast-iron pipes. 
With comparison of results obtained by Kutter's formula, with n = .013. 
(Condensed from Flynn on Water Power.) 

Values of a, and also the values of the factors c Vr and ac fr for use in 
the formulae Q = av; v — c V'r X V s , an d Q = ac Vr X V$« 



570 



HYDRAULICS. 



Q = discharge in cubic feet per second, a = area in square feet, v = veloc- 
ity in feet per second, r — mean hydraulic depth, J4 diam. for pipes running 
full, s = sine of slope. 

(For values of 4/s see page 558.) 







Clean Cast-iron 




Old Cast- 


iron Pipes 


Size Ol .ripe. 


Pipes. 


Value of 

ac \/r by 


Lined with Deposit. 














d= diam. 


a = area 


For 


For Dis- 


Kutter's 
Formula 


For 


For 


in 


square 
feet. 


Velocity, 


charge, 


when 


Velocity, 


Discharge, 


ft. in. 


c Vr- 


ac Vr- 


n = .013. 


cYr- 


ac |/r. 


% 


.00077 


5.251 


.00403 




3.532 


.00272 


Vq 


.00136 


6.702 


.00914 




4.507 


.00613 


H 


.00307 


9.309 


.02855 




6.261 


.01922 


1 


.00545 


11.61 


.06334 




7.811 


.04257 


VA 


.00852 


13.68 


.11659 




9.255 


.07885 




.01227 


15.58 


.19115 




10.48 


.12855 


m 


.01670 


17.32 


.28936 




11.65 


.19462 


2 


.02182 


18.96 


.41357 




12.75 


.27824 


W/2 


.0341 


21.94 


.74786 




14.76 


.50321 




.0491 


24.63 


1.2089 




16.56 


.81333 


4 


.0873 


29.37 


2.5630 




19.75 


1.7246 


5 


.136 


33.54 


4.5610 




22.56 


3.0681 


6 


.196 


37.28 


7.3068 


4.822 


25.07 


4.9147 


7 


.267 


40.65 


10 852 




27.34 


7.2995 


8 


.349 


43.75 


15.270 




29.43 


10.271 


9 


.442 


46.73 


20.652 


15.03 


31.42 


13.891 


10 


.545 


49.45 


26.952 




33.26 


18.129 


11 


.660 


52.16 


34.428 




35.09 


23.158 




.785 


54.65 


42.918 


33.50 


36.75 


28.867 


1 2 


1.000 


59.34 


63.435 




39 91 


42 668 


1 4 


1.396 


63.67 


88.886 




42.83 


59.788 


1 6 


1.767 


67.75 


119.72 


102.14 


45.57 


80.531 


1 8 


2.182 


71.71 


156.46 




48.34 


105.25 


1 10 


2.640 


75.32 


198.83 




50.658 


133.74 


2 


3.142 


78.80 


247.57 


224.63 


52.961 


166.41 


2 2 


3.687 


82.15 


302.90 




55.258 


203.74 


2 4 


4.276 


85.39 


365.14 




57.436 


245.60 


2 6 


4.909 


88.39 


433.92 


411.37 


59.455 


291.87 


2 8 


5.585 


91.51 


511.10 




61.55 


343.8 


2 10 


6.305 


94.40 


595.17 




63.49 


400.3 


3 


7.068 


97.17 


686.76 


674.09 


65.35 


461.9 


3 2 


7.875 


99.93 


786.94 




67.21 


529.3 


3 4 


8.726 


102.6 


895.7 




69 


602 


3 6 


9.621 


105.1 


1011.2 


1021.1 


70.70 


680.2 


3 8 


10 559 


107.6 


1136.5 




72.40 


764.5 


3 10 


11.541 


110.2 


1271.4 




74.10 


855.2 


4 


12.566 


112.6 


1414.7 


1463.9 


75.73 


951.6 


4 3 


14.186 


116.1 


1647.6 




78.12 


1108.2 


4 6 


15.904 


1196 


1901.9 


2007 


80.43 


1279.2 


4 9 


17.721 


1:22.8 


2176.1 




82.20 


1456.8 


5 


19.6:5 


126.1 


2476.4 • 


2659 


84.83 


1665.7 


5 3 


21.648 


129.3 


2799.7 




86.99 


1883.2 


5 6 


23.758 


132.4 


3146.3 


3429 


89.07 


2116.2 


5 9 


25.967 


135.4 


3516 




91.08 


2365 


6 


2S.274 


138.4 


3912.8 


4322 


93.08 


2631.7 


6 6 


33.183 


144.1 


4782.1 


5339 


96.93 


3216.4 


7 


38.485 


149.6 


5757.5 


6510 


100.61 


3872.5 


7 6 


44.179 


154 9 


6841.6 


7814 


104.11 


4601.9 


8 


50.266 


160 


8043 


9272 


107.61 


5409.9 


8 6 


56.745 


165 


9364.7 


10889 


111 


6299.1 


9 


63.617 


169.8 


10804 


12663 


114.2 


7267.3 


9 6 


70.882 


174.5 


12370 


14597 


117.4 


8320.6 


10 


78.540 


179.1 


14066 


16709 


120.4 


9460.9 



FLOW OP WATER IK CIRCULAR PIPES, ETC. 571 







Clean Cast-iron 




Old Cast-iron Pipet 




Pipes. 


Value of 

ac Vr by 


Lined with Deposit. 














d= diam. 


a = area 


For 


For Dis- 


Kutter's 


For 


For 


HI 


square 
feet. 


Velocity, 


ci] arge, 


when 


Velocity, 


Discharge, 


ft. in. 


c Vr. 


ac Vr- 


n = .013 


c Vr. 


ac Vr. 


10 6 


86.590 


183.6 


15893 


18996 


123.4 


10690 


11 


95.033 


187.9 


17855 


21464 


126.3 


12010 


11 6 


103.869 


192.2 


19966 


24139 


129.3 


13429 


12 


113.098 


196.3 


22204 


26981 


132 


14935 


12 6 


122 719 


200.4 


24598 


30041 


134.8 


16545 


13 


132.733 


204.4 


27134 


33301 


137.5 


18252 


13 6 


143.139 


208.3 


29818 


36752 


140.1 


20056 


14 


153.938 


212.2 


32664 


40432 


142.7 


21971 


14 6 


165.130 


216.0 


35660 


44322 


145.2 


23986 


15 


176.715 


219.6 


38807 


48413 


147.7 


26103 


15 6 


188.692 


223.3 


42125 


52753 


150.1 


28335 


16 


201.062 


226.9 


45621 


57343 


152.6 


30686 


16 6 


213.825 


230 4 


49273 


62132 


155 


33144 


1? 


226.981 


233.9 


53082 


67140 


157.3 


35704 


17 6 


240.529 


237.3 


57074 


72409 


159.6 


38389 


18 


254.470 


240.7 


61249 


77932 


161.9 


41199 


19 


283.529 


247.4 


70154 


89759 


166.4 


47186 


20 


314.159 


253.8 


79736 


102559 


170.7 


53633 



Flow of Water in Circular Pipes from % inch to 12 inches 

Diameter. 



Based on D'Arcy's formula for clean cast-ii 


on pipes. Q = 


ac Vr Vs' 


Value of 


Dia. 


Slope, or Head Divided by Length of Pipe. 


ac Vr- 


linlO. 


lin 20. 


1 in 40. 


1 in 60. 


1 in 80. 


1 in 

100. 


1 in 
150. 


lin 

200. 








Quan 


tity in 


cubic 


feet p 


er sec 


ond. 




.00403 


% 


.00127 


.00090 


.00064 


.00052 


.00045 


.00040 


.00033 


.00028 


.00914 


X, 


.00289 


.00204 


.00145 


.00118 


.00102 


.00091 


.00075 


.00065 


.02855 


34 


.00903 


.00638 


.00451 


.00369 


.00319 


.00286 


.00233 


.00202 


.06334 


1 


.02003 


.01416 


.01001 


.00818 


.00708 


.00633 


.00517 


.00448 


.11659 


114 03687 


.02607 


.01843 


.01505 


.01303 


.01166 


.00952 


.00824 


.19115 


\X4> .06044 


.04274 


.03022 


.02468 


.02137 


.01912 


.01561 


.01352 


.28936 


W A 


.09140 


.06470 


.04575 


.03736 


.03235 


.02894 


.02363 


.02046 


.41357 


2 


.13077 


.09247 


.06539 


.05339 


.04624 


.04136 


.03377 


.02927 


.74786 


2Vo 


.23647 


.16722 


.11824 


.09655 


.08361 


.07479 


.06106 


.05288 


1.2089 


3 


.38225 


.27031 


.19113 


.15607 


.13515 


.12089 


.09871 


.08548 


2.5630 


4 


.81042 


.57309 


.40521 


.33088 


.28654 


.25630 


.20927 


.18123 


4.5610 


5 


1.4422 


1.0198 


.72109 


.58882 


.50992 


.45610 


.37241 


.32251 


7.3068 


6 


2.3104 


1.6338 


1 . 1552 


.94331 


.81690 


.73068 


.59660 


.51666 


10.852 


7 


3.4314 


2.4265 


1.7157 


1.4110 


1.2132 


1 .0852 


.88607 


.76734 


15.270 


8 


4.8284 


3.4143 


2.4141 


1.9713 


1.7072 


1.5270 


1.2468 


1.0797 


20.652 


9 


6.5302 


4.6178 


3.2651 


2.6662 


2.3089 


2.0652 


1.6862 


1.4603 


26.952 


10 


8.5222 


6.0265 


4.2611 


3.4795 


3.0132 


2.6952 


2.2006 


1.9058 


34.428 


11 


10.886 


7.6981 


5.4431 


4.4447 


3.8491 


3.4428 


2.8110 


2 4344 


42.918 


12 


13.571 


9.5965 


6.7853 


5.5407 


4.7982 


4.2918 


3.5043 


3.0347 


Value of \ 


s = 


.3162 


.2236 


.1581 


.1291 


.1118 


.1 


.08165 


.07071 



572 



HYDRAULICS. 







Slope, or Head Divided by Length of Pipe. 


Value of 


Dia. 
in. 






ac Vr. 




1 in 


1 in 


lin 


1 in 


lin 


lin 


lin 






1 in 250. 


300. 


350. 


400. 


450. 


500. 


550. 


600. 


.00403 


% 


.00025 


.00023 


.00022 


.00020 


.00019 


.00018 


.00017 


.00016 


.00914 


v« 


.00058 


.00053 


.00049 


.00046 


.00043 


.00041 


.00039 


.00037 


.02855 


Z A 


.00181 


.00165 


.00153 


.00143 
.00317* 


.00134 


.00128 


.00122 


.00117 


.06334 


i 


.00400 


.00366 


.00339 


.00298 


.00283 


.00270 


.00259 


.11659 


1M 


.00737 


.00673 .00623 


.00583 


.00549 


.00521 


.00497 


.00476 


.19115 




.01209 


.01104 .01022 


.00956 


.00901 


.00855 


.00815 


.00780 


.28936 




.01830 


.01671 


.01547 


.01447 


.01363 


.01294 


.01234 


.01181 


.41357 


2 


.02615 


.02388 


.02211 


.02068 


.01948 


.01849 


.01763 


.01688 


.74786 


21/, 


.0473C 


.04318 


.03997 


.03739 


.03523 


.03344 


.03189 


.03053 


1.2089 


8 


.07645 


.06980 


.06462 


.06045 


.05695 


.05406 


.05155 


.04935 


2.5630 


4 


.16208 


.14799 


.13699 


.12815 


.12074 


.11461 


.10929 


.10463 


4.5610 


5 


.28843 


.26335 


.24379 


.22805 


.21487 


.20397 


.19448 


.19620 


7.3068 


6 


.46208 


.42189 


.39055 


.36534 


.34422 


.32676 


.31156 


.29830 


10.852 


7 


.68628 


.62660 


.58005 


.54260 


.51124 


.48530 


.46273 


.44303 


15.270 


8 


.96567 


.88158 


.81617 


.76350 


.71936 


.68286 


.65111 


.62340 


20.652 


9 


1.3060 


1.1924 


1.1038 


1.0326 


.97292 


.92356 


.88060 


.84310 


26.952 


10 


1.7044 


1.5562 


1.4405 


1.3476 


1.2697 


1 2053 


1 . 1492 


1.1003 


34.428 


11 


2.1772 


1.9878 


1.8402 


1.7214 


1.6219 


1.5396 


1.4680 


1.4055 


42.918 


12 


2.7141 


2.4781 


2.2940 


2.1459 


2.0219 


1.9193 


1.8300 


1.7521 


Value of V 


s = 


.06324 


.05774 


.05345 


.05 


.04711 


.04472 


.04264 


.04082 



For U. S. gals, per sec, multiply the figures in the table by 7.4805 

" " " " min., ',' " " 448.83 

" " " " horn, *'■ " " 26929.8 

" " " " 24hi^., " " " 646315. 

For any other slope the flow is proportional to the square i-oot of the 
slope ; thus, flow in slope of 1 in 100 is double that in slope of 1 in 400. 

Flow of Water in Pipes from % Inch to 12 Inches 
Diameter for a Uniform Velocity of 100 Ft. per Min. 



Diameter 


Area 


Flow in Cubic 


Flow in U. S 


Flow in U. S. 


in 


in 


Feet per 


Gallons per 


Gallons per 


Inches. 


Square Feet. 


Minute. 


Minute. 


Hour. 


% 


.00077 


0.077 


.57 


34 


M. 


.00136 


0.136 


1.02 


61 


u 


.00307 


0.307 


2.30 


138 


1 


.00545 


0.545 


4.08 


245 


M 


.00852 


0.852 


6.38 


383 


M 


.01227 


1.227 


9.18 


551 


m 


.01670 


1.670 


12.50 


750 


2 


.02182 


2.182 


16.32 


979 


®4 


.0341 


3.41 


25.50 


1,530 


3 


.0491 


4.91 


36.72 


2,203 


4 


.0873 


8.73 


65.28 


3,917 


5 


.136 


13.6 


102.00 


6,120 


6 


.196 


19.6 


146.88 


8,813 


7 


.267 


26.7 


199.92 


11,995 


8 


.349 


34.9 


261.12 


15,667 


9 


.442 


44.2 


330.48 


19,829 


10 


.545 


54.5 


408.00 


24,480 


11 


.660 


66.0 


493.68 


29,621 


12 


.785 


78.5 


587.52 


35,251 



Given the diameter of a pipe, to find the quantity in gallons it will deliver, 
the velocity of flow being 100 ft. per minute. Square the diameter in inches 
and multiply by 4.08. 



LOSS OF HEAD. 573 

If Q f = quantity in gallons per minute and d — diameter in inches, then 

g , = d* X .7854 X 100 X 7.4805 = ^^ 

V 
For any other velocity, V, in feet per minute, Q' = 4.08d a — - .0408d a F\ 

Given diameter of pipe in inches and velocity in feet per second, to find 
discharge in cubic feet and in gallons per minute. 

_ d a X -7854 X v X 60 _ 33-05^ cub i c feet per minute. 
* 144 

= .32725 x 7,4805 or 2.448d 2 v U. S. gallons per minute. 

To find the capacity of a pipe or cylinder in gallons, multiply the square 
of the diameter in inches by the length in inches and by .0034. Or multiply 
the square of the diameter in inches by the length in feet and by .0408. 



LOSS OF HEAD. 

The loss of head due to friction when water, steam, air, or gas of any kind 
flows through a straight tube is represented by the formula 



AlV* , /64.4 hd 

--flT^ whence V =j/ — — 



J d 2g' 



in which I = the length and d = the diameter of the tube, both in feet; v = 
velocity in feet per second, and / is a coefficient to be determined by experi- 
ment. According to Weisbach, / = .00644, in which case 



|/^ = 50, and , = S0j/f 



which is one of the older formulae for flow of water (Downing's). Prof. Un- 
win says that the value of / is possibly too small for tubes of small bore, 
and he would put/ = .006 to .01 for 4-inch tubes, and/ = .0084 to .012 for 2- 
inch tubes. Another formula by Weisbach is 



Rankine gives 




0+4)- 



From the general equation for velocity of flow of water v = c Yr Vs, = 

for round pipes c,i/ — if -, we have v" 2 = c 2 -- - and h = — — , in which 

c is the coefficient c of D'Arcy's, Bazin's, Kutter's, or other formula, as found 
by experiment. Since this coefficient varies with the condition of the inner 
surface of the tube, as well as with the velocity, it is to be expected that 
values of the loss of head given by different writers will vary as much as those 
of quantity of flow. Twotables for loss of head per 100 ft. in length in pipes 
of different diameters with different velocities are given below. The first 
is given by Clark, based on Ellis' and Howland's experiments; the second is 
from the Pelton Water-wheel Co.'s catalogue, authority not stated. The 
loss of head as given in these two tables for any given diameter and velocity 
differs considerably. Either table should be used with caution and the re- 
sults compared with the quantity of flow for the given diameter and head 
as given in the tables of flow based on Kutter's and D'Arcy's formulae. 



574 



HYDRAULICS. 



Relative Loss of Head by Friction for each 100 Feet 
Length of Clean Cast-iron Pipe. 





(Based on 


Ellis and Howland's experiments.) 






Velocity 




Diameter of Pipes in 


Inches. 






in Feet 
per 


S | 4 


5 | 6 | 7 | 8 


9 1 10 


n\ 


14 


Second. 














Loss of Head in Feet, per 100 Feet Long 








Feet 


Feet 


Feet 


Feet 


Feet 


Feet 


Feet 


Feet 


Feet 


Feet 


Feet 


of 


of 


of 


of 


or 


of 


of 


of 


of 


of 




Head 


Head 


Head 


Head 


Head 


Head 


Head 


Head 


Head 


Head 


2 


.97 


.55 


.41 


.32 


.27 


.23 


.19 


.18 


.15 


.12 


2.5 


1.49 


.92 


.64 


.50 


.43 


.36 


.30 


.27 


.23 


.19 


3 


1.9 


1.2 


.82 


.72 


.61 


.51 


.44 


.39 


.33 


.27 


3.5 


2.6 


1.6 


1.2 


1.0 


.7 


.71 


.61 


.52 


.45 


.37 


4 


3.3 


2.2 


1.7 


1.3 


.9 


.92 


.79 


.69 


.59 


.49 


4.5 








1.6 


1.2 


1.2 


1.01 


.87 


.75 


.61 


5 














1.2 


1.1 


.90 


.76 


5.5 














.92 


6 
























15 


18 


21 


24 


27 


30 


33 


36 


42 


48 


2 


.11 


.095 


.075 


.065 


.055 


.052 


.049 


.047 


.036 


.030 


2.5 


.17 


.147 


.117 


.109 


.088 


.085 


.076 


.067 


.056 


.046 


3 


.25 


.21 


.17 


.15 


.13 


.12 


.108 


.10 


.081 


.067 


3.5 


.34 


.29 


.23 


.20 


.18 


.16 


.15 


.14 


.111 


.092 


4 


.44 


.36 


.31 


.27 


.23 


.22 


.20 


.17 


.14 


.116 


4.5 


.50 


.46 


.39 


.34 


.30 


.28 


.25 


.22 


.18 


.15 


5 


.70 


.58 


.48 


.41 


.37 


.34 


.30 


.27 


.22 


.18 


5.5 


.84 


.70 


.59 


.50 


.44 


.39 


.36 


.32 


.27 


.22 


6 








.59 


.53 


.49 


43 


.4 


.32 


.27 



Loss of Head in Pipe by Friction.— Loss of head by friction in 
each 100 IVet in length of different diameters of pipe when discharging the 
following quantities of water per minute (Pelton Water-wheel Co.) : 



~ 








Inside Diameter of Pipe in Inches. 


43 


1 


2 


3 


4 


5 | 6 


_, C 


-v3 


ft 


-3 
o3 


ft 




u 

<D 
ft 


1 


ft 


-a 


01 

ft 


0> 


ft 


.■=, ° 


W 


•g 


w . 


-g 


W 


"g 


a 


-g 


w 


4) ® 
fa"3 


K 


© 


b™ 


*o % 




ei_, <D 
3<D 


£~ 




<D 0) 

fa^| 


O 01 


fa^ 


O 0> 


O 1) 


r ®.2 


o 


w fe 


o a 




« a 


fa 


o a 


fa 




fa 


g 3 


fa 


t> a 


_o 


©•■« 


■sa 


8-fl 


3i 


o-S 


?§ 


tie c 


3 § 


%n 


5» 


As 


•la 


> 


J 


o 


iJ 


o 


hJ 


O 


hJ 


o 




o 


hJ 


O 


V 


h 


Q 


h 


Q 


h 


5.89 


ft 


<y 


h 
.474 


« 


h 


Q 


2.0 


2.37 


.65 


1.185 


2.62 


.791 


.593 


10.4 


16.3 


.395 


23.5 


3.0 


4.89 


.99 


2.44 


3.92 


1.62 


8.83 


1.22 


15.7 


.978 


24.5 


.815 


35.3 


4.0 


8 20 


1.32 


4.10 


5.23 


2.73 


11.80 


2.05 


20.9 


1.64 


32.7 


1.37 


47.1 


5.0 


12.33 


1.65 


6.17 


6.54 


4.11 


14.70 


3.08 


26.2 


2 46 


40.9 


2.05 


58.9 


6.0 


17.23 


1.98 


8.61 


7.85 


5.74 


17.70 


4.31 


31.4 


3.45 


49" 1 


2.87 


70.7 


7.0 


22.89 


2.31 


11.45 


9.16 


7.62 


20.6 


5.72 


36.6 


4.57 


57.2 


2.81 


82.4 



LOSS OF HEAD. 



575 





Inside Diameter of Pipe in Inches. 




7 


8 


9 


10 


11 | 12 


V 


h 


Q 


h 


Q 


h 


Q 


h 


Q 


h 


Q 


h 


Q 


2 


.338 


32.0 


.296 


41.9 


.264 


53 


.237 


65.4 


.216 


79.2 


.198 


94.2 


3.0 


.698 


48.1 


.611 


62.8 


.544 


79.5 


.488 


98.2 


.444 


119 


.407 


141 


4.0 


1.175 


64.1 


1.027 


83.7 


.913 


106 


.822 


131 


.747 


158 


.685 


188 


5.0 


1.76 


80.2 


1.54 


105 


1.37 


132 


1.23 


163 


1.122 


198 


1.02S 


235 


6.0 


2.46 


96.2 


2.15 


125 


1.92 


159 


1.71 


196 


1.56 


237 


1.43 


283 


7.0 


3.26 


112.0 


2.85 


146 


2.52 


185 


2.28 


229 


2.07 


277 


1.91 


330 








Inside Diameter of Pipe in Inches. 






13 


14 


15 


16 


18 


20 


V 


h 


Q 


h 


Q 


h 


Q 


h 


Q 


h 


Q 


h 


Q 


2.0 


.183 


110 


.169 


128 


.158 


147 


.147 


167 


.132 


212 


119 


262 


3.0 


.375 


166 


.349 


192 


.325 


221 


.306 


251 


.271 


318 


.245 


393 


4.0 


.632 


221 


.587 


256 


.548 


294 


.513 


335 


.456 


424 


.410 


523 


5.0 


.949 


276 


.881 


321 


.822 


368 


.770 


419 


.685 


530 


.617 


654 


6.0 


1.325 


332 


1.229 


385 


1.148 


442 


1.076 


502 


.957 


636 


.861 


785 


7.0 


1.75 


387 


1.63 


449 


1.52 


515 


1.43 


586 


1.27 742 


1.143 


916 






Inside Diameter of Pipe in Inches. 






22 


24 


26 


28 


30 


36 


V 


h 


Q 


h 


Q 


h 


Q 


h 


Q 


h 


Q 


h 


Q 


2.0 


.108 


31G 


.098 


377 


.091 


442 


.084 


513 


.079 


589 


Ofifi 


848 


8.0 


.222 


475 


.204 


565 


.188 


663 


.174 


770 


.163 


883 


.135 


1273 


4.0 


.373 


633 


.342 


754 


.315 


885 


.293 


1026 


.273 


1178 


.228 


1697 


5.0 


.561 


792 


.513 


942 


.474 


1106 


.440 


1283 


.411 


1472 


.342 


2121 


6.0 .782 

7.01 1.040 


950 


.717 


1131 


.662 


1327 


.615 


1539 


.574 


1767 


.479 


2545 


1109 


.953 


1319 


.879 


1548 


.817 


1796 .762 


2061 


.636 


2868 



Example.— Given 200 ft. head and 600 ft. of 11 -inch pipe, carrying 119 cubic 
feet of water per minute. To find effective head : In right-hand column, 
under 11-inch pipe, find 119 cubic ft.; opposite this will be found the loss by- 
friction in 100 ft. of length for this amount of water, which is .444. Multiply 
this by the number of hundred feet of pipe, which is 6, and we have 
2.66 ft., which is the loss of head. Therefore the effective head is 200 — 2.66 
= 197.34. 

Explanation.— The loss of head by friction in pipe depends not only upon 
diameter and length, but upon the quantity of water passed through it. Th-> 
head or pressure is what would be indicated by a pressure-gauge attached 
to the pipe near the wheel. Headings of gauge should be taken while the 
water is flowing from the nozzle. 

To reduce heads in feet to pressure in pounds multiply by .433. To reduce 
pounds pressure to feet multiply by 2.309. 

Cox's Formula,- Weisbach's formula for loss of head caused by the 
friction of water in pipes is as follows : 

Friction-head = /o.0144 + ~^\ hll, 
\ VV I 5.367d 

where L — length of pipe in feet; 

V = velocity of the water in feet per second ; 
d = diameter of pipe in inches. 
William Cox {Amer. Mach., Dec. 28, 1893) gives a simpler formula which 
gives almost identical results : 

H = friction-head in feet = — — (1) 

Hd _ 4V* A-5V-2 
L " 1200" ( } 



576 



HYDRAULICS. 



He gives a table by means of which the value of 
obtained when V is known, and vice versa. 

4F 2 -f 5V - 2 



[F2 + 5F-2 . 



Values op - 



1200 



V 


0.0 


0.1 


0.2 


0.3 


0.4 


0.5 


0.6 


0.7 


0.8 


0.9 


1 


.00583 


.00695 


.00813 


.00938 


.01070 


.01208 


.01353 


.01505 


.01663 


.01828 


2 


.02000 


.02178 


.02363 


.02555 


.02753 


.02958 


.03170 


.03388 


.03613 


.03845 


3 


.04083 


.04328 


.04580 


.04838 


.05103 


.05375 


.05653 


.05938 


.06230 


.06528 


4 


.06833 


.07145 


.07463 


.07788 


.08120 


.08458 


.08803 


.09155 


.09513 


.09878 


5 


.10250 


.10628 


.11013 


.11405 


.11803 


.12208 


.12620 


.13038 


.13463 


.13895 


6 


.14333 


.14778 


.15230 


.15688 


.16153 


.16625 


.17103 


.17588 


.18080 


. 18578 


7 


.19083 


.19595 


.20113 


.20638 


.21170 


.21708 


.22253 


.22805 


.22363 


.23928 


8 


.24500 


.25078 


.25663 


.26255 


.26853 


.27458 


.28070 


.28688 


.29313 


.29945 


9 


.30583 


.31228 


.318S0 


.32538 


.33203 


.33875 


.34553 


.35238 


.35930 


.36628 


10 


.37333 


.38045 


.38763 


.39488 


.40220 


.40958 


.41703 


.42455 


.43213 


.43978 


11 


.44750 


.45528 


.46313 


.47105 


.47903 


.48708 


.49520 


.50338 


.51163 


.51995 


12 


.52833 


.53678 


.54530 


.55388 


.56253 


.57125 


.58003 


.58888 


.59780 


.60678 


13 


.61583 


.62495 


.63413 


.64338 


.65270 


.66208 


.67153 


.68105 


.69063 


.70028 


14 


.71000 


.71978 


.72963 


.73955 


.74953 


.75958 


.76970 


.7798S 


.79013 


.80045 


15 


.81083 


.82128 


.83180 


.84238 


.85303 


.86375 


.87453 


.88538 


.89630 


.90728 


16 


.91833 


.92945 


.94063 


.95188 


.96320 


.97458 


.98603 


.99755 


1.00913 


1.02078 


17 


1.03-250 


1 .04428 


1.05613 


1.06805 


1.08003 


1.09208 


1 . 10420 


1.11638 


1.12863 


1.14095 


18 


1.15333 


1.16578 


1.17830 


1.19088 


1.20353 


1.21625 


1.22903 


1.24188 


1.25480 


1.26778 


19 


1.28083 


1.29395 


1.30713 


1.32038 


1.33370 


1.34708 


1.36053 1.37405 


1.38763 


1.40128 


20 


1.41500 


1.42878 


1.44263 


1.45655 


1.47053 


1.48458 


1.49870 1.51288 


1.52713 


1.54145 


21 


1.55583 


1.57028 


1.58480 


1.59938 


1.61403 


1.62875 


1.64353 1.65838 


1.67330 


1.68828 



The use of the formula and table is illustrated as follows: 
Given a pipe 5 inches diameter and 1000 feet long, with 49 feet head, what 
will the discharge be? 

If the velocity Fis known in feet per second, the discharge is 0.32725d 2 F 
cubic foot per minute. 
By equation 2 we have 

4F2 + 5F-2 _Hd L _ 49x5 . 

1200 L 1000 

whence, by table, V = real velocity = 8 feet per second. 

The discharge in cubic feet per minute, if V is velocity in feet per second 
and d diameter in inches, is 0.32725d 2 F, whence, discharge 

= 0.32725 x 25 X 8 = 65.45 cubic feet per minute. 

The velocity due the head, if there were no friction, is 8.025 4/ff = 56.175 
feet per second, and the discharge at that velocity would be 
0.32725 x 56.175 x 8 = 460 cubic feet per minute. 

Suppose it is required to deliver this amount, 460 cubic feet, at a velocity 
of 2 feet per second, what diameter of pipe will be required and what will be 
the loss of head by friction? 



d = diameter 



J VX 0.32725 \\ 



2 X 0.32725 



= 4/703" = 26.5 inches. 



H-. 



-- 0.75 foot, 



Having now the diameter, the velocity, and the discharge, the friction-head 
is calculated by equation 1 and use of the table; thus, 
L 4F 2 -f5F-2 _ 1000 20 

" d 1200 26.5 ' 26.5 = 

thus leaving 49 — 0.75 = say 48 feet effective head applicable to power-pro- 
ducing: purposes. 

Problems of the loss of head may be solved rapidly by means of Cox's 
Pipe Computer, a mechanical device on the principle of the slide-rule, for 
sale by Keuffel & Esser, New York. 



LOSS OF HEAD. 



57^ 



Frictional Heads at Given Rates of Discharge in Clean 
Cast-iron Pipes for Each 1000 Feet of Length. 

(Condensed from Ellis and HowlancTs Hydraulic Tables.) 





4-inch 


6-inch 


8-inch 


10-inch 


12- 


nch 


14-inch 




Pipe. 


Pipe. 


Pipe. 


Pipe. 


Pipe. 


Pipe. 


o y-g 




jj 


a 6 


+j 


c 6- 


^j 




+j 


a ,n 


jj 


c 6 


^; 


= MS 














— o> 
















3 - 

"5 03 




S3 «8 

o - 
£73 




O - 
ST3 




o - 

•£T3 


'^ 


5 - 




o<2 

o .. 


& 5ft 


j££ 


E-S 


g£ 


&•* 




z* 






3>H 


■gjy 


>£ 




25 


.64 

1.28 


.59 
2.01 


.28 
.57 


.11 

.32 


.16 
.32 


.04 
.10 


.10 
.20 


.02 
.04 


.07 
.14 


.01 
.02 






50 


10 


.01 


100 


2.55 


7.36 


1.18 


1.08 


.64 


.29 


.41 


.11 


.28 


.05 


.21 


.03 


150 


3.83 


16.05 


1.70 


2.28 


.96 


.60 


.61 


.22 


43 


.10 


.31 


.05 


200 


5.11 


28.09 


2.27 


3.92 


1.28 


1.01 


.82 


.36 


.57 


.16 


,42 


.08 


250 


6.37 


43.47 


2.84 


6.00 


1.60 


1.52 


1.02 


.54 


.71 


.24 


52 


.12 


300 


7.66 


62.20 


3.40 


8.52 


1.91 


2.13 


1.23 


.75 


.85 


32 


.63 


.16 


350 


8.94 


84.26 


3.97 


.11.48 


2.23 


2.85 


1.43 


.99 


.99 


.43 


73 


.21 


400 


10.21 


109.68 


4.54 


14.89 


2.55 


3.68 


1.63 


1.27 


1.13 


.54 


S3 


.27 


500 


12.77 


170.53 


5.67 


23.01 


3.19 


5.64 


2.04 


1.93 


1.42 


.81 


1.04 


.40 


600 


15.32 


244.76 


6.81 


32.89 


3.83 


8.03 


2.45 


2.72 


1.70 


1.14 


1 25 


.55 


700 


17.87 


332.36 


7.94 


44.54 


4.47 


10.83 


2.86 


3.66 


1.98 


1.52 


1 46 


.73 


800 






9.08 
10.21 
11.35 
13.61 
15.88 
18.15 
20.42 
22.69 


57.95 
73.12 
90.05 
129.20 

175.38 
228 62 
288.90 
356.22 


5.09 
5.74 
6.38 
7.66 
8.94 
10.21 
11.47 
12.77 
15.96 


14.05 

17.68 
21.74 
31.10 
42.13 
54.84 
69.22 
85.27 
132.70 


3.27 
3.68 
4.08 
4.90 
5.72 
6.53 
7.35 
8.17 
10.21 


4.73 
5.93 

7.28 
10.38 
14.02 
18.22 
22.96 


2.27 
2.55 
2.84 
3.40 
3.97 
4.54 
5 11 


1.96 
2.45 
3.00 
4.26 
5.74 
7.44 
9.36 
11.50 
17.82 


1.67 

1 88 
2.08 
2.50 

2 91 
3.33 
3.75 
4.17 
5 21 


.94 


000 






1 17 


1000 






1 43 


1200 






2 02 


1400 






2 72 


1600 






3 51 


1800 






4 41 


2000 






28.25'5.67 

43.87,7.09 


5 41 


2500 






8.35 


3000 














12.25 


62.92,8.51 


25.51 


6.25 


11.93 


4000 
















1.... 




8.34 


21.00 




16-inch 


18-i 


nch 


20-inch 


24-inch 


30- 


inch 


36-inch 


03 


Pipe. 


Pi 


je. 


Pipe. 


Pipe. 


P 


pe. 


Pipe. 


Si? S 




^ 


a 6 


^j 


a 6 


4J 




^j 




^j 


c 6 


^ 


3 MS 


































i<2 
o .. 
•43^3 

£3 




c-8 

is 


1! 


.2 - - 

y o3 

•r v 


i. - 


o - 

£5 




o"l 
•.Co 
y <=3 
•r a> 


500 


.80 


.22 


.63 


.13 


.51 


.08 


.35 


.04 


.23 


.01 


16 


.01 


1000 


1.60 


.76 


1.26 


.44 


1.02 


.27 


.71 


.12 


45 


.04 


82 


.02 


1500 


2.39 


1.6c 


1.89 


.93 


1.53 


.56 


1.06 


.24 


68 


.OK 


.47 


.04 


2000 


3 19 


2.82 


2.52 


1.6C 


2.04 


.96 


1.42 


.41 


.91 


.15 


63 


.06 


2500 


3.99 


4.34 


3.15 


2 45 


2.55 


1.47 


1.77 


.62 


1.13 


.22 


.79 


09 


3000 


4.79 


6. IS 


3.78 


3.48 


3.06 


2.09 


2. IS 


.87 


1.36 


.3C 


.95 


.13 


3500 


5.59 


8.37 


4.41 


4.7C 


3.57 


2.81 


2.48 


1.16 


1.59 


.4C 


1.10 


.17 


4000 


6.38 


10.87 


5.04 


6.09 


4.08 


3.64 


2.84 


1.50 


1.82 


.52 


1.26 


.22 


4500 


7.18 


13. 7C 


5.67 


7.67 


4.59 


4.58 


3.19 


1.88 


2,04 


.64 


1.42 


.27 


5000 


7.98 


16.85 


6 30 


9.43 


5.11 


5.62 


3.55 


2.31 


2.27 


.78 


1 . 58 


33 


6000 






7.57 


13.49 


6.13 
7.15 


8.03 
10.86 


4.26 
4.96 
5.67 
6.38 


3.28 
4.43 
5.75 
7 25 


2.72 

3.18 
3.63 
4.08 
4 . 54 
5.44 

t; 3i j 


1.11 
1.49 
1.93 
2.43 
2.98 
4.25 
5.75 


1.89 
2 21 
2.52 
2.84 
3.15 
3.78 
4 41 


46 


7000 






.62 


8000 










80 


9000 














1 00 


10000 














1.23 


12000 


















1.74 


14000 


















.2.35 


16000 






















5.05 


3.04 


18000 






















5 68 


3 83 


20000 
























4.71 



578 



HYDRAULICS. 



Effect of Bends and Curves in Pipes.— Weisbach's rule for 



bends : Loss of head in feet = .131 + 1.847i--\ 2 X -^~ X -r|U 



in which r 



- radius of curvature of axis of pipe, v 

- the central angle, or angle subtended 



= internal radius of pipe in feet, R - 
— velocity in feet per second, and a - 
by the bend. 

Hamilton Smith, Jr., in his work on Hydraulics, says: The experimental 
data at hand are entirely insufficient' to permit a satisfactory analysis of 
this quite complicated subject; in fact, about the only experiments of value 
are those made by Bossut and Dubuat with small pipes. 

Curves.— If the pipe has easy curves, say with radius not less than 5 
diameters of the pipe, the flow will not be materially diminished, provided 
the tops of all curves are kept below the hydraulic grade-line and provision 
be made for escape of air from the tops of all curves. (Trautwine.) 

Hydraulic Grade-line.— In a straight tube of uniform diameter 
throughout, running full and discharging freely into the air, the hydraulic 
grade-line is a straight line drawn from the discharge end to a point imme- 
diately over the entry end of the pipe and at a depth below the surface 
equal to the entry and velocity heads. (Trautwine.) 

In a pipe leading from a reservoir, no part of its length should be above 
the hydraulic grade-line. 

Flow of Water in House-service Pipes. 

Mr. E. Kuichling, C.E., furnished the following table to the Thomson 
Meter Co.: 





„ 


Discharge, 


or Quantity 


capable of being delivered, in 




'3 
.£ a .2 


Cubic Feet per Minute, from the Pipe, 




under the conditions specified in the first column. 


Condition 




of 




Discharge. 


£-c£ 


Nominal Diameters of Iron or Lead Service-pipe in 




3 2 °3 

III 

Hi 


Inches. 




y z 


% 


H | I 


IX 


2 
33.34 


3 


4 


6 


Through 35 


30 


1.10 


1.92 


3.01 


6.13 


16.58 


88.16 


173.85 


444.63 


40 


1.27 


2.22 


3.48 


7.08 


19.14 


38.50 


101.80 


200.75 


513.42 




50 


1.42 


2.48 


3.89 


7.92 


21.40 


43.04 


113.82 


224.44 


574.02 




60 


1.56 


2.71 


4.26 


8.67 


23.44 


47.15 


124.68 245.87 


628.81 


back 


75 


1.74 


3.03 


4.77 


9.70 


26.21 


52.71 


139.39 274.89 


703.03 


100 


2.01 


3.50 


5.50 


11.20 


30.27 


60.87 


160.96 317.41 


811.79 




130 


2.29 


3.99 


6.28 


12.77 


34.51 


69.40 


183.52361.91 


925.58 


Through 
100 feet of 


30 


0.66 


1.16 


1.84 


3.78 


10.40 


21.30 


58.19118.13 


317.23 


40 


0.77 


1.34 


2.12 


4.36 


12.01 


24.59 


67. 191136.41 


366.30 


50 


0.86 


1.50 


2.37 


4.88 


13.43 


27.50 


75.13152.51 


409.54 




60 


0.94 


1.65 


2.60 


5.34 


14.71 


30.12 


82. 30! 167. 06 


448.63 


back 


75 


1.05 


1.84 


2.91 


5.97 


16.45 


33.68 


92.01 186.78 


501 .58 


100 


1.22 


2.13 


3.36 


6.90 


18.99 


38.89 


106.24 215.68 


579.18 




130 


1.39 


2.42 


3.83 


7.86 


21.66 


44.34 


121.14 245.91 


660.36 


Through 


30 


0.55 


0.96 


1.52 


3.11 


8.57 


17.55 


47.90 97.17 


260.56 


100 feet O! 


40 


0.66 


1.15 


1.81 


8 72 


10.24 


20.95 


57.20116.01 


311.09 


service- 


50 


0.75 


1.31 


2.06 


4.24 


11.67 


23.87 


65. 18 132. 2C 


354.49 


pipe and 


60 


0.83 


1.45 


2.29 


4.70 


12.94 


26.48 


72.28 146 61 


393.13 


15 feet 


75 


0.94 


1.64 


2.59 


5.32 


14.64 


29.96 


81. 79, 165. 9( 


444.85 


vertical 


100 


1.10 


1.92 


3.02 


6.21 


17. 1C 


35. 0C 


95.55! 193. 82 


519.72 


rise. 


130 


1.26 


2.20 


3.48 


7.14 


19.66 


40.23 


109.82 


222.75 597.31 


Through 


30 


0.44 


0.77 


1.22 


2.50 


6.80 


14.11 


38.63 


78.54 211.54 


100 feet o: 


40 


0.55 


0.97 


1.53 


3.15 


8.68 


17. 7E 


48.68 


98.98 266.59 


service- 


50 


0.65 


1.14 


1.79 


3.69 


10.16 


20.82 


56.98 


115.87:312.08 


pipe, and 


60 


0.73 


1.28 


2.02 


4.15 


11.45 


23.47 


64.22:130.59 351.73 


30 feet 


75 


0.84 


1.47 


2.32 


4.77 


13 15 


26.95 


73. 76 1149.99 403.98 


vertical 


100 


1.00 


1.74 


2.75 


5.65 


15.58 


31.93 


87.38 177.67 478.55 


rise. 


130 


1.15 


2.02 


3.19 


6.55 


18.07 


37.02 


101.33 


^06.04 


554.96 



FIRE-STREAMS. 



579 



In this table it is assumed that the pipe is straight and smooth inside; that 
the friction of the main and meter are disregarded; that the inlet from the 
main is of ordinary character, sharp, not flaring or rounded, and that the 
outlet is the full diameter of pipe. The deliveries given will be increased if, 
first, the pipe between the meter and the main is of larger diameter than the 
outlet; second, if the main is tapped, say for 1-inch pipe, but is enlarged 
from the tap to 1J4 or 1}4 inch; or, third, if pipe on the outlet is larger than 
that on the inlet side of the meter. The exact details of the conditions given 
are rarely met in practice; consequently the quantities of the table may be 
expected to be decreased, because the pipe is liable to be throttled at the 
joints, additional bends may interpose, or stop-cocks may be used, or the 
back-pressure may be increased. 

Air-bound. Pipes.— A pipe is said to be air-bcfcnd when, in conse- 
quence of air being entrapped at the hign points of vertical curves in the 
line, water will not flow out of the pipe, although the supply is higher than 
the outlet. The remedy is to provide cocks or valves at the high points, 
through which the air may be discharged. The valve may be made auto- 
matic by means of a float. 

Vertical Jets. (Molesworth.)— H = head of water, h = height of jet, 
d = diameter of jet, K — coefficient, varying with ratio of diameter of jet 
to head; then h = KH. 

IfiJ=dX300 600 1000 1500 1800 2800 3500 4500, 
K= .96 .9 .85 .8 .7 .6 .5 .25 

Water Delivered through Meters. (Thomson Meter Co.).— The 
best modern practice limits the velocity in water-pipes to 10 lineal feet per 
second. Assume this as a basis of delivery, and we find, for the several sizes 
of pipes usually metered, the following approximate results: 
Nominal diameter of pipe in inches: 

% % H 1 1^2 3 4 6 

Quantity delivered, in cubic feet per minute, due to said velocity: 

0.46 1.28 1.85 3.28 7.36 13.1 29.5 52.4 117.9 

Prices Charged for Water in Different Cities (National 
Meter Co.;: 
Average minimum price for 1000 gallons in 163 places 9.4 cents. 

" maximum " " " " " " " 28 " 

Extremes, 2J^ cents to 100 " 

FIRE-STREAMS. 

Discharge from Nozzles at Different Pressures. 

(J. T. Fanning, Am. Water-works Ass'n, 1892, Eng'g Neivs, July 14, 1892.) 



Nozzle 

diam., 

in. 


Height 

of 
stream, 

ft. 


Pressure 
at Play- 
pipe, 
lbs. 


Horizon- 
tal Pro- 
jection of 
Streams, 
ft. 


Gallons 

per 
minute. 


Gallons 
per 24 
hours. 


Friction 

per 100 

ft. Hose, 

lbs. 


Friction 
per 100 
ft. Hose, 

Net 
Head, ft. 


1 


70 


46.5 


59.5 


303 


292,298 


10.75 


24.77 


1 


80 


59.0 


67.0 


230 


331,200 


13.00 


31.10 


1. 


90 


79.0 


76.6 


267 


384,500 


17.70 


40.78 


1 


100 


130.0 


88.0 


311 


447.900 


22.50 


54.14 


m 


70 


44.5 


61.3 


249 


358,520 


15.50 


35.71 




80 


55.5 


69.5 


281 


404,700 


19.40 


44.70 


1^4 


90 


72.0 


78.5 


324 


466,600 


25.40 


58.52 


ty* 


100 


103.0 


89.0 


376 


541,500 


33.80 


77.88 


va 


70 


43.0 


66.0 


306 


440,613 


22.75 


52.42 


m 


80 


53.5 


72.4 


343 


493.900 


28.40 


65.43 


m 


90 


68.5 


81.0 


388 


558,800 


35.90 


82.71 


m 


100 


93.0 


92.0 


460 


662,500 


57.75 


86.98 


Ws 


70 


41.5 


77.0 


368 


530,149 


32.50 


74.88 


1% 

1% 


80 


51.5 


74.4 


410 


590,500 


40.00 


92.16 


90 


65.5 


82.6 


468 


674,000 


51.40 


118.43 


m 


100 


88.0 


92.0 


540 


777,700 


72.00 


165.89 



580 



HYDRAULICS. 



Friction Losses in Hose.— In the above table the volumes of 
water discharged per jet were for stated pressures at the play-pipe. 

In providing for this pressure due allowance is to be made for friction 
losses in each hose, according to the streams of greatest discharge which are 
to be used. 

The loss of pressure or its equivalent loss of head (h) in the hose may be 

found by the formula h = vK4m)^-^. 

In this formula, as ordinarily used, for friction per 100 ft. of 2^-in. hose 
there are the following constants : 2)4, in. diameter of hose d = .20833 ft.; 
length of hose I = 100 ft., and 2g = 64.4. The variables are : v = velocity in 
feet per second; h^ loss of head in feet per 100 ft. of hose; m = a coeffi- 
cient found by experiment ; the velocity v is found from the given dis- 
charges of the jets through the given diameter of hose. 

Head and Pressure Losses by Friction in 100 -ft. 
Lengths of Rubber-lined Smooth 2^-in. Hose. 



Discharge 


Velocity 


Coefficient, 


Head Lost, 


Pressure 


Gallons per 


per minute, 


per second, 


m. 


ft. 


Lost, lbs. 


24 hours. 


gallons. 


ft. 






per sq. in. 




200 


13.072 


.00450 


22.89 


9.93 


288,000 


250 


16.388 


.00446 


35.55 


15.43 


360,000 


300 


18.858 


.00442 


46.80 


20.31 


432,000 


347 


21.677 


.00439 


61.53 


26.70 


499,680 


350 


22.873 


.00439 


68.48 


29.73 


504,000 


400 


26.144 


.00436 


88.83 


38.55 


576,000 


450 


29.408 


.00434 


111.80 


48.52 


648,000 


500 


32.675 


.00432 


137.50 


59.67 


720,000 


520 


33.982 


.00431 


148.40 


64.40 


748,800 



These frictions are for given volumes of flow in the hose and the veloci- 
ties respectively due to those volumes, and are independent of size of 
nozzle. The changes in nozzle do not affect the friction in the hose if there 
is no change in velocity of flow, but a larger nozzle with equal pressure at 
the nozzle augments the discharge and velocity of flow, and thus materially 
increases the friction loss in the hose. 

Loss of Pressure (p) and Head (/*) in Rubber-lined 
Smooth 2Hj-in. Hose may be found approximately by the formulae 

p = jl and h = ^ , in which p = pressure lost by friction, in 

pounds per square inch; I = length of hose in feet; q — gallons of water 
discharged per minute: d = diam. of the hose in inches, 2J^j in.; h = friction- 
head in feet. The coefficient of d b would be decreased for rougher hose. 

The loss of pressure and head for a lj^-in. stream with power to reach a 
height of 80 ft. is, in each 100 ft. of 2^-in. hose, approximately 20 lbs., or 45 
ft. net, or, say, including friction in the hydrant, ^ ft. loss of head for each 
foot of hose. 

If we change the nozzles to 1J4 or 1% in. diameter, then for the same 80 ft. 
height of stream we inci'ease the friction losses on the hose to approxi- 
mately % ft. and 1 ft. head, respectively, for each foot-length of hose. 

These computations show the great difficulty of maintaining a high 
stream through large nozzles unless the hose is very short, especially for a 
gravity or direct-pressure system. 

This single 1^-in. stream requires approximately 56 lbs pressure, equiva- 
lent to 129 ft. head, at the play-pipe, and 45 to 50 ft. head for each 100 ft. 
length of smooth 2i^-in. hose, so that for 100, 200, and 300 ft. of hose we 
must have available heads at the hydrant or fire-engine of 1C6, 156, and 206 
ft., respectively. If we substitute 1 J^-in. nozzles for same height of stream 
we must have available heads at the hydrants or engine of 185, 255 and 325 
ft., respectively, or we must increase the diameter of a portion at least of 
the long hose and save friction-loss of head. 

Rated Capacities of Steam Fire-engines, which is perhaps 
one third greater than their ordinary rate of work at fires, are substantially 
as follows : 

3d size, 550 gals, per min., or 792,000 gals, per 24 hours. 
2d " 700 " " 1,008,000 

1st " 900 " " 1,296,000 

1 ext., 1,100 " " 1,584,000 



THE SIPHON". 



581 



Pressures required at Nozzle and at Pump, with Quantity 
and Pressure of Water Necessary to throw Water 
Various Distances through Different-sized Nozzles- 
using 234-inch Rubber Hose and Smooth Nozzles. 

(From Experiments of Ellis & Leshure, Farming's " Water Supply.") 



Size of Nozzles. 



1% Inch. 



Pressure at nozzle, lbs. per sq. in 

* Pressure at pump or hydrant with 

100 ft. 2)4 inch rubber hose 

Gallons per minute. . 

Horizontal distance thrown, feet 

Vertical distance thrown, feet 



135 
310 
193 



Size of Nozzles. 



1J4 Inch. 



1% Inch. 



Pressure at nozzle, lbs. per sq. in 

* Pressure at pump or hydrant with 

100 feet 2^}-inch rubber hose 

Gallons per minute 

Horizontal distance thrown, feet 

Vertical distance thrown, feet 



115 



14-2 



1G4 



80 100 

144 180 
413 462 
200 224 
146l 169 



* For greater length of 2^-inch hose the increased friction can be ob- 
tained by noting the differences between the above given "pressure at 
nozzle 1 ' and "pressure at pump or hydrant with 100 feet of hose." For 
instance, if it requires at hydrant or pump eight pounds more pressure 
than it does at nozzle to overcome the friction when pumping through 100 
feet of 2^-inch hose (using 1-inch nozzle, with 40-pound pressure at said 
nozzle) then it requires 16-pounds pressure to overcome the friction in 
forcing through 200 feet of same size hose. 

Decrease of Flow due to Increase of Length of Hose. 
{J. R. Freeman's Experiments, Trans. A. S. C. E. 1889.)— If the static pres- 
sure is 80 lbs. and the hydrant-pipes of such size that the pressure at the hy- 
drant is 70 lbs., the hose 2)4 in. nominal diam., and the nozzle \% in. diam., 
the height of effective fire-stream obtainable and the quantity in gallons per 
minute will be : 

Best Rubber- 
Linen Hose. lined Hose. 
Height, Gals. Height, Gals, 
feet. per min. feet, per min. 
73 261 81 282 
42 184 61 229 
27 146 46 192 



With 50 ft. of 2^-in. hose . . 
" 250 " " " " ., 

.4 50Q « U K «. _ 



With 500 ft. of smoothest and best rubber-lined hose, if diameter be 
exactly 2)4 in., effective height of stream will be 39 ft. (177 gals.); if diameter 
be 14 m « larger, effective height of stream will be 46 ft. (192 gals.) 

THE SIPHON. 

The Siphon is a bent tube of unequal branches,' open at both ends, and 
is used to convey a liquid from a higher to a lower level, over an intermedi- 
ate point higher than either. Its parallel branches being in a vertical plane 
and plunged into two bodies of liquid whose upper surfaces are at different 
levels, the fluid will stand at the same level both within and without each 
branch of the tube when a vent or small opening is made at the bend. If 
the air be withdrawn from the siphon through this vent, the water will rise 
in the branches by the atmospheric pressure without, and when the two 
columns unite and the vent is closed, the liquid will flow from the upper 
reservoir as long as the end of the shorter branch of the siphon is below the 
surface of the liquid in the reservoir. 

If the water was free from air the height of the bend above the supply 
level might be as great as 33 feet. 



582 HYDRAULICS. 

If A = area of cross-section of the tube in square feet, H= the difference 
in level between the two reservoirs in feet, D the density of the liquid in 
pounds per cubic foot, then ADH measures the intensity of the force which 
causes the movement of the fluid, and V= V'tyH = 8.02 VH is the theoretical 
velocity, in feet per second, which is reduced by the loss of head for entry 
and friction, as in other cases of flow of liquids through pipes. In the case 
of the difference of level being greater than 33 feet, however, the velocity of 
the water in the shorter leg is limited to that due to a height of 33 feet, or 
that due to the difference between the atmospheric pressure at the entrance 
and the vacuum at the bend. 

Leicester Allen {Am. Mack., Nov. 2, 1893) says: The supply of liquid to a 
siphon must be greater than the flow which would take place from the dis- 
charge end of the pipe, provided the pipe were filled with the liquid, the 
supply end stopped, and the discharge end opened when the discharge end 
is left free, unregulated, and unsubnierged. 

To illustrate this principle, let us suppose the extreme case of a siphon 
having a calibre of 1 foot, in which the difference of level, or between the 
point of supply and discharge, is 4 inches. Let us further suppose this 
siphon to be at the sea-level, and its highest point above the level of the 
supply to be 27 feet. Also suppose the discharge end of this siphon to he un- 
regulated, unsubnierged. It would be inoperative because the water in the 
longer leg would not be held solid by the pressure of the atmosphere against 
it, and it would therefore break up and run out faster than it could be re- 
placed at the inflow end under an effective head of only 4 inches. 

Long Siphons.— Prof. Joseph Torrey, in the Amer. Machinist, 
describes a long siphon which was a partial failure. 

The length of the pipe was 1792 feet. The pipe was 3 inches diameter, and 
rose at one point 9 feet above the initial level. The final level was 20 feet 
below the initial level. No automatic air valve was provided. The highest 
point in the siphon was about one third the total distance from the pond and 
nearest the pond. At this point a pump was placed, whose mission was to 
fill the pipe when necessary. This siphon would flow for about two hours 
and then cease, owing to accumulation of air in the pipe. When in full 
operation it discharged 43J^ g-allons per minute. The theoretical discharge 
from such a sized pipe witli the specified head is 55^ grallons per minute. 

Siphon on the Water-supply of Mount Vernon, N. Y. 
(Enifa News, May 4, 1893.)— A 12-inch siphon, 925 feet lone:, with a maximum 
lift of 22.12 feet and a 45° change in alignment, was put in use in 1892 by the 
New York City Suburban Water Co., which supplies Mount Vernon, N. Y. 

At its summit the siphon crosses a supply main, which is tapped to charge 
the siphon. 

The air-chamber at the siphon is 12 inches by 16 feet long. A J^-inch tap 
and cock at the top of the chamber provide an outlet for the collected air. 

It was found that the siphon with air-chamber as desc.ibed would run 
until 125 cubic feet of air had gathered, and that this took place only half as 
soon with a 14-foot lift as with the full lift of 22.12 feet. The siphon will 
operate about 12 hours without being recharged, but more water can be 
gotten over by charging every six hours. It can be kept running 23 hours 
out of 24 with only one man in attendance. With the siphon as described 
above it is necessary to close the valves at each end of the siphon to 
recharge it. 

It has been found by weir measurements that the discharge of the siphon 
before air accumulates at the summit is practically the same as through a 
straight pipe. 

MEASUREMENT OF FLOWING WATER. 

Piezometer.— If a vertical or oblique tube be inserted into a pipe con- 
taining water under pressure, the water will rise in the former, and the ver- 
tical height to which it rises will be the head producing the pressure at the 
point where the tube is attached. Such a tube is called a piezometer or 
pressure measure. If the water in the piezometer falls below its proper 
level it shows that the pressure in the main pipe has been reduced by an 
obstruction between the piezometer and the reservoir. If the water rises 
above its proper level, it indicates that the pressure there has been in- 
creased by an obstruction beyond the piezometer. 

If we imagine a pipe full of water to be provided with a number of pie- 
zometers, then a line joining the tops of the columns of water in them is 
the hydraulic grade-line. 



MEASUREMENT OF PLOWING WATER. 583 

Pitot Tube Gauge.— The Pitot tube is used for measuring the veloc- 
ity of fluids in motion. It has been used with great success in measuring 
the flow of natural gas. (S. W. Robinson, Report Ohio Geol. Survey, 1S90.) 
(See also Van No strand's Mag., vol. xxxv.) It is simply a tube so bent that 
a short leg extends into the current of fluid flowing from a tube, with the 
plane of the entering orifice opposed at right angles to the direction of the 
current. The pressure caused by the impact of the current is transmitted 
through the tube to a pressure' gauge of any kind, such as a column of 
water or of mercury, or a Bourdon spring-gauge. From the pressure thus 
indicated and the known density and temperature of the flowing gas is ob- 
tained the head corresponding to the pressure, and from this the velocity. 
In a modification of the Pitot tube described by Prof. Robinson, there are 
two tubes inserted into the pipe conveying the gas, one of which has the 
plane of the orifice at right angles to the current, to receive the static pres- 
sure plus the pressure due to impact; the other has the plane of its orifice 
parallel to the current, so as to receive the static pressure only. These 
tubes are connected to the legs of a U tube partly filled with mercury, which 
then registers the difference in pressure in the two tubes, from which the 
velocity may be calculated. Comparative tests of Pitot tubes with gas- 
meters, for measurement of the flow of natural gas, have shown an agree- 
ment within Sfo. 

The Venturi Meter, invented by Clemens Herschel, and described in 
a pamphlet issued by the Builders' Iron Foundry of Providence, R. I., is 
named from Venturi, who first called attention, in 1796, to the relation be- 
tween the velocities and pressures of fluids when flowing through converging 
and diverging tubes. 

It consists of two parts— the tube, through which the water flows, and the 
recorder, which registers the quantity of water that passes through the 
tube. 

The tube takes the shape of two truncated cones joined in their smallest 
diameters by a short throat-piece. At the up-stream end and at the throat 
there are air-chambers, at which points the pressuees are taken. 

The action of the tube is based on that property which causes the small 
section of a gently expanding frustum of a cone to receive, without material 
resultant loss of head, as much water at the smallest diameter as is dis- 
charged at the large end, and on that further property which causes the 
pressure of the water flowing through the throat to be less, by virtue of its 
greater velocity, than the pressure at the up-stream end of the tube, each 
pressure being at the same time a function of the velocity at that point and 
of the hydrostatic pressure which would obtain were the water motionless 
within the pipe. 

The recorder is connected with the tube by pressure-pipes which lead to 
it from the chambers surrounding the up-stream end and the throat of the 
tube. It may be placed in any convenient position within 1000 feet of the 
tube. It is operated by a weight and clockwork. 

The difference of pressure or head at the entrance and at the throat of the 
meter is balanced in the recorder by the difference of level hi two columns 
of mercury in cylindrical receivers, one within the other. The inner carries 
a float, the position of which is indicative of the quantity of water flowing 
through the tube. By its rise and fall the float varies the time of contact 
between an integrating drum and the counters by which the successive 
readings are registered. 

There is no limit to the sizes of the meters nor the quantity of water that 
may be measured. Meters with 21-inch, 36-inch, 48-inch, and even 20-foot 
tubes can be readily made. 

Measurement by Venturi Tubes. (Trans. A. S. C. E., Nov., 1887, 
and Jan., 1888.) — Mr. Herschel recommends the use of a Venturi tube, in- 
serted in the force-main of the pumping engine, for determining the quantity 
of water discharged. Such a tube applied to a 24-inch main has a total 
length of about 20 feet. At a distance of 4 feet from the end nearest the 
engine the inside diameter of the tube is contracted to a throat having a 
diameter of about 8 inches. A pressure-gauge is attached to each of two 
chambers, the one surrounding and communicating with the entrance or 
main pipe, the other with the throat. According to experiments made upon 
two tubes of this kind, one 4 in. in diameter at the throat and 12 in. at the en- 
trance, and the other about 36 in. in diameter at the throat and 9 feet at its 
entrance, the quantity of water which passes through the tube is very nearly 
the theoretical discharge through an opening having an area equal' to that 
of the throat, and a velocity which is that due to the difference in head Miown 



584 



HYDRAULICS. 



by the two gauges. Mr. Herschel states that the coefficient for these two 
widely-varying sizes of tubes and for a wide range of velocity through the 
pipe, was found to be within two per cent, either way, of 98%. In other 
words, the quantity of water flowing through the tube per second is ex- 
pressed within two per cent by the formula W — 0.98 X A X V%gh, in which 
A is the area of the throat of the tube, h the head, in feet, correspond- 
ing to the difference in the pressure of the water entering the tube and that 
found at the throat, and q = 32.16. 

measurement of Discharge of Pumping-engines toy 
Means of Nozzles. (Trans. A. S. M. E., xiii, 557). — The measurement 
of water by computation from its discharge through orifices, or through the 
nozzles of fire-hose, furnishes a means of determining the quantity of water 
delivered by apumping-engine which can be applied without much difficulty. 
John R. Freeman, Trans. A. S. C. E., Nov., 1889, describes a series of experi- 
ments covering a wide range of pressures and sizes, and the results showed 
that the coefficient of discharge for a smooth nozzle of ordinary good form 
was within one half of one per cent, either way, of 0.977 ; the diameter of 
the nozzle being accurately calipered, and the pressures being determined 
by means of an accurate gauge attached to a suitable piezometer at the base 
of the play-pipe. 

In order to use this method for determining the quantity of water dis- 
charged by a pumping-engine, it would be necessary to provide a pressure- 
box, to which the water would be conducted, and attach to the box as many 
nozzles as would be required to carry off the water. According to Mr. 
Freeman's estimate, four 134-inch nozzles, thus connected, with a pressure 
of 80 lbs. per square inch, would discharge the full capacity of a two-and a- 
half-million engine. He also suggests the use of a portable apparatus with 
a single opening for discharge, consisting essentially of a Siamese nozzle, 
so-called, the water being carried to it by three or more lines of fire-hose. 

To insure reliability for these measurements, it is necessary that the shut- 
off valve in the force-main, or the several shut-off valves, should be tight, 
so that all the water discharged by the engine may pass through the nozzles. 

Flow through Rectangular Orifices. (Approximate. Seep. 556.) 

Cubic Feet of Water Discharged per Minute through an Orifice One 
Inch Square, under any Head of Water from 3 to 72 Inches. 
For any other orifice multiply by its area in square inches. 
Formula, Q' — .624 Vh"X a. Q' — cu. ft. per min. ; a = area in sq. in. 





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K.S 
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H5.S 
43 


Gfl t 


53 


■° %z 


53 a 


3 


1.12 


2,20 


3.47 


3 95 


4.39 


4.78 


4 


1.27 


14 


2.28 


24 


2.97 


34 


3.52 


44 


4.00 


54 


4.42 


64 


4.81 


5 


1.40 


15 


2.36 


25 


3.03 


35 


3.57 


45 


4 05 


55 


4.46 


65 


4.85 


6 


1.52 


16 


2.43 


2G 


3.08 


36 


3.62 


46 


4.09 


56 


4.52 


66 


4.89 


7 


1.64 


17 


2.51 


27 


3.14 


37 


3 67 


47 


4.12 


57 


4 55 


67 


4.92 


8 


1.75 


18 


2.58 


28 


3.20 


38 


3.72 


48 


4.18 


58 


4.58 


68 


4.97 


9 


1.84 


19 


2.64 




3 25 


30 


3.77 


49 


4.21 


59 


4.63 


69 


5.00 


10 


1.94 


20 


2.71 


30 


3.31 


40 


3.81 


50 


4.27 


60 


4.65 


70 


5.03 


11 


2.03 


21 


2.78 


31 


3.36 


41 


3.86 


51 


4.30 


61 


4.72 


71 


5.07 


12 


2.12 


22 


2.84 


32 


3.41 


42 


3.91 


52 


4.34 


62 


4.74 


72 


5.09 



Measurement of an Open Stream by Velocity and Cross- 
section. — Measure the depth of the water at from 6 to 12 points across 
the stream at equal distances between. Add all the depths in feet together 
and divide by the number of measurements made; this will be the average 
depth of the stream, which multiplied by its width will give its area or cross- 
section. Multiply this by the velocity of the stream in feet per minute, and 
the result will be the discharge in cubic feet per minute of the stream. 

The velocity of the stream can be found by laying off 100 feet of the bank 
and throwing afloat into the middle, noting the time taken in passing over 
the 100 ft. Do this a number of times and take the average ; then, dividing 



MEASUREMENT OF FLOWING WATER. 



585 



this distance by the time gives the velocity at the surface. As the top of the 
stream flows faster than the bottom or sides — the average velocity being 
about 83$ of the surface velocity at the middle— it is convenient to measure 
a distance of 120 feet for the float and reckon it as 100. 





' >/-'-=- |! Mli< \\' Lik 



Fig. 130. 

OTiners' Incli Measurements. (Pelton Water Wheel Co.) 

The cut, Fig. 130, shows the form of measuring-box ordinarily used, and the 

following table gives the discharge in cubic feet per minute of a miner's inch 

of water, as measured under the various heads and different lengths and 

heights of apertures used in California. 



Length 


Openin 


gs 2 Inches High. 


Openi 


ngs 4 Inches High. 


of 














Opening 


Head to 


Head to 


Head to 


Head to 


Head to 


Head to 


in 


Centre, 


Centre, 


Centre, 


Centre, 


Centre, 


Centre, 


inches. 


5 inches. 


6 inches. 


7 inches. 


5 inches. 


6 inches. 


7 inches. 




Cu. ft. 


Cu. ft. 


Cu. ft. 


Cu. ft. 


Cu. ft. 


Cu. ft. 


4 


1.348 


1.473 


1.589 


1.320 


1.450 


1.570 


6 


1.355 


1.480 


1.596 


1.336 


1.470 


1.595 


8 


1.359 


1.484 


1.600 


1.344 


1.481 


1.608 


10 


1.361 


1.485 


1.602 


1.349 


1.487 


1.615 


12 


1.363 


1.487 


1.604 


1.352 


1.491 


1.620 


14 


1.364 


1.488 


1.604 


1.354 


1.494 


1.623 


16 


1.365 


1.489 


1.605 


1.356 


1.496 


1.626 


18 


1.365 


1-489 


1.606 


1.357 


1.498 


1.628 


20 


1.365 


1.490 


1.606 


1.359 


1.499 


1.630 


22 


1.366 


1.490 


1.607 


1.359 


1.500 


1.631 


24 


1.366 


1.490 


1.607 


1.360 


1.501 


1.632 


26 


1.366 


1.490 


1.607 


1.361 


1.502 


1.633 


28 


1.367 


1.491 


1.607 


1.361 


1.503 


1.634 


30 


1.367 


1.491 


1.608 


1.362 


1.503 


1.635 


40 


1.367 


1.492 


1.608 


1.363 


1.505 


1.637 


50 


1.368 


1.493 


1.609 


1.364 


1.507 


1.639 


60 


1.368 


1.493 


1.609 


1.365 


1.508 


1.640 


70 


1.368 


1.493 


1.609 


1.365 


1.508 


1.641 


80 


1.368 


1.493 


1.609 


1.366 


1.509 


1.641 


90 


1.369 


1.493 


1.610 


1.366 


1.509 


1.641 


100 


1.369 


1.494 


1.610 


1.366 


1.509 


1.642 



Note.— The apertures from which the above measurements were obtained 



586 



HYDRAULICS. 



were through material \y± inches thick, and the lower edge 2 inches above 
the bottom of the measuring-box, thus giving full contraction. 
Flow of Water Over Weirs. Weir Dam Measurement. 

(Pelton Water Wheel Co.)— Place a board or plank in the stream, as shown 




Fig. 131. 

in the sketch, at some point where a pond will form above. The length of 
the notch in the dam should be from two to four times its depth for small 
quantities and longer for large quantities. The edges of the notch should 
be bevelled toward the intake side, as shown. The overfall below the notch 
should not be less than twice its depth, that is, 12 inches if the notch is 6 
inches deep, and so on. 

In the pond, about 6 ft. above the dam, drive a stake, and then obstruct the 
water until it rises precisely to the bottom of the notch and mark the stake 
at this level. Then complete the dam so as to cause all the water to flow 
through the notch, and, after time for the water to settle, mark the stake 
again for this new level. If preferred the stake can be driven with its top 
precisely level with the bottom of the notch and the depth of the water be 
measured with a rule after the water is flowing free, but the marks are pre- 
ferable in most cases. The stake can then be withdrawn ; and the distance 
between the marks is the theoretical depth of flow corresponding to the 
quantities in the table. 

Francis's Formulae for Weirs. 

As given by As modified by 

Francis. Smith. 

Weirs with both end contractions I n — ^ qq7/,§ onofn h. ~\]$ 

suppressed j v ' v "*" 7 ' 

W s 6 u S ressed 116 ^ contraction j- Q = 3.33(J - .lh)h% 3.29Z^ 

Weirs with full contraction Q = 3.33(Z - .2h)h? 3.29^ - —Jh* 

The greatest variation of the Francis formulas from the values of c given by 
Smith amounts to 3*/£#. The modified Francis formulas, says Smith, will give 
results sufficiently exact, when great accuracy is not required, within the 
limits ©f h, from .5 ft. to 2 ft., I being not less than 3 h. 



MEASUREMENT OF FLOWING WATER. 



587 



Q — discharge in cubic feet per second, I = length of weir in feet, /(, =effec- 
tive head in feet, measured from the level of the crest to the level of still 
water above the weir. 

If Q' -. discharge in cubic feet per minute, and V and h' are taken in 
inches, the first of the above formulas reduces to Q' = 0.41'h' 71 . From this 
formula the following table is calculated. The values are sufficiently accu- 
rate for ordinary computations of water-power for weirs without end con- 
traction, that is, for a weir the full width of the channel of approach, and 
are approximate also for weirs with end contraction when I - at least 10/i, 
but about 6$ in excess of the truth when I = 4h. 

Weir Table. 

Giving Cubic Feet op Water per Minute that will Flow over a Weir 
one inch wide and from % to 20% inches deep. 

For other widths multiply by the width in inches. 



in. 


cu. ft. 





.00 


1 


.40 


2 


1.13 


3 


2.07 


4 


3.20 


5 


4.47 


6 


5.87 


7 


7.40 


8 


9.05 


9 


10.80 


10 


12.64 


11 


14.59 


12 


16.62 


13 


18.74 


14 


20.95 


15 


23.23 


16 


25.60 


17 


28.03 


18 


30.54 


19 


33.12 


20 


35.77 



^i". 


14 in. 


% in. 


^in. 


Y&in. 


34 in. 


%m. 


cu. ft. 


cu. ft. 


cu. ft. 


cu. ft. 


cu. ft. 


cu. ft. 


cu. ft. 


.01 


.05 


.09 


.14 


.19 


.26 


.32 


.47 


.55 


.64 


.73 


.82 


.92 


1.02 


1.23 


1.35 


1.46 


1.58 


1.70 


1.82 


1.95 


2.21 


2.34 


2.48 


2.61 


2.76 


2.90 


3.05 


3.35 


3.50 


3.66 


3.81 


3.97 


4.14 


4.30 


4.64 


4.81 


4.98 


5.15 


5.33 


5.51 


5.69 


6.06 


6.25 


6.44 


6.62 


6.82 


7.01 


7.21 


7.60 


7.80 


8.01 


8.21 


8.42 


8.63 


8.83 


9.26 


9.47 


9.69 


9.91 


10.13 


10.35 


10.57 


11.02 


11.25 


11.48 


11.71 


11.94 


12.17 


12.41 


12.88 


13.12 


13.36 


13.60 


13.85 


14.09 


14.34 


14.84 


15.09 


15 34 


15.59 


15.85 


16.11 


16.36 


16.88 


17.15 


17.41 


17.67 


17.94 


18.21 


18.47 


19.01 


19.29 


19.56 


19.84 


20.11 


20 39 


20.67 


21.23 


21.51 


21.80 


22.08 


22.37 


22.65 


22.94 


23.52 


23.82 


24.11 


24.40 


24.70 


25.00 


25.30 


25.90 


26.20 


26.50 


26.80 


27.11 


27.42 


27.72 


28.34 


28.65 


28.97 


29.28 


29.59 


29.91 


30.22 


30.86 


31.18 


31.50 


31.82 


32.15 


32.47 


32.80 


33.45 


33.78 


34.11 


34.44 


34.77 


35.10 


35.44 


36.11 


36.45 


36.78 


37.12 


37.46 


37.80 


38.15 



For more accurate computations, the coefficients of flow of Hamilton 
Smith, Jr., or of Bazin should be used. In Smith's hydraulics will be found 
a collection of results of experiments on orifices and weirs of various shapes 
made by many different authorities, together with a discussion of their 
several formulae. (See also Trautwine's Pocket Book.) 

Bazln'S Experiments. - M. Bazin (Annates des Pouts et Cliaussees, 
Oct., 1888, translated by Marichal and Trautwine, Proc. Engrs. Club of Phila., 
Jan , 1890), made an extensive series of experiments with a sharp-crested 
weir without lateral contraction, the air being admitted freely behind the 
falling sheet, and found values of m varying from 0.42 to 0.50, with varia- 
tions of the length of the weir from 19% to 78% in., of the height of the crest 
above the bottom of the channel from 0.79 to 2.46 ft., and of the head from 
1.97 to 23.62 in. From these experiments he deduces the following formula : 

Q =[0.425 + ^(jrf^) 2 ]^ V^H, 

in which P is the height in feet of the crest of the weir above the bottom of 
the channel of approach, L the length of the weir, H the head, both in feet, 
and Q the discharge in cu. ft. per sec. This formula, says M. Bazin, is en- 
tirely practical where errors of 2% to 3% are admissible. The following 
table is condensed from M. Bazin's paper : 



588 



WATER-POWER. 



Values of the Coefficient m in the Formula Q = mLH V'igH, for a 
Sharp-crested Weir without Lateral Contraction ; the Air being 
Admitted Freely Behind the Falling Sheet. 





Height of Crest of Weir Above Bed of Channel. 


Head, 






H. 














I 








Feet ...0.66 


0.98 


1.31 


1.64 


1.97 


2.62 


3.28 4.92 


6 56 


oo 




Inches 7.87 


11.81 


15,75 


19.69 


23.62 


31.50 


39.38 59.07 

I 


78.76 


GO 


Ft. 


Tn. 


m 


m 


m 


m 


VI 


m 


m | m 


m 


m 


.164 


1.97 


0.458 


0.453 


0.451 


0.450 


0.449 


0.449 


0.449 0.448 


0.448 


0.4481 


.230 


2.76 


0.455 


0.448 


0.445 


0.443 


0.442 


0.441 


0.440 0.440 


0.439 


0.4391 


.295 


3.54 


0.457 


0.447 


0.442 


0.440 


0.438 


0.436 


0.436 0.435 


0.434 0.4340 


.394 


4.72 


0.462 


0.448 


0.442 


0.438 


0.436 


0.433 


0.432 0.43C 


430 


0.4291 


.525 


6.30 


0.471 


0.453 


0.444 


0.438 


0.435 


0.431 


0.429! 0.427 


0.426 


0.4246 


.656 


7.87 


0.480 


0.459 


0.447 


0.440 


0.436 


0.431 


0.428 0.425 


0.423 


0.4215 


.787 


9.45 


0.488 


0.465 


0.452 


0.444 


0.438 


0.432 


0.428,0.424 


422 


0.4194 


.919 


11.02 


0.496 


0.472 


0.457 


0.448 


0.441 


0.433 


0.429 0.424 


0.422 


0.4181 


1.050 


12 60 




478 


0.462 


0.452 


0.444 


436 


0.430 


0.424 


0.421 


0.4168 


1.181 


14 17 




483 


0.467 


0.456 


0.448 


438 


0.432 


0.424 


0.421 


0.4156 


1.312 


15.75 




0.489 


0.472 


0.459 


0.451 


0.440 


0.433 




0.421 


0.4144 


1.444 


17.3V 




494 


0.476 


463 


0.454 


0.442 


0.435 0.425 


0.421 


0.4134 


1.575 


18.90 
30.47 






0.480 
0.483 


0.467 
470 


0.457 
0.460 


0.444 
0.446 


0.436 0.425 
0.438 0.426 


0.421 
0.421 


0.4122 


1.706 






0.4112 


1.837 


' 






0.487 
0.490 


0.473 
0.476 


0.463 
0.466 


0.448 
0.451 


0.4390.427 
0.441 0.427 


0.421 
0-421 


0.4101 


1.969 


23.62 






0.4092 








i r 





A comparison of the results of this formula with those of experiments, 
says M. Bazin, justifies us in believing that, except in the unusual case of a 
very low weir (which should always be avoided), the preceding table will 
give the coefficient m in all cases within 1%; provided, however, that the ar- 
rangements of the standard weir are exactly reproduced. It is especially 
important that the admission of the air behind the falling sheet be perfectly 
assured. If this condition is not complied with, m may vary within much 
wider limits. The type adopted gives the least possible variation in tho 
coefficient. 



WATER-POWER. 



Power of a Fall of Water— Efficiency.— The gross power of 
a fall of water is the product of the weight of water discharged in a unit of 
time into the total head, i.e., the difference of vertical elevation of the 
upper surface of the water at the points where the fall in question begins 
and ends. The term " head " used in connection with water-wheels is the 
difference in height from the surface of the water in the wheel-pit to the 
surface in the pen-stock when the wheel is running. 

If Q = cubic feet of water discharged per second, D = weight of a cubic 
foot of water = 62.36 lbs. at 60° F., H = total head in feet; then 



DQH = gross pow r er in foot-pounds per second, 
arid DQH -*- 550 =.ll%4 QH = gross horse-power. 

If Q' if. taken in cubic feet per minute, H> P. = ■ ' = 

oo,U00 



'H. 



A water-wheel or motor of any kind cannot utilize the whole of the head 
H, since there are losses of head at both the entrance to and the exit from 
i he wheel. There are also losses of energy due to friction of the water in 
its passage through the wheel, t'he ratio of the power developed by the 
wheel to the gross power of the fall is the efficiency of the wheel. For 75% 

efficiency, net horse-power = .00142Q' H = -^r . 



MILL-POWER. 589 

A head of water can be made use of in one or other of the following ways 
viz. : 

1st. By its weight, as in the water-balance and overshot-wheel. 

2d. By its pressure, as in turbines and in the hydraulic engine, hydraulic 
press, crane, etc. 

3d. By its impulse, as in the undershot- wheel, and in the Pelton wheel. 

4th. By a combination of the above. 

Horse-power of a Running Stream.— The gross horse power 
is, H. P. = QH X 62.30 -*- 550 = MMQH, in which Q is the discharge in cubic 
feet per second actually impinging on the float or bucket, and H — theoret- 
ic v 2 
ical head due to the velocity of the stream = — = rr- - , in which v is the 

2g 64.4 
velocity in feet per second. If Q' be taken in cubic feet per minute, 
H.P. = .00189Q'if. 

Thus, if the floats of an undershot-wheel driven by a current alone be 5 
feet x 1 foot, and the velocity of stream = 210 ft. per minute, or 3% ft. per 
sec, of which the theoretical head is .19 ft., Q = 5 sq. ft. X 210 = 1050 cu. ft. 
per minute ; H = .19 ft. ; H. P. = 1050 X .19 X. 00189 = .377 H. P. 

The wheels would realize only about .4 of this power, on account of friction 
and slip, or .151 H. P., or about .03 H.P. per square foot of float, which is 
equivalent to 33 sq. ft. of float per H. P. 

Current Motors. — A current motor could only utilize the whole power 
of a running stream if it could take all the velocity out of the water, so that 
it would leave the floats or buckets with no velocity at all ; or in other words, 
it would require the backing up of the whole volume of the stream until the 
actual head was equivalent to the theoretical head due to the velocity of the 
stream. As but a small fraction of the velocity of the stream can be taken 
up by a current motor, its efficiency is very small. Current motors may be 
used' to obtain small amounts of power from large streams, but for large 
powers they are not practicable. 

Horse-power of Water Flowing in a "Tube.— The head due to 
i> 2 f 

the velocity is — ; the head due to the pressure is - ; the head eiue to actual 

2g w 

height above the datum plane is h feet. The total head is the sum of these = 
v 9 f 

— 4- h -f — , in feet, in which v = velocity in feet per second,/ = pressure 
2g w 

in lbs. per sq. ft., to — weight of 1 cu. ft. of water = 62.36 lbs. If p = pres- 
sure in lbs. per sq. in., — = 2.309jp. In hydraulic transmission the velocity 

and the height above datum are usually small compared with the pressure- 
head. The work or energy of a given quantity of water underpressure = 
its volume in cubic feet X its pressure in lbs. per sq. ft.; or if Q = quantity 
in cubic feet per second, and p = pressure in lbs. per square inch, W = 

U4 P Q, and the H. P. = ^^ = .2618pQ. 

Maximum Efficiency of a Long Conduit.— A. L. Adams and 
R. G. Uenimel {Euy'y News, May 4, 1893), show by mathematical analysis that 
the conditions for securing the maximum amount of power through a long 
conduit of fixed diameter, without regard to the economy of water, is that 
the draught from the pipe should be such that the frictional loss in the pipe 
will be equal to one third of the entire static head. 

Mill-Power. — A "mill-power " is a unit used to rate a water-power for 
the purpose of renting it. The value of the unit is different in different 
localities. The following are examples (from Emerson): 

Holyoke, Mass. — Each mill-power at the respective falls is declared to be 
the right during 16 hours in a day to draw 38 cu. ft. of water per second at 
the upper fall when the head there is 20 feet, or a quantity proportionate to 
the height at the falls. This is equal to 86.2 horse-power as a maximum. 

Lowell, M ass. —The right to draw during 15 hours in the day so much water 
as shall give a power equal to 25 cu. ft. a second at the great fall, when the 
fall there is 30 feet. Equal to 85 H. P. maximum. 

Lawrence, Mass. — The right to draw during 16 hours in a day so much 
water as shall give a horse-power equal to 30 cu. ft. per second when the 
head is 25 feet. Equal to 85 H. P. maximum. 

Minneapolis, Minn.— 30 cu. ft. of water per second with head of 22 feet. 
Equal to 74.8 H. P. 

Manchester, N. H.— Divide 725 by the number of feet of fall minus 1, and 



590 WATER-POWER. 

the quotient will be the number of cubic feet per second in that fall. For 20 
feet fall this equals 38.1 cu. ft., equal to 86.4 H. P. maximum. 

Cohoes, N. Y.— " Mill-power " equivalent to the power given by 6 cu. ft. 
per second, when the fall is 20 feet. Equal to 13.6 H. P.,«maximum. 

Passaic, N. J.— Mill-power: The right to draw 8J^ cu. ft. of water per sec, 
fall of 22 feet, equal to 21.2 horse-power. Maximum rental $700 per year for 
each mill-power = $33.00 per H. P. 

The horse-power maximum above given is that due theoretically to the 
weight of water and the height of the fall, assuming the water-wheel to 
have perfect efficiency. It should be multiplied by the efficiency of the 
wheel, say 75# for good turbines, to obtain the H. P. delivered by the wheel. 

Value of a Water-power.— In estimating the value of a water- 
power, especially where such value is used as testimony for a plaintiff whose 
water-power has been diminished or confiscated, it is a common custom for 
the person making such estimate to say that the value is represented by a 
sum of money which, when put at interest, would maintain a steam-plant 
of the same power in the same place. 

Mr. Charles T. Main (Trans. A. S. M. E. xiii. 140) points out that this sys- 
tem of estimating is erroneous; that the value of a power depends upon a 
great number of conditions, such as location, quantity of water, fall or head, 
uniformity of flow, conditions which fix the expense of dams, canals, founda- 
tions of buildings, freight charges for fuel, raw materials and finished prod- 
uct, etc. He gives an estimate of relative cost of steam and water-power 
for a 500 H. P. plant from which the following is condensed: 

The amount of heat required per H. P. varies with different kinds of busi- 
ness, but in an average plain cotton-mill, the steam required for heating and 
slashing is equivalent to about 25$ of isteam exhausted from the high- 
pressure cylinaer of a compound engine of the power required to run that 
mill, the steam to be taken from the receiver. 

The coal consumption per H. P. per hour for a compound engine is taken 
at 1% lbs. per hour, when no steam is taken from the receiver for heating 
purposes. The gross consumption when 25$ is taken from the receiver is 
about 2.06 lbs. 

75% of the steam is used as in a compound engine at 1.75 lbs. = 1.31 lbs. 
25% " " " " high-pressure " 3.00 lbs. = .75 " 

2.06 " 
The running expenses per H. P. per year are as follows : 
2.06 lbs. coal per hour = 21.115 lbs. for 10J4 hours or one day = 6503.42 

lbs. for 308 days, which, at $3.00 per long ton = $8 71 

Attendance of boilers, one man @ $2.00, and one man @ $1.25 = 2 00 

" engine, ■« '■ " $3.50. 2 16 

Oil, waste, and supplies. 80 

The cost of such a steam-plant in New England and vicinity of 500 
H. P. is about $65 per H. P. Taking the fixed expenses as 4% on 
engine, 5% on boilers, and 2% on other portions, repairs at 2%, in- 
terest at 5%, taxes at 1 \^% on % cost, an insurance at y%% on exposed 
portion, the total average per cent is about 12J^#, or $65 X -12^ = 8 13 

Gross cost of power and low-pressure steam per H. P. $21 80 

Comparing this with water-power, Mr. Main says : "At Lawrence the cost 
of dam and canals was about $650,000, or $65 per H. P. The cost per H. P. 
of wheel-plant from canal to river is about $45 per H. P. of plant, or about 
$65 per H. P. used, the additional $20 being caused by making the plant 
large enough to compensate for fluctuation of power due to rise and fall <>f 
river. The total cost per H. P. of developed plant is then about $130 per H. P. 
Placing the depreciation on the whole plant at 2%, repairs at 1%, interest at 
5%, taxes and insurance at 1 %, or a total o£ 9%, gives: 

Fixed expenses per H. P. $130 X .09 = $11 70 
Running " " " (Estimated) 2 00 

$13 70 

" To this has to be added the amount of steam required for heating pur- 
poses, said to be about 25$ of the total amount used, but in winter months 
the consumption is at least 37^$. It is therefore necessary to have a boiler 
plant of about S7},4% of the size of the one considered with" the steam-plant. 



TURBINE WHEELS. 591 

costing about $20 X .375 = $7.50 per H. P. of total power used. The ex- 
pense of running this boiler-plant is, per H. P. of the the total plant per year: 

Fixed expenses 12^$ on $7.50 $0.94 

Coal 3.26 

Labor 1 .23 

Total $5.43 

Making a total cost per year for water-power with the auxiliary boiler plant 
$13.70 + $5.43 = $19.13 which deducted from" $21.80 make a difference in 
favor of water-power of $2.67, or for 10,000 H. P. a saving of $26,700 per 
year. 

" It is fair to say," says Mr. Main," that the value of this constant power is 
a sum of money which when put at interest will produce the saving ; or if 6% 
is a fair interest to receive on money thus invested the value would be 
$26,700-- .06 = $445,000." 

Mr. Main makes the following general statements as to the value of a 
water-power : "The value of an undeveloped variable power is usually noth- 
ing if its variation is great, unless it is to be supplemented by a steam-plant. 
It is of value then only when the cost per horse-power for the double-plant 
is less than the cost of steam-power under the same conditions as mentioned 
for a permanent power, and its valuecan be represented in the same man- 
ner as the value of a permanent power has been represented. 

" The value of a developed power is as follows: If the power can be run 
cheaper than steam, the value is that of the power, plus the cost of plant, 
less depreciation. If it cannot be run as cheaply as steam, considering its 
cost, etc., the value of the power itself is nothing, but the value of the plant 
is such as could be paid for it new, which would bring the total cost of run- 
ning down to the cost of steam-power, less depreciation." 

Mr. Samuel Webber, Iron Age, Feb. and March, 1893, writes a series of 
articles showing the development of American turbine wheels, and inci- 
dentally criticises the statements of Mr. Main and others who have made 
comparisons of costs of steam and of water-power unfavorable to the latter. 
Hesays : ' ' They have based their calculations on the cost of steam, on large 
compound engines of 1000 or more H. P. and 120 pounds pressure of steam 
in their boilers, and by careful 10-hour trials succeeded in figuring down 
steam to a cost of about $20 per H. P., ignoring the well-known fact that its 
average cost in practical use, except near the coal mines, is from $40 to $50. 
In many instances dams, canals, and modern turbines can be all completed 
for a cost of $100 per H. P. ; and the interest on that, and the cost of attend- 
ance and oil, will bring water-power up to but about $10 or $12 per annum; 
and with a man competent to attend the dynamo in attendance, it can 
probably be safely estimated at not over $15 per H. P." 

TURBINE WHEELS. 

Proportions of Turbines.— Prof. De Volson Wood discusses at 
length the theory of turbines in his paper on Hydraulic Reaction Motors, 
Trans. A. S. M. E. xiv. 266. His principal deductions which have an imme- 
diate bearing upon practice are condensed in the following : 
Notation. 

Q = volume of water passing through the wheel per second, 

h l = head in the supply chamber above the entrance to the buckets, 

/< 2 = head in the tail-race above the exit from the buckets, 

Z) — fall in passing through the buckets. 

H — h x + z-i — /i 2 , the effective head, 

l*. x = coefficient of resistance along the guides, 

ju., = coefficient of resistance along the buckets, 

r x = radius of the initial rim, 

r 2 = radius of the terminal rim, 

V = velocity of the water issuing from supply chamber, 

v x = initial velocity of the water in the bucket in reference to the bucket, 

v 2 = terminal velocity in the bucket, 

co = angular velocity of the wheel, 

a = terminal angle between the guide and initial rim = CAB, Fig. 132, 

Vi = angle between the initial element of bucket and initial rim = BAD. 

V 2 = OFI, the angle between the terminal rim and terminal element of 
the bucket. 

a == eb, Fig. 133 = the arc subtending one gate opening, 



592 



WATEft-POWEft. 



<*! = the arc subtending one bucket at entrance. (In practice a t is larger 
than «,) 

« 2 = gh, the arc subtending one bucket at exit, 

K = £>/, normal section of passage, it being assumed that the passages 
and buckets are very narrow, 

k l = bd, initial normal section of bucket, 
fc 2 = gi, terminal normal section, 
to?! = velocity of initial rim, 
wr 2 = velocity of terminal rim, 
= HFI, angle between the terminal rim and actual direction of the 
water at exit, 

Y — depth of K. y, of a,, and y 2 of K^ then 

K = Ya sin a; K x = y x a x sin y t ; 1T 2 == # 2 a 2 sin y 2 . 




Fig. 132. 



Fig. 133. 



Three simple systems are recognized, i\ < ?- 2 ,{called outward flow; r x > r 3 , 
called inward flow; r t = r 2 , called parallel flow. The first and second may 
be combined with the third, making a mixed system. 

Value of v? (the quitting angle).— The efficiency is increased as y 9 de- 
creases, and is greatest for y 2 = 0. Hence, theoretically, the terminal ele- 
ment of the bucket should be tangent to the quitting rim for best efficiency. 
This, however, for the discharge of a finite quantity of water, would 
require an infinite depth of bucket. In practice, therefore, this angle must 
have a finite value. The larger the diameter of the terminal rim the smaller 
may be this angle for a given depth of wheel and given quantity of water 
discharged. In practice y 2 is from 10° to 20°. 

In a wheel in which all the elements except y 2 are fixed, the velocity of 
the wheel for best effect must increase as the quitting angle of the bucket 
decreases. 

Values of a -{- y } must be less than 180°, but the best relation cannot be 
determined by analysis. However, since the water should be deflected from 
its course as much as possible from its entering to its leaving the wheel, the 
angle a for this reason should be as small as practicable. 

In practice, a cannot be zero, and is made from 20° to 30°. 

The value ?j = 1.4r a makes the width of the crown for internal flow about 
the same as for i\ =r 3 \/% for outward flow, being approximately 0.3 of the 
external radius. 

Values of v-! and /u. 2 . — The frictional resistances depend upon the construc- 
tion of the wheel as to smoothness of the surfaces, sharpness of the angles, 



TUKBLNfi WHEELS. 593 

regularity of the curved parts, and also upon the speed it is run. These 
values cannot be definitely assigned beforehand, but Weisbach gives for 
good conditions /u^ = /u. 2 = 0.05 to 0.10. 

They are not necessarily equal, and /u-i may be from 0.05 to 0.075, and /u. 2 
from 0.06 to 0.10 or even larger. 

Values of y t must be less than 180° — a. 

To be on the safe side, Vi may be 20 or 30 degrees less than 180° -2a, giving 

Vl = 180° - 2a - 25 (say) = 155 - 2a. 

Then if a = 30°, y 1 = 95°. Some designers make y± 90°; others more, and 
still others less, than that amount. Weisbach suggests that it be less, so 
that the bucket will be shorter and friction less. This reasoning appears to 
be correct for the inflow wheel, but not for the outflow wheel. In the Tre- 
mont turbines, described in the Lowell Hydraulic Experiments, this angle 
is 90°, the angle a 20°, and y 2 10°, which proportions insured a positive 
pressure in the wheel. Fourneyron made y x = 90°, and a from 30° to 33°, 
which values made the initial pressure in the wheel near zero. 

Form of Bucket —The form of the bucket cannot be determined analytic- 
ally. From the initial and terminal directions and the volume of the water 
flowing through the wheel, the area of the normal sections may be found. 

The normal section of the buckets will be : 

V Vi v 7 

The depths of those sections will be : 



K k x _ 



« 2 sin "y 3 



The changes of curvature and section must be gradual, and the general 
form regular, so that eddies and whirls shall not be formed. For the same 
reason the wheel must be run with the correct velocity to secure the best 
effect. In practice the buckets are made of two or three arcs of circles, 
mutually tangential. 

The Value of <■>.— So far as analysis indicates, the wheel may run at any 
speed; but in order that the stream shall flow smoothly from the supply 
chamber into the bucket, the velocity V should be properly regulated. 

If ix j = nz = 0.10, r 2 -^ n = 1.40, a = 25°, ?i = 90°, v 2 .= 12°, the velocity of 
the initial rim for outward flow will be f or m aximum efficiency 0.614 of the 
velocity due to the head, or wj-j = 0.614 V2 gH. 
The velocity due to the head would be V2yH = 1.414 VgH. 
For an inflow wheel for the case in w hich r^ = 2?- a 2 , and the other dimen 
sions as given above, wrj = 0.682 S/2gH. 

The highest efficiency of the Tremont turbine, found experimentally, was 
0.79375, and the corresponding velocity, 0.62645 of that due to the head, and 
for all velocities above and below this value the efficiency was less. 

In the Tremont wheel a = 20° instead of 25°, and y ? = 10° instead of 12°. 
These would make the theoretical efficiency and velocity of the wheel some- 
what, greater. Experiment showed that the velocit}' might be considerably 
larger or smaller than this amount without much diminution of the efficiency. 
It was found that if the velocity of the initial (or interior) rim was not less 
than 44$ nor more than 75$ of that due to the fall, the efficiency was 75$ or 
more. This wheel was allowed to run freely without any brake except its 
own friction, and the velocity of the initial rim was observed to be 
1.335 V2gH, half of which is 0.6675 V^gH, which is not far from the velocity 
giving maximum effect; that is to say, when the gate is fully raised the coeffi- 
cient of effect is a maximum when the wheel is moving with about half its 
maximum velocity. 

Number of Buckets.— Successful wheels have been made in which the dis- 
tance between the buckets was as small as 0.75 of an inch, and others as 
much as 2.75 inches. Turbines at the Centennial Exposition had buckets 
from i\4, inches to 9 inches from centre to centre. If too large they will not 
work properly. Neither should they be too deep. Horizontal partitions 
are sometimes introduced. These secure more efficient working in case the 
gates are only partly opened. The form and number of buckets for com- 
mercial purposes are chiefly the result of experience. 



594 WATER-POWEtfc. 

Ratio of Radii.— Theory does not limit the dimensions of the wheel. In 
practice, 

for outward flow, r 2 ■*- r 2 is from 1.25 to 1.50; 
for inward flow, r 2 -s- r x is from 0.66 to 0.80. 

It appears that the inflow-wheel has a higher efficiency than the outward- 
flow wheel. The inflow-wheel also runs somewhat slower for best effect. 
The centrifugal force in the outward-flow wheel tends to force the water 
outward faster than it would otherwise flow ; while in the inward-flow wheel 
it has the contrary effect, acting as it does in opposition to the velocity in 
the buckets. 

It also appears that the efficiency of the outward-flow wheel increases 
slightly as the width of the crown is less and the velocity for maximum 
efficiency is slower ; while for the inflow-wheel the efficiency slightly in- 
creases for increased width of crown, and the velocity of the outer rim at the 
same time also increases. 

Efficiency. — The exact value of the efficiency for a particular wheel must 
be found by experiment. 

It seems hardly possible for the effective efficiency to equal, much less 
exceed, 86$, and all claims of 90 or more per cent for these motors should be 
discarded as improbable. A turbine yielding from 75$ to 80$ is extremely 
good. Experiments with higher efficiencies have been reported. 

The celebrated Tremont turbine gave 79*4$ without the " diffuser," which 
might have added some 2$. A Jonval turbine (parallel flow) was reported 
as yielding 0.75 to 0.90, but Morin suggested corrections reducing it to 0.63 to 
0.71. Weisbach gives the results of many experiments, in which the effi- 
ciency ranged from 50$ to 84$. Numerous experiments give E = 0.60 to 0.65. 
The efficiency, considering only the energy imparted to the wheel, will ex- 
ceed by several per cent the efficiency of the wheel, for the latter will in- 
clude the friction of the support and leakage at the joint between the sluice 
and wheel, which are not included in the former ; also as a plant the resist- 
ances and losses in the supply-chamber are to be still further deducted. 

The Crowns.— The crowns may be plane annular disks, or conical, or 
curved. If the partitions forming the buckets be so thin that they may be 
discarded, the law of radial flow will be determined bv the form of the 
crowns. If the crowns be plane, the radial flow (or radial component) will 
diminish, for the outward flow-wheel, as the distance from the axis increases 
—the buckets being full — for the angular space will be greater. 

Prof. Wood deduces from the formulae in his paper the tables on page 595. 

It appears from[these tables: 1. That the terminal angle, a, has frequently 
been made too large in practice for the best efficiency. 

2. That the terminal angle, a, of the guide should be for the inflow less 
than 10° for the wheels here considered, but when the initial angle of the 
bucket is 90°, and the terminal angle of the guide is 5° 28', the gain of effi- 
ciency is not 2$ greater than when the latter is 25°. 

3. That the initial angle of the bucket should exceed 90° for best effect for 
outflow-wheels. 

4. That with the initial angle between 60° and 120° for best effect on inflow 
wheels the efficiency varies scarcely 1$. 

5. In the outflow-wheel, column (9) shows that for the outflow for best 
effect the direction of the quitting water in reference to the earth should be 
nearly radial (from 76° to 97°), but for the inflow wheel the water is thrown 
forward in quitting. This shows that the velocity of the rim should some- 
what exceed the relative final velocity backward in the bucket, as shown in 
columns (4) and (5). 

6. In these tables the velocities given are in terms of Vtyh, and the co- 
efficients of this expression will be the part of the head which would produce 
that velocity if the water issued freely. There is only one case, column (5), 
where the coefficient exceeds unity, and the excess is so small it may be dis- 
carded; and it may be said that in a properly proportioned turbine with the 
conditions here given none of the velocities will equal that due to the head 
in the supply-chamber when running at best effect. 

7. The inflow turbine presents the best conditions for construction for 
producing a given effect, the only apparent disadvantage being an increased 
first cost due to an increased depth, or an increased diameter for producing 
a given amount of work. The larger efficiency should, however, more than 
neutralize the increased first cost. 






TURBINE WHEELS. 



595 



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596 



WATER-POWER. 



Tests of Turbines.— Emerson says that in testing turbines it is a rare 
thing to find two of the same size which can be made to do their best at the 
same speed. The best speed of one of the leading wheels is invariably wide 
from the tabled rate. It was found that a 54-in. Leffel wheel under 12 ft. 
head gave much better results at 78 revolutions per minute than at 90. 

Overshot wheels have been known to give 75$ efficiency, but the average 
performance is not over 60$. 

A fair average for a good turbine wheel may be taken at 75$. In tests of 18 
wheels made at the Philadelphia Water-works in 1859 and 1860, one wheel 
gave less than 50$ efficiency, two between 50$ and 60$, six between 60$ and 
70$, seven between 71$ and 77$, two 82^. and one 87.77$. (Emerson.) 

Tests of Turbine Wheels at the Centennial Exhibition, 
1876. (From a paper by R. H. Thurston on The Systematic Testing of 
Turbine Wheels in the United States, Trans. A. S. M. E., viii. 359.)— In 1876 
the judges at the International Exhibition conducted a series of trials of 
turbines. Many of the wheels offered for tests were found to be more or 
less defective in fitting and workmanship. The following is a statement of 
the results of all turbines entered which gave an efficiency of over 75$. 
Seven other wheels were tested, giving results between 65$ and 75$. 



Maker's Name, or Name the 
Wheel is Known By. 


3 CO 

* 5 

^ O 
B m ® 

Ph 

87.68 
83.79 
83.30 
82.13 
81.21 
78.70 
79.59 
77.57 
77.43 
76.94 
76.16 
75.70 
75.15 


£ 03 

go U 
Ph 


3.2 

OQ 

« . 

a P 
o°5 

Ph 


3 w 
-^ . 

Ph 


3.2? 
B P 

3*5 

^NOD-B 

Ph 


J c8 
Ph 
75.35 


3.22 
OQ 
ce — 

BO S, 

Ph 






86.20 


82.41 
70.79 








Geyelin (single) 

Thos. Tait 




71.66 


Yi.bV 

81.24 


70.40 
55.90 
68.60 
79.92 


66.35 

51.03 

67.23 




55.00 










Tyler Wheel 

Geyelin (duplex) 


69.59 




74.25 
73.88 

74.89 






62.75 








69.92 












70.87 
62.06 


71.74 




York Manufacturing Co. . . 

W. F. Mosser & Co 


67.08 
71.90 


67.57 
70.52 




66.04 





The limits of error of the tests, says Prof. Thurston, were very uncertain; 
they are undoubtedly considerable as compared with the later work done in 
the permanent flume at Holy oke— possibly as much as 4$ or 5$. 

Experiments with "draught-tubes," or "suction-tubes," which were 
actually " diff users " in their effect, so far as Prof. Thurston has analyzed 
them, indicate the loss by friction which should be anticipated in such 
cases, this loss decreasing as the tube increased in size, and increasing as 
its diameter approached that of the wheel— the miuimum diameter tried. 
It was sometimes found very difficult to free the tube from air completely, 
and next to impossible, during the interval, to control the speed with the 
brake. Several trials were often necessary before the power due to the full 
head could be obtained. The loss of power by gearing and by belting was 
variable with the proportions and arrangement of the gears and pulleys, 
length of belt, etc.. but averaged not far from 30$ for a single pair of bevel- 
gears, uncut and dry, but smooth for such gearing, and but 10$ for the same 
gears, well lubricated, after they had been a short time in operation. The 
amount of power transmitted was, however, small, and these figures are 
probably much higher than those representing ordinary practice. Intro- 
ducing a second pair— spur-gears— the best figures were but little changed, 
although the difference between the case in which the larger gear was the 
driver, and the case in which the small wheel was the driver, was perceiv- 
able, and was in favor of the former arrangement. A single straight belt 
gave a loss of but 2$ or 3$, a crossed belt 6$ to 8$, when transmitting 14 



TURBINE WHEELS. 



597 




horsepower with maximum tightness and transmitting power. A " quarter 
turn ■" wasted about 10;? as a maximum, and a "quarter twist" about 5%. 

Dimensions of Turbines.— For dimensions, power, etc., of stand- 
ard makes of turbines consult the catalogues of different manufacturers. 
The wheels of different makers vary greatly in their proportions for any 
given capacity. 

The Pelton Water-wheel.— Mr. Ross E. Browne (Eng'g News, Feb. 
20, 1892) thus outlines the principles upon which this water-wheel is 
constructed : 

The function of a water-wheel, operated by a jet of water escaping from 
a nozzle, is to convert the energy of the jet, due to its velocity, into useful 
work In order to utilize this energy fully the wheel-bucket, after catching 
the jet, must bring it to rest before discharging it, without inducing turbu- 
lence or agitation of the particles. 

This cannot be fully effected, and unavoidable difficulties necessitate the 
loss of a portion of the energy. The principal losses occur as follows: 
First, in sharp or angular diversion of the jet in entering, or in its course 
through the bucket, causing impact, or the conversion of a portion of the 
energy into heat instead of useful work. Second, in the so-called frictional 
resistance offered to the motion of the water by the wetted surfaces of the 
buckets, causing also the conversion of a portion of the energy into heat 
instead of useful work, Third, in the velocity of the water, as it leaves the 
bucket, representing energy which has not been converted into work. 

Hence, in seeking a high efficiency: 1. The bucket-surface at theentrance 
should be approximately parallel to the relative course of the jet, and 
the bucket should be curved in such 
a manner as to avoid sharp angular de- 
flection of the stream. If, for example, 
a jet strikes a surface at an angle and 
is sharply deflected, a portion of the 
water is backed, the smoothness of the 
stream is disturbed, and there results 
considerable loss by impact and other- 
wise. The entrance and deflection in 
the Pelton bucket are such as to avoid 
these losses in the main. (See Fig. 136.) 

2. The number of buckets should be small, and the path of the jet in the 
bucket short; in other words, the total wetted surface should be small, as 
the loss by friction will be proportional to this. 

3. The discharge end of the bucket should be as nearly tangential to the 
wheel periphery as compatible with the clearance of the bucket which 
follows; and great differences of velocity in the parts of the escaping water 
should be avoided. In order to bring the water to rest at the discharge end 
of the bucket, it is shown, mathematically, that the velocity of the bucket 
should be one half the velocity of the jet. 

A bucket, such as shown in Fig. 135, will cause the heaping of more or less 
dead or turbulent water at the point indicated by dark 
shading. This dead water is subsequently thrown from 
the wheel with considerable velocity, and represents a 
large loss of energy. The introduction of the wedge in 
the Pelton bucket (see Fig. 134) is an efficient means of 
avoiding this loss. 

A wheel of the form of the Pelton conforms closely in 
construction to each of these requirements. 
In a test made by the proprietors of the Idaho mine, 
Fig 136 near Grass Valley, Cal., the dimensions and results were 

as follows : Main supply-pipe, 22 in. diameter, 6900 ft. 
long, with a head of 386^ feet above centre of nozzle. The loss by friction 
in the pipe was 1.8 ft., reducing the effective head to 384.7 ft. The Pelton 
wheel used in the test was 6 ft. in diameter and the nozzle was 1.89 in. 
diameter. The work done was measured by a Prony brake, and the mean 
of 13 tests showed a useful effect of 87.3#. 

The Pelton wheel is also used as a motor for small powers. A test by 
M. E. Cooley of a 12-inch wheel, with a %-inch nozzle, under 100 lbs. pressure, 
gave 1.9 horse-power. The theoretical discharge was .0935 cubic feet per 
second, and the theoretical horse-power 2.45; the efficiency being 80 per 
cent. Two other styles of water-motor tested at the same time each gave 
efficiencies of 55 per cento 



Fig. 134. 



Fig. 135. 




598 



WATER-POWER. 



Pelton Water-wheel Tables. (Abridged.) 

The smaller figures uuder those denoting the various heads give the 

spouting velocity of the water in feet per minute. The cubic-feet measure- 
ment is also based on the flow per minute. 



Head 
in ft. 


Size of 
Wheels. 


6 

in 

No.l 

.05 

1.67 
684 


12 
in. 

No. 2 


18 
in. 
No. 3 


18 
in. 

No. 4 


24 

in. 

No. 5 


3 

ft. 


4 
ft. 


5 
ft. 


6 
ft. 


20 

2151.97 


Horse-power. 
Cubic feet.... 
Revolutions.. 


.12 
3.91 
342 


.20 
6.62 

228 


.37 

11.72 

228 


.66 

20.83 

171 


1 50 

46.93 

114 


2.64 
83.32 

85 


4.18 

130.36 

70 


6.00 

187.72 

57 


30 

2635.62 


Horse-power. 
Cubic feet .... 
Revolutions.. 


.10 
2.05 

837 


.23 
4.79 

418 


.38 
8.11 
279 


.69 
14.36 

279 


1.22 

25.51 
209 


2.76 

57.44 

139 


4.88 

102.04 

104 


7.69 

159.66 

83 


11.04 

229.76 

69 


40 

3043.39 


Horse-power. 
Cubic feet. . . . 
Revolutions.. 


.15 
2.37 
969 


.35 
5.53 

484 


.59 
9.37 
323 


1.06 

16.59 

323 


1.89 

29.46 

242 


4.24 

66.36 

161 


7.58 

107.84 

121 


11.85 

184.36 

96 


16.96 

265.44 

80 


50 

3402.61 


Horse-power. 
Cubic feet.... 
Revolutions.. 


.2! 

- 
1083 


.49 

6.18 
541 


.84 

10.47 

361 


1.49 

18.54 
361 


2.65 

32.93 

270 


5.98 

74.17 

180 


10.60 

131.72 

135 


16.63 

206.13 

108 


23.93 

296.70 

90 


60 


Horse-power. 

Cubic feet 

Revolutions.. 

Horse-power. 

Cubic feet 

Revolutions.. 


.28 

11 85 

.35 
3.13 

1281 


.65 
6.77 
592 

.82 
7.31 
640 


1.10 

11.47 
395 

1.39 

12.39 
427 


1.96 

20.31 

395 

2.47 

21.94 

427 


3.48 

36.08 

296 


7.84 

81.25 

197 


13.94 

144.32 

148 


21.77 

225.80 

118 


31.36 

325.00 

98 


70 

4026.00 


4.39 

38.97 

320 


9.88 

87.76 

213 


17.58 

155.88 

160 


27.51 

243.89 

130 


39.52 

351.04 

106 


80 
4303.99 


Horse-power. 

Cubic feet 

Revolutions.. 


.43 

3.35 
1308 


1.00 

7.82 
684 


1.70 

13 25 

456 


3.01 

23.46 

456 


5.36 

41.66 

342 


12.04 

93 84 

228 


21.44 

166.64 

171 


33.54 

260.73 

137 


48.16 

375.36 

114 


90 

4565.04 


Horse-power. 

Cubic feet 

Revolutions. . 


.51 
3.55 
1152 


1.20 
8.29 
726 


2.03 
14.05 

481 


3.60 

24.88 
484 


6.39 

44.19 

363 


14.40 

99.52 

242 


25.59 

176.75 

181 


40.04 

276.55 

145 


57.60 

398.08 

121 


100 

4812.00 


Horse-power. 

Cubic feet 

Revolutions.. 


.60 
3.74 
1530 


1.40 
8.74 
765 


2.32 

14.81 
510 


4.21 

26.22 
510 


7.49 

46.58 

382 


16.84 

104.88 

255 


29.93 

186.32 

191 


46.85 
291.51 

152 


67.36 
419.52 

127 


120 

5271.30 


Horse-power. 

Cubic feet 

Revolutions.. 

Horse-power. 
Cubic feet. .. 
Revolutions.. 


.79 
4.10 
1677 

.09 
4 . 43 
1812 


1 84 
9.57 

838 

2.33 

10.31 

906 


3.12 

16.21 

559 


5.54 

28.72 
559 


9.85 

51.02 

419 


22.18 

114.91 

279 


39.41 

204.10 

209 


61.66 

319.33 

167 


88.75 

459.64 

139 


140 

5693.65 


3.94 

17.53 

604 

4.82 

18.74 

646 


6.99 

31.03 

604 


12.41 

55.11 

453 


27.96 

124.12 

302 


49.64 

220.44 

226 


77.71 
344.92 

181 


111.85 

496.48 

151 


160 

6086 74 


Horse-power. 

Cubic feet 

Revolutions.. 


1.22 
4.73 
1938 


2.84 

1.1.05 

969 


8.54 

33.17 

646 


15.17 

58.92 

484 


34.16 

132.68 

323 


60.68 

235.68 

242 


94.94 

368.73 
193 


136.65 

530.75 

161 


180 

6455.97 


Horse power. 
Cubic feet.. . 
Revolutions. . 


1.45 


3.39 
11.72 
1024 


5.75 

19.87 

683 


10.19 

35.18 

683 


18.10 

62.49 

513 


40.77 

140.74 

342 


72.41 

249.97 

256 


113.30 

391.10 

206 


163.08 

562.96 

171 


200 

6805.17 


Horse-power. 

Cubic feet 

Revolutions.. 


1.70 
2100 


3.97 

12.36 
1080 


6.74 

20.94 

720 


11.93 

37.08 

720 


21.20 

65.87 

540 


47.75 

148.35 

360 


84.81 

263.49 

270 


132.70 

412 25 

216 


191.00 

593.40 

180 


250 

7608.44 


Horse power. 
Cubic feet. . . . 
Revolutions. . 




5.56 
13.82 
1209 


9.42 

23.42 
800 


16.68 

41.46 

806 


29.63 

73.64 

605 


66.74 

165.86 

403 


118.54 

291.59 

302 


185.4? 

460.91 

241 


266.96 

663.45 

202 



POWER OF OCEAN WAVES. 59§ 

Pelton Water-wlieel Tables.— Continued. 



Head 
in ft. 


Size of 
Wheels. 


in. 
No.l 


12 
in. 

No. 2 


18 
in. 
No. 3 


18 
in. 
No. 4 


24 
in. 

No. 5 


3 

ft. 


4 

ft. 1 


5 

ft. 


6 
ft. 


300 

8334.62 


Horse-pow'r 
Cubic feet... 
Revolutions 


3.13 
6.48 
2G52 


7.31 
15.13 
1326 


12.38 
25.66 

884 


21.93 

45.42 
884 


38. 95 1 87.73 

80.67 181.69 

663 j 442 


155.83 

322.71 

331 


243.82 

504.91 

265 


350.94 

726.76 

221 


350 

9002.43 


Horse-pow'r 
Cubic feet... 
Revolutions 


3.94 
7.00 


9.21 
16.35 
1432 


15.61 

27.71 

955 


27.64 

49.06 

955 


49.09 110.56 

87.14 196.25 

716 477 


196.38 

348.57 

358 


307.25 

545.36 

285 


442.27 

785.00 

238 


400 

9624.00 


Horse-pow'r 
Cubic feet... 
Revolutions 


4.82 

7.40 


11.25 
17.48 
1531 


19.0 
29.63 
1021 


33.77 
52.45 
1021 


59.98 

93.16 

765 


135.08 

209.80 

510 


239.94 

372.64 

382 


375.40 

583.02 

306 


540.35 

839.20 

255 


450 

10207.79 


Horse-pow'r 
Cubic feet... 
Revolutions 


5.75 
7.94 
3-219 


13.43 
18.54 
1624 


22.76 
31.42 
1083 


40.29 
55.63 
1083 


71.57 

98.81 

812 


161.19 

222.52 

541 

188.80 

234.56 

571 


286.31 

395.24 

406 


447.95 

618.38 

324 


644.78 

890.11 

270 


500 

10759.96 


Horse-pow'r 
Cubic feet... 
Revolutions 


6.74 

8.37 
3420 


15.73 
19.54 
1713 


26.66 
33.12 
1142 


47.20 
58.64 
1142 


83.83 

104.15 

856 


335.34 

416.62 

428 


524.66 

651.83 

342 


755.20 

938.25 

285 


600 


Horse-pow'r 








62.04 
64.24 
1251 


110.19 

114.09 

938 


248.16 

256.95 

625 


440.77 

456.38 

469 


689.63 

714.05 

375 


992.65 










1027.80 


11786.94 


Revolutions 




— '- 




312 


650 


Horse-pow'r 






69.95 
66.86 
1302 


124.25 

118.75 

976 


279.82 

267.44 

651 


497.01 
475.02 

488 


777.62 

743.21 
390 


1119.29 










1069.77 


12268.24 


Revolutions 








325 


700 


Horse-pow'r 








78.18 
69.38 
1351 


138.86 

123.23 

1013 


675 


555.46 

492.95 

506 


869.06 

771.26 

405 


1250.92 










1110.16 


12731.34 


Revolutions 








337 


750 

13178.19 


Horse-pow'r 
Cubic feet... 
Revolutions 








86.70 
71.82 
1399 


154.00 

127.56 

1049 


346.83 
- 
699 


616.03 

510.25 

524 


963 . 82 

798.33 

419 


1387.34 

1149.13 

319 


800 

13610.40 


Horse-pow'r 
Cubic feet... 
Revolutions 








95.52 
74.17 
1444 


169.66 

131.74 

1083 


382.09 

296.70 

722 


678.66 

526.99 

542 


1061.81 

824.51 

433 


1528.36 

1186.81 

361 












900 


Horse-pow'r 
Cubic feet... 
Revolutions 








113.98 
78.67 
1532 


202.45 

139.74 

1149 


455.94 

314.70 

766 


809.82 

558.96 

574 


1267.02 

874.53 

459 


1823.76 


14436.00 






11111 


1258.81 
383 


1000 


Horse-pow'i 






133 50 


237.12 

147.30 

1210 


534.01 
80? 


948.48 

589.19 

605 


1483.9? 
921 . S3 


2136 04 










82.93 
1615 


132(i.91 


15216.89 


Revolutions 








484' 403 



THE POWER OF OCEAN WAVES. 

Albert W. Stahl, U. S. N. (Trans. A. S. M. E., xiii. 438), gives the following 
formulae and table, based upon a theoretical discussion of wave motion: 

The total energy of one whole wave-length of a wave iJfeet high, L feet 
long, and one foot in breadth, the length being the distance between succes- 
sive crests, and the height the vertical distance between the crest and the 

trough, is E= 8LH* (l - 4.935 ^) foot-pounds. 

The time required for each wave to travel through a distance equal to its 

own length is P = a/ r— ^~ seconds, and the number of waves passing any 



600 



WATER-POWER. 



60 



given point in one minute is N 

of an indefinite series of such waves, expressed in horse-power per foot of 
breadth, is 



04/ —7— • Hence the total energy 



T~L(\ 



By substituting various values for H -f- L, within the limits of such values 
actually occurring in nature, we obtain the following table of 

Total Energy of Deep-sea Waves in Terms op Horse-power per Foot 
op Breadth. 



Ratio of 
Length of 
Waves to 






Length of Waves in Feet. 




















Height of 
Waves. 


25 


50 


75 


100 


150 


200 


[300 


400 


50 


.04 


.23 


.64 


1.31 


3.62 


7.43 


20.46 


42.01 


40 


.06 


.36 


1.00 


2.05 


5.65 


11.59 


31.95 


65.58 


30 


.12 


.64 


1.77 


3.64 


10.02 


20.57 


56.70 


116.38 


20 


.25 


1.44 


3.96 


8.13 


21 79 


45.98 


120.70 


260.08 


15 


.42 


2.83 


6.97 


14.31 


39.43 


80.94 


223.06 


457.89 


10 


.98 


5.53 


15.24 


31.29 


86.22 


177.00 


487.75 


1001.25 


5 


3.30 


18.68 


51 48 


105.68 


291.20 


597.78 


1647.31 


3381.60 



The figures are correct for trochoidal deep-sea waves only, but they give 
a close approximation for any nearly regular series of waves in deep water 
and a fair approximation for waves in shallow water. 

The question of the practical utilization of the energy which exists in 
ocean waves divides itself into several parts : 

fe 1. The various motions of the water which may be utilized for power 
purposes. 

2. The wave motor proper. That is, the portion of the apparatus in direct 
contact with the water, and receiving and transmitting the energy thereof ; 
together with the mechanism for transmitting this energy to the machinery 
for utilizing the same. 

C. Regulating devices, for obtaining a uniform motion from the irregular 
and more or less spasmodic action of the waves, as well as for adjusting the 
apparatus to the state of the tide and condition of the sea. 

4. Storage arrangements for insuring a continuous and uniform output of 
power during a calm, or when the waves are comparatively small. 

The motions that may be utilized for power purposes are the following: 
1. Vertical rise and fall of particles at and near the surface. 2. Horizontal 
to-and-fro motion of particles at and near the surface. 3. Varying slope of 
surface of wave. 4. Impetus of waves rolling up the beach in the form of 
breakers. 5. Motion of distorted verticals. All of these motions, except the 
last one mentioned, have at various times been proposed to be utilized for 
power purposes; and the last is proposed to be used in apparatus described 
by Mr. Stahl. 

The motion of distorted verticals is thus defined: A set of particles, origi- 
nally in the same vertical straight line when the water is at rest, does not 
remain in a vertical line during the passage of the wave; so that the line 
connecting a set of such particles, while vertical and straight in still water, 
becomes distorted, as well as displaced, during the passage of the wave, its 
upper portion moving farther and more rapidly than its lower portion. 

Mr. StahPs paper contains illustrations of several wave-motors designed 
upon various principles. His conclusions as to their practicability is as fol- 
lows: " Possibly none of the methods described in this paper may ever prove 
commercially successful; indeed the problem may not be susceptible of a 
financially successful solution. My own investigations, however, so far as I 
have yet been able to carry them, incline me to the belief that wave-power 
can and will be utilized on a paying basis." 

Continuous Utilization of Tidal Power. (P. Decoeur, Proc. 
Inst. C. E. 1890.)— In connection with the training-walls to be constructed in 



PUMPS AND PUMPING ENGINES. 601 

the estuary of the Seine, it is proposed to construct large basins, by means 
of which the power available from the rise and fall of the tide could be util- 
ized. The method proposed is to have two basins separated by a bank rising 
above high water, within which turbines would be placed. The upper basin 
would be in communication with the sea during the higher one third of the 
tidal range, rising, and the lower basin during the lower one third of the 
tidal range, falling. If H be the range in feet, the level in the upper 
basin would never fall below %H measured from low water, and the 
level in the lower basin would never rise above X /&H. The available head 
varies between 0.53Hand 0.80H, the mean value being %H. If S square feet 
be the area of the lower basin, and the above conditions are fulfilled, a 
quantity l/'dSH cu. ft. of water is delivered through the turbines in the space 
of 9*4 hours. The mean flow is, therefore, SH -f- 99,900 cu. ft. per sec , and, 
the mean fall being ?£H, the available gross horse-power is about l/306"i? 2 , 
where S' is measured in acres. This might be increased by about one third 
if a variation of level in the basins amounting to %H were permitted. But 
to reach this end the number of turbines would have to be doubled, the 
mean head being reduced to J^i?, and it would be more difficult to transmit 
a constant power from the turbines. The turbine proposed is of an improved 
model designed to utilize a large flow with a moderate diameter. One has 
been designed to produce 300 horse-power, with a minimum head of 5 ft. 3 
in. at a speed of 15 revolutions per minute, the vanes having 13 ft. internal 
diameter. The speed would be maintained constant by regulating sluices. 



PUMPS AND PUMPING ENGINES. 

Theoretical Capacity of a Pump.— Let Q' = cu. ft. per min.; 

G' = Amer. gals, per min. = 7.4805(5'; d = diam. of pump in inches; I — 
stroke in inches; N = number of single strokes per min. 

77- d 2 IN 
Capacity in cu. ft. per min. = Q' = - ■ — — - . -- - .0004545iVd 2 Z; 

,7 NdW 
Capacity in gals, per min. G' = - . —— = .OOMNdH; 

Capacity in gals, per hour = .204iVd 2 Z. 

Diameter required for ^^^ — Aaq./Q^ _ ir- -.c . / & 
given capacity per min. \ 1/ jVZ 1/ jyi ' 

If v — piston speed in feet per min., d = 13.54 j/ jL == 4.95a / . 

If the piston speed is 100 feet per min.: 

Nl = 1200, and d = 1.354 V~Q' = .495 V~G'\ G' = 4.08cZ 2 per min. 

The actual capacity will be from 60% to 95% of the theoretical, according to 
the tightness of the piston, valves, suction-pipe, etc. 

Theoretical Horse-power required to raise "Water to a 
given Height.— Horse-power == 

Volume in cu. ft. per min. X pressure per sq. ft. _ Weight x height of lift 
33,000 ~ " 33,000 ' 

Q' == cu. ft. per min.; G' — gals, per min.; W — wt. in lbs.; P = pressure 
in lbs. per sq. ft.; p — pressure in lbs. per sq. in.; H = height of lift in ft.: 
W- 62.36Q', P= U4p,p = .433iJ, H = 2.309p, G' = 7.4805Q'. 



Q'P _ 
33,000 


Q'H X 144 X .433 Q'H G'H _ 
33,000 ~ 529.2 ~ 3958.7' 


WH 

33,000 ~ 


Q' X 62.36 X 2.309p Q'p G'p 
33,000 ~ 229.2 ~ 1714.5' 



For the actual horse-power require/1 an allowance must be made for the 
friction, slips, etc., of engine, pump, valves, and passages, 



602 



WATER-POWER. 



Depth of Suction.— Theoretically a perfect pump will draw water 
from a height of nearly 34 feet, or the height corresponding to a perfect 
vacuum (14.7 lbs. X 2.309 = 33.95 feet); but since a perfect vacuum cannot be 
obtained, on account of valve-leakage, air contained in the water, and the 
vapor of the water itself, the actual height is generally less than 30 feet. 
When the water is warm the height to which it can be lifted by suction de- 
creases, on account of the increased pressure of the vapor. In pumping hot 
water, therefore, the water must flow into the pump by gravity. The fol- 
lowing table shows the theoretical maximum depth of suction for different 
temperatures, leakage not considered: 



Temp. 
F. 


Absolute 
Pressure 
ofVapor, 
lbs. per 
sq. in. 


Vacuum 

in 
Inches of 
Mercury. 


Max. 

Depth 

of 

Suction, 

feet. 


Temp. 
F. 


Absolute 
Pressure 
ot Vapor, 
lbs. per 
sq. in. 


Vacuum 

in 
Inches of 
Mercury. 


Max. 

Depth 

of 

Suction, 

feet. 


101.4 
126.2 
144.7 
153.3 
162.5 
170.3 
177.0 


1 

2 
3 
4 
5 
6 
7 


27.88 
25.85 
23.81 
21.77 
19.74 
17.70 
15.66 


31.6 
29.3 
27.0 
24.7 
22.4 
20.1 
17.8 


183.0 

188.4 
193.2 
197.6 
201.9 
205.8 
209.6 


8 
9 
10 
11 
12 
13 
14 


13.63 
11.59 
9.55 
7.51 
5.48 
3.44 
1.40 


15.5 
13.2 
10.9 
8.5 
6.2 
3.9 
1.6 



Amount of Water raised by a Single-acting Lift-pump. 

—It is common to estimate that the quantity of water raised b}' a 
single-acting bucket-valve pump per minute is equal to the number of 
strokes in one direction per minute, multiplied by the volume traversed by 
the piston in a single stroke, on the theory that the water rises in the pump 
only when the piston or bucket ascends; but the fact is that the column of 
water does not cease flowing when the bucket descends, but flows on con- 
tinuously through the valve in the bucket, so that the discharge of the 
pump, if it is operated at a high speed, may amount to nearly double that 
calculated from the displacement multiplied by the number of single strokes 
in one direction. 

Proportioning the Steam-cylinder of a Direct-acting 
Pump.— Let 

A — area of steam-cylinder; a — area of pump-cylinder; 

D = diameter of steam-cylinder; d = diameter of pump-cylinder; 

P = steam-pressure, lbs. per sq. in. ; p = resistance per sq. in. on pumps; 

H= head = 2.309p; p = ASSH: 

_, „ . . .., work done in pump-cylinder 

E — efficiency of the pump = ■ r—j , ^. „ — t^-t- • 

work done by the steam-cyhnder 



•^ — T7TD 



a}) _ 
EP' 



EAP 



p y EP' \ p EA 



EAP 



H = 2.B09EP — ; If E = 75*, H = 1 .732P - 



A - JL - - 433g . 
a~ EP ~ EP ' 

E is commonly taken at 0.7 to 0.8 for ordinary direct-acting pumps. For 
the highest class of pumping-engines it may amount to 0.9. The steam- 
pressure Pis the mean effective pressure, according to the indicator-dia- 
gram; the water-pressure p is the mean total pressure acting on the pump 
plunger or piston, including the suction, as could be shown by an indicator- 
diagram of the water-cylinder. The pressure on the pump-piston is fre- 
quently much greater than that due to the height of the lift, on account of 
the friction of the valves and passages, which increases rapidly with velocity 
of flow. 

Speed of Water through Pipes and Pump-passages.— 
The speed of the water is commonly from 100 to 200 feet per minute. If 200 
feet per minute is exceeded, the loss from friction may be considerable. 



The diameter of pipe required is 4. 



_ / gallons per minute 
\ velocity in feet per minute' 



For a velocity of 200 feet per minute, diameter =,35 x ^'gallons per roiq. 



PUMPS. 



003 



Sizes of Direct-acting Pumps.— The two following tables are se- 
lected from catalogues of manufacturers, as representing the two common 
types of direct-acting pump, viz., the single-cylinder and the duplex. 
Both types are now made by most of the leading manufacturers. 

The Deane Direct-acting Pump. 

Standard Sizes for Ordinary Service. 



a 

| = 

55 - 


£ 


6 


6 


o3 






fl 


a 


ft 


M 








o 


p 


: S 


Capacity 
per Minute 


■B 

"5) 

B en 


■S . 


S 

a a? 


Ift 


o 


ho 


O Z 


■- 


a 


CD 


a5 


at G 


lven 


32 


> 03 
03 O 
03 


cS ft 

5 -a 

oft 


.2 ft 


a 


w 




■- 1. 
i ■- 


°.a 
a 


ft 

O 

"5 


ft 

03 

o 


Speed. 
Stks. Gals. 


os 
u 


cc« 


w 
o 

0) 


5 

o 


Q 


5 


*3 


O 


OQ 






S 


H 


w 


w 


33 


w 


4 


3i o 


5 


.14 


1 to 300 


130 


18 


33 


9k 


k 


Va 


2 


Ik 


4 


4 ~ 


5 


.27 


1 to 300 


130 


35 


33 


9k 


k 


% 


2 


ik 


5 


4 


7 


.39 


1 to 300 


125 


49 


45k 


15 


H 




3 


2k 


m 


5 


7 


.51 


1 to 275 


125 


64 


45k 


15 


% 




3 


2k 


5Va 


m 


7 


.72 


1 to 275 


125 


90 


45k 


15 


H 




3 


2k 


7 




10 


1.64 


1 to 250 


110 


180 


58 


17 


l 


Ik 


5 


4 


7k 


7L> 


10 


1.91 


1 to 250 


110 


210 


58 


17 


l 


Ik 


5 


4 


<k 


8 


10 


2.17 


1 to 250 


110 


239 


58 


17 


l 


Ik 


5 


4 


8 


6 


V2 


1.47 


1 to 250 


100 


147 


67 


20k 


l 


m 


4 


4 


8 


7 


12 


2.00 


1 to 250 


100 


200 


67 


-0k 


l 


ik 


5 


4 


8 


8 


12 


2.61 


1 to 250 


100 


261 


68 


30 


l 


Ik 


5 


5 


8 


10 


12 


4.08 


1 to 250 


100 


408 


68 


20k 


l 


Ik 


8 


8 


10 


8 


12 


2.61 


1 to 250 


100 


261 


68k 


30 


Ik 


2 


5 


5 


10 


10 


12 


4.08 


1 to 250 


100 


408 


68k 


30 


Ik 


2 


8 


8 


10 


12 


12 


5.87 


1 to 250 


100 


587 


68k 


30 


Ik 


2 


8 


8 


12 


10 


12 


4.08 


1 to 250 


100 


408 


64 


24 


2 


2k 


8 


8 


12 


10 


18 


6.12 


1 to 200 


70 


428 


68k 


30 


2 


2k 


8 


8 


12 


12 


12 


5.87 


1 to 250 


100 


587 


64 


28k 


2 


2k 


8 


8 


13 


12 


18 


8.80 


1 to 175 


70 


616 


88 


28^ 


o 


2V 2 


8 


8 


12 


14 


18 


12.00 


1 to 175 


70 


840 


88 


28k 


2 


2Va 


8 


8 


14 


10 


12 


4.08 


1 to 250 


100 


408 


69 


30 


2 


2k 


8 


8 


14 


10 


18 


6.12 


1 to 175 


70 


428 


93 


25 


2 


^2 


8 


8 


14 


10 


24 


8.16 


1 to 150 


50 


408 


112 


26 


2 


2k 


8 


8 


14 


12 


12 


5.8^ 


1 to 250 


100 


587 


69 


30 


2 


2k 


8 


8 


14 


12 


18 


8.80 


1 to 175 


70 


616 


88 


28k 


2 


2k 


8 


8 


14 


12 


24 


11.75 


1 to 150 


50 


587 


112 


26 


2 


2Y> 


10 


8 


14 


14 


24 


15.99 


1 to 150 


50 


800 


112 


34 


2 


2k 


12 


10 


14 


16 


16 


13.92 


1 to 175 


80 


1114 


84 


34 


2 


2k 

2k 


12 


10 


14 


16 


24 


20.88 


1 to 150 


50 


1044 


112 


38 


2 


12 


10 


16 


14 


18 


12.00 


1 to 175 


70 


840 


89 


27 


2 


2k 


8 


8 


16 


14 


24 


15.99 


1 to 150 


50 


800 


109 


34 


2 


2k 


12 


10 


16 


16 


16 


13.92 


1 to 175 


80 


1114 


85 


34 


2 


2k 


12 


10 


16 


1Q 


24 


20.88 


1 to 150 


50 


1044 


115 


34 


2 


2k 


12 


10 


16 


18 


24 


26.43 


1 to 125 


50 


1322 


115 


40 


2 


2k 


14 


12 


18 


16 


24 


20.88 


1 to 125 


50 


1044 


118 


38 


3 


3k 


12 


10 


18 


18 


24 


26.43 


1 to 125 


50 


1322 


118 


40 


3 


3k 


14 


12 


18 


20 


24 


32.64 


1 to 125 


50 


1632 


118 


40 


3 


3k 


16 


14 


20 


18 


24 


26.43 


1 to 125 


50 


1322 


118 


40 


3 


3k 


14 


12 


20 


20 


24 


32.64 


1 to 125 


50 


1632 


118 


40 


3 


3k 


16 


14 


20 


22 


24 


39.50 


1 to 125 


50 


1975 


120 


40 


3 


3k 


18 


14 



Efficiency of Small Direct-acting Pumps.— Chas. E. Emery, in 
Reports of Judges of Philadelphia Exhibition, 1876, Group xx., says : "Ex- 
periments made with steam-pumps at the American Institute Exhibition of 
1867 showed that average sized steam-pumps do not, on the average, utilize 
more than 50 per cent of the indicated power in the steam-cylinders, the re- 
mainder being absorbed in the friction of the engine, but more particularly 
in the passage of the water through the pump. Again, all ordinary steam- 
pumps for miscellaneous uses require that the steam -cylinder shall have 
three to four times the area of the water-cylinder to give sufficient power 



604 



WATER-POWER. 



when the steam is accidentally low; hence as such pumps usually work 
against the atmospheric pressure, the net or effective pressure forms a 
small percentage of the total pressure, which, with the large extent of 
radiating surface exposed and the total absence of expansion, makes the 
expenditure of steam ve\y large. One pump tested required 120 pounds 
weight of steam per indicated horse-power per hour, and it is believed that 
the cost will rarely fall below 60 pounds ; and as only 50 per cent of the in- 
dicated power is utilized, it may be safely stated that ordinary steam-pumps 
rarely require less than 120 pounds of steam per hour for each horse-power 
utilized in raising water, equivalent to a duty of only 15,000,000 foot-pounds 
per 100 pounds of coal. With larger steam-pumps, particularly when they 
are proportioned for the work to be done, the duty will be materially in- 
creased. " 

The Worthlngton Duplex Pump. 

Standard Sizes for Ordinary Service. 













ki 


=1 . 


Sizes of Pipes for 










<H,a 


rO fi 


•'-"CO 

t) ® 


Short Lengths. 


u 






u 


o.-B 


l£ 


0»+J 0) 


To be increased as 


35 

O 


S 
ft 




a be 


§31 


3_ 

53 * 


•5 P-cr 1 

ESS 


length 


increases. 


O 










i 

o 
u 


O 


V 

M 
O 




ZO 

0<M 

<L> 
S <D 

1! 


Jr, OS'S 

is 3 ^ 

^£0 


°*5 

<p g xh 
S ® $ 

> to-s 
8*0 


g = a 
2 1* 


ft 
ft 

S 

0> 


ft 
'ft 

| 


© 
ft 
"ft 

a 



a 
'ft 

5 


I 

03 


g 

c3 


tx 
c 

ID 


fao 




-S'3'fe 

c3^^ 


.2 ss 


oj 

-f 


3 


£ 


3 


s 


J 


5 


ft 





q 






CO 


ft 


3 


2 


3 


.04 


100 to 250 


8 to 20 


W/a 


1J4 


l 


*H 


Wa 


4 


.10 


100 to 200 


20 to 40 


4 


Vz 


i J i 


2 


i« 


514 


Wz 


5 


.20 


100 to 200 


40 to 80 


5 




2^ 


m 


6 


4 


6 


.33 


100 to 150 


70 to 100 


5% 


1 




3 


2 


W% 


Q& 


G 


.42 


100 to 150 


85 to 125 


6% 


m 


2 


4 


3 


TA 


5 


G 


.51 


100 to 150 


100 to 150 


7 


n/ 2 


2 


4 


3 


w* 


4^ 


10 


.69 


75 to 125 


100 to 170 


6% 




2 


4 


3 


9 


5'/4 


10 


.93 


75 to 125 


135 to 230 


7^ 


■i 


^% 


4 


3 


10 


6 


10 


1.22 


75 to 125 


180 to 300 


8fc& 


2 




5 


4 


10 


7 


10 


1.66 


75 to 135 


245 to 410 


9% 


2 


2^ 


6 


5 


12 


7 


10 


1.66 


75 to 125 


245 to 410 


9% 


2J^ 


3 


6 


5 


14 


7 


10 


1.66 


75 to 125 


215 to 410 


9% 




3 


6 


5 


12 


Wz 


10 


2.45 


75 to 125 


365 to 610 


12 




3 


6 


5 


14 


m 


10 


2.45 


75 to 125 


365 to 610 


12 




3 


6 


5 


16 


8^2 


10 


2.45 


75 to 125 


365 to 610 


12 




3 


6 


5 


18)4 


32 


10 


2.45 


75 to 125 


365 to 610 


12 


3 


o% 


6 


5 


20 


10 


2.45 


75 to 125 


365 to 610 


12 


4 


5 


6 


5 


12 


1014 


10 


3.57 


75 to 125 


530 to 890 


1414 


•2 l/o 


3 


8 


7 


14 


1014 


10 


3.57 


75 to 125 


530 to 890 


1414 




3 


8 


7 


16 


1014 
10& 


10 


3.57 


75 to 125 


530 to 890 


14J4 




3 


8 


7 


18V6 


10 


3.57 


75 to 125 


530 to 890 


14J4 


3 


31 -a 


8 


7 


20 


io# 


10 


3.57 


75 to 125 


530 to 890 


14J4 


4 


5 


8 


7 


14 


12 


10 


4.89 


75 to 125 


730 to 1220 


17 




3 


10 


8 


16 


12 


10 


4.89 


75 to 125 


730 to 1220 


17 




3 


10 


8 


18^ 


12 


10 


4.89 


75 to 125 


730 to 1220 


17 


3 


3^ 


10 


8 


20 


12 


10 


4.89 


75 to 125 


730 to 1220 


17 


4 


5 


10 


8 


\&y 2 


14 


10 


6.66 


75 to 125 


990 to 1660 


19% 


3 




12 


10 


20 


14 


10 


6.66 


75 to 122 


990 to 1660 


1 


5 


12 


10 


17 


10 


15 


5.10 


50 to 100 


510 to 1020 


14 


3 




10 


8 


20 


12 


15 


7.34 


50 to 100 


730 to 1460 


17 


4 


5 


12 


10 


20 


15 


15 


11.47 


50 to 100 


1145 to 2290 


21 










25 


15 


15 


11.47 


50 to 100 


1145 to 2290 


21 





















PUMPS. 



605 



Speed of Piston.— A piston speed of 100 feet per minute is commonly 
assumed as correct in practice, but for short-stroke pumps this gives too 
high a speed of rotation, requiring too frequent a reversal of the valves.' 
For long stroke pumps, 3 feet and upward, this speed may be considerably 
exceeded, if valves and passages are of ample area. 

Number of Strokes required to Attain a Piston Speed 

from 50 to 125 Feet per Minute for Pumps having 

Strokes from 3 to 18 Inclies in Length. 



s! . 


Length of Stroke in Inches. 


<M - 

73 „ r 


3 


* 


5 | 6 


» 


8 


» 


12 


» 


» 


$> 9 & 
&5 p. 


Number of Strokes per Minute. 


50 


200 


150 


120 


100 


86 


75 


60 


50 


40 


33 


55 


220 


165 


132 


110 


94 


82.5 


66 


55 


44 


37 


60 


240 


180 


144 


120 


103 


90 


72 


60 


48 


40 


65 


260 


195 


156 


130 


111 


97.5 


78 


65 


52 


43 


70 


280 


210 


168 


140 


120 


105 


84 


70 


56 


47 


75 


300 


225 


180 


150 


128 


112.5 


90 


75 


60 


50 


80 


320 


240 


192 


160 


137 


120 


96 


80 


64 


53 


85 


340 


255 


204 


170 


146 


127.5 


102 


85 


68 


57 


90 


360 


270 


216 


180 


154 


135 


108 


90 


72 


60 


95 


380 


285 


228 


190 


163 


142.5 


114 


95 


76 


63 


100 


400 


300 


240 


200 


171 


150 


120 


100 


80 


67 


105 


420 


315 


252 


210 


180 


157.5 


126 


105 


84 


70 


110 


440 


330 


264 


220 


188 


165 


132 


110 


88 


73 


115 


460 


345 


276 


230 


197 


172.5 


138 


115 


92 


77 


120 


480 


360 


288 


,240 


206 


180 


144 


120 


96 


80 


125 


500 


375 


300 


250 


214 


187.5 


150 


125 


100 


83 



Piston Speed of Pumping-engines. (John Birkinbine, Trans. 
A. I. M. E., v. 459.)— In dealing with such a ponderous and unyielding sub- 
stance as water there are many difficulties to overcome in making a pump 
work with a high piston speed. The attainment of moderately high speed 
is, however, easily accomplished. Well-proportioned pumping-engines of 
large capacity, provided with ample water-ways and properly constructed 
valves, are operated successfully against heavy pressures at a speed of 250 ft. 
per minute, without "thug.'" concussion, or injury to the apparatus, and 
there is no doubt that the speed can be still further increased. 

Speed of Water through Valves.— If areas through valves and 
water passages are sufficient to give a velocity of 250 ft. per min. or less, 
they are ample. The water should be carefully guided and not too abruptly 
deflected. (F. W. Dean. Eng. News, Aug. 10, 1893.) 

Boiler-feed Pumps.— Practice has shown that 100 ft. of piston speed 
per minute is the limit, if excessive wear and tear is to be avoided. 

The velocity of water through the suction-pipe must not exceed 200 ft. 
per minute, else the resistance of the suction is too great. 

The approximate size of suction-pipe, where the length does not exceed 
25 ft. and there are not more than two elbows, may be found as follows : 

7/10 of the diameter of the cylinder multiplied by 1/100 of the piston speed 
in feet. For duplex pumps of small size, a pipe one size larger is usually 
employed. The velocity of flow in the discharge-pipe should not exceed 
500 ft. per minute. The volume of discharge and length of pipe vary so 
greatly in different installations that where the water is to be forced more 
than 50 ft. the size of discharge-pipe should be calculated for the particular 
conditions, allowing no greater velocity than 500 ft. per minute. The size of 
discharge-pipe is calculated in single-cylinder pumps from 250 to 400 ft. per 
minute. Greater velocity is permitted in the larger pipes. , 

In determining the proper size of pump for a steam-boiler, allowances 
must be made for a supply of water sufficient to cover all the demands of 
engines, steam-heating, etc., up to the capacity of generator, and should not 
be calculated simply according to the requirements of the engine. In prac- 
tice engines use all the way from 12 up to 50, or more, pounds of steam per 
H.P. per hour when being worked up to capacity. When an engine is over- 
loaded or underloaded more water per H.P. will be required than when 
operating at its rated capacity. The average run of horizontal tubular 



606 



WATER-POWER. 



boilers will evaporate from 2 to 3 lbs. of water per sq. ft. of heating-surface 
per hour, but may be driven up to 6 lbs. if the grate-surface is too large or 
the draught too great for economical working. 

Pump- Valves.— A. F. Nagle (Trans. A. S. M. E., x. 521) gives a number 
of designs with dimensions of double-beat or Cornish valves used in large 
pumping-engines, with a discussion of the theory of their proportions. The 
following is a summary of the proportions of the valves described. 

Summary of Valve Proportions. 



Location of Engine. 



Providence high-ser- 
vice engine 



Providence Cornish- 
engine 

St. Louis Water Wks, 

Milwaukee " " 

Chicago " " 



wood seats , 

Chicago Water Wks 









1 lb. 
reduced to 
.66 lb. 



1.41 
1.31 



1.16 

.96 



®|5 $ 


a r 


CO «D »- 


3 o* 


73 < 








sw J£t3 


Hi h 






»-i a 


3 a ce 


o a 




o* 3 «8 


CO c«5 




£^.S 


16% 


377 lbs. 


12 


680 


67 


250 


88 


120 


75 


151 


85 


140 


94 


132 


75 


151 



Good 

Some noise 

Some noise at 

high speed. 

Noisy 



Mr. Nagle says : There is one feature in which the Cornish valves are 
necessarily defective, namely, the lift must always be quite large, unless great 
power is sacrificed to reduce it. It is undeniable that a small lift is prefer- 
able to a great one, and hence it naturally leads to the substitution of 
numerous small valves for one or several large ones. To what extreme re- 
duction of size this view might safely lead must be left to the judgment of 
the engineer for the particular case in hand, but certainly, theoretically, we 
must adopt small valves. Mr. Corliss at one time carried the theory so 
far as to make them only 1% inches in diameter, but from 3 to 4 inches is 
the more common practice now. A small valve presents proportionately a 
larger surface of discharge with the same lift than a larger valve, so that 
whatever the total area of valve-seat opening, its full contents can be dis- 
charged with less lift through numerous small valves than with one large 
one. 

Henry R. Worthington was the first to use numerous small rubber valves 
in preference to the larger metal valves. These valves work well under all 
the conditions of a city pumping-engine. A volute spring is generally used 
to limit the rise of the valve. 

In theLeavitt high-duty sewerage-engine at Boston (Am. Machinist. May 
31, 1884), the valves are of rubber, %-inch thick, the opening in valve-seat 
being 13^ x 4^ inches. The valves have iron face and back-plates, and 
form their own hinges. 

CENTRIFUGAL PUMPS. 

Relation of Height of Iiift to Velocity.— The height of lift 
depepds only on the tangential velocity of the circumference, every tangen- 
tial velocity giving a constant height of lift— sometimes termed "head "— 
whether the pump is small or large. The quantity of water discharged is in 
proportion to the area of the discharging orifices at the circumference, or in 
proportion to the square of the diameter, when the breadth is kept the same. 
R. H. Buel (App. Cyc. Mech., ii, 606) gives the following: 

Let Q represent the quantity of water, in cubic feet, to be pumped per 
minute, h the height of suction in feet, h' the height of discharge in feet, and 
d the diameter of suction-pipe, equal to the diameter of discharge-pipe, in 



CENTRIFUGAL PUMPS. 



f V'2g 



607 



, g being the accel- 



feet; then, accordingto Fink, d = .-„- 

f \2g (h + h') 

eration due to gravity. 

If the suction takes place on one side of the wheel, the inside diameter of 
the wheel is equal to 1 .2d, and the outside to 2 Ad. If the suction takes place 
at both sides of the wheel, the inside diameter of the wheel is equal to 0.85d, 
and the outside to 1.7d. Then the suction-pipe will have two branches, the 
area of each equal to half the area of d. The suction-pipe should be as short 
as possible, to prevent air from entering the pump. The tangenti al velocity 
of the outer edge of wheel for the delivery Q is equal to 1.25 \'2y{h -\-h') 
feet per second. 

The arms are six in number, constructed as follows : Divide the central 
angle of 60°, which incloses the outer edges of the two arms, into any num- 
ber of equal parts by dividing the radii, and divide the breadth of the wheel 
in the same manner by drawing concentric circles. The intersections of the 
several radii with the corresponding circles give points of the arm. 

In experiments with AppokTs pump, a velocity of circumference of 500 
ft. per min. raised the water 1 ft. high, and maintained it at that level 
without discharging any; and double the velocity raised the water to four 
times the height, as the centrifugal force was proportionate to the square 
of the velocity; consequently, 

500 ft. per min. raised the water 1 ft. without discharge. 
1000 " lv " " •' 4 " 

2000 " " " " " 16 " 

4000 " " " " " 64 " 

The greatest height to which the water had been raised without discharge, 
in the experiments with the 1-ft. pump, was 67.7 ft., with a velocity of 4153 
ft. per min., being rather less than the calculated height, owing probably to 
leakage with the greater pressure. A velocity of 1128 ft. per min. raised the 
water 5}^ ft. without any discharge, and the maximum effect from the 
j^ower employed in raising to the same height 5J^ ft. was obtained at the 
"* velocity of 1678 ft. per min., giving a discharge of 1400 gals, per min. from 
the 1-ft. pump. The additional velocity required to effect a discharge of 
1400 gals, per min., through a 1-ft. pump working at a dead level without any 
height of lift, is 550 ft. per min. Consequently, adding this number in each 
case to the velocity given above, at which no discharge takes place, the fol- 
lowing velocities are obtained for the maximum effect to be produced in 
each case : 

1050 ft. per min., velocity for 1 ft. height of lift. 
1550 " '' " " 4 " 

2550 " " " " 16 " " " 

4550 " " " " 64 " " " 

Or, in general terms, the velocity in feet per minute for the circumference 
of the pum p to be driven, to rai se the water to a certain height, is equal to 
550 -f 500 Vheight of lift in feet. 

Lawrence Centrifugal Pumps, Class B— For Lifts from 
15 to 35 ft. 





Size of Pipes. 


Economical 


Total 


Horse-power 








Capacity, 


Capacity, 


per Ft. Lift, 






Dis- 
charge. 


in gallons 


in gallons 


for smaller 




Suction. 


per min. 


per min. 


quantity. 


No. iy % 


2 in. 


l^in. 


20 to 50 


150 


.024 


" 2 


2V, 


2 


60 to 80 


300 


.035 


" 3 


w% 


3 


80 to 160 


650 


.055 


" 4 


V/z 


4 


160 to 350 


1,250 


.075 


" 5 


6 


5 


330 to 600 


1,850 


.175 


" 6 


6 


6 


500 to 900 


2,600 


.22 


" 8 


8 


8 


1,100 to 2,000 


4,750 


.45 


" 10 


10 


10 


1,600 to 3,000 


7,500 


.62 


" 12 


12 


12 


2,000 to 3,000 


10,000 


1.00 


" 14 


14 


14 


3,000 to 5,000 


14,000 


1.25 


" 15 


15 


15 


3,500 to 7,000 


16,000 


1.40 


" 18 


18 


18 


6.000 to 11,000 


22,000 


2.40 



608 



WATER-POWER. 



Table of Diameters and Width of Pulley s. Width of Belts, 
and Number of Revolutions per Minute Necessary to 
raise Minimum Quantity of Water to Different Heights 
with Different Sizes of Pumps of Class B. 





u 

So 




«M 




Height 


in Feet and Revolutions per 




dJ 


■S 3 


o 








Minute. 








o a 


33 


6 


8 


10 


12 


16 


20 


25 


30 


35 


6 § 


Ins. 


Ins. 


Ins. 


Ins. 
























v& 


5 


5 


3 


40 


465 


515 


560 


605 


6S0 


745 


820 


885 


945 


1M 


2 


5 


5 


4 


60 


425 


475 


515 


560 


625 




750 


810 


870 


2 


3 


Wz 


7 


6 


80 


390 


435 


475 


510 


575 


. 


! 


750 


800 


3 


4 


m 


7 


7 


160 


365 


405 


445 


475 


535 


590 


645 


700 


745 


4 


5 


12 


11 


8 


330 




355 


!9l 


415 


470 


520 


570 


610 


750 


5 


6 


14 


11 


9 


500 


285 


315 


345 


370 


415 


460 


500 


540 




6 


8 


16 


12 


10 


1100 


215 


240 


260 


280 


310 


340 


375 


410 


435 


8 


10 


18 


12 


10 


1600 


170 


190 


210 


225 


250 


275 


300 




350 


10 


12 


22 


14 


12 


2000 


150 


165 


185 


195 






265 


285 


310 


12 


14 


24 


14 


13 


3000 


135 


150 


165 


175 


195 


215 


240 


295 


275 


14 


15 


28 


15 


14 


3500 


125 


145 


155 


165 


190 


210 


230 




360 


15 


18 


28 


16 


14 


6000 


110 


120 


130 


135 


160 


175 


190 


255 


220 


18 



Efficiencies of Centrifugal and Reciprocating Pumps.— 

W. O. Webber (Trans. A. S. M. E., vii. 598) gives diagrams showing the 
relative efficiencies of centrifugal and reciprocating pumps, from which the 
following figures are taken for the different lifts stated : 
Lift, feet: 

2 5 10 15 20 25 30 35 40 50 60 80 100 120 160 200 240 280 
Efficiency reciprocating pump: 

30 .45 .55 .61 .66 .68 .71 .75 .77 .82 .85 .87 .90 .89 .88 .85 

Efficiency centrifugal pump: 

.50 .56 .64 .68 .69 .68 .66 .62 .58 .50 .40 

The term efficiency here used indicates the value of W. H. P. -=- 1. H. P., 
or horse-power of the water raised divided by the indicated horse-power of 
the steam-engiue,and does not therefore show the full efficiency of the pump, 
but that of the combined pump and engine. It is, however, a very simple 
way of showing the relative values of different kinds of pumping-engines 
having their motive power forming a part of the plant. 

The highest value of this term, given by Mr. Webber, is .9164 for a lift of 
170 ft., and 3615 gals, per min. This was obtained in a test of the Leavitt 
pumping engine at Lawrence, Mass., July 24, 1879. 

With reciprocating pumps, for higher lifts than 170 ft., the curve of effi 
ciencies falls, and from 200 to 300 ft. lift the average value seemo about 
.84. Below 170 ft. the curve also falls reversely and slowly, until at about 90 
ft. its descent becomes more rapid, and at 35 ft. .727 appears the best 
recorded performance. There are not any very satisfactory records below 
this lift, but some figures are given for the yearly coal consumption and 
total number of gallons pumped by engines in Holland under a 16-ft. lift, 
from which an efficiency of .44 has been deduced. 

With centrifugal pumps, the lift at which the maximum efficiency is ob- 
tained is approximately 17 ft. At lifts from 12 to 18 ft. some makers of 
large experience claim now to obtain from 65$ to 70$ of useful effect, but 
.613 appears to be the best done at a public test under 14.7 ft. head. 

The drainage-pumps constructed some years ago for the Haarlem Lake 
were designed to lift 70 tons per min. 15 ft., and they weighed about 150 
tons. Centrifugal pumps for the same work weigh only 5 tons. The weight 
of a centrifugal pump and engine to lift 10,000 gals, per min. 35 ft. high is 
6 tons. 

The pumps placed by Gwynne at the Ferrara Marshes, Northern Italy, in 
1865, are, it is believed, capable of handling more water than other set of 
pumping-engines in existence. The work performed by these pumps is the 
lifting of 2000 tons per min.— over 600.000,000 gals, per 24 hours— on a mean 
lift of about 10 ft. (maximum of 12.5 ft.). (See Engineering, 1876.) 

The efficiency of centrifugal pumps seems to increase as the size of pump 



DUTY TEIALS OF PUMPIKG-ENGINES. 



609 



increases, approximately as follows: A 2" pump (this designation meaning 
always the size of discharge-outlet in inches of diameter), giving an effi- 
ciency of 38#, a 3" pump 45#, and a 4" pump 52#, a 5" pump 60$, and a 6" 
pump 64$ efficiency. 

Tests of Centrifugal Pumps. 

W. O. Webber, Trans. A. S. M. E., ix. 237. 



Berlin. 

Schwartz- 

kopff. 



Size 

Diam. discharge . 

" suction ... 

" disk 

Rev. per minute. 
Galls, per minute 
Height in feet.. . . 

Water H.P 

Dynam'eter H.P. 
Efficiency 



An- 
drews. 


An- 
drews. 


An- 
drews. 


Heald 

& 
Sisco. 


Heald 

& 
Sisco. 


Heald 

& 
Sisco. 


No. 9. 


No. 9. 


No. 9. 


No. 10. 


No. 10. 


No. 10. 


$Vs" 


9^" 


$H" 


10" 


10" 


10" 


9H" 


9M" 


Q%" 


12" 


12" 


12" 


26" 


26" 


26" 


30.5" 


30.5" 


30.5" 


191.9 


195.5 


200.5 


188.3 


202.7 


213.7 


1513.12 


2023.82 


2499.33 


1673.37 


2044.9 


2371.67 


12.25 


12.62 


13.08 


12.33 


12.58 


13.0 


4.69 


6.47 


8.28 


5.22 


6.51 


7.81 


10.09 


12.2 


14.38 


8.11 


10.74 


14.02 


46.52 


53.0 


57.57 


64.5 


60.74 


55.72 



No. 9. 
Wi" 
10.3" 
20.5" 
500 
1944.8 
16.46 

"ii 



Vanes of Centrifugal Pumps.— For forms of pump vanes, see 
paper by W. O. Webber, Trans. A. S. M. E., ix. 228, and discussion thereon 
by Profs. Thurston, Wood, and others. 

The Centrifugal Pump used, as a Suction Dredge.— The 
Andrews centrifugal pump was used by Gen. Gillmore, U. S. A., in 1871, in 
deepening tbe channel over the bar at the mouth of tbe St. John's River, 
Florida. The pump was a No. 9, with suction and discharge pipes each 9 
inches diam. It was driven at 300 revolutions per minute by belt from an 
engine developing 26 useful horse-power. 

Although 200 revolutions of the pump disk per minute will easily raise 
3000 gallons of clear water 12 ft. high, through a straight vertical 9-inch 
pipe, 300 revolutions were required to raise 2500 gallons of sand and water 
11 ft. high, through two inclined suction-pipes having two turns each, dis- 
charged through a pipe having one turn. 

The proportion of sand that can be pumped depends greatly upon its 
specific gravity and fineness. The calcareous and argillaceous sands flow 
more freely than the silicious, and fine sands are less liable to choke the 
pipe than those that are coarse. When working at high speed, 50$ to 55$ of 
sand can be raised through a straight vertical pipe, giving for every 10 cubic 
yards of material discharged 5 to 5% cubic yards of compact sand. With 
the appliances used on the St. John's bar, the proportion of sand seldom 
exceeded 45$, generally ranging from 30$ to 35$ when working under the 
most favorable conditions. 

In pumping 2500 gallons, or 12.6 cubic yards of sand and water per minute, 
there would therefore be obtained from 3.7 to 4.3 cubic yards of sand. Dur- 
ing the early stages of the work, before the teeth under the drag had been 
properly arranged to aid the flow of sand into the pipes, the yield was con- 
siderably below this average. (From catalogue of Jos. Edwards & Co., 
Mfrs. of the Andrews Pump, New York.) 

DUTY TRIALS OF PUMPING-JENGINES. 

A committee of the A. S. M. E. (Trans., xii. 530) reported in 1891 on a 
standard method of conducting duty trials. Instead of the old unit of 
duty of foot-pounds of work per 100 lbs. of coal used, the committee recom- 
mend a new unit, foot-pounds of work per million heat-units furnished by 
the boiler. The variations in quality of coal make the old standard unfit as 
a basis of duty ratings. The new unit is the precise equivalent of 100 lbs. of 
coal in cases where each pound of coal imparts 10,000 heat-units to the 
water in the boiler, or where the evaporation is 10,000 -e- 965.7 = 10 355 lbs. of 
water from and at 212° per pound of fuel. This evaporative result is readily 
obtained from all grades of Cumberland bituminous coal, used in horizontal 
return tubular boilers, and, in many cases, from the best grades of anthra- 
cite coal, 



610 WATER-POWER. 

The committee also recommend that the work done be determined by 
plunger displacement, after making a test for leakage, instead of by meas- 
urement of flow by weirs or other apparatus, but advise the use of such 
apparatus when practicable for obtaining additional data. The following 
extracts are taken from the report. When important tests are to be made 
the complete report should be consulted. 

The necessary data having been obtained, the duty of an engine, and other 
quantities relating to its performance, may be computed by the use of the 
following formulae: 

„ _ , Foot-pounds of work done . nnn .... 

1. Duty = rf — — — X 1,000,000 

Total number of heat-units consumed 

= A(P± P +£XLXN x 1)000)0()0 (foot . pounds) , 

C X 144 

3. Percentage of leakage = ~ - - x 100 (per cent). 

■A. X -Li X iv 

3. Capacity = number of gallons of water discharged in 24 hours 
A X L X NX 7.4805 X 24 AxLx NX 1.24675 



D X 144 
4. Percentage of total frictions, 

rw. - ^p±p+ 



- (gallons). 



D X 60 X 33,000 
~ i-~~ I.H.P. 

= L 1 - isXM.E.P.Xi,XiyJ X 10 ° (per CeQt); 

or, in the usual case, where the length of the stroke and number of strokes 
of the plunger are the same as that of the steam-piston, this last formula 
becomes: 

tA(P + v-V- s) "1 
1 - / x MEF ' X 10 ° ( ' per cent )* 

In these formulae the letters refer to the following quantities: 
A = Area, in square inches, of pump plunger or piston, corrected for area 

of piston rod or rods; 
P = Pressure, in pounds per square inch, indicated by the gauge on the 

force main; 
p — Pressure, in pounds per square inch, corresponding to indication of the 
vacuum-gauge on suction -main (or pressure -gauge, if the suction- 
pipe is under a head). The indication of the vacuum-gauge, in 
inches of mercury, may be converted into pounds by dividing it by 
2.035; 
s = Pressure, in pounds per square inch, corresponding to distance be- 
tween the centres of the two gauges. The computation for this 
pressure is made by multiplying the distance, expressed in feet, by 
the weight of one cubic foot of water at the temperature of the 
pump-well, and dividing the product by 144; 
L = Average length of stroke of pump-plunger, in feet; 
N = Total number of single strokes of pump-plunger made during the trial; 
As = Area of steam-cylinder, in square inches, corrected for area of piston- 
rod. The quantity As X M.E.P., in an engine having more than one 
cylinder, is the sum of the various quantities relating to the respec- 
tive cylinders; 
Ls = Average length of stroke of steam -piston, in feet; 
JVs = Total number of single strokes of steam-piston during trial; 
M-E.P. = Average mean effective pressure, in pounds per square inch, 
measured from the indicator-diagrams taken from the steam-cylin- 
der; 
I.H.P. = Indicated horse-power developed by the steam-cylinder; 
C = Total number of cubic feet of water which leaked by the pump-plunger 

during the trial, estimated from the results of the leakage test; 
D = Duration of trial in hours; 



DUTY TUtALS OF PUMPING-ENGI^ES. 611 

H— Total number of heat-units (B. T. U.) consumed by engine = weight of 
water supplied to boiler by main feed-pump x total beat of steam 
of boiler pressure reckoned from temperature of main feed-water -f 
weight of water supplied by jacket-pump X total heat of steam of 
• boiler-pressure reckoned from temperature of jacket-water -f- weight 
of any other water supplied X total heat of steam reckoned from its 
temperature of supply. The total heat of the steam is corrected for 
the moisture or superheat which the steam may contain. No allow- 
ance is made for water added to the feed water, which is derived 
from auy source, except the engine or some accessory of the engine. 
Heat added to the water by the use of a flue -heater at the boiler is 
not to be deducted. Should heat be abstracted from the flue by 
means of a steam reheater connected with the intermediate re- 
ceiver of the engine, this heat must be included in the total quantity 
supplied by the boiler. 
Leakage Test of Pump.— The leakage of an inside plunger (the 
only type which requires testing) is most satisfactorily determined by mak- 
ing the test with the cylinder-head removed. A wide board or plank may 
be temporarily bolted to the lower part of the end of the cylinder, so as to 
hold back the water in the manner of a dam, and an opening made in the 
temporary head thus provided for the reception of an overflow-pipe. The 
plunger is blocked at some intermediate point in the stroke (or, if this posi- 
tion is not practicable, at the end of the stroke), and the water from the 
force main is admitted at full pressure behind it. The leakage escapes 
through the overflow-pipe, and it is collected in barrels and measured. The 
test should be made, if possible, with the plunger in various positions. 

In the case of a pump so planned that it is difficult to remove the cylinder- 
head, it may be desirable to take the leakage from one of the openings 
which are provided for the inspection of the suction-valves, the head being 
allowed to remain in place. 

It is assumed that there is a practical absence of valve leakage. Exami- 
nation for such leakage should be made, and if it occurs, and it is found to 
be due to disordered valves, it should be remedied before making the plunger 
test. Leakage of the discharge valves will be shown by water passing down 
into the empty cylinder at either end when they are under pressure. Leak- 
age of the suction-valves will be shown by the disappearance of water which 
covers them. 

If valve leakage is found which cannot be remedied the quantity of water 
thus lost should also be tested. One method is to measure the amount of 
water required to maintain a certain pressure in the pump cylinder when 
this is introduced through appe temporarily erected, no water being al- 
lowed to enter through the discharge valves of the pump. 

Table of Data and Results.— In order that uniformity may be se- 
cured, it is suggested that the data and results, worked out in accordance 
with the standard method, be tabulated in the manner indicated in the fol- 
lowing scheme : 

DUTY TRIAL OF ENGINE. 

DIMENSIONS. 

1. Number of steam-cylinders 

2. Diameter of steam-cylinders ins. 

3. Diameter of piston -rods of steam-cylinders ins. 

4. Nominal stroke of steam-pistons , .... ft. 

5. Number of water-plungers '. 

6. Diameter of plungers ins. 

7. Diameter of piston-rods of water-cylinders ins. 

8. Nominal stroke of plungers ft. 

9. Net area of steam-pistons „ sq. ins. 

10. Net area of plungers sq. ins. 

11. Average length of stroke of steam-pistons during trial ft. 

12. Average length of stroke of plungers during trial ft. 

(Give also complete description of plant.) 

TEMPERATURES. 

13. Temperature of water in pump-well degs. 

14. Temperature of water supplied to boiler by main feed-pump. . degs. 

15. Temperature of water supplied to boiler from various other 

sources degs. 



612 WATER-POWER. 

FEED-WATER. 

16. Weight of water supplied to boiler by main feed-pump lbs. 

17. Weight of water supplied to boiler from various other sources, lbs. 

18. Total weight of feed-water supplied from all sources . lbs. 

PRESSURES. 

19. Boiler pressure indicated by gauge lbs. 

20. Pressure indicated by gauge on force main lbs. 

21. Vacuum indicated by gauge on suction main ins. 

22. Pressure corresponding to vacuum given in preceding line lbs. 

23. Vertical distance between the centres of the two gauges ins. 

24. Pressure equivalent to distance between the two gauges lbs. 

MISCELLANEOUS DATA. 

25. Duration of trial hrs. 

26. Total number of single strokes during trial 

27. Percentage of moisture in steam supplied to engine, or number 

of degrees of superheating % or deg„ 

28. Total leakage of pump during trial, determined from results of 

leakage test lbs. 

29. Mean effective pressure, measured from diagrams taken from 

steam-cylinders M.E.P. 

PRINCIPAL RESULTS. 

30. Duty , ft. lbs. 

31. Percentage of leakage % 

32. Capacity gals. 

33. Percentage of total friction % 

ADDITIONAL RESULTS. 

34. Number of double strokes of steam-piston per minute 

35. Indicated horse-power developed by the various steam-cylinders I.H.P. 

36. Feed- water consumed by the plant per hour lbs. 

37. Feed-water consumed by the plant per indicated horse-power 

per hour, corrected for moisture in steam lbs. 

38. Number of heat units consumed per indicated horse-power 

per hour B.T.U. 

39. Number of heat units consumed per indicated horse-power 

per minute B.T.U. 

40. Steam accounted for by indicator at cut-off and release in the 

various steam-cylinders lbs. 

41. Proportion which steam accounted for by indicator bears to 

the feed-water consumption 

42. Number of double strokes of pump per minute 

43. Mean effective pressure, measured from pump diagrams ...... M.E.P. 

44. Indicated horse-power exerted in pump-cylinders I.H.P. 

45. Work done (or duty) per 100 lbs. of coal ft. lbs. 

SAMPLE DIAGRAM TAKEN FROM STEAM-CYLINDERS. 

(Also, if possible, full measurement of the diagrams, embracing pressures 
at the initial point, cut off, release, and compression ; also back pressure, 
and the proportions of the stroke completed at the various points noted.) 

SAMPLE DIAGRAM TAKEN FROM PUMP-CYLINDERS. 

These are not necessary to the main object, but it is desirable to give 
them. 

DATA AND RESULTS OF BOILER TEST. 

(In accordance with the scheme recommended by the Boiler-test Com- 
mittee of the Society.) 

VACUUM PUMPS-AIR-LIFT PUMP. 

Tlie Pulsometer.-In the pulsometer the water is raised by suction 
into the pump-chamber by the condensation of steam within it, and is then 
forced into the delivery-pipe by the pressure of a new quantity of steam on 
the surface of the water. Two chambers are used which work alternately, 
one raising while the other is discharging. 

Test of a Pulsometer.— A test of a pulsometer is described by De Volson 
Wood in Trans. A. S. M. E. xiii. It had a 3^-inch suction-pipe, stood 40 ii 
high, and weighed 695 lbs. 

The steam-pipe was 1 inch in diameter. A throttle was placed about 2 feet 



VACUUM PUMPS— AIR-LIFT PUMP. 



612 



from the pump, and pressure gauges placed on both sides of the throttle, 
and a mercury well and thermometer placed beyond the throttle. The wire 
drawing due to throttling caused superheating. 

The pounds of steam used were computed from the increase of the tern 
perature of the water in passing through the pump. 
Pounds of steam x loss of heat = lbs. of water sucked in x increase of temp. 

The loss of heat in a pound of steam is the total heat in a pound of satu- 
rated steam as found from "steam tables " for the given pressure, plus the 
heat of superheating, minus the temperature of the discharged water ; or 



Pounds of steam = 



lbs. water X increase of temp. 
H - 0.48* - T. 
The results for the four tests are given in the following table : 



Data and Results. 



Strokes per minute 

Steam press. in pipe before throttPg 
■ Steam press, in pipe after throttPg 
Steam temp, after throttling,deg.F. 
Steam am'nt of superheat'g.deg.F. 
Steam used asdet'd from temp., lbs. 

Water pumped, lbs. 

Water temp. before entering pump, 

Water temp., rise of 

Water head by gauge on lift, ft ... . 
Water head by gauge on suction. . . 

Water head by gauge, total (H) 

Water head by measure, total (h) 
Coeff . of friction of plant (h) -=- (H) 

Efficiency of pulsometer 

Effic. of plant exclusive of boiler. . . 
Effic. of plant if that of boiler be 0.7 
Duty,if 1 lb.evaporates 10 lbs.w^ater 



Number of Test. 



71 

114 

19 

270.4 

3.1 

1617 
404,786 

75.15 
4.47 

29.90 

12.26 

42.16 

32.8 
0.777 
0.012 
0.0)93 
0.1065 



3.4 
931 

186.362 

80.6 
5.5 

54.05 

12.26 

66.31 

57.80 
0.877 
0.0155 
0.0136 
0.0095 



10.5li,4Ul)j 1:1391,000 



43.8 
309.0 
17.4 

1518 

228,425 
76.3 
7.49 
54.05 
19.67 

66^6 
0.911 
0.0126 
0.0115 
0.0080 
11,059,000 



26.1 

270.1 

1.4 

1019.9 

248.053 

70.25 

4.55 



49.57 
41.60 
0.839 
0.0138 
0.0116 
0.0081 
12,036,300 



Of the tw r o tests having the highest lift (54.05 ft.), that was more efficient 
which had the smaller suction (12.26 ft.), and this was also the most efficient 
of the four tests. But, on the other hand, the other two tests having the 
same lift (,29.9 ft.), that was the more efficient which had the greater suction 
(19.67), so that no law in this regard was established. The pressures used, 
19, 30, 43.8, 26.1, follow the order of magnitude of the total heads, but are 
not proportional thereto. No attempt was made to determine what press- 
ure would give the best efficiency for any particular head. The pressure used 
was intrusted to a practical runner, and he judged that when the pump was 
running regularly and well, the pressure then existing was the proper one. 
It is peculiar that, in the first test, a pressure of 19 lbs. of steam should pro- 
duce a greater number of strokes and pump over 50$ more water than 26.1 
lbs., the lift being the same, as in the fourth experiment. 

Chas. E. Emery in discussion of Prof. Wood's paper says, referring to 
tests made by himself and others at the Centennial Exhibition in 1876 (see 
Report of Judges, Group xx.), says that a vacuum-pump tested by him in 
1871 gave a duty of 4.7 millions; one tested by J. F. Flagg, at the Cincinnati 
Exposition in 1875, gave a maximum duty of 3.25 millions. Several vacuum 
and small steam-pumps, compared later on the same basis, were reported 
to have given duties of 10 to 11 millions, the steam-pumps doing no better 
than the vacuum-pumps. Injectors, when used for lifting water not re- 
quired to be heated, have an efficiency of 2 to 5 millions; vacuum-pumps 
vary generally between 3 and 10; small steam-pumps between 8 and 15 ; 
larger steam-pumps, between 15 and 30, and pumping-engiues between 30 
and 140 millions. 

A very high record of test of a pulsometer is given in Enc/'g. Nov. 24, 1893, 
p. 639, viz. : Height of suction 11.27 ft. ; total height of lift, 102.6 ft. ; hori- 
zontal length of delivery-pipe, 118 ft. ; quantity delivered per hour, 26,188 
British gallons. Weight of steam used per H. P. per hour, 92.76 lbs. ; work 



614 WATER-POWER. 

done per pound of steam 21,345 foot-pounds, equal to a duty of 21,345,000 
foot-pounds per 100 lbs. of coal, if 10 lbs of steam were generated per 
pound of coal. 

The Jet-pump.— This machine works by means of the tendency of a 
stream or jet of fluid to drive or carry contiguous particles of fluid along 
with it. The water-jet pump, in its present form, was invented by Prof. 
James Thomson, and first described in 1852. In some experiments on a 
small scale as to the efficiency of the jet-pump, the greatest efficiency was 
found to take place when the depth from which the water was drawn by the 
suction-pipe was about nine tenths of the height from which the water fell 
to form the jet ; the flow up the suction-pipe being in that case about one 
fifth of that of the jet, and the efficiency, consequently, 9/10 X 1/5 = 0.18. 
This is but a low efficiency; but it is probable that it may be increased by 
improvements in proportions of the machine. (Rankine, S. E.) 

Tlte Injector when used as a pump has a very low efficiency. (See 
Injectors, under Steam-boilers.) 

Air-lift Pump.— The air-lift pump consists of a vertical water-pipe 
with its lower end submerged in a well, and a smaller pipe delivering air 
into it at the bottom. The rising column in the pipe consists of air mingled 
with water, the air being in bubbles of various sizes, and is therefore lighter 
than a column of water of the same height; consequently the water in the 
pipe is raised above the level of the surrounding water. This method of 
raising water was proposed as early as 1797, by Loeseher, of Freiberg, and 
was mentioned by Collon in lectures in Paris in 1876, but its first practical 
application probably was by Werner Siemens in Berlin in 1885. Dr. J. G. 
Pohle experimented on the principle in California in 1886, and U. S. patents 
on apparatus involving it were granted to Pohle and Hill in the same year. 
A paper describing tests of the air-lift pump made by Randall, Browne and 
Behr was read before the Technical Society of the Pacific Coast in Feb. 1890. 

The diameter of the pump-column was 3 in., of the air-pipe 0.9 in., and 
of the air-discharge nozzle % in. The air-pipe had four sharp bends and a 
length of 35 ft. plus the depth of submersion. 

The water was pumped from a closed pipe-well (55 ft. deep and 10 in. in 
diameter). The efficiency of the pump was based on the least work theo- 
retically required to compress the air and deliver it to the receiver. If the 
efficiency of the compressor be taken at 70$, the efficiency of the pump and 
compressor together would be 70$ of the efficiency found for the pump 
alone. 

For a given submersion (70 and lift (II), the ratio of the two being kept 
within reasonable limits, (H) being not much greater than (h), the efficiency 
was greatest when the pressure in the receiver did not greatly exceed the 
head due to the submersion. The smaller the ratio H-t-h, the higher was 
the efficiency. 

The pump, as erected, showed the following efficiencies : 

For H+h= 0.5 1.0 1.5 2.0 

Efficiency = 50$ 40$ 30$ 25$ 

The fact that there are absolutely no moving parts makes the pump 
especially fitted for handling dirty or gritty water, sewage, mine water, 
and acid or alkali solutions in chemical or metallurgical works. 

In Newark, N. J., pumps of this type are at work having a total capacity 
of 1,000,000 gallons daily, lifting water from three 8-in. artesian wells. The 
Newark Chemical Works use an air-lift pump to raise sulphuric acid of 1.72° 
gravity. The Colorado Central Consolidated Mining Co., in one of its mines 
at Georgetown, Colo., lifts water in one case 250 ft., using a series of lifts. 

For a full account of the theory of the pump, and details of the tests 
above referred to, see Eng^g News, June 8, 1893. 

THE HYDRAULIC RAM. 

Efficiency.— The hydraulic ram is used where a considerable flow of 
water with a moderate fall is available, to raise a small portion of that flow 
to a height exceeding that of the fall. The following are rules given by 
Eytelwein as the results of his experiments (from Rankine): 

Let Q be the whole supply of water in cubic feet per second, of which q is 
lifted to the height h above the pond, and Q — q runs to waste at the depth 
H below the pond; L, the length of the supply -pipe, from the pond to the 
waste-clack ; D, its diameter in feet; then 

D - 4/(1.63Q); L = H+h + fiX 2 feet; 
Volume of air vessel = volume of feed pipe; 



THE HYDRAULIC RAM. 



615 



Efficiency, 



qh 



-™i/h 



(Q-q)H- 
1 -*- ( 1 + Tqtt) nearly, when — does not exceed 12. 



D'Aubisson gives 



Clark, using five sixths of the values given by D , Aubisson , s formula, gives: 
Ratio of lift to fall. ... 4 6 8 10 12 14 16 18 20 22 24 26 
Efficiency per cent 72 61 52 44 37 31 25 19 14 9 4 

Prof. R. C. Carpenter (Eng'g Mechanics, 1894) reports the results of four 
tests of a ram constructed by Runisey & Co., Seneca Falls. The ram was 
fitted for pipe connection for l^-inch supply and i^-inch discharge. The 
supply-pipe used was 1}^ inches in diameter, about 50 feet long, with 3 elbows, 
so that it was equivalent to about - 65 feet of straight pipe, so far as resist- 
ance is concerned. Each run'was made with a different stroke for the waste 
or clack-valve, the supply and delivery head being constant; the object of 
the experiment was to find that stroke of clack-valve which would give the 
highest efficiency. 



Length of stroke, per cent 

Number of strokes per minute 

Supply head, feet of water 

Delivery head, feet of water. . 
Total water pumped, pounds. . 
Total water supplied, pounds.. 
Efficiency, per cent , 



100 


80 


60 


52 


56 


61 


5.67 


5.77 


5.58 


19.75 


19.75 


19.75 


.297 


296 


301 


1615 


1567 


1518 


64.9 


66 


74.9 



66 
5.65 
19.75 
297.5 
1455.5 
70 



The efficiency, 74.9, the highest realized, was obtained when the clack-valve 
travelled a distance equal to 60$ of its full stroke, the full travel being 15/16 
of one inch. 

Quantity of Water Delivered, by the Hydraulic Ram, 

(Chadwick Lead Works.)— From 80 to 100 feet conveyance, one seventh of 
supply from spring can be discharged at an elevation five times as high as 
the fall to supply the ram; or, one fourteenth can be raised and discharged 
say ten times as high as the fall applied. 

Water can be conveyed by a ram 3000 feet, and elevated 200 feet. The 
drive-pipe is from 25 to 50 feet long. 

The following table gives the capacity of several sizes of rams, the dimen- 
sions of the pipes to be used, and the size of the spring or brook to which 
they are adapted: 





Quantity of Water 

Furnished per 

Min. by the Spring 

or Brook to which 

the Ram is 

Adapted. 


Caliber of 
Pipes. 


Weight of Pipe (Lead), if Wrought 
Iron, then of Ordinary Weight. 


Size of 
Ram. 


03 

> 


& 

5 


Drive-pipe 

for head 

or fall not 

over 10 ft. 


Discharge- 
pipe for not 
over 50 ft. 
rise. 


Discharge- 
pipe for 
over 50 ft. 
and not ex- 
ceeding 
100 ft. in 
height. 


No. 2 
" 3 
" 4 
" 5 
" 6 
" 7 
"10 


Gals, per min. 
U to 2 

iy 2 " 4 

3 " 7 
6 " 14 

12 " 25 
20 " 40 
25 " 75 


inch. 
% 
1 

2 


inch. 

Vs 

y* 
y% 

.* 

2 


per foot. 

2 lbs. 

3 " 
5 " 
8 " 

13 " 
13 " 
21 " 


per foot. 
10 ozs. 
12 " 
12 " 

lib. 4 " 

2 " 

3 " 


per foot, 
lib. 
1 " 4 ozs. 

1 " 4 ozs. 

2 " 

3 " 

4 " 
8 " 



616 WATER-POWER. 

HYDRAULIC-PRESSURE TRANSMISSION, 

Water under high pressure (700 to 2000 lbs. per square inch and upwards) 
affords a very satisfactory method of transmitting power to a distance, 
especially for the movement of heavy loads at small velocities, as by cranes 
and elevators. The system consists usually of one or more pumps capable 
of developing the required pressure; accumulators, which are vertical cylin- 
ders with heavily-weighted plungers passing through stuffing-boxes in the 
upper end, by which a quantity of water may be accumulated at the pres- 
sure to which the plunger is weighted ; the distributing-pipes; and the presses, 
cranes, or other machinery to oe operated. 

The earliest important use of hydraulic pressure probably was in the 
Bramah hydraulic press, patented in 1796. Sir W. G. Armstrong in 1846 was 
one of the pioneers in the adaptation of the hydraulic system to cranes. The 
use of the accumulator by Armstrong led to the extended use of hydraulic 
machinery. Recent developments and applications of the system are largely 
due to Ralph Tweddell, of London, and Sir Joseph Whitworth. Sir Henry 
Bessemer, in his patent of May 13, 1856, No. 1292, first suggested the use of 
hydraulic pressure for compressing steel ingots while in the fluid state. 

The Gross Amount of Energy of the water under pressure stored 
in the accumulator, measured in foot-pounds, is its volume in cubic feet X 
its pressure in pounds per square foot. The horse-power of a given quantity 

l44r>Q 
steadily flowing is H.P. = ^ = .26l8pQ, in which Q is the quantity flowing 

in cubic feet per second andp the pressure in pounds per square inch. 

The loss of energy due to velocity of flow in the pipe is calculated as fol- 
lows (R. Gr. Blaine, Eng'g, May 22 and June 5, 1891): 

According to D'Arcy, every pound of water loses -=— times its kinetic 

energy, orenergy due to its velocity in passing along a straight pipe L feet 
in length and D feet diameter, where A is a variable coefficient. For clean 

cast-iron pipes it may be taken as A = .005 ( 1 -f- — - ), or for diameter in 

inches — d. 

d= l£ 1 2 3 45 6 7 8 9 10 12 

A = .015 .01 .0075 .00667 .00625 .006 .00583 .00571 .00563 .00556 .0055 .00542 

The loss of energy per minute is 60 x 62.36Q X -jr- .5-, andthehorse- 

«. /. • «. • • ur -6363AL(H.P.)3 . ... ? . 
power wasted in the pipe is W = 3 -=-g — — , m which A varies with the 

diameter as above, p = pressure at entrance in pounds per square inch. 
Values of .6363A for different diameters of pipe in inches are: 
d= Y 2 1 2 3 4 5 6 7 8 9 10 12 

.00954 .00636 .00477 .00424 .00398 .00382 .00371 .00363 .00358 .00353 .00350 .00345 

Efficiency of Hydraulic Apparatus.- The useful effect of a 
direct hydraulic plunger or ram is usually taken at 93$. The following is 
given as the efficiency of a ram with chain-and-pulley multiplying gear 
properly proportioned and well lubricated: 

Multiplying.... 2 to 1 4 to 1 6 to 1 8 to 1 lOtol 12 to 1 14 to 1 16 to 1 
Efficiency*.... 80 76 72 67 63 59 54 50 

With large sheaves, small steel pins, and wire rope for multiplying gear 
the efficiency has been found as high as 66* for a multiplication of 20 to 1. 

Henry Adams gives the following formula for effective pressure in cranes 
and hoists: 

P — accumulator pressure in pounds per square inch; 

m — ratio of multiplying power; 

E = effective pressure in pounds per square inch, including all allowances 
for friction ; 

E = P(.84 - .02m). 

J. E. Tuit (Eng^g, June 15, 1888) describes some experiments on the fric- 
tion of hydraulic jacks from 3J4 to 13^-inch diameter, fitted with cupped 
leather packings. The friction loss varied from 5.6* to 18.8$ according to 
the condition of the leather, the distribution of the load on the ram, etc. 
The friction increased considerably with eccentric loads. With hemp pack- 
ing a plunger, 14 inch diameter, showed a friction loss of from 11.4* to 3.4$, 
the load being central, and from 15.0* to 7.6* with eccentric load, the per- 
centage of loss decreasing in both cases with increase of lo&cl. 



HYDRAULIC-PRESSURE TRANSMISSION". 017 

Thickness of Hydraulic Cylinders.- -From a table used by Sir 
W. W. Armstrong we take the following, tor cast-iron cylinders, for an in- 
terior pressure ot 10U0 lbs. per square inch: 

Diam. of cylinder, iuches.. 2 4 6 8 10 12 16 20 24 
Thickness,' inches 0.832 1.146 1.552 1.875 2.222 2.578 3.19 3.69 4.11 

For any other pressure multiply by the ratio of that pressure to 1000. 
These figures correspond nearly to the formula t — 0.175cZ -f 0.4S, in which 
t = thickness and d — diameter in inches, up to 16 inches diameter, but for 
20 inches diameter the addition 0.48 is reduced to 0.19 and at 24 inches it 
disappears. For formulae for thick cylinders see page 287, ante. 

Cast iron should not be used for pressures exceeding 2000 lbs. per square 
inch. For higher pressures steel castings or forged steel should be used. 
For working pressures of 750 lbs. per square inch the test pressure should 
be 2500 lbs. per square inch, and for 1500 lbs. the test pressure should not be 
less than 3500 lbs. 

Speed, of Moisting by Hydraulic Pressure.— The maximum 
allowable speed for warehouse cranes is 6 feet per second; for platform 
cranes 4 feet per second; for passenger and wagon hoists, heavy loads, 2 
feet per second. The maximum speed under any circumstances should 
never exceed 10 feet per second. 

The Speed of Water Through Valves should never be greater 
than 100 feet per second. 

Speed of Water Through Pipes.— Experiments on water at 1600 
lbs. pressure per square inch flowing into a flanging-machine ram, 20-inch 
diameter, through a J^-inch pipe contracted at one point to ^-incli, gave a 
velocity of 114 feet per second in the pipe, and 456 feet at the reduced sec- 
tion. Through a J^-inch pipe reduced to ^s-inch at one point the velocity 
was 213 feet per second in the pipe and 381 feet at the reduced section In a 
J^-inch pipe without contraction the velocity was 355 feet per second. 

For many of the above notes the author is indebted to Mr. John Piatt, 
consulting engineer, of New York. 

High-pressure Hydraulic Presses in Iron-works are de- 
scribed by R. M. Daeleu, of Germany, in Trans. A. I. M. E. 1892. The fol- 
lowing distinct arrangements used in different systems of high-pressure 
hydraulic work are discussed and illustrated: 

1. Steam-pump, with fly-wheel and accumulator. 

2. Steam pump, without fly-wheel and with accumulator. 

3. Steam-pump, without fly-wheel and without accumulator. 

In these three systems the valve-motion of the working press is operated 
in the high-pressure column. This is avoided in the following: 

4. Single-acting steam-intensifier without accumulator. 

5. Steam-pump with fly-wheel, without accumulator and with pipe-circuit. 

6. Steam-pump with fly-wheel, without accumulator and without pipe- 
circuit. 

The disadvantages of accumulators are thus stated: The weighted plungers 
which formerly served in most cases as accumulators, cause violent shocks 
in the pipe-line when changes take place in the movement of the water, 
so that in many places, in order to avoid bursting from this cause, the pipes 
are made exclusively of forged and bored steel. The seats and cones of the 
metallic valves are cut by the water (at high speed), and in such cases only 
the most careful maintenance can prevent great losses of power. 

Hydraulic Power in London.— The general principle involved 
is pumping water into mains laid in the streets, from which service-pipes 
are carried into the houses to work lifts or three-cylinder motors when 
rotatory power is required. In some cases a small Pelton wheel has been 
tried, working under a pressure of over 700 lbs. on the square inch. Over 55 
miles of hydraulic mains are at present laid (1892). 

The reservoir of power consists of capacious accumulators, loaded to a 
pressure of 800 lbs. per square inch, thus producing the same effect as if 
large supply-tanks were placed at 1700 feet above the street-level. The 
water is taken from the Thames or from wells, and all sediment is removed 
therefrom by filtration before it reaches the main engine-pumps. 

There are over 1750 machines at work, and the supply is about 6,500,000 
gallons per week. 

It is essential that the water used should be clean. The storage-tank ex- 
tends over the whole boiler-house and coal-store. The tank is divided, and 
a certain amount of mud is deposited here. It then passes through the sur- 
face condenser of the engines, and it is turned into a set of filters, eight in 
number. The body of the filter is a cast-iron cylinder, containing a layer of 



618 WATER-POWER* 

granular filtering material resting upon a false bottom; under this is the dis- 
tributing arrangement, affording passage for the air, and under this the real 
bottom of the tank. The dirty water is supplied to the filters from an over- 
head tank. After passing through the filters the clean effluent is pumped 
into the clean-water tank, from which the pumping-engines derive their 
supply. The cleaning of the filters, which is done at intervals of 24 hours, is 
effected so thoroughly in situ that the filtering material never requires to be 
removed. 

The engine-house contains six sets of triple-expansion engines. The 
cylinders are 15-inch, 22-inch, 36 inch X 24-inch. Each cylinder drives a 
single plunger-pump with a 5-inch ram, secured directly to the cross-head, 
the connecting-rod being double to clear the pump. The boiler-pressure is 
150 lbs. on the square inch. Each pump will deliver 300 gallons of water per 
minute under a pressure of 800 lbs. to the square inch, the engines making 
about 61 revolutions per minute. This is a high velocity, considering the 
heavy pressure; but the valves work silently and without perceptible shock. 

The consumption of steam is 14.1 pounds per horse per hour. 

The water delivered from the main pumps passes into the accumulators. 
The rams are 20 inches in diameter, and have a stroke of 23 feet. They are 
each loaded with 110 tons of slag, contained in a wrought-iron cylindrical 
box suspended from a cross-head on the top of the ram. 

One of the accumulators is loaded a little more heavily than the other, so 
that they rise and fall successively; the more heavily loaded actuates a stop- 
valve on the main steam-pipe. If the engines suppty more water than is 
wanted, the lighter of the two rams first rises as far as it can go; the other 
then ascends, and when it has nearly reached the top, shuts off steam and 
checks the supply of water automatically. 

The mains in the public streets are so constructed and laid as to be per- 
fectly trustworthy and free from leakage. 

Every pipe and valve used throughout the system is tested to 2500 lbs. per 
square inch before being placed on the ground and again tested to a reduced 
pressure in the trenches to insure the perfect tightness of the joints. The 
jointing material used is gutta-percha. 

The average rate obtained by the company is about 3 shillings per thou- 
sand gallons. The principal use of the power is for intermittent work in cases 
where direct pressure can be employed, as, for instance, passenger elevators, 
cranes, presses, warehouse hoists, etc. 

An important use of the hydraulic power is its application to the extin- 
guishing of fire by means of Greathead's injector hydrant. By the use of 
these hydrants a continuous fire-engine is available. 

Hydraulic Riveting-machines.— Hydraulic riveting was intro- 
duced in England by Mr. R. H. Tweddell. Fixed riveters were first used about 
1868. Portable riveting-machines were introduced in 1872. 

The riveting of the large steel plates in the Forth Bridge was done by small 
portable machines working with a pressure of 1000 lbs. per square inch. In 
exceptional cases 3 tons per inch was used. (Proc. Inst. M. E., May, 1889.) 

An application of hydraulic pressure invented by Andrew Higginson, of 
Liverpool, dispenses with the necessity of accumulators. It consists of a 
three-throw pump driven by shafting or worked by steam, and depends 
partially upon the work accumulated in a heavy fly-wheel. The water in its 
passage from the pumps and back to them is in constant circulation at a 
very feeble pressure, requiring a minimum of power to preserve the tube of 
water ready for action at the desired moment, when by the use of a tap the 
current is stopped from going back to the pumps, and is thrown upon the 
piston of the tool to be set in motion. The water is now confined, and the 
driving-belt or steam-engine, supplemented by the momentum of the heavy 
fly-wheel, is employed in closing up the rivet, or bending or forging the ob- 
ject subjected to its operation. 

Hydraulic Forging.— In the production of heavy forgings from 
cast ingots of mild steel it is essential that the mags of metal should be 
operated on as equally as possible throughout its entire thickness. When 
employing a steam-hammer for this purpose it has been found that the ex- 
ternal surface of the ingot absorbs a large proportion of the sudden impact 
of the blow, and that a comparatively small effect only is produced on the 
central portions of the ingot, owing to the resistance offered by the inertia 
of the mass to the rapid motion of the falling hammer— a disadvantage that 
is entirely overcome by the slow, though powerful, compression of the 
hydraulic forging-press, which appears destined to supersede the steam- 
hammer for the production of massive steel forgings. 



HYDRAULIC-PRESSURE TRANSMISSION. 619 

It) the Allen forging-press the force-pump and the large or main cylinder 
of the press are in direct and constant communication. There are no inter- 
mediate valves of any kind, nor has the pump any clack-valves, but it 
simply forces its cylinder full of water direct into the cylinder of the press, 
and receives the same water, as it were, back again on the return stroke. 
Thus, when both cylinders and the pipe connecting them are full, the large 
ram of the press rises and falls simultaneously with each stroke of the 
pump, keeping up a continuous oscillating motion, the ram, of course, 
travelling the shorter distance, owing to the larger capacity of the press 
cylinder. (Journal Iron and Steel Institute, 1891. See also illustrated article 
in "' Modern Mechanism/' page 668.) 

For a very complete illustrated account of the development of the hy- 
draulic forging-press, see a paper by R. H. Tweddell in Proc. Inst. C. E., vol. 
cxvii. 1893-4. 

Hydraulic Forging-press.— A 2000-ton forging-press erected at 
the Couillet forges in Belguim is described in Eng. and M. Jour., Nov. 25, 1893. 

The press is composed essentially of two parts— the press itself and the 
compressor. The compressor is formed of a vertical steam-cylinder and a 
hydraulic cylinder. The piston-rod of the former forms the piston of the 
latter. The hydraulic piston discharges the water into the press proper. 
The distribution is made by a cylindrical balanced valve; as soon as the 
pressure is released the steam-piston falls automatically under the action of 
gravity. During its descent the steam passes to the other face of the piston 
to reheat Ihe cylinder, and finally escapes from the upper end. 

When steam enters under the piston of the compressor-cylinder the pis- 
ton rises, and its rod forces the water into the press proper. The pressure 
thus exerted on the piston of the latter is transmitted through a cross-head 
to the forging which is upon the anvil. To raise the cross-head two small 
single-acting steam-cylinders are used, their piston-rods being connected to 
the cross-head ; steam acts only on the pistons of these cylinders from below. 
The admission of steam to the cylinders, which stand on top of the press 
frame, is regulated by the same lever which directs the motions of the com- 
pressor. The movement given to the dies is sufficient for all the ordinary 
purposes of forging. 

A speed of 30 blows per minute has been attained. A double press on the 
same system, having two compressors and giving a maximum pressure of 
6000 tons, has been erected in the Krupp works, at Essen. 

The Aiken Intensifies {Iron Age, Aug. 1890.)— The object of the 
machine is to increase the pressure obtained by the ordinary accumulator 
which is necessary to operate powerful hydraulic machines requiring very 
high pressures, without increasing the pressure carried in the accumulator 
and the general hydraulic system. 

The Aiken Intensifier consists of one outer stationary cylinder and one 
inner cylinder which moves in the outer cylinder and on a fixed or stationary 
hollow plunger. When operated in connection with the hydraulic bloom- 
shear the method of working is as follows: The inner cylinder having been 
filled with water and connected through the hollow plunger with thehydrau- 
lic cylinder of the shear, water at the ordinary accumulator-pressure is ad- 
mitted into the outer cylinder, which being four times the sectional area of 
the plunger gives a pressure in the inner cylinder and shear cylinder con- 
nected therewith of four times the accumulator-pressure— that is, if the ac- 
cumulator-pressure is 500 lbs. per square inch the pressure in the intensifier 
will be 2000 lbs. per square inch. 

Hydraulic Engine driving an Air-compressor and a 
Forging-hammer. {Iron Age, May 12, 1892.)— The great hammer in 
Terni, near Rome, is one of the largest in existence. Its falling weight 
amounts to 100 tons, and the foundation belonging to it consists of a block 
of cast iron of 1000 tons. The stroke is 16 feet \% inches; the diameter of 
the cylinder 6 feet S}4 inches: diameter of piston-rod 13% inches; total height 
of the hammer, 62 feet 4 inches. The power to work the hammer, as well as 
the two cranes of 100 and 150 tons respectively, and other auxiliary appli- 
ances belonging to it, is furnished by four air-compressors coupled together 
and driven directly by water-pressure engines, by means of which the air is 
compressed to 73.5 pounds per square inch. The cylinders of the water- 
pressure engines, which are provided with a bronze lining, have a 13%-inch 
bore. The stroke is 47% inches, with a pressure of water on the piston 
amounting to 264.6 pounds per square inch. The compressors are bored out 
to 31^2 inches diameter, and have 47%-inch stroke. Each of the four cylin- 
ders requires a power equal to 280 horse-power. The compressed air is de- 



620 FUEL. 

livered into huge reservoirs, where a uniform pressure is kept up by means 
of a suitable water-column. 

The Hydraulic Forging Plant at Bethlehem, Pa., is de- 
scribed in a paper by R. W. Davenport, read before the Society of Naval 
Engineers and Marine Architects, 1893. It includes two hydraulic forging- 
presses complete, with engines and pumps, one of 1500 and one of 4500 tons 
capacity, together with two Whitworth hydraulic travelling forging-cranes 
and other necessary appliances for each press ; and a complete fluid-compres- 
sion plant, including a press of 7000 tons capacity and a 125 ton hydraulic 
travelling crane for serving it (the upper and lower heads of this press 
weighing respectively about 135 and 120 tons). 

A new forging-press has been designed by Mr. John Fritz, for the Bethle- 
hem Works, of 14,000 tons capacity, to be run by engines and pumps of 15,000 
horse-power. The plane is served by four open-hearth steel furnaces of a 
united capacity of 120 tons of steel per heat. 

Some References on Hydraulic Transmission.— Reuleaux's 
"Constructor;" "Hydraulic Motors, Turbines, and Pressure-engines, 1 ' G. 
Bodmer, London, 1889; Robinson's "Hydraulic Power and Hydraulic Ma- 
chinery," London, 1888; Colyer's " Hydraulic Steam, and Hand- power Lift- 
ing and Pressing Machinery," London, 1881. See also Engineering (London), 
Aug. 1, 1884, p. 99; March 13, 1885, p. 262; May 22 and June 5, 1891, pp. 612, 
665; Feb. 19, 1892, p. 25; Feb. 10, 1893, p. 170. 

FUEL. 
Theory of Combustion of Solid Fuel. (From Rankine, some- 
what altered.) — The ingredients of every kind of fuel commonly used may 
be thus classed: (1) Fixed or free carbon, which is left in the form of char- 
coal or coke after the volatile ingredients of the fuel have been distilled 
away. These ingredients burn either wholly in the solid state (C to C0 2 ), or 
part in the solid state and part in the gaseous state (CO + O = C0 2 ), the lat- 
ter part being first dissolved by previously formed carbonic acid by the re- 
action C0 2 + C = 2CO. Carbonic oxide, CO, is produced when the supply 
of air to the fire is insufficient. 

(2) Hydrocarbons, such as olefiant gas, pitch, tar, naphtha, etc., all of 
which must pass into the gaseous state before being burned. 

If mixed on their first issuing from amongst the burning carbon with a 
large quantity of hot air, these inflammable gases are completely burned with 
a transparent blue flame, producing carbonic acid and steam. When mixed 
with cold air they are apt to be chilled and pass off uuburned. When 
raised to a red heat, or thereabouts, before being mixed with a sufficient 
quantity of air for perfect combustion, they disengage carbon in fine pow- 
der, and pass to the condition partly of marsh gas, and partly of free hydro- 
gen; and the higher the temperature, the greater is the proportion of carbon 
thus disengaged. 

If the disengaged carbon is cooled below the temperature of ignition be- 
fore coming in contact with oxygen, it constitutes, while floating in the gas, 
smoke, and when deposited on' solid bodies, soot. 

But if the disengaged carbon is maintained at the temperature of ignition, 
and supplied with oxygen sufficient for its combustion, it burns while float- 
ing in the inflammable gas. and forms red, yellow, or white flame. The flame 
from fuel is the larger the more slowly its combustion is effected. The 
flame itself is apt to be chilled by radiation, as into the heating surface of a 
steam-boiler, so that the combustion is not completed, and part of the gas 
and smoke pass off unburned. 

(3) Oxygeu or hydrogen either actually forming water, or existing in 
combination with the other constituents.in the proportions which form water. 
Such quantities of oxygen and hydrogen are to left be out of account in deter- 
mining the heat generated by the combustion. If the quantity of water 
actually or virtually present in each pound of fuel is so great as to make its 
latent heat of evaporation worth considering, that heat is to be deducted 
from the total heat of combustion of the fuel. 

(4) Nitrogen, either free or in combination with othar constituents. This 
substance is simply inert. 

(5) Sulphuret of iron, which exists in coal and is detrimental, as tending 
to cause spontaneous combustion. 

(6) Other mineral compounds of various kinds, which are also inert, and 
form the ash left after complete combustion of the fuel, and also the clinker 

j material produced by fusion of the ash, which tends to choke the 



FUEL. 



621 



Total Heat of Combustion of Fuels. (Rankine.)— The follow- 
ing table shows the total heat of combustion with oxygen of one pound of 
each of the substances named in it, in British thermal units, and also in 
lbs. of water evaporated from 212 p . It also shows the weight of oxygen re- 
quired to combine with each pound of the combustible and the weight of 
air necessary in order to supply that oxygen. The quantities of heat are 
given on the authority of MM. Favre and Silbermann. 



Hydrogen gas 

Carbon imperfectly burned so as 

to make carbonic oxide 

Carbon perfectly burned so as to 

make carbonic acid 

defiant gas, 1 lb 



Various liquid hydrocarbons, 1 lb. 

Carbonic oxide, as much as is made 
by the imperfect combustion of 
1 lb of carbon, viz., 2^ lbs. . . 



Lbs. Oxy- 
gen per 
lb. Com- 
bustible. 



Lb. Air 

(about). 



VA 



3 3/7 



12 
15 3/7 



Total Brit- 
ish Heat- 
units. 



Evapora- 
tive Power 
from 212° 
F., lbs. 



4,400 

14,500 

21,344 

from 21,700 

to 19,000 

10,000 



64.2 

4.55 

15.0 

22.1 

from 22££ 

to 20 

10.45 



The imperfect combustion of carbon, making carbonic oxide, produces 
iess than one third of the heat which is yielded by the complete combustion. 

The total heat of combustion of any compound of hydrogen and carbon 
is nearly the sum of the quantities of heat which the constituents would pro- 
duce separately by their combustion. (Marsh-gas is an exception.) 

In computing the total heat of combustion of compounds containing oxy- 
gen as well as hydrogen and carbon, the following principle is to be 
observed: When hydrogen and oxygen exist in a compound in - the proper 
proportion to form water (that is, by weight one part of hydrogen to eight 
of oxygen), these constituents have no effect on the total heat of combus- 
tion. If hydrogen exists in a greater proportion, only the surplus of hydro- 
gen above'that which is required by the oxygen is to be taken into account. 

The following is a general formula (Dulong's) for the total heat of combus- 
tion of any compound of carbon, hydrogen, and oxygen : 

Let C, H, and O be the fractions of one pound of the compound, which 
consists respectively of carbon, hydrogen, and oxygen, the remainder being 
nitrogen, ash, and other impurities. Let h be the total heat of combustion 
of one pound of the compound in British thermal units. Then 



h = 14,500 i C + 4.28(if - ~ ) j • 



I 

The following table shows the composition of those compounds which are 
of importance, either as furnishing oxygen for combustion, as entering into 
the composition, or as being produced by the combustion of fuel : 



O S3 •§ 

I'll 

flip. 

>-~ r 

m 3 



111 



PI 






Air 

Water 

Ammonia 

Carbonic oxide 

Carbonic acid 

defiant gas 

Marsh-gas or fire-damp. . 

Sulphurous acid 

Sulphuretted hydrogen. . 
Sulphuret of carbon 



H 2 

NH, 
CO 

co 2 

CHo 
CH 4 

SOo 
SH 2 

s 2 c 



N77 
H2 
H3 
C12 
C12 
C12 
C 12 
S 32 
S32 
S64 



+ 23 
+ 16 
+ N14 
+ 16 
+ 32 
+ H2 
+ H4 
+ 32 
+ H2 
+ C12 



N 79 + O 21 
H2 +0 
H3 fN 
C + O 
C + 02 
C +H2 
C +H4 



622 



Since each lb. of C requires 2%$ lbs. of O to burn it to C0 2 , and air contains 
23$ of O, by weight, 2% -^0.23 or 11.6 lbs. of air are required to burn 1 lb. of C. 

Analyses of Gases of Combustion. — The following are selected 
from a large number o£ analyses of gases from locomotive boilers, to show 
the range of composition under different circumstances (P. H. Dudley, 
Trans. A. I. M. E., iv. 250) : 



Test. 


co 2 


CO 


o 


N 




1 
2 
3 
4 
5 
6 

8 
9 


13.8 
11.5 
8.5 
2.3 
5.7 
8.4 
12 
3.4 
6 


2.5 

i'.k 
1 


2.5 
6. 

8. 
17.2 

14.7 
8.4 
4.4 

16.8 
13.5 


81.6 

82.5 

83. 

80.5 

79.6 

82. 

82.6 

76.8 

81.5 


No smoke visible. 

Old fire, escaping gas white, engine working hard. 
Fresh fire, much black gas, " " '.' 
Old fire, damper closed, engine standing still. 

" " smoke white, engine working hard. 
New fire, engine not working hard. 
Smoke black, engine not working hard. 

" dark, blower on, engine standing still. 

" white, engine working hard. 



In analyses on the Cleveland and Pittsburgh road, in every instance 
when the smoke was the blackest, there was found the greatest percentage 
of uuconsumed ox} r gen in the product, showing that something besides the 
mere presence of oxygen is required to effect the combustion of the volatile 
carbon of fuels. 

J. C. Hoadley (Trans. A. S. M. E., vi. 749) found as the mean of a great 
number of analyses of flue gases from a boiler using anthracite coal: 
C0 2 , 13.10; CO, 0.30; O, 11.94; N, 74.66. 

The loss of heat due to burning C to CO instead of to C0 2 was 2.13$. The 
surplus oxygen averaged 113.3$ of the O required for the C of the fuel, the 
average for different weeks ranging from 88.6$ to 137$. 

Analyses made to determine the CO produced by excessively rapid firing 
gave results from 2.54$ to 4.81$ CO and 5.12$ to 8.01$ C0 2 ; the ratio of C in 
the CO to total carbon burned being from 43.80$ to 48.55$, and the number of 
pounds of air supplied to the furnace per pound of coal being from 33 2 to 
19.3 lbs. The loss due to burning C to CO was from 27.84$ to 30.86$ of the 
fnll power of the coal. 

Temperature of the Fire. (Rankine, S. E., p. 283.)— By temper- 
ature of the fire is meant the temperature of the products of combustion at 
the instant that the combustion is complete. The elevation of that temper- 
ature above the temperature at which the air and the fuel are supplied to 
the furnace may be computed by dividing the total heat of combustion of 
one lb. of fuel by the weight and by the mean specific heat of the whole 
products of combustion, and of the air employed for their dilution under 
constant pressure. The specific heat under constant pressure of these prod- 
ucts is about as follows : 

Carbonic-acid gas, 0.217 ; steam, 0.475 ; nitrogen (probably), 0.245 ; air, 
0.238; ashes, probably about 0.200. Using: these data, the following results 
are obtained for pure carbon and for olefiant gas burned, respectively, first, 
in just sufficient air, theoretically, for their combustion, and, second, when, 
an equal amount of air is supplied in addition for dilution. 



Fuel. 


Products undiluted. 


Products diluted. 


Carbon. 


Olefiant 
Gas. 


Carbon. 


Olefiant 
Gas. 


Total heat of combustion, per lb. . . 
Wt. of products of combustion, lbs. . 

Their mean specific heat 

Specific heat x weight 

Elevation of temperature, F 


14,500 

13 
0.237 

3.08 
4580° 


21,300 
•16.43 
0.257 
4.22 
5050° 


14,500 

0?238 
5.94 
2440° 


21,300 
31.86 

0.248 

7.9 
2710° 



[The above calculations are made on the assumption that the specific 
heats of the gases are constant, but they probably increase with the in- 
crease of temperature (see Specific Heat), in which case the temperatures 
would be less than those above given. The temperature would be further 



CLASSIFICATION OF FUEL. 



623 



reduced by the heat rendered latent by the conversion into steam of any 
water present in the fuel.] 

Rise of Temperature in Combustion of Gases. (Evg'g, 
March 12 and April 2, 1886.)— It is found that the. temperatures obtained 
by experiment fall short of those obtained by calculation. Three theo- 
ries have been given to account for this : 1. The cooling effect of the 
sides of the containing vessel; 2. The retardation of the evolution of heat 
caused by dissociation; 3. The increase of the specific heat of the gases at 
very high temperatures. The calculated temperatures are obtainable only 
on the condition that the gases shall combine instantaneously and simulta- 
neously throughout their whole mass. This condition is practically impos- 
sible in experiments. The gases formed at the beginning of an explosion 
dilute the remaining combustible {gases and tend to retard or check the 
combustion of the remainder. 

CLASSIFICATION OF SOLID FUELS. 

Gruner classifies solid fuels as follows (2£ug'<; and M'g Jour., July, 1874) : 

._.-., Ratio — Proportion of Coke or 

fcameofFuel. " Charcoal yielded by 

or O + N* . the Dry p ure FueL 
H 

Pure cellulose 8 0.28 @, 0.30 

Wood (cellulose and encasing matter) 7 .30 @ .35 

Peat and fossil fuel 6 @, 5 .35 @, .40 

Lignite, t or brown coal 5 .40 @ .50 

Bituminous coals 4 @ 1 .50 @, .90 

Anthracite 1 @ 0.75 .90 @, .92 

The bituminous coals he divides into five classes as below: 



Name of Type. 


Elementary 
Composition. 


Ratio — 

O+N*. 
01 — 


Propor- 
tion of 
Coke 
yielded 
by Dis- 
tilla- 
tion. 


Nature 


C. 


H. 


O. 


Appear- 
ance of 
Coke. 


1. Long flaming dry { 

coal, \ 

2. Long flaming fat ) 

or coking coals, >- 
or gas coals, ) 

3. Caking fat coals, ) 

or blacksmiths' \ 
coals, ) 

4. Short flaming fat ) 

or caking coals. > 
coking coals, ) 

5. Lean or anthra- } 
citic coals, j 


75@80 
80@85 

84@89 

88@91 
90@,93 


5.5@4.5 

5.8@5 

5@4.5 

5.5@4.5 
4.5@4 


19.5@15 
14.2@10 

11 @5.5 

6.5@5.5 
5.5@3 


4@3 
3@2 

2@1 

1 
1 


0.50@,.60 
.60©.68 

.68®. 74 

.74@.82 
.82@.90 


j Pulveru- 
1 lent. 
\ Melted, 
•< but 
( friable, 
f Melted; 
| some- 
■{ what 
com- 
t pact, 
f Melted; 
J very 
j com- 
[ pact. 
\ Pulveru- 
1 lent. 



* The nitrogen rarely exceeds 1 per cent of the weight of the fuel. 
tNot including bituminous lignites, which resemble petroleums. . 

Rankine gives the following: The extreme differences in the chemical 
composition and properties of different kinds of coal are very great. The 
proportion of free carbon ranges from 30 to 93 per cent; that of hydrocar- 
bons of various kinds from 5 to 58 per eent; that of water, or oxygen and 
hydrogen in the proportions which form water, from an inappreciably 
small quantity to 27 per cent; that of ash, from IJ^ to 26 per cent. 

The numerous varieties of coal may be divided into principal classes as 
follows: 1, anthracite coal; 2, semi-bituminous coal; 3, bituminous coal; 
4, long flaming or cannel coal; 5, lignite or brown coal. 



624 - FUEL. 

Diminution of H and O in Series from Wood to Anthracite. 

(Groves and Thorp's Chemical Technology, vol. i., Fuels, p. 58.) 

Substance. Carbon. Hydrogen. Oxygen. 

Woody fibre 52.65 5.25 42.10 

Peat from Vulcaire 59.57 5.96 34.47 

Lignite from Cologne 66.04 5.27 28.69 

Earthy brown coal 73.18 5.88 21.14 

Coal from Belestat, secondary 75.06 5.84 19.10 

Coal from Rive de GHer 89.29 5.05 5.66 

Anthracite, Mayenne, transition formation 91.58 3.96 4.46 

Progressive Change from Wood to Graphite. 

(J. S. Newberry in Johnson's Cyclopedia.) 

Wood. Loss. I£ L„ss.J^, Loss. ^" Loss. Grajh- 

Carbon 49.1 18.65 30.45 12.35 18.10 3.57 14.53 1.42 13.11 

Hydrogen... 6.3 3.25 3.05 1.85 1.20 0.93 0.27 0.14 0.13 

Oxygen 44.6 24.40 20.20 18.13 2.07 1.32 0.65 0.65 0.00 

100.0 46.30 53.70 32.33 21.37 5.82 15.45 2.21 13.24 

Classification of Coals, as Anthracite, Bituminous, etc.— 
Prof. Persifer Frazer (Trans. A. I. M. E., vi, 430) proposes a classifica- 
tion of coals according to their " fuel ratio," that is, the ratio the fixed car- 
bon bears to the volatile hydrocarbon. 

In arranging coals under this classification, the accidental impurities, such 
as sulphur, earthy matter, and moisture, are disregarded, and the fuel con- 
stituents alone are considered. 

Carbon Fixed Volatile 

Ratio. Carbon. Hydrocarbon. 

I. Hard dry anthracite. 100 to 12 100. to 92.31$ 0. to 7.69$ 

II. Semi -anthracite 12 to 8 92.31to88.89 7.69 to 11.11 

IH. Semi-bituminous. ... 8 to 5 88.89 to 83.33 11.11 to 16.67 

IV. Bituminous 5 to 83.33 to 0. 16.67tol00 

It appears to the author that the above classification does not draw the 
line at the proper point between the semi-bituminous and the bituminous 
coals, viz., at a ratio of C -*- V.H.C. = 5, or fixed carbon 83.33$, volatile hy- 
drocarbon 16.67$, since it would throw many of the steam coals of Clearfield 
and Somerset counties, Penn., and the Cumberland, Md., and Pocahontas, 
Va., coals, which are practically of one class, and properly rated as 
semi-bituminous coals, into the bituminous class. The dividing: line be- 
tween the semi -anthracite and semi-bituminous coals, C ■*■ V.H.C. = 9, 
would place several coals known as semi-anthracite in the semi-bituminous 
class. The following is proposed by the author as a better classification : 

Carbon Ratio. Fixed Carbon. Vol. H.C. 

I. Hard dry anthracite.. 100 to 12 100 to 92.31$ to 7.69$ 

II. Semi-anthracite 12 to 7 92.31to87.5 7.69to 12.5 

III. Semi-bituminous 7 to 3 87.5 to 75 12.5 to 25 

IV. Bituminous 3. to 75 to 25 to 100 

Rhode Island Graphitic Anthracite.— A peculiar graphite is 
found at Cranstdn, near Providence, R. I. It resembles both graphite and 
anthracite coal, and has about the following composition (A. E. Hunt, Trans. 
A. I. M. E., xvii., 678): Graphitic carbon, 78$; volatile matter, 2.60$; silica, 
15.06$; phosphorus, .045$. It burns with extreme difficulty. 

ANALYSES OF COALS. 

Composition of Pennsylvania Anthracites. (Trans. A. I. 

M. E., xiv., 706.)— Samples weighing 100 to 200 lbs. were collected from lots 
of 100 to 200 tons as shipped to market, and reduced by proper methods to 
laboratory samples. Thirty-three samples were analyzed by McCreath, giv- 
ing results as follows. They show the mean character of the coal of the more 
important coal-beds in the Northern field in the vicinity of Wilkesbarre, in 
the Eastern Middle (Lehigh) field in the vicinity of Hazleton, in the Western 



ANALYSES OF COALS. 



625 



Middle field in the vicinity of Shenandoah, and in the Southern field between 
Mauch Chunk and Tamaqua. 



Name of 
Bed. 


o . 






3§ 

^6 


< 


% 

CO 


*2 c °3 


q 

O 


Wharton. . . 


E. Middle 


3.71 


3.08 


86.40 


6.22 


.58 


3.44 


28.07 


Mammoth.. 


E. Middle 


4.12 


3.08 


86.38 


5.92 


.49 


3.45 


27.99 


Primrose . . 


W. Middle 


3.54 


3.72 


81.59 


10.65 


.50 


4.36 


21.93 


Mammoth . 


W. Middle 


3.16 


3.72 


81.14 


11.08 


.90 


4.38 


21.83 


Primrose F 


Southern 


3.01 


4.13 


87.98 


4.38 


.50 


4.48 


21.32 


BuckMtn.. 


W. Middle 


3.04 


3.95 


82.66 


9.88 


.41) 


4.56 


20.93 


Seven Foot 


W. Middle 


3.41 


3.98 


80.87 


11.23 


.51 


4.69 


20.32 


Mammoth . 


Southern 


3.09 


4.28 


83.81 


8.18 


.64 


4.85 


19.62 


Mammoth . 


Northern 


3.42 


4.38 


83.27 


8.20 


.73 


5.00 


19.00 


B. Coal Bed 


Loyalsock 


1.30 


8.10 


83.34 


6.23 


1.03 


8.86 


10.29 



The above analyses were made of coals of all sizes (mixed). When coal is 
screened into sizes for shipment the purity of the different sizes as regards 
ash varies greatly. Samples from one mine gave results as follows: 
Screened Analyses. 

Name of Through Over Fixed 

Coal. inches. inches. Carbon. Ash. 

Egg 2.5 1.75 88.49 5.66 

Stove 1.75 1.25 83.67 10.17 

Chestnut 1.25 .75 80.72 12.67 



Buckwheat. . 



.50 



.50 



) 05 
76.92 



14.6 
16.62 



Sulphur. 
0.24 



1.04 



Sulphur. 
0.91 
0.59 



Bernice Basin, Pa., Coals. 

Water. Vol. H.C. Fixed C. Ash. 

Bernice Basin, Pullivan andj * 6 '\f 8 ? 52 3 £ 7 

Lycoming Cos.; range of 8. . j j m g 5g g9 39 QM 

This coal is on the dividing-line between the anthracites and semi-anthra- 
cites, and is similar to the coal of the Lykens Valley district. 

More recent analyses (Trans. A. I. M. E., xiv. 721) give : 

Water. Vol. H.C. Fixed Carb. Ash. 

Working seam 65 9.40 83.69 5.34 

60 ft. below seam .... 3.67 15.42 71.34 8.97 

The first is a semi-anthracite, the second a semi-bituminous. . 

Space Occupied by Anthracite Coal. (J. C. I. W., vol. hi.)— The 
cubic contents of 2240 lbs. of hard Lehigh coal is a little over 36 feet ; an 
average Schuylkill W. A., 37 to 38 feet ; Shamokin, 38 to 39 feet; Lor berry, 
nearly 41. 

According to measurements made with Wilkesbarre anthracite coal from 
the Wyoming Valley, it requires 32.2 cu. ft. of lump, 33.9 cu. ft. broken, 
34.5 cu. ft. egg, 34.8 cu. ft. of stove, 35.7 cu. ft, of chestnut, and 36.7 cu. ft. 
of pea, to make one ton of coal of 2240 lbs.; while it requires 28.8 cu. ft. of 
lump, 30.3 cu. ft. of broken, 30.8 cu. ft. of egg, 31.1 cu. ft. of stove, 31.9 cu. 
ft. of chestnut, and 32.8 cu. ft. of pea, to make one ton of 2000 lbs. 

Composition of Anthracite and Semi-bituminous Coals. 
(Trans. A. I. M. E., vi. 430.)— Hard dry anthracites, 16 analyses by Rogers, 
show a range from 94.10 to 82.47 fixed carbon, 1.40 to 9.53 volatile matter, 
and 4.50 to 8.00 ash, water, and impurities. Of the fuel constituents alone, 
the fixed carbon ranges from 98.53 to 89.63, and the volatile matter from 1.47 
to 10.37, the corresponding carbon ratios, or C -*- Vol. H.C. being from 67.02 
to 8.64. 

Semi-anthracites. — 12 analyses by Rogers show a range of from 90.23 to 
74.55 fixed carbon, 7.07 to 13.75 volatile matter, and 2.20 to 12.10 water, ash, 
and impurities. Excluding the ash, etc., the range of fixed carbon is 92.75 
to 84.42, and the volatile combustible 7.27 to 15.58, the corresponding carbon 
ratio being from 12.75^0 5.41. 



626 FUEL. 

Semi-bituminous Coals. — 10 analyses of Penna. and Maryland coals give 
fixed carbon 68.41 to 84.80, volatile matter 11.2 to 17.28, and ash, water, and 
impurities 4 to 13.99. The percentage of the fuel constituents is fixed carbon 
79.84 to 88.80, volatile combustible 11.20 to 20.16, and the carbon ratio 11.41 to 



American Semi-bituminous and Bituminous Coals. 

(Selected chiefly from various papers in Trans. A. I. M. E.) 



Moist- 
ure. 



Vol. 
Hydro- 
arbon 



Fixed 
Carbon 



Sul- 
phur. 



Penna. Semi-bituminous : 
Broad Top, extremes of 5 

Somerset Co., extremes of 5 

Blair Co., average of 5 

Cambria Co. , average of 7, ) 
lower bed, B. j 

Cambria Co. , 1 , | 

upper bed, C. \ 

Cambria Co., South Fork, 1 

Centre Co., 1 

Clearfield Co., average of 9, I 

upper bed, C. f " " 

Clearfield Co., average of 8, j 

lower bed, D. j — 

Clearfield Co., range of 17 anal. . 

Bituminous : 

Jefferson Co., average of 26 

Clarion Co., average of 7 

Armstrong Co., 1 

Connellsville Coal 

Coke from Conn'ville (Standard) 

Youghiogheny Coal 

Pittsburgh, Ocean Mine 



jl.27 
(1.89 
1.07 

0.74 
1.14 

0.60 
0.70 

0.81 
0.41 
to 
1.94 

1.21 
1.97 



.49 
1.03 



13.84 
17.38 
14.33 
18.51 
26.72 

21.21 



15.51 
22.60 

23.94 

21.10 
20.09 

to 
25.19 

32.53 

38.60 
42.55 
30.10 
0.01 



78.46 
76.14 



74.08 
66.69 



60.99 
54.15 
49.69 
59.61 

87.46 
59.05 
57.33 



6.00 
4.81 



10.62 
9.45 



5.84 
5.40 



2.65 
to 
7.65 

3.76 
4.10 
4.58 
8.23 
11.32 
2.61 



2.20 
1.98 



2.69 
1.42 

0.42 
0.43 



1.00 
1.19 

2.00 



The percentage of volatile matter in the Kittaning lower bed B and the 
Freeport lower bed D increases with great uniformity from east to west; thus: 

Volatile Matter. Fixed Carbon. 

Clearfield Co, bed D 20.09 to 25.19 68.73 to 74.76 

" B 22.56 to 26.13 64.37 to 69.63 

ClarionCo., "B 35.70 to 42.55 47.51 to 55.44 

" D 37.15 to 40.80 51.39 to 56.36 

Connellsville Coal and Coke. (Trans. A. I. M. E., xiii. 332.) — 
The Connellsville coal-field, in the southwestern part of Pennsylvania, is a 
strip about 3 miles wide and 60 miles in length. The mine workings are 
confined to the Pittsburgh seam, which here has its best development as to 
size, and its quality best adapted to coke-making. It generally affords 
from 7 to 8 feet of coal. 

The following analyses by T. T. Morrell show about its range of composi- 
tion : 

Moisture. Vol. Mat. Fixed C. Ash. Sulphur. Phosph's. 
Herold Mine .... 1.26 28.83 60.79 8.44 .67 .013 

Kintz Mine 0.79 31.91 56.46 9.52 1.32 .02 

In comparing the composition of coals across the Appalachian field, in the 
western section of Pennsylvania, it will be noted that the Connellsville 
variety occupies a peculiar position between the rather dry semi-bituminous 
coals eastward of it and the fat bituminous coals flanking it on the west. 

Beneath the Connellsville or Pittsburgh coal-bed occurs an interval of 
from 400 to 600 feet of "barren measures," separating it from the lower 
productive coal measures of Western Pennsylvania. The following tables 



ANALYSES OE COALS. 



627 



show the great similarity in composition in the coals of these upper and 
lower coal-measures in the same geographical belt or basin. 

Analyses froni the Upper Coal-measures (Penna.) in a 
Westward Order. 

Localities. Moisture. Vol. Mat. Fixed Carb. Ash. Sulphur. 

Anthracite 1.35 3.45 89.06 5.81 0.30 

Cumberland, Md 0.89 15.52 74.28 9.29 0.71 

Salisbury, Pa 1.66 22.35 68.77 5.96 1.24 

Connellsville, Pa 31.38 60.30 7.24 1.09 

Green sbnrg, Pa 1.02 33.50 61.34 3.28 0.86 

Irwin's, Pa 1.41 37.66 54.44 5.86 0.64 

Analyses from tlie Lower Coal-measures in a Westward 
Order. 

Localities. Moisture. Vol. Mat. Fixed Carb. Ash. Sulphur. 

Anthracite 1.35 3.45 89.06 5.81 0.30 

Broad Top 0.77 18.18 73.34 6.69 1.02 

Bennington 1.40 27.23 61.84 6.93 2.60 

Johnstown 1.18 16.54 74.46 5.96 1.86 

Blairsville 0.92 24.36 62.22 7.69 4.92 

Armstrong Co 0.96 38.20 52.03 5.14 3.66 

Pennsylvania and Ohio Bituminous Coals. Variation 
in Character of Coals of the same Beds in different Dis- 
tricts.— From 50 analyses in the reports of the Pennsylvania Geological 
Survey, the following are selected. They are divided into different groups, 
and the extreme analysis in each group is given, ash and other impurities 
being neglected, and the percentage in 100 of combustible matter being 
alone considered. 



Waynesburg coal-bed, upper bench. . . . 

Jefferson township, Greene Co 

Hopewell township, Washington Co.. 
Waynesburg coal-bed, lower bench 

Morgan township, Greene Co 

Pleasant Valley, Washington Co 

Sewickley coal-bed. 

Whitely Creek. Greene Co 

Gray's Bank Creek, Greene Co 

Pittsburgh coal-bed: 

Upper bench, Washington Co 

Lower bench, " " 

Main bench, Greene Co 

Frick & Co., Washington Co., average 

Lower bench, Greene Co 

Somerset Co., semi-bituminous (showing 

decrease of vol. mat. to the eastward) 
Beaver Co., Pa 

Diehl's Bank, Georgetown 

Bryan's Bank, Georgetown 

Ohio. 
Pittsburgh coal-bed in Ohio: 
Jefferson Co., Ohio 

Belmont Co., Ohio 

Harrison Co., Ohio 

Pomeroy Co., Ohio 



"No. of 


Fixed 


Vol. 


Analyses 


Carbon 


H. C. 


5 








59.72 


40.28 




53.22 


46.78 


9 








60.69 


39.31 




54.31 


45.69 


3 








64.39 


35.61 




60.35 


39.65 




j 60.87 


39.13 




1 59.11 


40.89 


5 


j 63.54 


36.46 


j 50.97 


49.03 


3 


j 61.80 


38.20 


j 54.33 


45.67 




66.44 


33.56 


1 


57.83 


42.17 


\ 8 


(79.73 

| 75.47 


20.27 


24.53 


' 


40.68 


59.32 




62.57 


37.43 




61.45 


38.55 




j 63.46 


36.54 




)66.14 


33.86 




I 63.46 


36.54 




(64.93 


35.07 




( 60.92 


39.08 




"(62.33 


37.67 



Carbon 
Ratio. 



1.48 
1.13 



1.54 

1.19 



1.80 
1.52 



1.74 
1.04 
1.61 
1.19 
1.98 
1.37 



0.68 
1.66 



1.59 
1.73 
1.95 
1.73 
1.85 
1.55 
1.65 



628 



Analyses of Southern and Western Coals. 





Moisture. 


Vol. Mat. 


Fixed C. 


Ash. 


Sul- 
phur. 


Ohio. 


j 5.00 
1 7.40 

j 95 
| 1.23 

j from 0.67 
| to 2.46 
1.48 
j 0.40 
1 1.79 
( 1 .57 
"I 1.56 

] from 

1 to .. 

( 0.52 
| 0.62 

J from 0.76 
I to 0.94 
j 0.34 
1 1.35 

(from 0.80 
| to 2.01 

(from 1.26 
| to 1.32 
j from 3.60 
1 to 7.06 

j from 

1 to .... 

j from 70 

1 to 1.83 

1.75 

2.74 

94 

1.60 

1.30 

1.20 

3.01 

.12 

1.59 

2.00 

1.78 


32.80 
29.20 

19.13 
15.47 

27.28 
38.60 
32.24 
18.60 
23.96 
9.64 
14.26 
21.33 
30.50 
26.06 
23.90 
18.48 

17.57 
18.19 
29.59 
25.35 

31.44 
36.27 

35.15 
39.44 
30.60 
38.70 
40.201 
63.301 

32.33 

41.29 
26.62 
26.50 
23.72 
29.30 
21.80 

23.05 

42.76 
26.11 

38.33 
32.90 
30.60 


53.15 
60.45 

72.70 
73.51 

46.70 

67.83 
58.89 
71.00 
59.98 
79.93 
81.61 
54.97 
70.80 
63.75 
74.20 
75.22 

75.89 
79.40 
69.00 
70.67 

54.80 
63.50 

60.85 
52.48 
58.80 
53.70 
59.80 coke 
33.70 coke 

46.61 
61.66 
60.11 
67.08 
63.94 
61.00 
74.20 

60.50 

48.30 
71.64 
54.64 
53.08 
66.58 


9.05 
2.95 

6.40 
9.09 

2.00 

15.76 
7.72 
10.00 
14.28 
8.86 
2.24 
3.35 
22.60 
10.06 
1.38 
5.68 

1.11 
4.92 
1.07 
2.10 

1.73 
8.25 

1.23 
5.52 
3.40 
6.50 
8.81 
4.80 

16.94 
1.11 

11.52 
3.68 

11.40 
7.80 
2.70 

15.16 

3.21 
2.03 
5.45 

11.34 
1.09 


0.44 


Maryland. 


0.93 
0.78 


Virginia. 
South of James River, 23 anal- 
yses, range 


0.70 

0.58 
2.89 
1.45 

0.23 


North of James River, eastern 

outcrop, 
Carbonite or Natural Coke 

Western outcrop, 11 analyses, 
range 


Pocahontas Flat-top* 
(Castner & Curran's Circular) 
West Virginia (New River.) 

Quinnimont,t 3 analyses 


0.52 
0.28 

0.23 
0.30 


Virginia and Kentucky. 
Big Stone Gap Field, % 9 anal- 
yses, range 

Kentucky. 
Pulaski Co., 3 analyses, range 

Muhlenberg Co., 4 analyses, 
range 

Kentucky Cannel Coals, § 5 an- 
alyses, range 

Tennessee. 

Scott Co., Range of several. 1.. 

Roane Co., Rockwood 

Hamilton Co., Melville 


0.08 

0.56 
1.72 

0.40 
1.00 
0.79 
3.16 
0.96 
1.32 

3.37 

0.77 

1.49 

91 

1 19 


Sewanee Co., Tracy City 

Kelly Co., Whiteside 




Georgia. 


84 


Alabama. 
Warren Field: 
Jefferson Co., Birmingham.. 
" Black Creek.. 

Tuscaloosa Co 

Cahaba Field, 1 Helena Vein . 
Bibb Co ("Coke Vein.... 


2.72 
.10 

1.33 
.68 
.04 



* Analyses of Pocahontas Coal by John Pattinson, F.C.S., 

C. H. O. N. S. Ash. Water. Coke. J°{/ 

Lumps... 86.51 4.44 4.95 0.66 0.61 1.54 1.29 78.8 21.2* 
Small ... 83.13 4.29 5.33 0.66 0.56 4.63 1.40 79.8 20.2 

Calorific value, by Thomson's Calorimeter: Lumps = 15.4 lbs. of water 
evaporated from and at 212°; small = 14.7 lbs. 

t These coals are coked in beehive ovens, and yield from 63$ to 64$ of coke. 

JThis field covers about 120 square miles in Virginia, and about 30 square 
miles in Kentucky. 

§ The principal use of the cannel coals is for enriching illuminating-gas. 

|| Volatile matter including moisture. 

1 Single analyses from Morgan, Rhea, Anderson, and Roane counties fall 
within this range. 



ANALYSES OF COALS. 



G29 



Alabama Coals. (W. B. Phillips, Eng. & M. J., June 3, 1 





Location. 


Proximate. 


Ultimate. 


Name of 
Seam. 


ill 


c 
o 

is 


d 

o 

3 
o 


bJ3 
O 


O 


■a 
<v 

o 


- 
ft 
W 


,d 

< 


o 


Wadsworth 

Pratt 

Brookwood 
Woodstock. 
Underwood 

Pratt 

Milldale.... 

Cab aba 
Field 


Helena 

Pratt mines.. 
Brookwood.. 
Blocton 

Pratt mines.. 
Brookwood . . 
Blue Creek . . 
Coalburg — 


34.30 
33.45 

27.80 
34.80 
35.65 
31.55 
30.50 
25.80 
32.55 

30.15 


60.50 
03.20 
58.70 
liO.liO 
57.30 
64.95 
oo.3<) 

09.90 

65.57 
52.90 


73.23 

75.82 
72.47 
72.75 
70.82 
75.05 
73.96 
72.08 
74.59 

60.37 


7.98 
10.52 
10.38 

8.61 
10.19 

9.91 
10.50 
10.77 
10.58 

10.70 


11.92 
7.51 
1.60 

11.12 
9.95 
8.95 
9.57 
9.83 
9.48 

9.00 


1.07 
1.73 
0.40 
1.48 

1.31 
1.62 
1 . 62 
1.3!) 
1.31 

1.26 


0.60 
1.07 
1.65 
1.44 
0.6S 
0.97 
1.15 
1.03 
1.32 

1.72 


3.50 
2.00 
11.90 
2.65 
5.25 
2.35 
2.20 
2.80 
1.90 

16.30 


1.70 
1.35 
1.60 
1.95 
1.80 
1.15 
1.00 
1.50 
0.82 

0.65 



Texas. 

Eagle Mine 

Sabinas Fiel d, Vein I. . . 

" II... 

"III... 

" IV... 

Indiana. 

Block coal, average.*. ., 

" " Lafayette ... 

" " Sand Creekt 

Illinois.^ 

La Salle 

Streator 

Danville 

Lincoln 

Barclay 

Carbondale 

Du Quoin 

Mt. Carbon , 

Staunton 



3.54 
1.91 
1.37 
0.84 
0.45 

2.10 
13.05 



7.20 
11.00 
5.78 
8 45 
10.80 
6.36 
8.86 
6.12 
6.27 



30.84 
20.04 
16.42 
29.35 
21.6 

37.35 
32.34 



50.69 
62.71 
68.18 
50.18 
45 . 75 

57.95 



39.40 
38.88 
32.55 
43.70 
34.99 
27.32 
26.40 
23.54 
24.68 
57.11 



3.95 
5.30 
3.00 



Sul- 
phur. 



15.35 
13.02 



2.60 
5.81 

4.50 

8.43 
8.60 
3.65 
6.15 
12.06 
17.10 
7.40 
7.00 
2.70 
10. 'Si 



* Indiana Block Coal (J. S. Alexander, Trans. A. I. M. E., iv. 100).— The 
typical block coal of the Brazil (Indiana) district differs in chemical com- 
position but little from the coking coals of Western Pennsylvania. The 
physical difference, however, is quite marked; the latter has a cuboid struc- 
ture made up of bituminous particles lying against each other, so that under 
the action of heat fusion throughout the mass readily takes place, while 
block coal is formed of alternate layers of rich bituminous matter and a 
charcoal-like substance, which is not only very slow of combustion, but so 
retards the transmission of heat that agglutination is prevented, and the 
coal burns away layer by layer, retaining its form until consumed. 

t Analysis by E. T. Cox: C, 72.94; H, 4.50; O, 11.77; N, 1.79; ash, 4.50; 
moisture, 4.50. 

t The Illinois coals are extremely variable in character. The above anal- 
yses are given in D. L. Barnes's paper on " American Locomotive Practice," 
Trans. A. S. C. E. 1893. except the last, the Staunton coal, which is by Hunt 
and Clapp (Trans. A. S. M. E., v. 266). The Staunton coal is remarkable for 
the high percentage of volatile matter, but it is excelled in this respect by 



630 



Iowa. * 

Hiteman 

Keb 

Flaglers , 

Chisholm 

Missouri.* 

Brookfield 

Mendota 

Hamilton 

Lingo 

Nebraska.* 
Hastings 

Wyoming.* 
Cambria 



Goose Creek. . 



Deek Creek. . 
Sheridan ... 



Colorado.:}: 
Sunshine, Colo, average. . 
Newcastle, " " 

El Moro, " " 

Crested Buttes, " 

Utah (Southern). 

Castledale 

Cedar City 

Oregon. 
Coos Bay 



Yaquina Bay . . 
John Day River.. 



Vancouver Island. 
Comox Coal 



Moisture. 


Vol. Mat. 


Fixed C, 


Ash. 


4.99 
9.81 
9.84 
9.18 


35.27 
37.49 
40.16 
40.42 


25.37 

44.75 
37.69 
39.58 


34.37 
7.95 
12.31 

10.82 


4.34 
9.03 
5.06 
7.33 


40.27 
37.48 
34.24 
38.29 


50.60 
46.24 
47.69 
47.24 


4.79 
7.25 
13.01 
7.14 


0.21 


27.82 


60.88 


11.09 


4.2 
2.5 
9.7 
13.92 
12.8 
6.04 


40.6 
37.4 
40.2 
36.78 
35.0 
42.37 


41.5 
37.9 
46.3 
42.03 

47.7 
35.57 


13.7 
22.2 
3.8 
7.27 
3.6 
16.02 


2.8 
1.7 
1.32 
1.10 


36.3 
37.95 
38.23 
23.20 


48^6 
55.86 
72.60 


23.8 
11.6 
3.59 
3.10 


3.43 
3.50 


42.81 
43.66 


47.81+ 
43.11+ 


9.73 
5.95 


15.45 
17.27 
13.03 
4.55 
6.54 


41.55 
44.15 
46.20 
40.00 
34.45 


34.95 
32.40 
32.60 
48.19 
52.41 


8.05 
6.18 
7.10 
7.26 
5.95 


1.7 




68.27 


2.86 



Sul- 
phur. 



2.53 
1.37 
1.07 
.60 
.65 



the Boghead coal of Linlithgowshire, Scotland, an analysis of which by Dr. 
Penny is as follows: Proximate— moisture 0.84; vol. 67.95; fixed C, 9.54, ash, 
21.4; Ultimate— C,63. 94; H, 8.86; O, 4.70; N, 0.96; which is remarkable for the 
high percentage of H. 

* The analyses of Iowa, Missouri, Nebraska, and Wyoming coals are 
selected from a paper on The Heating Value of Western Coals, by Win. 
Forsyth, Mech. Engr. of the C, B. & Q. R. R., Eng'g News, Jan. 17, 1895. 

+ Includes sulphur, which is very high. Coke from Cedar City analyzed : 
Water and volatile matter, 1.42; fixed carbon, 76.70; ash, 16.61; sulphur, 5.27. 

% Colorado Coals.— The Colorado coals are of extremely variable com- 
position, ranging all the way from lignite to anthracite. G. C. Hewitt 
(Trans. A. I. M. E., xvii. 377) says : The coal seams, where unchanged 
by heat and flexure, carry a £gnite containing from h% to 20$ of water. In 
the south-eastern corner of the field the same have been metamorphosed so 
that in four miles the same seams are an anthracite, coking, and dry coal. 
In the basin of Coal Creek the coals are extremely fat, and produce a hard, 
bright, sonorous coke. North of coal basin half a mile of development 
shows a gradual change from a good coking coal with patches of dry coal to 
a dry coal that will barely agglutinate in a" beehive oven. In another half 
mile the same seam is dry. In this transition area, a small cross-fault 
makes the coal fat for twenty or more feet on either side. The dry seams 
also present wide chemical and physical changes in short distances. A soft 
and loosely bedded coal has in a hundred feet become compact and hard 
without tire intervention of a fault. A couple of hundred feet has reduced 
the water of combination from 12$ to 5$. 

Western Arkansas and Indian Territory. (H. M. Chance, 
Trans. A. I. M. E. 1890.)— The Choctaw coal-field is a direct westward exten- 



ANALYSES OF COALS. 



631 



sion of the Arkansas coal-field, but its coals are not like Arkansas coals, ex- 
cept in the country immediately adjoining the Arkansas line. 

The western Arkansas coals are dry semi-bituminous or semi-anthracitic 
coals, mostly uon -coking, or with quite feeble coking properties, ranging 
from 14% to 16% in volatile matter, the highest percentage yet found, accord- 
ing to Mr. Winslow's Arkansas report, being 17 655. 

In the Mitchell basin, about 10 miles west from the Arkansas line, coal 
recently opened shows )9% volatile matter; the Mayberry coal, about 8 miles 
farther west, contains 23% volatile matter; and the Bryan Mine coal, about 
the same distance west, shows 26% volatile matter. About 30 miles farther 
west, the coal shows from 38$ to 4iy s % volatile matter, which is also about 
the percentage in coals of the McAlester and Lehigh districts. 
Western Lignites. (R. W. Raymond, Trans. A. I. M. E., vol. ii. 1873.) 



c. 


H. 


N. 


59.72 


5. OS 


1.01 


64.84 


4,34 


1.29 


69.84 


3.90 


1.93 


64.99 


3.76 


1.74 


69.14 


4.36 


1.25 


56.24 


3.oS 


0.42 


55.79 


3.2(i 


0.61 


67.67 


4. (;t; 


1.58 


67.58 


7.42 




60.72 


4 . 30 





S. 


Mois- 


4 


ture. 


< 


3.92 


8.94 


5.64 


1.60 


9 41 


3.00 


0.7; 


9.17 


3.40 


1.07 


11.56 


1 68 


1.03 


8.06 


6 62 


0.S1 


13.28 


4.05 


0.C3 


16.52 


4.18 


0.92 


3.08 


9.28 


0.63 


5.18 


5.77 


2.08 


14.68 


3.80 



Calorific 
Power, 
calories. 



Monte Diabolo 

Weber Canon, Utah. . . 

Echo Canon , Utah 

Carbon Station, Wyo . 

Coos Bay, Oregon 

Alaska 



Canon City, Colo 

Baker Co., Ore 



15.69 
15.52 
10.99 
15.20 
9.54 
21.82 
19.01 
12.80 
13.42 
14.42 



5912 
6400 
5738 
6578 
4565 
4610 
6428 
7330 
5602 



The calorific power is calculated by Dulong's formula, 

8080C + 34462(h - -^), 

deducting the heat required to vaporize the moisture and combined water, 
that is, 537 calories for each unit of water. 1 calorie = 1.8 British thermal 
units. 
Analyses of Foreign Coals. (Selected from D. L. Barnes's paper 
on American Locomotive Practice, A. S. C. E., 1893.) 



Great Britain : 
South Wales. . 



Lancashire, Eng.. 
Derbyshire, " .. 
Durham, " .-, 
Scotland 



Staffordshire, Eng 
South America: 

Chili, Conception Bay 
" Chiroqui 

Patagonia 

Brazil 

Canada: 

Nova Scotia 

Cape Breton 

Australia 

Australian lignite 

Sydney, South Wales, 

Borneo 

Van Diemen's Land 



Volatile 


Fixed 


Ash. 


Matter. 


Carbon. 


8.5 


88.3 


3.2 


6.2 


92.3 


1.5 


17.2 


80.1 


2.7 


17.7 


79.9 


2.4 


15 05 


86.8 


1.1 


17.1 


63.1 


19.8 


17.5 


80.1 


2.4 


20.4 


78.6 


1.0 


21.93 


70.55 


7.52 


24.11 


38.98 


36.91 


24.35 


62 25 


13.4 


40.5 


57.9 


1.6 


26.8 


60.7 


12.5 


26.9 


67.6 


5.5 


15.8 


64.3 


10.0 


14.98 


82.39 


2.04 


26.5 


70.3 


14.2 


6.16 


63.4 


30.45 



Semi-bit. coking coal. 
Boghead cannel gas coal. 
Semi-bit. steam-coal. 



An analysis of Pictou, N. S., coal, in Trans. A. I. M. E., xiv. 560, is: Vol., 
29.63; carbon. 56.98; ash, 13.39; and one of Sydney, Cape Breton, coal is: 
vol., 34.07; carbon, 61.43; ash, 4.50. 



632 FUEL. 

Nixon's Navigation Welsh. Coal is remarkably pure, and con- 
tains not more than 3 to 4 per cent of ashes, giving 88 per cent of hard and 
lustrous coke. The quantity of fixed carbon it contains would classify it 
among the dry coals, but on account of its coke and its intensity of com- 
bustion it belongs to the class of fat, or long flaming coals. 

Chemical analysis gave the following results: Carbon, 90.27; hydrogen, 
4.39; sulphur, .69; nitrogen. .49; oxygen (difference), 4.16. 

The analysis showed the following composition of the volatile parts: Car- 
bon, 22.53; hydrogen, 34.96 ; O + N + S, 42.51. 

The heat of combustion was found to be, as a result of several experi- 
ments, 8864 calories for the unit of weight. Calculated according to its 
composition, the heat of combustion would be 8805 calories = 15,849 British 
thermal units per pound. 

This coal is generally used in trial-trips of steam-vessels in Great Britain. 

Sampling Coal for Analysis,— J. P. Kimball, Trans. A. I. M. E., 
xii. 317, says : The unsuitable sampling of a coal-seam, or the improper 
preparation of the sample in the laboratory, often gives rise to errors in de- 
terminations of the ash so wide in range as to vitiate the analysis for all 
practical purposes ; every other single determination, excepting moisture, 
showing its relative part of the error. The determination of sulphur and 
ash are especially liable to error, as they are intimately associated in the 
slates. 

Wm. Forsyth, in his paper on The Heating Value of Western Coals (Eng'g 
News, Jan. 17, 1895), says : This trouble in getting a fairly average sample of 
anthracite coal has compelled the Reading R. R. Co. , in getting their samples, 
to take as much as 300 lbs. for one sample, drawn direct from the chutes, as 
it stands ready for shipment. 

The directions for collecting samples of coal for analysis at the C, B. & Q. 
laboratory are as follows : 

Two samples should be taken, one marked " average," the other " select." 
Each sample should contain about 10 lbs., made up of lumps about the size- 
of an orange taken from different parts of the dump or car, and so selected 
that they shall represent as nearly as possible, first, the average lot; second, 
the best coal. 

An example of the difference between an "average" and a "select" 
sample, taken from Mr. Forsyth's paper, is the following of an Illinois coal: 
Moisture. Vol. Mat. Fixed Carbon. Ash. 

Average 1.36 27.69 35.41 35.54 

Select 1.90 34.70 48.23 15.17 

The theoretical evaporative power of the former was 9.13 lbs. of water 
from and at 212° per lb. of coal, and that of the latter 11.44 lbs. 

Relative Value of Fine Sizes of Anthracite.— For burning 
on a grate coal-dust is commercially valueless, the finest commercial an- 
thracites being sold at the following rates per ton at the mines, according 
to a recent address by Mr. Eckley B. Coxe (1893): 

Size. Ransre of Size. Price at Mines. 

Chestnut 1]4 to % inch $2.75 

Pea %to9/16 1.25 

Buckwheat 9/16 to % 0.75 

Rice %to3/16 0.25 

Barley 3/16 to 2/32 0.10 ' 

But when coal is reduced to aa impalpable dust, a method of burning it 
becomes possible to which even the finest of these sizes is wholly una- 
dapted; the coal may be blown in as dusc. mixed with its proper proportion 
of air. and no grate at all is then required. 

Pressed Fuel. (E. F. Loiseau, Trans. A. I. M. E., viii. 314.)— Pressed 
fuel has been made from anthracite dust by mixing the dust with ten per 
cent of its bulk of dry pitch, which is prepared by separating from tar at a 
temperature of 572° F. the volatile matter it contains. The mixture is kept 
heated by steam to 212°, at which temperature the pitch acquires its ce- 
menting properties, and is passed between two rollers, on the periphery of 
which are milled out a series of semi-oval cavities. The lumps of the mix- 
ture, about the size of an egg, drop out under the rollers on an endless belt 
which carries them to a screen in eight minutes, which time is sufficient to 
cool the lumps, and they are then ready for delivery. 

The enterprise of making the pressed fuel above described was not com- 
mercially successful, on account of the low price of other coal. In France, 
however, " briquettes " are regularly made of coal-dust (bituminous and 
semi-bituminous). 



RELATIVE VALUE OF STEAM COALS. 633 

RELATIVE VALUE OF STEAM COALS. 

The heating value of a coal may be determined, with more or less approx- 
imation to accuracy, by three different methods. 

1st, by chemical analysis ; 2d. by combustion in a coal calorimeter ; 3d, 
by actual trial in a steam-boiler. The first two methods give what may be 
called the theoretical heating value, the third gives the practical value. 

The accuracy of the first two methods depends on the precision of the 
method of analysis or calorimetry adopted, and upon the care and skill of 
the operator. The results of the third method are subject to numerous 
sources of variation and error, and may be taken as approximately true 
only for the particular conditions under which the test is made. Analysis 
and caloi'imetiy give with considerable accuracy the heating value which 
may be obtained under the conditions of perfect combustion and complete 
absorption of the heat produced. A boiler test gives the actual result under 
conditions of more or less imperfect combustion, and of numerous and va- 
riable wastes. It may give the highest practical heating value, if the condi- 
tions of grate-bars, draft, extent of heating surface, method of firing, etc., 
are the best possible for the particular coal tested, and it may give results 
far beneath the highest if these conditions are adverse or uusuitable to the 
coal. 

The results of boiler tests being so extremely variable, their use for the 
purpose of determining the relative steaming values of different coals has 
frequently led to false conclusions. A notable instance is found in the 
record of Prof. Johnson's tests, made in 1844, the only extensive series of 
tests of American coals ever made. He reported the steaming value of the 
Lehigh Coal & Navigation Co.'s coal to be far the lowest of all the anthra- 
cites, a result which is easily explained bj T an examination of the conditions 
under which he made the test, which were entirely unsuited to that coal. 
He also reported a r esult for Pi ttsburgh coal which is far beneath that now 
obtainable in every-day practiceTBis low result being chiefly due to the use 
of an improper furnace. 

In a paper entitled Proposed Apparatus for Determining the Heating 
Power of Different Coals (Trans. A. I. M. E., xiv. 727) the author described 
tjad illustrated an apparatus designed to test fuel on a large scale, avoiding 
he errors of a steam-boiler test. It consists of a fire-brick furnace enclosed 
n a water- casing, and two cylindrical shells containing a great number of 
tubes, which are surrounded by cooling water and through which the gases 
of combustion pass while being cooled. \ No steam is generated in the ap- 
paratus, but water is passed through it and allowed to escape at a tempera- 
ture below 200° F. The product of the weight of the water passed through 
the apparatus by its increase in temperature is the measure of the heating 
value of the fuel. 

There has been much difference of opinion concerning the value of chemi- 
cal analysis as a means of approximating the heating power of coal. It 
was found by Scheurer-Kestner and Meunier-Dollfus, in their extensive series 
of tests, made in Europe in 1868, that the heating power as determined by 
calori metric tests was greater than that given to chemical analysis accord- 
ing to Dulong's law. 

Recent tests made in Paris by M. Mahler, however, show a much closer 
agreement of analysis and calorimetric tests. A brief description of these 
tests, translated from the French, may be found in an article by the author 
in The Mineral Industry, vol. i. page 97. 

Dulong's law may be expressed by the formula, 

Heating Power in British Thermal Units = 14,500C + 62,500 (h - ^-),* 

in which C, H, and O are respectively the percentage of carbon, hydrogen, 
and oxygen, each divided by 100. A study of M. Mahler's calorimetric tests 
shows that the maximum difference between the results of these tests and 
the calculated heating power by Dulong's law in any single case is only a 
little over 3$, and the results of 31 tests show that Dulong's formula gives' an 
average of only 47 thermal units less than the calorimetric tests, the 
average total heating value being over 14,000 thermal units, a difference of 
less than 4/10 of 1%. 

* Mahler gives Dulong's formula with'Berthelot's figure for the heating 
value of carbon, in British thermal units, 

Heating Power = 14,650C + G2,025 (h - 1 9+^ ~ * ^ t 



634 



Mahler's calorimetric apparatus consists of a strong steel vessel or 
" bomb " immersed in water, proper precaution being taken to prevent radi- 
ation. One gram of the coal to be tested is placed in a platinum boat within 
this bomb, oxygen gas is introduced under a pressure of 20 to 25 atmospheres, 
and the coal ignited explosively by an electric spark. Combustion is com- 
plete and instantaneous, the heat is radiated into the surrounding water, 
weighing 2200 grams, and its quantity is determined by the rise in tempera- 
ture of this water, due corrections being made for the heat capacity of the 
apparatus itself. The accuracy of the apparatus is remarkable, duplicate 
tests giving results varying only about 2 parts in 1000. 

The close agreement of the results of calorimetric tests when properly 
conducted, and of the heating power calculated from chemical analysis, in- 
dicates that either the chemical or the calorimetric method may be ac- 
cepted as correct enough for all practical purposes for determining the total 
heating power of coal. The results obtained by either method may be 
taken as a standard by which the results of a boiler test are to be com- 
pared, and the difference between the total heating power, and the result of 
the boiler test is a measure of the inefficiency of the boiler under the con- 
ditions of any particular test. 

In practice with good anthracite coal, in a steam-boiler properly propor- 
tioned, and with all conditions favorable, it is possible to obtain in the 
steam 80$ of the total heat of combustion of the coal. This result was nearly 
obtained in the tests at the Centennial Exhibition in 1876, in five different 
boilers. An efficiency of 70$ to 75$ may easily be obtained in regular prac- 
tice. With bituminous coals it is difficult to obtain as close an approach to 
the theoretical maximum of economy, for the reason that some of the vola- 
tile combustible portion of the coal escapes unburned, the difficulty increas- 
ing rapidly as the content of volatile matter increases beyond 20$. With 
most coals of the Western States it is with difficulty that as much as 60$ or 
65$ of the theoretical efficiency can be obtained without the use of gas-pro- 
ducers. 

The chemical analysis heretofore referred to is the ultimate analysis, or 
the percentage of carbon, hydrogen, and oxygen of the dry coal. It is found, 
however, from a study of Mahler's tests that the proximate analysis, which 
gives fixed carbon, volatile matter, moisture, and ash, may be relied on as 
giving a measure of the heating value with a limit of error of only about 3$. 
After deducting the moisture and ash, and calculating the fixed carbon as a 
percentage of the coal dry and free from ash, the author has constructed the 
following table : 

AppRoxtMATE Heating Value of Coals. 



Percentage 


Heating 


Equiv. Water 


Percentage 


Heating 


Equiv. Water 
Evap. from 


F. C. in 


Value 


Evap. from 


F. C. in 


Value 


Coal Dry 


B.T.U. 


and at 212° 


Coal Dry 


B.T.U. 


and at 212° 


and Free 


per lb. 


per lb. 


and Free 


per lb. 


per lb. 


from Ash. 


Comb'le. 


Combustible. 


from Ash. 


Comb'le. 


Combustible. 


100 


14500 


15.00 


68 


15480 


16.03 


97 


14760 


15.28 


63 


15120 


15.65 


94 


15120 


15.65 


60 


14580 


15.09 


90 


15480 


16.03 


57 


14040 


14.53 


87 


15660 


16.21 


54 


13320 


13.79 


80 


15840 


16.40 


51 


12600 


13.04' 


72 


15660 


16.21 


50 


12240 


12.67 



Below 50$ the law of decrease of heating-power shown in the table appar- 
ently does not hold, as some cannel coals and lignites show much higher 
heating-power than would be predicted from their chemical constitution. 

The use of this table may be shown as follows: 

Given a coal containing moisture 2$, ash 8$, fixed carbon 61$, and volatile 
matter 29$. what is its probable heating value ? Deducting moisture and 
ash we find the fixed carbon is 61/90 or 68$ of the total of fixed carbon and 
volatile matter. One pound of the coal dry and free from ash would, by the 
table, have a heating value of 15,480 thermal units, but as the ash and moist- 
ure, having no heating value, are 10$ of the total weight of the coal, the 
coal would have 90$ of the table value, or 13,932 thermal units. This divided 
by 966, the latent heat of steam at 212° gives an equivalent evaporation per 
lb. of coal of 14.42 lbs, 



RELATIVE VALUE OF STEAM COALS. 635 

The heating value that can be obtained in practice from this coal would 
depend upon the efficiency of the boiler, and this largely upon the difficulty 
of thoroughly burning its volatile combustible matter in the boiler furnace. 
If a boiler efficiency of 65$ could be obtained, then the evaporation per lb. of 
coal from and at 212° would be 14.42 x .65 = 9.3? lbs. 

With the best anthracite coal, in which the combustible portion is, say, 97$ 
fixed carbon and 3$ volatile matter, the highest result that can be expected 
in a boiler-test with all conditions favorable is 12.2 lbs. of water evaporated 
from and at 212° per lb u of c ombustible, which is 80$ of 15.28 lbs. the theo- 
retical heating: -power. IvVith the best semi-bituminous coals, such as Cum- 
berland and Pocahonta^rtffwhich the fixed carbon is 80$ of the total com- 
bustible, 12 5 lbs., or 76$ of the theoretical 16.4 lbs., may be obtained. For 
Pittsburgh coal, with a fixed carbon ratio of 68$, 11 lbs., or 69$ of the theo- 
retical 16.03 lbs., is about the best practically obtainable with the best boilers. 
With some good Ohio coals, with a fixed carbon ratio of 60$, 10 lbs., or 66$ 
of the theoretical 15.09 lbs., has been obtained, under favorable conditions, 
with a fire-brick arch' over the furnace. With coals mined west of Ohio, 
with lower carbon ratios, the boiler efficiency is not apt to be as high as 60$. 

From these figures a table of probable maximum boiler-test results from 
coals of different fixed carbon ratios may be constructed as follows: 

Fixed carbon ratio 97 80 68 60 54 50 

Evap. from and at 212° per lb. combustible, maximum in boiler- tests: 

12.2 12.5 11 10 8.3 7.0 

Boiler efficiency, per cent 80 76 69 66 60 55 

Loss, chimney radiation, imperfect combustion, etc : 

20 24 31 34 40 45 

The difference between the loss of 20$ with anthracite and the greater 
losses with the other coals is chiefly due to imperfect combustion of the 
bituminous coals, the more highly volatile coals sending up the chimney the 
greater quantity of smoke and un burned hydrocarbon gases. It is a measure 
of the inefficiency of the boiler furnace and of the inefficiency of heating- 
surface caused by the deposition of soot, the latter being primarily caused 
by the imperfection of the ordiuary furnace and its unsuitability to the 
proper burning of bituminous coal. If in a boiler-test with an ordinary fur- 
nace lower results are obtained than those in the above table, it is an indica- 
tion of unfavorable conditions, such as bad firing, w r rong proportions of 
boiler, defective draft, and the like, which are remediable. Higher results 
can be expected only with gas-producers, or other styles of furnace espe- 
cially designed for smokeless combustion. 

Kind of Furnace Adapted for Different Coals. (From the 
author's paper on "The Evaporative Power of Bituminous Coals," Trans. 
A. S. M. E., iv, 257.)— Almost any kind of a furnace will be found well 
adapted to burning anthracite coals and semi-bituminous coals containing 
less than 20$ of volatile matter. Probably the best furnace for burning 
those coals which contain between 20$ and 40$ volatile matter, including the 
Scotch, English, Welsh, Nova Scotia, and the Pittsburgh and Monongahela 
river coals, is a plain grate-bar furnace with a fire-brick arch thrown over 
it, for the purpose of keeping the«combustion-chamber thoroughly hot. The 
best furnace for coals containing over 40$ volatile matter will be a furnace 
surrounded hy fire-brick with a large combustion-chamber, and some spe- 
cial appliance for introducing very hot air to the gases distilled from the 
coal, or, preferably, a separate gas-producer and combustion-chamber, with 
facilities for heating both air and gas before they unite in the combustion - 
chamber. The character of furnace to be especially avoid- d in burning all 
bituminous coals containing over 20$ of volatile matter is the ordinary fur- 
nace, in w-liich the boiler is set directly above the grate bars, and in which the 
heating-surfaces of the boiler are directly exposed to radiation from the 
coal on the grate. The question of admitting air above the grate is still un- 
settled. The London Engineer recently said: " All our experience, extending 
over many years, goes to show that when the production of smoke is pre- 
vented by special devices for admitting air, either there is an increase in the 
consumption of fu4 or a diminution in the production of steam. * * * The 
best smoke-preventer yet devised is a good fireman." 

Downward-draught Furnaces.— Recent experiments show that 
with bituminous coal considerable saving may be made by causing the 
draught to go downwards from the freshly-fired coal through the hot coal 
on the grate. Similar good results are also obtained by the upward draught 
by feeding the fresh coal under the bed of hot coal instead of on top. (See 
Boilers.) 



636 



Calorimetric Tests of American Coals. -From a number of 
tests of American and foreign coals, made with an oxygen calorimeter, by 
Geo. H. Barrus (Trans. A. S. M. E., vol. xiv. 816), the following are selected, 
showing the range of variation: 





Total Heat 


Total Heat 


Percentage 


of Com- 


reduced to 


of Ash. 


bustion. 


Fuel free 




B. T. U. 


from Ash. 


J 6.1 


14,217 


15,141 


) 8.6 


12,874 


14,085 


j 3.2 

"1 6.2 


14.603 


15,086 


13,608 


14,507 


j 3.5 
1 5.7 


13,922 


14,427 


13,858 


14.696 


7.8 


13,180 


14.295 


7.7 


13,581 


14,714 


5.9 


12,941 


13,752 


10.2 


11,664 


12,988 


17.7 


10,506 


12,765 


8.7 


12,420 


13,602 


6.8 


12,122 


13,006 


j 10.5 


11,521 


12,873 


\ 9.1 


13,189 


14,509 



Semi-bituminous. 
George's Cr'k, CumbeiTd, Md.,10 tests 

Pocahontas, Va., 5 tests 

New River, Va., 6 tests 

Elk Garden, Va., 1 test 

Welsh, 1 test 

Bituminous. 

Youghiogheny, Pa., lump 

" " slack 

Frontenac, Kansas 

Cape Breton, (Caledonia) 

Lancashire, Eng , 

Anthracite, 11 tests 



Evaporative Power of Bituminous Coals. 

(Tests with Babcock & Wilcox Boilers, Trans. A. S. M. E., iv. 267.) 



Name of Coal. 



. Welsh 

. Anthracite scr's 1/5 

Powelton, Pa., 
Semi-bit. 4/5, 
. Pittsbg'h fine slack 
" 3d Pool lump 
. Castle Shannon, nr 

Pittsb'gh, % nut, 

% lump, 
. 111. " run of mine '' 
' Ind. block, " very 

good " 
. Jackson, O., nut .. 
1 Staunton, 111., nut.. 
. Renton screenings. I 
' Wellington scr'gs.. I 
' Black Diam. scr'gs I 
' Seattle screenings. ( 
' Wellington lump. . ' 
' Cardiff lump 

1 South Paine lump. 
' Seattle lump 











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Dura- 
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Test. 


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eg 


Y6)4 hrs 


40 


1679 


7.5 


6.3 


2.07 


11.53 


12.46 


146 


j-10Mh 
4 hrs 


60 


3126 


8.8 


17.6 


4.32 


11.32 


12.42 


272 


33.7 


1679 


12.3 


21.9 


4.47 


8.12 


9.29 


146 


10 " 


43.5 


2T6U 


4.8 


27.5 


4.76 


10.47 


11.00 


240 


U2J4h 


69.1 


4784 


10.5 


27.9 


4.13 


10.00 


11.17 


416 


6 days. 




1190 






1.41 


9.49 




104 


^d'ys 




1190 






2.95 


9.47 




104 


8 hrs. 


48 


335S 


9.0 


32 1 


4.11 


8.93 


9.88 


292 


8 " 


60 


;;m5s 


17.7 


25.1 


2.27 


5.09 


6.19 


292 


5 h 50 m 


21.2 


1.-04 


13.8 


81.6 


2.95 


6.88 


7.98 


136 


6 h 30 m 


21.2 


1504 


18.3 


27 


2.93 


7.89 


9.66 


136 


5 h 58 m 




1501 


19.3 36.4 


3.11 


6.29 


7.80 


136 


6 h 24 m 




150-4 


13.4 


31.3 


2.91 


6.86 


7.92 


136 


6h 19 m 


21.2 


1504 


18.8 


2S.2 


3.52 


9.02 


10.46 


136 


6 h 47 m 


21.2 


1564 


11.726.7 


3.69 


10.07 


11.40 


130 


7 h 23 m 




1504 


19.l|25.e 


3.35 


9.62 


11.89 


130 


6h 35 m 


21.2 




18.9|28.9 


3.53 


8.96 


10.41 


136 


6h 5 m 


21.2 




9.5 


34.1 


8.57 


7.68 


8.49 


130 



637 



!. Cincinnati, O. ; 
San Francisco, 



Place of Test: 1. London. England; 2. Peacedale, R. I. ; 

4. Pittsburgh, Pa.; 5. Chicago, 111.; 6. Springfield, O. ; 

Cal. 

In all the above tests the furnace was supplied with a fire-brick arch for 
preventing: the radiation of heat from the coal directly to the boiler. 

Weathering of Coal. (I. P. Kimball, Trans. A. I. M. E., viii. 204.)— 
The practical effect of the weathering of coal, while sometimes increasing 
its absolute weight, is to diminish the quantity of carbon and disposable 
hydrogen and to increase the quantity of oxygen and of indisposable hy- 
drogen. Hence a reduction in the calorific value. 

An excess of pyrites in coal tends to produce rapid oxidation and mechan- 
ical disintegration of the mass, with development of heat, loss of coking 
power, and spontaneous ignition. 

The only appreciable results of the weathering of anthracite within the 
ordinary limits of exposure of stocked coal are confined to the oxidation of 
its accessory p3rites. In coking coals, however, weathering reduces and 
finally destroys the coking power, while the pyrites are converted from the 
state of bisulphide into comparatively innocuous sulphates. 

Richters found that at a temperature of 158° to 180° Fahr., three coals lost 
in fourteen days an average of 3.6$ of calorific power. (See also paper by 
R. P. Rothwell, Trans. A. I. M. E., iv. 55.) 

COKE. 

Coke is the solid material left after evaporating the volatile ingredients of 
coal, either by means of partial combustion in furnaces called coke ovens, 
or by distillation in the retorts of gas-w r orks. 

Coke made in ovens is preferred to gas coke as fuel. It is of a dark-gray 
color, with slightly metallic lustre, porous, brittle, and hard. 

The proportion of coke yielded by a given weight of coal is very different 
for different kinds of coal, ranging from 0.9 to 0.35. 

Being of a porous texture, it readily attracts and retains water from the 
atmosphere, and sometimes, if it is kept without proper shelter, from 0.15 to 
0.20 of its gross weight consists of moisture. 

Analyses of Coke. 
(From report of John R. Procter, Kentucky Geological Survey.) 



Where Made. 


Fixed 
Carbon 


Ash. 


Sul- 
phur. 


Connellsville, Pa. (Average of 3 samples) 

Chattanooga, Tenn. " "4 " 

Birmingham, Ala. " "4 " 

Pocahontas, Va. " "3 " 

New River, W. Va. " "8 " 

Big Stone Gap, Ky. " "7 " 


88.96 
80.51 
87.29 
92.53 
92.38 
93.23 


9.74 
16.34 
10.54 
5.74 
7.21 
5.69 


0.810 

1.595 
1.195 
0.597 
0.562 
0.749 



Experiments in Coking. Connellsville Region. 
(John Fulton, Amer. Mfr., Feb. 10, 1893.) 



<u 


<D 


a 
> 
O 


t3 

o 


i 

s 
< 


o © 

si 


|i 

r=> o 

o 


q 

E-1 


Per cent of Yield. 


a 


6 


< 


it 


"S CD 
I 


3 s 


?? 
^ 

fU 




h. 


m. 


lb. 


lb. 


lb. 


lb. 


lb. 












1 


67 


00 


12,420 


99 


385 


7,518 


7,903 


00.80 


3 10 


60.53 


63.63 


35.57 


•2 


08 


00 


11,090 


90 


359 


6,580 


6,939 


00.81 


3.24 


59.33 


62.57 


36.62 


3 


45 


00 


9,120 


77 


272 


5,418 


5,690 


00.84 


2.98 


59.41 


62.39 


36 77 


4 


45 


00 


9,020 

41,650 


74 


349 


5,334 
24.850 


5,683 


00.82 


3.87 


59.13 


63.00 


36.18 




340 


1365 


26,215 


00.82 


3.28 


59.66 


62.94 


36.24 



These results show, in a general average, that Connellsville coal carefully 
coked in a modern beehive oven will yield 66.17$ of marketable coke, 2.30'^ 
of small coke or braize, and 0.82$ of ash. 



638 FUEL. 

The total average loss in volatile matter expelled from the coal in coking 
amounts to 30.71$. 

The modern beehive coke oven is 12 feet in diameter and 7 feet high at 
crown of dome. It is used in making 48 and 12 hour coke. 

In making these tests the coal was weighed as it was charged into the 
oven; the resultant marketable coke, small coke or braize and ashes 
weighed drv as they were drawn from the oven. 

Coal Washing.— In making coke from coals that are high in ash and 
sulphur, it is advisable to crush and wash the coal before coking it. A coal- 
washing plant at Brookwood, Ala., has a capacity of 50 tons per hour. The 
average percentage of ash in the coal during ten days' run varied from 14$ to 
21$, in the washed coal from 4 8$ to 8.1$. and in the coke from 6.1$ to 10.5$. 
During three months the average reduction of ash was 60.9$. (Eng. and 
Mining Jour., March 25, 1893.) 

Recovery of By-products in Coke Manufacture.— In Ger- 
many considerable progress has been made in the recovery of by-products. 
The Hoffman-Otto oven has been most largely used, its principal feature 
being that it is connected with regenerators. In 1884 40 ovens on this 
system were running, and in 1892 the number had increased to 1209. 

A Hoffman-Otto oven in Westphalia takes a charge of 6*4 tons of dry coal 
and converts it into coke in 48 hours. The product of an oven annually is 
1025 tons in the Ruhr district, 1170 tons in Silesia, and 960 tons in the Saar dis- 
trict. The yield from dry coal is 75$ (o 77$ of coke, 2.5$ to 3$ of tar, and 1.1$ 
to 1.2$ of sulphate of ammonia in the Ruhr district; 65$ to 70$ of coke, 4$ to 
4.5$ of tar, and l$to 1.25$ of sulphate of ammonia in the Upper Silesia region 
and 68$ to 72$ of coke, 4$ to 4.3$ of tar and 1.8$ to 1.9$ of sulphate of ammonia 
in the Saar district. A group of 60 Hoffman ovens, therefore, yields annually 
the following: 

Poke Tar Sulphate 

District. <**£ Tar Ammonia , 

tons. 

Ruhr 51,300 1860 780 

Upper Silesia 48,000 3000 840 

Saar 40,500 2400 492 

An oven which has been introduced lately into Germany in connection 
with the recovery of by-products is the Semet-Solvay, which works hotter 
than the Hoffman -Otto, and for this reason 73$ to 77$ of gas coal can be 
mixed with 23$ to 27$ of coal low in volatile matter, and yet yield a good 
coke. Mixtures of this kind yield a larger percentage of coke, but, oh the 
other hand, the amount of gas is lessened, and therefore the yield of tar and 
ammonia is not so great. 

In the manufacture of coke from soft coal in retort ovens, particularly in 
those constructed so as to save the by-products formed in the coking oper- 
ations, the coke has the disadvantage of being more porous, softer, with 
more easily crushed" cell-walls than when the same coal is coked in the 
ordinary beehive-oven. 

References: F. W. Luerman, Verein Deutscher Eisenhuettenleute 1891, 
Iron Age, March 31, 1892 ; Amer: Mfr., April 28, 1893. An excellent series 
of articles on the manufacture of coke, by John Fulton, of Johnstown, Pa., 
is published in the Colliery Engineer, beginning in January, 1893. 

Making Hard Coke.— J. J. Fronheiser and C. S Price, of the Cam- 
bria Iron Co., Johnstown. Pa., have made an improvement in coke manu- 
facture by which coke of any degree of hardness may be turned out. It is 
accomplished by first grinding the coal to a coarse powder and mixing it 
with a hydrate of lime (air or water slacked caustic lime) before it is 
charged into the coke-ovens. The caustic lime or other fluxing material 
used is mechanically combined with the coke, filling up its cell-walls. It has 
been found that about 5$ by weight of caustic lime mixed with the fine coal 
gives the best results. However, a larger quantity of lime can be added to 
coals containing more than 5$ to 7$ of ash (Amer. Mfr.) 

Generation of Steam from the "Waste Meat and Gases 6f 
Coke-ovens. (Erskine Ramsey, Amer. Mfr., Feb. 16, 1894.)— The gases 
from a number of adjoining ovens of the beehive type are led into a long 
horizontal flue, and thence to a combustion-chamber under a battery of 
boilers. Two plants are in satisfactory operation at Tracy City, Tenn., and 
two at Pratt Mines. Ala. 

A Bushel of Coal.— The weight of a bushel of coal in Indiana is 70 lbs., 
in Penna. 76 lbs.; in Ala., Colo., Ga., 111., Ohio, Tenn., and W. Va. it is 80 lbs. 

A Bushel of Coke is almost uniformly 40 lbs., but in exceptional 



WOOD AS FUEL. 639 

cases, when the coke is very light, 38, 36, and 33 lbs. are regarded as a bushel. 
In others, from 42 to 50 lbs are given as the weight of a bushel ; in this case 
the coke would be quite heavy. 

Products of the Distillation of Coal.— S. P. Sadler's Handbook 
of Industrial Organic Chemistry gives a diagram showing over 50 chemical 
products that are derived from distillation of coal. The first derivatives are 
coal-gas, gas-liquor, coal-tar, and coke. From the gas-liquor are derived 
ammonia and sulphate, chloride and carbonate of ammonia. The coal-tar 
is split up into oils lighter than water or crude naphtha, oils heavier than 
water — otherwise dead oil or tar. commonly called creosote, — and pitch. 
From the two former are derived a variety of chemical products. 

From the coal-tar there comes an almost endless chain of known combina- 
tions. The greatest industry based upon their use is the manufacture of 
dyes, and the enormous extent to which this has grown can be judged from 
the fact that there are over GOO different coal-tar colors in use, and many more 
which as yet are too expensive for this purpose. Many medicinal prepara- 
tions come from the series, pitch for paving purposes, and chemicals for 
- the photographer, the rubber manufacturers and tanners, as well as for 
preserving timber and cloths. 

The composition of the hydrocarbons in a soft coal is uncertain and quite 
complex; but the ultimate analysis of the average coal shows that it ap- 
proaches quite nearly to the composition of CH 4 (marsh-gas). (W. H. 
Blauvelt, Trans. A. I. M. E., xx. 625.) 

WOOD AS FUEL. 

Wood, when newly felled, contains a proportion of moisture which varies 
very much in different kinds and in different specimens, ranging between 
30$ and 50$, and being on an average about 40%. After 8 or 12 months' ordi- 
nary drying in the air the proportion of moisture is from 20 to 25$. This 
degree of dryness, or almost perfect dryness if required, can be produced 
by a few days' drying in an oven supplied with air at about 240° F. When 
coal or coke is used as the fuel for that oven. 1 lb. of fuel suffices to expel 
about 3 lbs. of moisture from the wood. This is the result of experiments 
on a large scale by Mr. J. R. Napier. If air- dried wood were used as 
fuel for the oven, from 2 to 2% lbs. of wood would 'probably be required to 
produce the same effect. 

The specific gravity of different kinds of wood ranges from 0.3 to 1.2. 

Perfectly dry wood'contains about 50$ of carbon, the remainder consisting 
almost entirely of oxygen and hydrogen in the proportions which form 
water. The coniferous family contain a small quantity of turpentine, which 
is a hydrocarbon. The proportion of ash in w r ood is from 1$ to 5$. The 
total heat of combustion of all kinds of wood, when dry, is almost ex- 
actly the same, and is that due to the 50$ of carbon. 

The above is from Rankine; but according to the table by S. P. Sharpless 
in Jour. 0. 1. W., iv. 36, the ash varies from 0.03$ to 1.20$ in American woods, 
and the fuel value, instead of being the same for all woods, ranges from 
3667 (for white oak) to 5546 calories (for long-leaf pine) = 6600 to 9883 British 
thermal units for dry wood, the fuel value of 0.50 lbs. carbon being 7272 
B. T. TJ. 

Heating Value of Wood.— The following table is given in several 
books of reference, authority and quality of coal referred to not stated. 

The weight of one cord of different woods (thoroughly air-dried) is about 
as follows : 

Hickory or hard maple 4500 lbs. equal to 1800 lbs. coal. (Others give 2000.) 

White oak 3850 " " 1540 " " ( " 1715.) 

Beech, red and black oak.. 3250 " " 1300 " " ( " 1450.) 

Poplar, chestnut, and elm.. 2350 " " 940 " " ( " 1050.) 

The average pine 2000 " " 800 " " ( ' 925.) 

Referring to the figures in the last column, it is said : 

From the above it is safe to assume that 2*4 lbs. of dry wood are equal to 
1 lb. average quality of soft coal and that the full value of the same weight 
of different woods is very nearly the same — that is, a pound of hickory is 
worth no more for fuel than a pound of pine, assuming both to be dry. It 
is important that the wood be dry, as each 10$ of water or moisture in wood 
will detract about 12$ from its value as fuel. 

Taking an average wood of the analysis C 51$, H 6.5$, O 42.0$, ash 0.5%, 
perfectly dry, its fuel value per pound, according to Dulong's formula, Y — 



640 



[l4,500 C + 62,000 (H -^)], is 8170 British thermal units. If the wood, as 

ordinarily dried in air, contains 25$ of moisture, then the heating value of a 
pouud of such wood is three quarters of 8170 — 6127 heat-units, less the 
heat required to heat and evaporate the J4 lb. of water from the atmospheric 
temperature, and to heat the steam made from this water to the tempera- 
ture of the chimney gases, say 150 heat-units per pound to heat the water to 
212°, 966 units to evaporate it at that temperature, and 100 heat-units to 
raise the temperature of the steam to 420° F., or 1 216 in all = 304 for 14 lb., 
which subtracted from the 6127, leaves 5821 heat-units as the net fuel value 
of the wood per pound, or about 0.4 that of a pound of carbon. 

Composition of Wood. 

(Analysis of Woods, by M. Eugene Chevandier.) 









Composition. 






Carbon. 


Hydrogen. 


Oxygen. 


Nitrogen. 


Ash. 




49.36$ 
49.64 
50.20 
49.37 
49.96 


6.01$ 

5.92 

6.20 

6.21 

5.96 


42.69$ 
41.16 
41.62 
41.60 
39.56 


0.91$ 
1.29 
1.15 
0.96 
0.96 


1.06$ 


Oak .. 


1.97 


Birch 


0.81 
1.86 


Willow.......... 


3.37 






Average 


49.70$ 


6.06$ 


41.30$ 


1.05$ 


1.80$ 



The following table, prepared by M. Violette, shows the proportion of 
water expelled from wood at gradually increasing temperatures: 



Temperature. 


Water Expelled from 100 Parts of Wood. 


Oak. 


Ash. 


Elm. 


Walnut. 


257° Fahr 


15.26 
17.93 
32.13 
35.80 
44.31 


14.78 
16.19 
21.22 
27.51 
33.38 


15.32 
17.02 
36.94? 
33.38 
40.56 


15.55 


302° Fahr 

347° Fahr 


17.43 
21.00 




41.77? 


437° Fahr 


36.56 







The wood operated upon had been kept in store during two years. When 
wood which has been strongly dried by means of artificial heat is left ex- 
posed to the atmosphere, it reabsorbs about as much water as it contains 
in its air-dried state. 

A cord of ivood = 4 X 4 X 8 = 128 cu. ft. About 56$ solid wood and 44$ 
interstitial spaces. (Marcus Bull, Phila.. 1829. J. C. I. W., vol. i. p. 293.) 

B. E. Fernow gives the per cent of solid wood in a cord as determined offi- 
cially in Prussia (J. C. I. W., vol. iii. p. 20): 

Timber cords, 74.07$ = 80 cu. ft. per cord; 
Firewood cords (over 6" diam.), 69.44$ = 75 cu. ft. per cord; 
"Billet" cords (over 3" diam.j, 55.55$ = 60 cu. ft. per cord; 
" Brush " woods less than 3" diam., 18.52$; Roots, 37.00$. 

CHARCOAL. 

Charcoal is made by evaporating the volatile constituents of wood and 
peat, either by a partial combustion of a conical heap of the material to be 
charred, covered with a layer of earth, or by the combustion of a separate 
portion of fuel in a furnace, in which are placed retorts containing the ma- 
terial to be charged. 

According to Peelet, 100 parts by weight of wood when charred in a heap 
yield from 17 to 22 parts by weight" of charcoal, and when charred in a 
retort from 28 to 30 parts. 

This has reference to the ordinary condition of the wood used in charcoal- 
making, in which 25 parts in 100 consist of moisture. Of the remaining 75 
parts the carbon amounts to one half, or 37^$ of the gross weight of the 
wood, Hence it appears that on an average nearly half of the carbon in the 



CHARCOAL. 



641 



wood is lost during the partial combustion in a heap, and about one quarter 
during the distillation in a retort. 

To char 100 parts by weight of wood in a retort, 12^> parts of wood must 
be burned in the furnace. Hence in this process the whole expenditure of 
wood to produce from 28 to 30 parts of charcoal is 112^ parts; so tbat if the 
weight of charcoal obtained is compared with the whole weight of wood 
expended, its amount is from 25$ to 27%; and the proportion lost is on an 
average U% s- 37J^ = 0.3, nearly. 

According to Peclet, good wood charcoal contains about 0.07 of its weight 
of ash. The proportion of ash in peat charcoal is very variable, and is es- 
timated on an average at about 0.18. (Rankine.) 

Much infoi'mation concerning charcoal may be found in the Jom*nal of the 
Charcoal-iron Workers' Assn., vols. i. to vi. From this source the following 
notes have been taken: 

Yield of Charcoal from a Cord of Wood.— From 45 to 50 
bushels to the cord in the kiln, and from 30 to 35 in the meiler. Prof. Egle*- 
ton in Trans. A. I. M. E., viii. 395, says the yield from kilns in the Lake 
Champlain region is often from 50 to 60 bushels for hard wood and 50 for 
soft wood; the average is about 50 bushels. 

The apparent yield per cord depends largely upon whether the cord is a 
full cord of 128 cu. ft. or not. 

In a four months' test of a kiln at Goodrich, Tenn., Dr. H. M. Pierce found 
results as follows: Dimensions of kiln— inside diameter of base, 28 ft. 8 in.; 
diam. at spring of arch, 26 ft. 8 in. ; height of walls, 8 ft. ; rise of arch, 5 ft.; 
capacity, 30 cords. Highest yield of charcoal per cord of wood (measured) 
59.27 bushels, lowest 50.14 bushels, average 53.65 bushels. 

No. of charges 12, length of each turn or period from one charging to 
another 11 days. (J. C. I. W., vol. vi. p. 26.) 

Results from Different Methods of Charcoal-making. 





Character of Wood used. 


Yield. 


|8o 

I- O 
*°° 

63.4 
54.2 
66.7 
62.0 
59.5 

43.9 
45.0 

35 


22 


Coaling Methods. 


- Z 

- -j 
c ■_ 


l f "z 

> ~ 


•z%£ 


Odelstjerna's experiments 
Mathieu's retorts, fuel ex- 
cluded 

Mathieu's retorts, fuel in- 


Birch dried at 230 F 

( Air dry, av. good yel- ) 

< low pine weighing > 
{ abt. 28 lbs. percu. ft. ) 

j Good dry fir and pine, ) 
) mixed. j 
j Poor wood, mixed fir | 
j and pine j 
I Fir and white-pine 

< wood, mixed. Av. 25 j- 
( lhs. per cu. ft. \ 

Av. good yellow pine 

< weighing abt. 25 lbs. > 
( per cu. ft. j 


77.0 
65.8 
81.0 

70.0 

72.2 

52.5 

54.7 

42.9 


35.9 

28.3 

24.2 
27.7 
25 8 

24 7 

IS 3 
22.0 

17.1 


15.7 
15.7 


Swedish ovens, av. results 

Swedish ovens, av. results 
Swedish meiiers excep- 


13.3 

13.3 
13.3 


Swedish meiiers. av. results 
American kilns, av. results 
American meiiers, av. re- 
sults 


13.3 
17 5 

17.5 



Consumption of Charcoal in Blast-furnaces per Ton of 
Pig Iron ; average consumption according to census of 1880, 1.14 tons 
charcoal per ton of pig. The consumption at the best furnaces is much 
below this average. As low as S53 ton, is recorded of the Morgan furnace; 
Bay furnace, 0.858; Elk Rapids. 0.884. (1892.) 

Absorption of Water and of Oases by Charcoal. -Svedlius, 
in his hand-book for charcoal-burners, prepared for the Swedish Govern- 
ment, says: Fresh charcoal, also reheated charcoal, contains scarcely 
any water but when cooled it absorbs it very rapidly, so that after 
twenty-four hours, it may contain 4% to 8% of water. After the lapse of a 
few weeks the moisture of charcoal may not increase perceptibly, and may 
be estimated at 10$ to 15#, or an average of 12$. A thoroughly charred 
piece of charcoal ought, then, to contain about 84 parts carbon, 12 parts 
water, 3 parts ash, and 1 part hydrogen. 



642 FUEL. 

M. Saussure, operating with blocks of fine boxwood charcoal, freshly- 
burnt, found that by simply placing such blocks in contact with certain 
gases they absorbed them in the following proportion: 



Volumes. 

Carbonic oxide 9.42 

Oxygen 9.25 

Nitrogen 6.50 

Carburetted hydrogen ...... 5.00 

Hydrogen 1.75 



Volumes. 

Ammonia 90.00 

Hydrochloric-acid gas 85.00 

Sulphurous acid 65.00 

Sulphuretted hydrogen 55.00 

Nitrous oxide (laughing-gas). . 40.00 
Carbonic acid . . 35.00 

It is this enormous absorptive power that renders of so much value a 
comparatively slight sprinkling of charcoal over dead animal matter, as a 
preventive of the escape of odors arising from decomposition. 

In a box or case containing one cubic foot of charcoal may be stored 
without mechanical compression a little over nine cubic feet of oxygen, 
representing a mechanical pressure of one hundred and twenty -six pounds 
to the square inch. From the store thus preserved the oxygen can be 
drawn by a small hand-pump. 

Composition of Charcoal Produced at Various Tempera- 
tures. (By M. Violette.) 



Temperature of Car 
bonization. 



Cent. 
150° 
200 
250 
300 
350 
432 



392 
482 
592 

662 

810 
1873 



Composition of the Solid Product. 




Hydro- 
gen. 



Per cent. 
6.12 
3.99 
4.81 
4.25 
4.14 
4.96 
2.30 



Per cent. 
46.29 
43.98 
28.97 
21.96 
18.44 
15.24 
14.15 



Per cent. 
0.08 
0.23 
0.63 
0.57 
0.61 
1.61 
1.60 



Per cent. 
47.51 
39.88 
32.98 
24.61 
22.42 
15.40 
15.30 



The wood experimented on was that of black alder, or alder buckthorn, 
which furnishes a charcoal suitable for gunpowder. It was previously 
dried at 150 deg. C. = 302 deg. F. 



MISCELLANEOUS SOLID FUELS. 

Dust Fuel— Dust Explosions. —Dust when mixed in air burns with 
such extreme rapidity as in some cases to cause explosions. Explosions of 
flour-mills have been attributed to ignition of the dust in confined passages. 
Experiments in England in 1876 on the effect of coal-dust in carrying flame in 
mines showed that in a dusty passage the flame from a blown-out shot may 
travel 50 yards. Prof. F. A. Abel (Trans. A. I. M. E , xiii. 260) says that coal- 
dust in mines much promotes and extends explosions, and that it may read- 
ily be brought into operation as a fiercely burning agent which will carry 
flame rapidly as far as its mixture with air extends, and will operate as an 
explosive agent though the medium of a very small proportion of fire-damp 
in the air of the mine. The explosive violence of the combustion of dust is 
largely due to the instantaneous heating and consequent expansion of the 
air. (See also paper on " Coal Dust as an Explosive Agent," by Dr. R. W. 
Raymond, Trans. A. I. M. E. 1894.) Experiments made in Germany in 1893. 
show that pulverized fuel may be burned without smoke, and with high 
economy. The fuel, instead of being introduced into the fire-box in the 
ordinary manner, is first reduced to a powder by pulverizers of any con- 
struction. In the place of the ordinary boiler fire-box there is a combustion 
chamber in the form of a closed furnace lined with fire-brick and provided 
with an air-injector similar in construction to those used in oil-burning fur- 
naces. The nozzle throws a constant stream of the fuel into the chamber. 
This nozzle is so located that it scatters the powder throughout the whole 



MISCELLANEOUS SOLID FUELS. 643 

space of the fire-box. When this powder is once ignited, and it is very 
readily done by first raising the lining to a high temperature by an open 
fire, the combustion continues in an intense and regular manner under the 
action of the current of air which carries it in. (Mfrs. Record, April, 1893.) 

Powered fuel was used in the Crompton rotary puddhng-furnace at 
Woolwich Arsenal. England, in 1873. (Jour. I. & S. I., i. 1873, p. 91.) 

Peat or Turf, as usually dried in the air, contains from 25% to 30$ of 
water, which must be allowed for in estimating its heat of combustion. This 
water having been evaporated, the analysis of M. Regnault gives, in 100 
parts of perfectly dry peat of the best quality: C 58$, H 6%, O 31%, Ash 5%. 

In some examples of peat the quantity of ash is greater, amounting to 7% 
and sometimes to 11%. 

The specific gravitj 7 of peat in its ordinary state is about 0.4 or 0.5. It can 
be compressed by machinery to a much greater density. (Rankine.) 

Clark (Steam-engine, i. 61) gives as the average composition of dried Irish 
peat: C 59j6, H 6%, O 30, N 1.25%, Ash 4. 

Applying Dulong's formula to this analysis, we obtain for the heating value 
of perfectly dry peat 10,260 heat-units per pound, and for air-dried peat con- 
taining 25% of moisture, after making allowance for evaporating the water, 
7391 heat-units per pounds. 

Sawdust as Fuel.— The heating power of sawdust is naturally the 
same per pound as that of the wood from which it is derived, but if allowed 
to get wet it is more like spent tan (which see below). The conditions neces- 
sary for burning sawdust are that plenty of room should be given it in the 
furnace, and sufficient air supplied on the surface of the mass. The same 
applies to shavings, refuse lumber, etc. Sawdust is frequently burned in 
saw-mills, etc., by being blown into the furnace by a fan-blast. 

Horse-manure has been successfully used as fuel by the Cable Rail- 
way Co. of Chicago. It was mixed with soft coal and burned in an ordinary 
urnace provided with a fire-brick arch. 

"Wet Tan Bark as Fuel.— Tan, or oak bark, after having been used 
in the processes of tanning, is burned as fuel. The spent tan consists of the 
fibrous portion of the bark. According to M. Peclet, five parts of oak bark 
produce four parts of dry tan; and the heating power of perfectly dry tan, 
containing 15% of ash, is 6100 English units; whilst that of tan in an ordinary 
state of dryness, containing 30% of water, is only 4284 English units. The 
weight of water evaporated from and at 212° by one pound of tan, equiva- 
lent to these heating powers, is, for perfectly dry tan, 5.46 lbs., for tan with 
30% moisture, 3.84 lbs. Experiments by Prof. R.H. Thurston (Jour. Frank. 
Inst., 1874) gave with the Crockett furnace, the wet tan containing 59% of 
water, an evaporation from and at 212° F. of 4.24 lbs. of water per pound 
of the wet tan, and with the Thompson furnace an evaporation of 3.19 lbs. 
per pound of wet tan containing 55% of water. The Thompson furnace con- 
sisted of six fire-brick ovens, each 9 feet x 4 feet 4 inches, containing 234 
square feet of grate in all, for three boilers with a total heating surface of 
2000 square feet, a ratio of heating to grate surface of 9 to 1. The tan was 
fed through holes in the top. The Crockett furnace was an ordinary fire- 
brick furnace, 6x4 feet, built in front of the boiler, instead of under it, the 
ratio of heating surface to grate being 14.6 to 1. According to Prof. Thurs- 
ton the conditions of success in burning wet fuel are the surrounding of the 
mass so completely with heated surfaces and with burning fuel that it may 
be rapidly dried, and then so arranging the apparatus that thorough com- 
bustion may be secured, and that the rapidity of combustion be precisely 
equal to and never exceed the rapidity of desiccation. Where this rapidity 
of combustion is exceeded the dry portion is consumed completely, leaving 
an uncovered mass of fuel which refuses to take fire. 

Straw as Fuel. (Eng'g Mechanics, Feb., 1893, p. 55.)— Experiments in 
Russia showed that winter-wheat straw, dried at 230° F., had the following 
composition: C, 46.1; H, 5.6; N, 0.42; O, 43.7; Ash, 4.1. Heating value in 
British thermal units: dry straw, 6290; with 6% water, 5770; with 10% water, 
5448. With straws of other grains the heating value of dry straw ranged 
from 5590 for buckwheat to 6750 for flax. 

Clark (S. E., vol. 1, p. 62) gives the mean composition of wheat and barley 
straw as C, 36; H. 5; O, 38; O, 0.50; Ash, 4.75; water, 15.75, the two straws 
varying less than 1%. The heating value of straw of this composition, accord- 
ing to Dulong's formula, and deducting the heat lost in evaporating the 
water, is 5155 heat units. Clark erroneously gives it as 8144 heat units. 

Bagasse as Fuel in Sugar Manufacture.- -Bagasse is the name 
given to refuse sugar-cane, after the juice has been extracted. Prof. L. A. 



644 FUEL. 

Becuel, in a paper read before the Louisiana Sugar Chemists' Association, in 
1892, says; " With tropical cane containing 12.5$ woody fibre, a juice contain- 
ing 16.13$ solids, and 83.37$ water, bagasse of, say, 66$ and 72$ mill extrac- 
tion would have the following percentage composition: 

Woody Combustible Wotm , 

Fibre. Salts. water. 

66$ bagasse 37 10 53 

72% bagasse 45 9 46 

"Assuming that the woody fibre contains 51$ carbon, the sugar and other 
combustible matters an average of 42.1$, and that 12,906 units of heat are 
generated for every pound of carbon consumed, the 66$ bagasse is capable 
of generating 297,834 heat units as against 345,200, or a difference of 47,366 
units in favor of the 72$ bagasse. 

"Assuming the temperature of the waste gases to be 450° F., that of the 
surrounding atmosphere and water in the bagasse at 86° F., and the quan- 
tity of air necessary for the combustion of one pound of carbon at 24 lbs., 
the lost heat will be as follows: In the waste gases, heating air from 86° to 
450° F., and in vaporizing the moisture, etc., the 66$ bagasse will require 
112,546 heat units, and 116,150 for the 72$ bagasse. 

" Subtracting these quantities from the above, we find that the 66$ bagasse 
will produce 185,288 available heat units, or nearly 38% less than the 72$ 
bagasse, which gives 299,050 units. Accordingly, one ton of cane of 2000 lbs. 
at 66$ mill extraction will produce 680 lbs. bagasse, equal to 125,995,840 avail- 
able heat units, while the same cane at 72$ extraction will produce 560 lbs. 
bagasse, equal to 167.468,000 units. 

"A similar calculation for the case of Louisiana cane containing 10$ woody 
fibre, and 16$ total solids in the juice, assuming 75$ mill extraction, shows 
that bagasse from one ton of cane contains 157,395,640 heat units, from 
which 56,146,500 have to be deducted. 

" This would make such bagasse worth on an average nearly 92 lbs. coal 
per ton of cane ground. Under fairly good conditions, 1 lb. coal will evap- 
orate 7% l° s - water, while the best boiler plants evaporate 10 lbs. Therefore, 
the bagasse from 1 ton of cane at 75$ mill extraction should evaporate from 
689 lbs. to 919 lbs. of water. The juice extracted from such cane would un- 
der these conditions contain 1260 lbs. of water. If we assume that the 
water added during the process of manufacture is 10$ (by weight) of the 
juice made, the total water handled is 1410 lbs. From the juice represented 
in this case, the commercial massecuite would be about 15$ of the weight of 
the original mill juice, or say 225 lbs. Said mill juice 1500 lbs., plus 10$, 
equals 1650 lbs. liquor handled; and 1650 lbs., minus 225 lbs., equals 1425 lbs., 
the quantity of water to be evaporated during the process of manufacture. 
To effect a 7^-lb. evaporation requires 190 lbs. of coal, and 1421^ lbs. for a 10- 
lb. evaporation. 

" To reduce 1650 lbs. of juice to syrup of, say, 27° Baume. requires the evap- 
oration of 1770 lbs. of water, leaving 480 lbs. of syrup: If this work be ac- 
complished in the open air, it will require about 156 lbs. of coal at 7\i lbs. 
boiler evaporation, and 117 at 10 lbs. evaporation. 

" With a double effect the fuel required would be from 59 to 78 lbs., and 
with a triple effect, from 36 to 52 lbs. 

" To reduce the above 480 lbs. of syrup to the consistency of commercial 
massecuite means the further evaporation of 255 lbs. of water, requiring 
the expenditure of 34 lbs. coal at 7J^ lbs. boiler evaporation, and 25^ lbs. 
with a 10-lb. evaporation. Hence, to manufacture one ton of cane into sugar 
and molasses, it will take from 145 to 190 lbs. additional coal to do the work 
by the open evaporator process; from 85 to 112 lbs. with a double effect, and 
only 7J^ lbs. evaporation in the boilers, while with 10 lbs. boiler evaporation 
the bagasse alone is capable of furnishing 8% more heat than is actually re- 
quired to do the work. With triple-effect evaporation depending on the ex- 
cellence of the boiler plant, the 1425 lbs. of w T ater to be evaporated from the 
juice will require between 62 and 86 lbs. of coal. These values show that 
from 6 to 30 lbs. of coal can be spared from the value of the bagasse to run 
engines, grind cane, etc. 

"It accordingly appears." says Prof. Becuel, "that with the best boiler 
plants, those taking up all the available heat generated, by using this heat 
economically the bagasse can be made to supply all the fuel required by our 
sugar-houses." 



PETROLEUM. 



645 



PETROLEUM. 
Products of the Distillation of Crude Petroleum. 

Crude American petroleum of sp. gr. 0.800 may be split up by fractional 
distillation as follows (Robinson's Gas and Petroleum Engines): 



Temp, of ' 

Distillation 

Fahr. 


Distillate. 


Percent- 
ages. 


Specific 
Gravity. 


Flashing 
Point. 
Deg. F. 


113° 


Rhigolene. |_ 


traces. 
1.5 
10. 
2.5 
2. 


.590 to .625 

.636 to .657 
.680 to .700 
.714 to .718 
.725 to .737 




113 to 140° 
140 to 158° 


Chymogene. ) 

Gasolene (petroleum spirit)... 
Benzine, naphtha C, benzolene. 

( Benzine, naphtha B 

^ " " A 




158 to 248° 
248° 
to 


14 
32 


347° 






338° and 1 

upwards, j 

482° 


Kerosene (lamp-oil) 

Lubricating oil 

Paraffine wax 

Residue and Loss 


50. 
15. 
2. 
16. 


.802 to .820 
.850 to .915 


100 to 122 
230 









Lima Petroleum, produced at Lima, Ohio, is of a dark green c 
very fluid, and marks 48° Baume at 15° C. (sp. gr., 0.792). 

The distillation in fifty parts, each part representing 2% by volume, 
the following results : 



Per 


Sp. 


Per 


Sp. 


Per 


Sp. 


Per 


Sp. 


Per 


Sp. 


Per 


Sp. 


cent. 


Gr. 


cent. 


Gr. 


cent. 


Gr. 


cent. 


Gr. 


cent. 


Gr. 


cent. 


Gr. 


2 


680 


18 


0.720 


34 


0.764 


50 


0.802 


68 


0.820 


88 


0.815 


4 


683 


20 


.728 


36 


.768 


52! 




70 


.825 


90 


.815 


6 


685 


22 


.730 


38 


.772 


to^ 


.806 


72 


.830 




s 


8 


690 


24 


.735 


40 


.778 


58^ 




73 


.830 


92) 


10 


694 


26 


.740 


42 


.782 


60 


.800 


76 


.810 


to V 


3 


12 


698 


28 


.742 


44 


.788 


62 


.804 


78 


.820 


100 ) 


s 


14 


700 


30 


.746 


46 


.792 


64 


.808 


82 


.818 




a; 


16 


706 


32 


.760 


48 


.800 


66 


.812 


86 


.816 




03 



RETURNS. 

16 per cent naphtha, 70° Baume. 6 per cent paraffine oil. 

68 " burning oil. 10 " residuum. 

The distillation started at 23° O, this being due to the large amount of 
naphtha present, and when 60$ was reached, at a temperature of 310° C, 
the hydrocarbons remaining in the retort were dissociated, then gases 
escaped, lighter distillates were obtained, and, as usual in such cases, the 
temperature decreased from 310° O. down gradually to 200° C, until 75$ of 
oil was obtained, and from this point the temperature remained constant 
until the end of the distillation. Therefore these hydrocarbons in statu 
moriendi absorbed much heat. {Jour. Am. Chem. Soc.) 

Value of Petroleum as Fuel.— Thos. Urquhart, of Russia (Proc. 
Inst. M. E., Jan. 1889), gives the following table of the theoretical evapora- 
tive power of petroleum in comparison with that of coal, as determined by 
Messrs. Favre & Silbermann: 



Fuel. 


Specific 
Gravity 

at 
32° F., 
Water 
= 1.000. 


Chem. Com p. 


Heating- 
power, 
British 

Thermal 
Units. 


Theoret. 
Evap., lbs. 


C. 


H. 


O. 


lb. Fuel, 
from and 
at 212° F. 


Penna. heavy crude oil 

Caucasian light crude oil. . 
" heavy " " .. 


S. G. 

0.886 
0.884 
0.938 
0.928 

1.380 


p. c. 

84.9 
86.3 
86.6 
87.1 

80.0 


P o C „ 

13.7 
13.6 
12.3 
11.7 

5.0 


p. c. 

1.4 

0.1 

1.1 

1.2 

8.0 


Units. 

20,736 
22,027 
20,138 ' 
19,832 

14,112 


lbs. 

21.48 

22.79 

20.85 

20.53 


Good English Coal, Mean 
of 98 Samples 


14.61 









646 FUEL. 

In experiments on Russian railways with petroleum as fuel Mr. Urquhart 
obtained an actual efficiency equal to 82$ of the theoretical heating-value 
The petroleum is fed to the furnace by means of a spray-injector driven by 
steam. An induced current of air is cariied in around the injector-nozzle, 
and additional air is supplied at the bottom of the furnace. 

Oil vs. Coal as Fuel. (Iron Age, Nov. 2, 1893.)— Test by the Twin 
City Rapid Transit Company of Minneapolis and St. Paul. This test showed 
that with the ordinary Lima oil weighing 6 6/10 pounds per gallon, and 
costing 2J4 cents per gallon, and coal that gave an evaporation of 7^£ lbs. of 
water per pound of coal, the two fuels were equally economical when the 
price of coal was $3 85 per ton of 2000 lbs. With the same coal at $2.00 per 
ton, the coal was 37$ more economical, and with the coal at $4.85 per ton, 
the coal was 20$ more expensive than the oil. These results include the 
difference in the cost of handling the coal, ashes, and oil. 

In 1892 there were reported to the Engineers 1 Club of Philadelphia some 
comparative figures, from tests undertaken to ascertain the relative value 
of coal, petroleum, and gas. 

Lbs. Water, from 
and at 212° F. 

1 lb. anthracite coal evaporated 9 . 70 

1 lb. bituminous coal 10.14 

1 lb. free oil, 36° gravity 16 48 

1 cubic foot gas, 20 C. P 1.28 

The gas used was that obtained in the distillation of petroleum, having 
about the same fuel-value as natural or coal-gas of equal candle-power. 

Taking the efficiency of bituminous coal as a basis, the calorific energy of 
petroleum is more than 60$ greater than that of coal; whereas, theoretically, 
petroleum exceeds coal only about 45$— the one containing 14,500 heat-units, 
and the other 21,000. 

Crude Petroleum vs. Indiana Block Coal for Steam- 
raising at tne South Chicago Steel Works. (E. C. Potter, 
Trans. A. I. M. E., xvii, 807.)— With coal, 14 tubular boilers 16 ft. X 5 ft. re- 
quired 25 men to operate them ; with fuel oil, 6 men were required, a saving 
of 19 men at $2 per day, or $38 per day. 

For one week's work 2731 barrels of oil were used, against 848 tons of coal 
required for the same work, showing 322 barrels of oil to be equivalent to 1 
ton of coal. With oil at 60 cents per barrel and coal at $2.15 per ton, the rel- 
ative cost of oil to coal is as $1.93 to $2.15. No evaporation tests were 
made. 

Petroleum as a Metallurgical Fuel.— C. E. Felton (Trans. A. I. 
M. E., xvii, 809) reports a series of trials with oil as fuel in steel-heating and 
open-hearth steel-furnaces, and in raising steam with results as follows: 1. 
In a run of six weeks the consumption of oil, partly refiued (the paraffine 
and some of the naphtha being removed), in heating 14-inch ingots in Siemens 
furnaces was about 6^ gallons per ton of blooms. 2. In melting in a 30-ton 
open-hearth furnace 48 gallons of oil were used per ton of ingots. 3. In a 
six weeks' trial with Lima oil from 47 to 54 gallons of oil were required per 
ton of ingots. 4. In a six months' trial with Siemens heating-furnaces the 
consumption of Lima oil was 6 gallons per ton of ingots. Under the most 
favorable circumstances, charging hot ingots and running full capacity, 4^ 
to 5 gallons per ton were required. 5. In raising steam in two 100-H.P. 
tubular boilers, the feed-water being supplied at 160° F., the average evap- 
oration was about 12 pounds of water per pound of oil, the best 12 hours' 
work being 16 pounds. 

In all of the trials the oil was vaporized in the Archer producer, an apparat- 
us for mixing the oil and superheated steam, and heating the mixture to a 
high temperature. From 0.5 lb. to 0.75 lb. of pea-coal was used per gallon 
of oil in the producer itself. 

FUEL. GAS. 

The following notes are extracted from a paper by W. J. Taylor on " The 
Energy of Fuel " (Trans. A. I. M. E., xviii. 205): 

Carbon Gas.— In the old Siemens producer, practically, all the heat of 
primary combustion— that is, the burning of solid carbon to carbon monox- 
ide, or about 30$ of the total carbon energy— was lost, as little or no steam 
was used in the producer, and nearly all the sensible heat of the gas was 
dissipated in its passage from the producer to the furnace, which was usu- 
ally placed at a considerable distance. 

Modern practice has improved on this plan, by introducing steam with the 



FUEL GAS. 647 

air blown into the producer, and by utilizing the sensible heat of the gas in 
the combustion-furnace. It ought to be possible to oxidize one out of every 
four lbs. of carbon with oxygen derived from water- vapor. The thermic 
reactions in this operation are as follows: 

Heat-units. 
4 lbs. C burned to CO (3 lbs. gasified with air and 1 lb. with water) 

develop 17,600 

1.5 lbs. of water (which furnish 1.33 lbs. of oxygen to combine with 1 

lb. of carbon) absorb by dissociation 10,333 

The gas, consisting of 9.333 lbs. CO, 0.167 lb. H, and 13.39 lbs. N, heated 

600°, absorbs 3,748 

Leaving for radiation and loss 3,519 

?7,600 
The steam which is blown into a producer with the air is almost all con- 
densed into finely-divided water before entering the fuel, and consequently 
is considered as water in these calculations. 

The 1.5 lbs. of water liberates .167 lb. of hydrogen, which is delivered to 
the gas, and yields in combustion the same heat that it absorbs in the pro- 
ducer by dissociation. According to this calculation, therefore, 60$ of the 
heat of primary combustion is theoretically recovered by the dissociation of 
steam, and, even if all the sensible heat of the gas be counted, with radia- 
tion and other minor items, as loss, yet the gas must carry 4 X 14,500 — 
(37'48 + 3519) = 50,733 heat-units, or 87$ of the calorific energy of the carbon. 
This estimate shows a loss in conversion of 13$, without crediting the gas 
with its sensible heat, or charging it with the heat required for generating 
the necessary steam, or taking into account the loss due to oxidizing some 
of the carbon to C0 2 . In good producer-practice the proportion of C0 2 in 
the gas represents from 4$ to 7$ of the C burned to C0 2 , but the extra heat 
of this combustion should be largely recovered in the dissociation of mora 
water-vapor, and therefore does not represent as much loss as it would indi- 
cate. As a conveyer of energy, this gas has the advantage of carrying 4.46 
lbs. less nitrogen than would be present if the fourth pound of coal had 
been gasified with air; and in practical working the use of steam reduces 
the amount of clinkering in the producer. 

Anthracite Gas.— In anthracite coal there is a volatile combustible 
varying in quantity from 1.5$ to over 7$. The amount of energy derived 
from the coal is shown in the following theoretical gasification made with 
coal of assumed composition: Carbon, 85$; vol. HC, 5$; ash, 10$; 80 lbs. car- 
bon assumed to be burned to CO; 5 lbs. carbon burned to C0 2 ; three fourths 
of the necessary oxygen derived from air, and one fourth from water. 

, Products. , 

Process. Pounds. Cubic Feet. Anal, by Vol. 

80 lbs. C burned to CO 186.66 2529.24 33.4 

5 lbs. C burned to C0 2 18.33 157.64 2.0 

5 lbs. vol. HC (distilled) ... , 5.00 116.60 1.6 

120 lbs. oxygen are required, of which 

30 lbs. from H 2 liberate H 3.75 712.50 9.4 

90 lbs. from air are associatied with N 301 .05 4064.17 53.6 



514.79 7580.15 100.0 

Energy in the above gas obtained from 100 lbs. anthracite: 

186.66 lbs. CO 807,304 heat-units. 

5.00 " CH 4 117,500 

3.75 " H 232,500 

1,157,304 

Total energy in gas per lb 2,248 " 

" " 100 lbs. of coal.. 1,349,500 " 

Efficiency of the conversion 86$. 

The sum of CO and H exceeds the results obtained in practice. The sen- 
sible heat of the gas will probably account for this discrepancy, and, there- 
fore, it is safe to assume the possibility of delivering at least 82$ of the 
energy of the anthracite. 

Bituminous Gas.— A theoretical gasification of 100 lbs. of coal, con- 
taining 55$ of carbon and 32$ of volatile combustible (which is above the 
average of Pittsburgh coal), is made in the following table. It is assumed 
that 50 lbs, of C are burned to CO and 5 lbs. to C0 2 ; one fourth of the O is 



648 FUEL. 

derived from steam and three fourths from air; the heat value of the 
volatile combustible is taken at 20,000 heat-units to the pound. In comput- 
ing' volumetric proportions all the volatile hydrocarbons, fixed as well as 
condensing, are classed as marsh-gas, since it is only by some such tenta- 
tive assumption that even an approximate idea of the volumetric composi- 
tion can be formed. The energy, however, is calculated from weight: 

, Products. — , 

Process. Pounds. Cubic Feet. Anal, by Vol. 

50 lbs. C burned to CO 116.66 1580.7 27.8 

5 lbs. C burned to C0 2 18.33 157.6 2.7 

32 lbs. vol. HC (distilled) 32.00 746.2 13.2 

80 lbs. O are required, of which 20 lbs., 

derived from H 2 0, liberate H 2.5 475.0 8.3 

60 lbs. O, derived from air, are asso- 
ciated with N 200.70 2709.4 47.8 

370.19 5668.9 99.8 

Energy in 116.66 lbs. CO 504,554 heat-units. 

" 32. 00 lbs. vol. HC... 640,000 
" 2.50 lbs. H 155,000 



1 299 554 

Energy in coal 1,437,'500 

Per cent of energy delivered in gas 

Heat-units in 1 lb. of gas , 



Water-gas.— Water-gas is made in an intermittent process, by blowing 
up the fuel-bed of the producer to a high state of incandescence (and in 
some cases utilizing the resulting gas, which is a lean producer-gas), then 
shutting off the air and forcing steam through the fuel, which dissociates 
the water into its elements of oxygen and hydrogen, the former combining 
with the carbon of the coal, and the latter being liberated. 

This gas can never play a very important part in the industrial field, owing 
to the large loss of energy entailed in its production, yet there are places 
and special purposes where it is desirable, even at a great excess in cost per 
unit of heat over producer-gas; for instance, in small high-temperature fur- 
naces, where much regeneration is impracticable, or where the " blow-up " 
gas can be used for other purposes instead of being wasted. 

The reactions and energy required in the production of 1000 feet of water- 
gas, composed, theoretically, of equal volumes of CO and H, are as follows: 

500 cubic feet of H weigh 2.635 lbs. 

500 cubic feet of CO weigh 36.89 " 

Total weight of 1000 cubic feet 39.525 lbs. 

Now, as CO is composed of 12 parts C to 16 of O, the weight of C in 36.89 
lbs. is 15.81 lbs. and of O 21.08 lbs. When this.oxygen is derived from water 
it liberates, as above, 2.635 lbs. of hydrogen. The heat developed and ab- 
sorbed in these reactions (roughly, as we will not take into account the en- 
ergy required to elevate the coal from the temperature of the atmosphere 
to say 1800°) is as follows: 

Heat-units. 
2.635 lbs. H absorb in dissociation from water 2.635 X 62,000.. = 163,370 

15.81 lbs. C burned to CO develops 15.81 X 4400 = 69,564 

Excess of heat- absorption over heat-development = 93,806 

If this excess could be made up from C burnt to C0 2 without loss by radi- 
ation, we would only have to burn an additional 4.83 lbs. C to supply this 
heat, and we could then make 1000 feet of water-gas from 20.64 lbs. of car- 
bon (equal 24 lbs. of 85% coal). This would be the perfection of gas-making, 
as the gas would contain really the same energy as the coal; but instead, we 
require in practice more than double this amount of coal, and do not deliver 
more than 50% of the energy of the fuel in the gas, because the supporting 
heat is obtained in an indirect way and with imperfect combustion. Besides 
this, it is not often that the sum of the CO and H exceed 90%, the balance be- 
ing CO a and N. But water-gas should be made with much less loss of en- 
ergy by burning the "blow-up" (producer) gas in brick regenerators, the 
stored -up heat of which can be returned to the producer by the air used in 
blowing-up. 

The following table shows what may be considered average volumetric 



FUEL GAS. 



649 



analyses, and the weight and energy of 1000 cubic feet, of the four types of 
gases used for heating and illuminating purposes: 





Natural 
Gas. 


Coal- 
gas. 


Water- 
gas. 


Producer-gas. 


CO 


0.50 
2.18 
92.6 
0.31 
0.26 
3.61 
0.34 


6.0 
46.0 
40.0 
4.0 
0.5 
1.5 
0.5 
1.5 
32.0 
735,000 


45.0 
45.0 
2.0 

"4.0 ' 
2.0 
0.5 
1.5 
45.6 

322,000 


Anthra. 
27.0 
12.0 
1.2 

'"2.5 
57.0 
0.3 


Bitu. 

27.0 


H 


12.0 


CH 4 


2.5 




0.4 


co 2 

N 


2.5 
56.2 





0.3 








3(45.6 

1,100,000 


65.6 
137,455 


65.9 


Heat units in 1000 cubic feet 


156,917 



Natural Gas in Ohio and. Indiana. 

(Eng. and M. J., April 81, 1894.) 





Ohio. 


Indiana. 


Description. 


Fos- 
toria. 


Findlay 


St. 
Mary's. 


Muncie. 


Ander- 
son. 


Koko- 

1110. 


Mar- 
ion. 


Hydrogen 


1.89 
92.84 
.20 
.55 
.20 
.35 
3.82 
.15 


1.64 
93.35 
.35 
.41 
.25 
.39 
3.41 
.20 


1.94 
93.85 
.20 
.44 
.23 
.35 

2.98 
.21 


2.35 
92.67 
.25 
.45 
.25 
.35 
3.53 
.15 


1.86 
93.07 
.47 
.73 
.26 
.42 
3.02 
.15 


1.42 
94.16 
.30 
.55 
.29 
.30 
2.80 
.18 


1.20 
93.57 


Olefiant gas 

Carbon monoxide.. 

Carbon dioxide 

Oxygen 

Nitrogen 

Hydrogen sulphide 


.15 
.60 
.30 
.55 
3.42 
.20 



Approximately 30,000 cubic feet of gas have the heating power of one 
ton of coal. 

Producer-gas from One Ton of Coal. 

(W. H. Blauvelt, Trans. A. I. M. E., xviii. 614.) 



Analysis by Vol. 


Per 
Cent. 


Cubic Feet. 


Lbs. 


Equal to — 


CO 

H 

CH 4 


25.3 
9.2 
3.1 
0.8 
3.4 

58.2 


33,213.84 
12,077.76 
4,069.68 
1,050.24 
4,463.52 
76,404.96 


2451.20 
63.56 
174.66 
77.78 
519.02 
5659.63 


1050.51 lbs. C+ 1400.7 lbs. O. 
63.56 " H. 
174.66 " CH 4 . 


N (by difference . 


77.78 " C 2 H 4 . 
141.54 " C + 377.44 lbs. O. 
7350.17 " Air. 




100.0 


131,280.00 


8945.85 





Calculated upon this basis, the 131,280 ft. of gas from the ton of coal con- 
tained 20,311,162 B.T.U., or 155 B.T.U. per cubic ft., or 2270 B.T.U. per lb. 

The composition of the coal from which this gas was made was as follows: 
Water. 1.26$; volatile matter, 36.22$; fixed carbon, 57.98$; sulphur, 0.70$; 
ash, 3.78$. One ton contains 1159.6 lbs. carbon and 724.4 lbs. volatile com- 
bustible, the energy of which is 31,302,200 B.T.U. Hence, in the processes of 
gasification and purification there was a loss of 35.2$ of the energy of the 
coal. 

The composition of the hydrocarbons in a soft coal is uncertain and quite 
complex; but the ultimate analysis of the average coal shows that it ap- 
proaches quite nearly to the composition of CH 4 (marsh-gas). 

Mr. Blauvelt emphasizes the folio wing. points as highly important in soft- 
coal producer-practice: 



650 £UeL. 

First. That a large percentage of the energy of the coal is lost when the 
gas is made in the ordinary low producer and cooled to the temperature of 
the air before being used. To prevent these sources of loss, the producer 
should be placed so as to lose as little as possible of the sensible heat of the 
gas, and prevent condensation of the hydrocarbon vapors. A high fuel-bed 
should be carried, keeping the producer cool on top, thereby preventing the 
breaking-down of the hydrocarbons and the deposit of soot, as well as keep- 
ing the carbonic acid low. 

Second. That a producer should be blown with as much steam mixed with 
the air as will maintain incandescence. This reduces the percentage of 
nitrogen and increases the hydrogen, thereby greatly enriching the gas. 
The temperature of the producer is kept down, diminishing the loss of heat 
by radiation through the walls, and in a large measure preventing clinkers. 
The Combustion of Producer-gas. (H. H. Campbell, Trans. 
A. I. M. E., xix, 128.)— The combustion of the components of ordinary pro- 
ducer-gas may be represented by the following formulae: 

C 2 H 4 + 60 = 2C0 2 + 2H 2 ! 2H + O = H 2 ; 
CH 4 + 40 = C0 2 + 2H 2 ; CO -f O = C0 2 . 
Average Composition by Volume op Producer-gas: A, made with Open 
Grates, no Steam in Blast; B, Open Grates, Steam-jet in Blast. 10 
Samples op Each. 

C0 2 . O. C 2 H 4 . CO. H. CH 4 . N. 

Amin 3.6 0.4 0.2 20.0 5.3 3.0 58.7 

A max 5.6 0.4 0.4 24.8 8.5 5.2 64.4 

A average... 4.84 0.4 0.34 22.1 6.8 3.74 61.78 

B min 4.6 0.4 .0.2 20.8 6.9 2.2 57.2 

B max 6.0 0.8 0.4 24.0 9.8 3.4 62.0 

B average... 5.3 0.54 0.36 22.74 8.37 2.56 60.13 

The coal used contained carbon 82#, hydrogen 4.7%. 
The following are analyses of products of combustion : 

C0 2 . O. CO. CH 4 . H. N. 

Minimum 15.2 0.2 trace, trace. trace. 80.1 

Maximum 17.2 1.6 2.0 0.6 2.0 83.6 

Average 16.3 0.8 0.4 0.1 0.2 82.2 

Use of Steam in Producers and in Boiler-furnaces. (R. 
W. Raymond, Trans. A. I. M. E., xx. 635.)— No possible use of steam can 
cause a gain of heat. If steam be introduced into a bed of incandescent 
carbon it is decomposed into hydrogen and oxygen. 

The heat absorbed by the reduction of one pound of steam to hydrogen is 
much greater in amount than the heat generated by the union of the 
oxygen thus set free with carbon, forming either carbonic oxide or carbonic 
acid. Consequently, the effect of steam alone upon a bed of incandescent 
fuel is to chill it. In every water-gas apparatus, designed to produce by 
means of the decomposition of steam a fuel -gas relatively free from nitro- 
gen, the loss of heat in the producer must be compensated by some reheat- 
ing device. 
This loss may be recovered if the hydrogen of the steam is subsequently 
, burned, to form steam again. Such a combustion of the hydrogen is con- 
t templated, in the case of fuel-gas, as secured in the subsequent use of that 
' gas. Assuming the oxidation of H to be complete, the use of steam will 
'. cause neither gain nor loss of heat, but a simple transference, the heat 
f absorbed by steam decomposition being restored by hydrogen combustion. 
. In practice, it may be doubted whether this restoration is ever complete. 
< But it is certain that an excess of steam would defeat the reaction alto- 
gether, and that there must be a certain proportion of steam, which per- 
mits the realization of important advantages, without too great a net loss in 
heat. 

The advantage to be secured (in boiler furnaces using small sizes of 
anthracite) consists principally in the transfer of heat from the lower side 
of the fire, where it is not wanted, to the upper side, where it is wanted. 
The decomposition of the steam below cools the fuel and the grate-bars, 
whereas a blast of air alone would produce, at that point, intense combus- 
tion (forming at first C0 2 ), to the injury of the grate, the fusion of part of 
the fuel, etc. 

The proportion of steam most economical is not easily determined. The 
temperature of the steam itself, the nature of the fuel mixture, and the use 
or non-use of auxiliary air- supply, introduced into the gases above or 



ILLUMINATING-GAS. 



651 



beyond the fire -bed, are factors affecting the problem. (See paper by R. J. 
Foster on the Use of the McClave Grate and Argand Steam Blower, etc., in 
Trans. A. I. M. E., xx. 625.) 

Gas-fuel for Small Furnaces. E. P. Reichhelm (Am. Mach., 
Jan. 10,1895) discusses the use of gaseous fuel for forge fires, for drop - 
forging, in annealing-ovens and furnaces for melting brass and copper, 
for case-hardening, muffle-furnaces, and kilns. Under ordinary conditions, 
in such furnaces he estimates that the loss by draught, radiation, and the 
heating of space not occupied by work is, with coal, 80$, with petroleum 70$, 
and with gas above the grade of producer-gas 25%. He gives the following 
table of comparative cost of fuels, as used in these furnaces : 



Kind of Gas. 


o 

o 3 o 


No. of Heat- 
units in Fur- 
naces after 
Deducting 

25$ Loss. 


bCr-T 

< 


Cost of 1,000,- 
000 Heat- 
units Ob- 
tained in Fur- 
naces. 


Natural gas 

Coal-gas, 20 candle-power 

Carburetted water-gas 


1,000,000 
675,000 
646,000 
690,000 
313,000 
377,000 
185,00( 
150,00f 
306,36. r 


750,000 
506,250 
484,500 
517,500 
234,750 
282,750 
138,750 
112,500 
229,774 






$1.25 
1.00 
.90 
.40 
.45 
.20 
.15 
.15 


$2.46 
2.06 
1.73 


Water-gas from coke. 

Water-gas from bituminous coal 

Water-gas and producer-gas mixed. . . 


1.70 
1.59 
1.44 
i.33 


Naphtha-gas, fuel 2% gals, per 1000 ft. 


.65 


Coal, $4 per ton, per 1 ,000,000 heat-units 
Crude petroleum, 3 cts. per gal., per 1,C 


utilizec 
00,000 he 






.73 


at-units. 




.73 



Mr. Reichhelm gives the following figures from practice in melting brass 
with coal and with naphtha converted into gas: 1800 lbs. of metal require 
1080 lbs. of coal, at $4.65 per ton, equal to $2.51, or, say, 15 cents per 100 lbs. 
Mr, T.'s report : 2500 lbs. of metal require 47 gals, of naphtha, at 6 cents per 
gal., equal to $2.82, or, say, 11J4 cents per 100 lbs. 



ILLUMINATING-GAS. 



Coal-gas is made by distilling bituminous coal in retorts. The retort 
is usually a long horizontal semi-cylindrical or a shaped chamber, holding 
from 160 to 300 lbs. of coal. The retorts are set in " benches " of from 
3 to 9, heated by one fire, which is generally of coke. The vapors distilled 
from the coal are converted into a fixed gas by passing through the retort, 
which is heated almost to whiteness. 

The gas passes out of the retort through an " ascension-pipe " into a long 
horizontal pipe called the hydraulic main, where it deposits a portion of 
the tar it contains: thence it goes into a condenser, a series of iron tubes 
surrounded by cold water, where it is freed f rom condensable vapors, as 
ammonia-water, then into a washer, where it is exposed to jets of water, 
and into a scrubber, a large chamber partially filled with trays made of 
wood or iron, containing coke, fragments of brick or paving-stones, which 
are wet with a spray of water. By the washer and scrubber the gas is freed 
from the last portion of tar and ammonia and from some of the sulphur 
compounds. The gas is then finally purified from sulphur compounds by 
passing it through lime or oxide of iron. The gas is drawn from the hy- 
draulic main and forced through the washer, scrubber, etc., by an exhauster 
or gas pump. 

The kind of coal used is generally caking bituminous, but as usually this 
coal is deficient in gases of high illuminating power, there is added to it a 
portion of cannel coal or other enricher. 

The following table, abridged from one in Johnson's Cyclopedia, shows 
the analysis, candle power, etc., of some gas-coals and enrichers: 



652 



ILLUMINATING-GAS. 



Gas-coals, etc. 



i - 


,a 




e . 


> 


Coke per 


-2 






©JS^ 




ton of 2240 




o 




^o^ 


&M 


lbs. 


S 






sjg 


-CO 








o 
> 


M 
£ 


< 


|o-S 


6" 


lbs. 


bush. 


36.76 


51.93 


7.07 










36.00 


58.00 


6.00 


10,642 


16.62 


1544 


40 


37.50 


56.90 


5.60 




18.81 


1480 


36 


40.00 


53.30 


6.70 


10,765 


20.41 


1540 


36 


43.00 


40.00 


17.00 


9,800 


34.98 


1320 


32 


46.00 


41.00 


13.00 


13,200 


42.79 


1380 


32 


53.50 


44.50 


2.00 


15,000 


28.70 


1056 


44 



73 OS 

m 



Pittsburgh, Pa 

Westmoreland, Pa 

Sterling, O 

Despard, W. Va... 

Darlington, O 

Petonia, W. Va — 
Grahamite, W. Va. 



6420 
3993 
2494 
2806 
4510 



The products of the distillation of 100 lbs. of average gas-coal are about as 
follows. They vary according to the quality of coal and the temperature of 
distillation. 

Coke, 64 to 65 lbs.; tar, 6.5 to 7.5 lbs.; ammonia liquor, 10 to 12 lbs.; puri- 
fied gas, 15 to 12 lbs.; impurities and loss, 4.5$ to 3.5$. 

The composition of the gas by volume ranges about as follows: Hydro- 
gen, 38$ to 48$; carbonic oxide, 2$ to 14$; marsh-gas (Methane, CH 4 ), 43$ to 
31$; heavy hydrocarbons (C«H 2 «, ethylene, propylene, benzole vapor, etc.), 
7.5$ to 4.5$; nitrogen, 1$ to 3$. 

In the burning of the gas the nitrogen is inert; the hydrogen and carbonic 
oxide give heat but no light. The luminosity of the flame is due to the de- 
composition by heat of the heavy hydrocarbons into lighter hydrocarbons 
and carbon, the latter being separated in a state of extreme subdivision. 
By the heat of the flame this separated carbon is heated to intense white- 
ness, and the illuminating effect of the flame is due to the light of incandes- 
cence of the particles of carbon. 

The attainment of the highest degree of luminosity of the flame depends 
upon the proper adjustment of the proportion of the heavy hydrocarbons 
(with due regard to their individual character) to the nature of the diluent 
mixed therewith. 

Investigations of Percy F. Frankland show that mixtures of ethylene and 
hydrogen cease to have any luminous effect when the proportion of ethy- 
lene does not exceed 10$ of the whole. Mixtures of ethylene and carbonic 
oxide cease to have any luminous effect when the proportion of the former 
does not exceed 20$, while all mixtures of ethylene and marsh-gas have more 
or less luminous effect. The luminosity of a mixture of 10$ ethylene and 90$ 
marsh-gas being equal to about 18 candles, and that of one of 20$ ethylene 
and 80$ marsh-gas about 25 candles. The illuminating effect of marsh-gas 
alone, when burned in an argand burner, is by no means inconsiderable. 

For further description, see the Treatises on Gas by King. Richards, and 
Huemes; also AppletoiTs Cyc. Mech., vol. i. p. 900. 

"Water-gas.— Water-gas is obtained by passing. steam through a bed of 
coal, coke, or charcoal heated to redness or beyond. The steam is decom- 
posed, its hydrogen being liberated and its oxygen burning the carbon of 
the fuel, producing carbonic-oxide eas. The chemical reaction is. C + H 2 
= CO + 2H, or 2C + 2H 2 = C -4- C0 2 + 4H, followed by a splitting up of 
the C0 2 , making 2CO + 4H. By weight the normal gas CO + 2H is com- 
posed of C + O + H = 28 parts CO and 2 parts H, or 93.33$ CO and 6.67$ H; 

12 + 16 + 2 
by volume it is composed of equal parts of carbonic oxide and hydrogen. 
Water-gas produced as above described has great heating-power, but no 
illuminating-power. It may, however, be used for lighting by causing it to 
heat to whiteness some solid substance, as is done in the Welsbach incan- 
descent light. 

An illuminating-gas is made from water-gas by adding to it hydrocarbon 
gases or vapors, which are usually obtained from petroleum or some of its 
products. A history of the development of modern illuminating water-gas 
processes, together with a description of.the most recent forms of apparatus, 
is given by Alex. C. Humphreys, in a paper on " Water-gas in the United 
States," read before the Mechanical Section of the British Association for 
Advancement of Science, in 1S89. After describing many earlier patents, he 
states that success in the manufacture of water-gas may be said to date 



ANALYSES OF WATER-GAS AND COAL-GAS COMPARED. 653 



from 1874, when the process of T. S. C. Lowe was introduced. All the later 
most successful processes are the modifications of Lowe's, the essential 
features of which were " an apparatus consisting of a generator and super- 
heater internally fired; the superheater being heated by the secondary 
combustion from the generator, the heat so stored up in the loose brick of 
the superheater being used, in the second part of the process, in the fixing 
or rendering permanent of the hydrocarbon gases; the second part of the 
process consisting in the passing of steam through the generator fire, and 
the admission of oil or hydrocarbon at some point between the fire of the 
generator and the loose filling of the superheater.' 1 

The water-gas process thus has two periods: first the '" blow," during 
which air is blown through the bed coal in the generator, and the partially 
burned gaseous products are completely burned in the superheater, giving 
up a great portion of their heat to the fire-brick work contained in it, and 
then pass out to a chimney; second, the "run" during which the air blast 
is stopped, the opening to the chimney closed, and steam is blown through 
the incandescent bed of fuel. The resulting water-gas passing into the car- 
buretting chamber in the base of the superheater is there charged with hy- 
drocarbon vapors, or spray (such as naphtha and other distillates or crude 
oil) and passes through the superheater, where the hydrocarbon vapors be- 
come converted into fixed illuminating gases. From the superheater the 
combined gases are passed, as in the coal-gas process, through washers, 
scrubbers, etc., to the gas-holder. In this case, however, there is no am- 
monia to be removed. 

The specific gravity of water-gas increases with the increase of the heavy 
hydrocarbons which give it illuminating power. The following figures, taken 
from different authorities, are given by F. H. Shelton in a paper on Water- 
gas, read before the Ohio Gas Light Association, in 1894: 
Candle-power ... 19.5 20. 22.5 24. 25.4 26.3 28.3 29.6 .30 to 31.9 

Sp.gr. (Air = 1).. .571 .630 .589 .60 to .67 .64 .602 .70 .65 .65 to .71 

Analyses of Water-gas and Coal-gas Compared, 

The following analyses are taken from a report of Dr. Gideon E. Moore 
on the Granger Water-gas, 1885: 





Composition by Volume. 


Composition by Weight. 




Water-gas. 


Coal-gas. 
Heidel- 
berg. 


Water-gas. 


Coal- 




Wor- 
cester. 


Lake. 


Wor- 
cester. 


Lake. 


gas. 




2.64 
0.14 
0.06 
11.29 

0.00 
1.53 
28.26 
18.88 
37.20 


3.85 
0.30 
0.01 
12.80 
0.00 
2.63 
23.58 
20.95 
35.88 


2.15 
3.01 
0.65 
2.55 

1.21 
1.33 

8.88 
34.02 
46.20 


0.04402 
0.00365 
0.00114 
0.18759 


0.06175 
0.00753 
0.00018 
0.20454 


04559 


Carbonic acid 


0.09992 
01569 




05389 


Propylene 

Benzole vapor 

Carbonic oxide.. . 

Marsh-gas 

Hydrogen 


03834 


0.07077 
0.46934 
0.17928 
0.04421 


0.11700 
0.37664 
0.19133 
0.04103 


0.07825 
0.18758 
0.41087 
0.06987 




100.00 


100.00 


100.00 


1.00000 


1.00000 


1.00000 




0.5S25 
0.5915 


0.6057 
0.6018 


0.4580 


























B. T. U. from 1 cu. 


650.1 
597.0 

5311. 2°F. 


688.7 
646.6 

5281. 1°F. 


642.0 

577.0 

5202. 9°F. 








ft. : Water liquid. 








































Av. candle-power. 


22.06 


26.31 











The heating values (B. T. U.) of the gases are calculated from the analysis 
by weight, by using the multipliers given below (computed from results of 



654 



ILLUMINATING-GAS. 



J. Thomsen), and multiplying the result by the weight of 1 cu. ft. of the gas 
at 62° F., and atmospheric pressure. 

The flame temperatures (theoretical) are calculated on the assumption of 
complete combustion of the gases in air, without excess of air. 

The candle-power was determined by photometric tests, using a pressure 
of l^-in. water-column, a candle consumption of 120 grains of spermaceti 
per hour, and a meter rate of 5 cu. ft. per hour, the result being corrected 
for a temperature of 62° F. and a barometric pressure of 30 in. It appears 
that the candle-power may be regulated at the pleasure of the person in 
charge of the apparatus, the range of candle-power being from 20 to 29 
candles, according to the manipulation employed. 

Calorific Equivalents of Constituents of Illuminating- 
gas. 
Heat-units from 1 lb. Heat-units from 1 lb. 

Water Water - Water Water 

Liquid. Vapor. Liquid. Vapor. 

Ethylene 21,524.4 20,134.8 Carbonic oxide. . 4,395.6 4,395.6 

Propylene 21,222.0 19,834.2 Marsh- gas 24,021.0 21,592.8 

Benzole vapor.... 18,954.0 17,847.0 Hydrogen 61,524.0 51,804.0 

Efficiency of a Water-gas Plant.— The practical efficiency of an 
illuminating water-gas setting is discussed in a paper by A. G. Glasgow 
(Proc. Am. Gaslight Assn., 1890), from which the following is abridged : 

The results refer to 1000 cu. ft. of unpurified carburetted gas, reduced to 
60° F. The total anthracite charged per 1000 cu. ft. of gas was 33.4 lbs., ash 
and unconsumed coal removed 9.9 lbs., leaving total combustible consumed 
23.5 lbs., which is taken to have a fuel-value of 14500 B. T. U. per pound, or 
a total of 340,750 heat- units. 





Composi- 
tion by 
Volume. 


Weight 

per 

100 cu. ft. 


Composi- 
tion by 
Weight. 


Specific 
Heat. 


fC0 2 + H a S.. 

i C n H 2n 

CO 

I. Carburetteu J CH 4 

Water-gas. J H 

I N 


3.8 
14.6 
28.0 
17.0 
35.6 

1.0 


.465842 
1.139968 
2.1868 
.75854 
.1991464 
.078596 

4.8288924 


.09647 
.23607 
.45285 
.15710 
.04124 
.01627 

1.00000 


.02088 
.08720 
.11226 
.09314 
.14041 
.00397 


1 


100.0 


.45786 


fC0 2 


3.5 
43.4 
51.8 

1.3 


.429065 

3.389540 

.289821 

.102175 


.1019 
.8051 
.0688 
.0242 


.02205 


1 CO 


.19958 
.23424 




.00591 






1 


100.0 


4.210601 


1.0000 


.46178 


f co a 


17.4 
3.2 

79.4 


2.133066 

.2856096 

6.2405224 


.2464 
.0329 
.7207 


.05342 




.00718 


escaping from -J N 


.17585 




100.0 


8.6591980 


1.0000 


.23645 


fCO a 


9.7 

17.8 
72.5 


1.189123 
1.390180 
5.698210 


.1436 
.1680 
.6884 


.031075 




.041647 


IV. Generator J ^ 


.167970 


blast- gases. ] 




1 


100.0 


8.277513 


1.0000 


.240692 



The heat energy absorbed by the apparatus is 23.5 X 14,500 = 340,750 heat- 
units = A. Its disposition is as follows : 
E, the energy of the CO produced; 

C, the energy absorbed in the decomposition of the steam ; 

D, the difference between the sensible heat of the escaping illuminating- 
gases and that of the entering oil; 

E, the heat carried off by the escaping blast products; 

F, the heat lost by radiation from the shells; 



EFFICIENCY OF A WATER-GAS PLANT. 655 

G, the heat carried away from the shells by convection (air-currents); 
H, the heat rendered latent in the gasification of the oil; 
J, the sensible heat in the ash and unconsumed coal recovered from the 
generator. 
The heat equation is A = B+C+D+E+ F+ G + H+I; A being 

known. A comparison of the CO in Tables I and II show that-r^- , or 64.5£ 

of the volume of carburetted gas is pure water-gas, distributed thus : CO a , 
2.3£; CO, 28.0^; H, 33.4^; N, 0.8^; = 64.5^. 1 lb. of CO at 60° F. = 13.531 cu. 
ft. CO per 1000 cu. ft. of gas = 280 -h 13.531 = 20.694 lbs. Energy of the CO 
= 20.694 x 4395.6 = 91,043 heat-units, = B. 1 lb. of H at 60° F. = 189.2 cu. 
ft. H per M of gas = 334-^-189.2 = 1.7653 lbs. Energy of the H per lb. 
(according to Thomsen, considering the steam generated by its combustion 
to be condensed to water at 75° F.) = 61,524 B. T. U. In Mr. Glasgow's ex- 
periments the steam entered the generator at 331° F. ; the heat required to 
raise the product of combustion of 1 lb. of H, viz., 8.9S lbs. H 2 0, from water 
at 75° to steam at 331° must therefore be deducted from Thomsen's figure, or 
61,524 - (8.98 X 1140.2) = 51,285 B. T. U. per lb. of H. Energy of the H, then, 
is 1.7653 X 51,285 = 90,533 heat-units, = C. The heat lost due to the sensible 
beat in the illuminating-gases, their temperature being 1450° F., and that of 
the entering oil 235° F., is 48.29 (weight) X .45786 sp. heat X 1215 (rise of tem- 
perature) = 26,864 heat-units = D. 

(The specific heat of the entering oil is approximately that of the issuing 
gas.) 

The heat carried off in 1000 cu. ft. of the escaping blast products is 86.592 
(weight) X .23645 (sp. heat) X 1474° (rise of temp.) = 30,180 heat-units: the 
temperature of the escaping blast gases being 1550° F., and that of the 
entering air 76° F. But the amount of the blast gases, by registra- 
tion of an anemometer, checked by a calculation from the analyses of the 
blast gases, was 2457 cubic feet for every 1000 cubic feet" of carburetted gas 
made. Hence the heat carried off per M. of carburetted gas is 30,180 x 
2.457 = 74,152 heat-units = E. 

Experiments made by a radiometer covering four square feet of the shell 
of the apparatus gave figures for the amount of heat lost by radiation 
= 12,454 heat-units = F, and by convection = 15,696 heat-units = G. 

The heat rendered latent by the gasefication of the oil was found by taking 
the difference between all the heat fed into the carburetter and super- 
heater and the total heat dissipated therefrom to be 12,841 heat-units = H. 
The sensible heat in the ash and unconsumed coal is 9.9 lbs. X 1500° x .25 
(sp. ht.) = 3712 heat-units — I. 

The sum of all the items B+ C + D + E+F-\- G-\- H+I= 327,295 heat- 
units, which substracted from the heat energy of the combustible consumed, 
340,750 heat-units, leaves 13,455 heat-units, or 4 percent, unaccounted for. 

Of the total heat energy of the coal consumed, or 340,750 heat-units, the 
energy wasted is the sum of items D, E, F, G, and I, amounting to 132,878 
heat-units, or 39 per cent; the remainder, or 207,872 heat-units, or 61 per 
cent, being utilized. The efficiency of the apparatus as a heat machine is 
therefore 61 per cent. 

Five gallons, or 35 lbs. of crude petroleum were fed into the carburetter 
per 1000 cu. ft. of gas made; deducting 5 lbs. of tar recovered, leaves 30 lbs. 
X 20,000 = 600,000 heat-units as the net heating value of the petroleum used. 
Adding this to the heating value of the coal, 340,750 B. T. U., gives 940,750 
heat-units, of which there is found as heat energy in the carburetted gas, as 
in the table below, 764,050 heat units, or 81 per cent, which is the commer- 
cial efficiency of the apparatus, i.e., the ratio of the energy contained in 
the finished product to the total energy of the coal and oil consumed. 

The heating power per M. of the 
uncarburetted gas is 
C0 2 35.0 

CO 434.0 X .078100 X 4395.6 = 148991 
H 518.0 X .005594 X 61524.0 = 178277 
N 13.0 



The heating power per M. cu. ft. of 
the carburetted gas is 
C0 2 38.0 

C 3 H 6 * 146.0 X .117220 X 21222.0 = 363200 
CO 280.0 X .078100 X 4395.6 = 96120 
CH 4 170.0 X .044620 X 24021.0 = 182210 
H 356.0 X 005594 X 61524.0 = 122520 
N 10.0 

1000.0 764050 



1000.0 327268 



* The heating value of the illuminants C n H 2 » is assumed to equal that 
of C 3 H 6 . 



656 ILLUMINATING-GAS. 

The candle-power of the gas is 31, or 6.2 candle-power per gallon of oil 
used. The calculated specific gravity is .6355, air being 1. 

For description of the operation of a modern carburetted water-gas 
plant, see paper by J. Stelfox, Eng'g, July 20, 1894, p. 89. 

Space required for a Water-gas Plant.— Mr. Shelton, taking 
15 modern plants of the form requiring the most floor-space, figures the 
average floor-space required per 1000 cubic feet of daily capacity as follows: 

Water-gas Plants of Capacity Require an Area of Floor-space for 

in 24 hours of each 1000 cu. ft. of about 

100,000 cubic feet .4 square feet. 

200,000 " " 3.5 " 

400,000 " " 2.75" 

600,000 " " 2 to 2.5 sq.ft. 

7 to 10 million cubic feet 1.25 to 1.5 sq. ft. 

These figures include scrubbing and condensing rooms, but not boiler and 
engine rooms. In coal-gas plants of the most modern and compact forms one 
with 16 benches of 9 retorts each, with a capacity of 1,500,000 cubic feet per 
24 hours, will require 4.8 sq. ft. of space per 1000 cu. ft. of gas, and one of 6 
benches of 6 retorts each, with 300,000 cu. ft. capacity per 24 hours wdl re- 
quire 6 sq. ft. of space per 1000 cu. ft. The storage-room required for the 
gas-making materials is: for coal-gas, 1 cubic foot of room for every 232 
cubic feet of gas made; for water-gas made from coke, 1 cubic foot of room 
for every 373 cu. ft. of gas made; and for water-gas made from anthracite, 
1 cu. ft. of room for every 645 cu. ft. of gas made. 

The comparison is still more in favor of water-gas if the case is considered 
of a water-gas plant added as an auxiliary to an existing coal-gas plant; 
for, instead of requiring further space for storage of coke, part of that 
already required for storage of coke produced and not at once sold can be 
cut off, by reason of the water-gas plant creating a constant demand for 
more or less of the coke so produced. 

Mr. Shelton gives a calculation showing that a water-gas of .625 sp. gr. 
would require gas-mains eight per cent greater in diameter than the same 
-quantity coal-gas of .425 sp. gr. if the same pressure is maintained at the 
holder. The same quantity may be carried in pipes of the same diameter 
if the pressure is increased in proportion to the specific gravity. With the 
same pressure the increase of candle-power about balances the decrease of 
flow. With five feet of coal-gas, giving, say, eighteen candle-power, 1 cubic 
foot equals 3.6 candle-power; with water-gas of 23 candle-power, 1 cubic 
foot equals 4.6 candle-power, and 4 cubic feet gives 18.1 candle-power, or 
more than is given by 5 cubic feet of coal-gas. Water-gas may be made 
from oven-coke or gas-house coke as well as from anthracite coal. A water- 
gas plant may be conveniently run in connection with a coal-gas plant, the 
surplus retort coke of the latter being used as the fuel of the former. 

In coal-gas making it is impracticable to enrich the gas to over twenty 
candle-power without causing too great a tendency to smoke, but water-gas 
of as high as thirty candle-power is quite common. A mixture of coal-gas 
and water-gas of a higher C.P. than 20 can be advantageously distributed. 

Fuel-value of. Illuminating-gas.— E. G. Love (School of Mines 
Qtly, January, 1892) describes F. W. Hartley's calorimeter for determining 
the calorific power of gases, and gives results obtained in tests of the car- 
buretted water-gas made by the municipal branch of the Consolidated Co. 
of New York. The tests were made from time to time during the past two 
years, and the figures give the heat-units per cubic foot at 60° F. and 30 
inches pressure: 715. 692, 725, 732, 691, 738, 735, 703, 734, 730, 731, 727. Average, 
721 heat units. Similar tests of mixtures of coal- and water-gases made by 
other branches of the same company give 694, 715, 684, 692, 727, 665, 695, and 
686 heat-units per foot, or an average of 694.7. The average of all these 
tests was 710.5 heat-units, and this we may fairly take as representing the 
calorific power of the illuminating gas of New York. One thousand feet of 
this gas, costing $1.25, would therefore yield 710,500 heat-units, which would 
be equivalent to 568,400 heat-units for $1.00. 

The common coal-gas of London, with an illuminating power of 16 to 17 
candles, has a calorific power of about 668 units per foot, and costs from 60 
to 70 cents per thousand. 

The product obtained by decomposing steam by incandescent carbon, as 
effected in the Motay process, consists of about 40$ of CO, and a little over 
W% of H. 



FLOW OF GAS IK PIPES. 657 

This mixture would have a heating-power of about 300 units per cubic foot, 
and if sold at 50 cents per 1000 cubic feet would furnish 600,000 units for $1.00, 
as compared with 568,400 units for $1.00 from illuminating gas at $1 .25 per 1000 
cubic feet. This illuminating-gas if sold at $1.15 per thousand would there- 
fore be a more economical heating agent than the fuel-gas mentioned, at 50 
cents per thousand, and be much more advantageous than the latter, in that 
one main, service, and meter could be used to furnish gas for both lighting 
and heating. 

A large number of fuel-gases tested by Mr. Love gave from 184 to 470 heat- 
units per foot, with an average of 309 units. 

Taking the cost of heat from illuminating-gas at the lowest figure given 
by Mr. Love, viz., $1.00 for 600,000 heat-units, it is a very expensive fuel, equal 
to coal at $40 per ton of 2000 lbs., the coal having a calorific power of only 
12,000 heat-units per pound, or about 83$ of that of pure carbon : 

600,000 : (12,000 X 2000) :: $1 : $40. 

FLOW OF GAS IN PIPES. 

The rate of flow of gases of different densities, the diameter of pipes re- 
quired, etc., are given in King's Treatise on Coal Gas, vol. ii. 374, as follows: 

If d = diameter of pipe in inches, 
. Q = quantity of gas in cu. ft. pe: 
hour, 
I — length of pipe in yards, 
h = pressure in inches of water, 
s = specific gravity of gas, air be 
ingl, 



= »/ Q*sl 
Y (1350) 2 /i ' 

ft - _ Q* sl 

f (1350)W_'_ 



Molesworth gives Q = 1000 ju — 



si 



/ d 6 h 
J. P. Gill, Am. Gas-light Jour. 1894, gives Q = 1291 A/ - 



+ d) 

This formula is said to be based on experimental data, and to make allow- 
ance for obstructions by tar, water, and other bodies tending to check the 
flow of gas through the pipe. 

A set of tables in Appleton's Cyc. Mech. for flow of gas in 2, 6, and 12 in. 
pipes is calculated on the supposition that the quantity delivered varies 
as the square of the diameter instead of as d 2 x Vd, or Yd 6 . 

These tables give a flow in large pipes much less than that calculated by 
the formulae above given, as is shown by the following example. Length of 
pipe 100 yds., specific gravity of gas .042, pressure 1-in. water-column. 

2 -in. Pipe. 6-in. Pipe. 12-in. Pipe. 
Q=12S0a/^-j- 1178 18,368 103,912 



, - nn /d*h 

) = 1000|/--, 



Q = 1291 A/ , " • 1116 16,327 93,845 

y s(l -J- a) 

Table in App. Cyc 1290 11,657 46,628 

An experiment made by Mr. Clegg, in London, with a 4-in. pipe, 6 miles 
long, pressure 3 in. of water, specific gravity of gas .398, gave a discharge 
into the atmosphere of 852 cu. ft. per hour, after a correction of 33 cu. ft. 
was made for leakage. 

Substituting this value, 852 cu. ft., for Q in the formula Q = C Vd 6 h ■+- si, 
we find C, the coefficient, = 997, which corresponds nearly with the formula 
given by Molesworth. 



658 



ILLUMINATING-GAS. 



Services -for Lamps. (Molesworth.) 



Lamps. 



4.. 
6.. 
10.. 



Ft. from 
Main. 
. . . . 40 
.... 40 
. . . . 50 
.... 100 



Require 
Pipe-bore. 



Lamps. 
15.... 
20.... 
25 ... . 



Ft. from 
Main. 



150 
180 



Require 
Pipe-bore. 
1 in. 
1*4 in. 
l^in. 
1% in. 



(In cold climates no service less than % in. should be used.) 

Maximum Supply of Gas through Pipes in en. ft. per 
Hour, Specific Gravity oeing taken at .45, calculated 
from the Formula Q = 1000 \/tVh -=- si. (Molesworth.) 

Length of Pipe = 10 Yards. 







Pressure by the Water-gauge in Inches. 






of Pipe in 






















































.1 


.2 


.3 


.4 


.5 


.6 


.7 


.8 


.9 


1.0 


% 


13 


18 


22 


26 


29 


31 


34 


36 


38 


41 


14 


26 


37 


46 


53 


59 


64 


70 


74 


79 


83 


H 


73 


103 


126 


145 


162 


187 


192 


205 


218 


230 


1 


149 


211 


25S 


298 


333 


365 


394 


422 


447 


471 


VA 


260 


368 


451 


521 


582 


638 


689 


737 


781 


823 


•m 


411 


581 


711 


821 


918 


1006 


1082 


1162 


1232 


1299 


2 


843 


1192 


1460 


1686 


1886 


2066 


2231 


2385 


2530 


2667 



Length of Pipe = 100 Yards. 





Pressure by the "Water-gauge in Inches. 




.1 

R 


.2 
12 


.3 

14 


.4 


.5 


.75 


1.0 


1.25 
29 


1.5 


2 


2.5 


% 


17 


19 


23 


26 


32 


36 


42 


¥4. 


23 


32 


42 


46 


51 


63 


73 


81 


89 


103 


115 


1 


47 


67 


82 


94 


105 


129 


149 


167 


183 


211 


236 


m 


82 


116 


143 


165 


184 


225 


260 


291 


319 


368 


412 


m 


130 


184 


225 


260 


290 


356 


411 


459 


503 


581 


649 




267 


377 


462 


533 


596 


730 


843 


943. 


1033 


1193 


1333 


2y 2 


466 


659 


K07 


932 


1042 


1276 


1473 


1647 


1804 


2083 


2329 


3 


735 


1039 


1270 


1470 


1643 


2012 


2323 


2598 


2846 


3286 


3674 


3^ 




152H 


1S71 


2161 


2416 


2958 


3416 


3820 


4184 


4831 


5402 


4 








3017 


3373 


4131 


4770 


5333 


5842 


6746 


7542 



Length of Pipe = 1000 Yards. 







Pressure by the Water-gauge in 


Inches. 






.5 


.75 


1.0 


1.5 


2.0 


2.5 


3.0 


1 


33 


41 


47 


58 


67 


75 


82 


m 


92 


113 


130 


159 


184 


205 


226 


2 


189 


231 


267 


327 


377 


422 


462 


2H 


329 


403 


466 


571 


659 


737 


807 


3 


520 


636 


735 


900 


1039 


1162 


1273 


4 


1067 


1306 


1508 


1847 


2133 


2385 


2613 


5 


1863 


2282 


2635 


3227 


3727 


4167 


4564 


6 


2939 


3600 


4157 


5091 


5879 


6573 


7200 



659 



Length of Pipe = 5000 Yards. 



Diameter 




Pressure by the Water-gauge in Inches. 


of Pipe 










in 












Inches. 


1.0 


1.5 


2.0 


2.5 


3.0 


2 


119 


146 


169 


189 


207 


3 


329 


402 


465 


520 


569 


4 


675 


826 


955 


1067 


1168 


5 


1179 


1443 


1667 


1863 


2041 


6 


1859 


2277 


2629 


2939 


3220 


7 


2733 


3347 


3865 


4321 


4734 


8 


3816 


4674 


5397 


6034 


6610 


9 


5123 


6274 


7245 


8100 


8873 


10 


6667 


8165 


9428 


10541 


11547 


12 


10516 


12880 


14872 


16628 


18215 



Mr. A. C. Humphreys says his experience goes to show that these tables 
give too small a flow, but it is difficult to accurately check the tables, on ac- 
count of the extra friction introduced by rough pipes, bends, etc. For 
bends, one rule is to allow 1/42 of an inch pressure for each right-angle bend. 

Where there is apt to be trouble from frost it is well to use no service of 
less diameter than % in., no matter how short it may be. In extremely cold 
climates this is now often increased to 1 in., even for a single lamp. The best 
practice in the U. S. now condemns any service less than % in. 

STEAM. 



The Temperature of Steam in contact with water depends upon 
the pressure under which it is generated. At the ordinary atmospheric 
pressure (14.7 lbs. per sq. in.) its temperature is 212° F. As the pressure is 
increased, as by the steam being generated in a closed vessel, its tempera- 
ture, and that of the water in its presence, increases. 

Saturated Steam is steam of the temperature due to its pressure- 
not superheated. 

Superheated Steam is steam heated to a temperature above that due 
to its pressure. 

I>ry Steam is steam which contains no moisture. It may be either 
saturated or superheated. 

"Wet Steam is steam containing intermingled moisture, mist, or spray. 
It has the same temperature as dry saturated steam of the same pressure. 

Water introduced into the presence of superheated steam will flash into 
vapor until the temperature of the steam is reduced to that due its pres- 
sure. Water in the presence of saturated steam has the same temperature 
as the steam. Should cold water be introduced, lowering the temperature 
of the whole mass, some of the steam will be condensed, reducing the press- 
ure and temperature of the remainder, until an equilibrium is established. 

Temperature and Pressure of Saturated Steam.— The re- 
lation between the temperature and the pressure of steam, according to 
Regnault's experiments, is expressed by the formula (Buchanan's, as given 

by Clark) t = _ <fW .», T- — \ — 371.85, in which p is the pressure in pounds 

6.1993o44 — logp 
per square inch and t the temperature of the steam in Fahrenheit degrees. 
It applies with accuracy between 120° F. and 446° F., corresponding to pres- 
sures of from 1.68 lbs. to 445 lbs. per square inch. (For other formulas see 
Wood's and Peabody's Thermodynamics.) 

Total Heat of Saturated Steam (above 32° F.).— According to 
Regnault's experiments, the formula for total heat of steam is H = 1091.7 + 
,30b(t — 32°), in which t is temperature Fahr., and H the heat-units. (Ran- 
kine and many others; Clark gives 1091.16 instead of 1091.7.) 

Latent Heat of Steam.— The formula for latent heat of steam, as 
given by Raukine and others, is L = 1091.7 — .695(£ — 32°). Clausius's for- 
mula, in Fahrenheit units, as given by Clark, is L = 1092.6 — ,708(i — 32°). 



660 STEAM. 

The total heat in steam (above 32°) includes three elements: 

1st. The heat required to raise the temperature of the water to the tem- 
perature of the steam. 

2d. The heat required to evaporate the water at that temperature, called 
internal latent heat. 

3d. The latent heat of volume, or the external work done by the steam in 
making room for itself against the pressure of the superincumbent atmos- 
phere (or surrounding steam if inclosed in a vessel). 

The sum of the last two elements is called the latent heat of steam. In 
Buel's tables (Weisbach, vol. ii., Dubois's translation) the two elements are 
given separately. 

Latent Heat of Volume of Saturated Steam. (External 
Work.)— The following formulas are sufficiently accurate for occasional use 
within the given ranges of pressure (Clark, S. E.): 

From 14.7 lbs. to 50 lbs. total pressure per square inch. . . 55.900 + .0772£. 
From 50 lbs. to 200 lbs. total pressure per square inch.... 59.191 -j- .0655*. 

Heat required to Generate 1 lb. of Steam from water at 32° F. 

Heat-units. 

Sensible heat, to raise the water from 32° to 212° = 180.9 

Latent heat, 1, of the formation of steam at 212° = 894.0 

2, of expansion against the atmospheric 
pressure, 21 16.4 lbs. per sq. f t. X26.36 cu. ft. , 
= 55,786 foot-pounds -*- 778 = 71.7 965.7 

Total heat above 32° F 1146.6 

The Heat Unit, or British Thermal Unit.— The definition of 
the heat-unit used in this work is that of Rankine, accepted by most modern 
writers, viz., the quantity of heat required to raise the temperature of 1 lb. 
of water 1° F. at or near its temperature of maximum density (39.1° F.). 
Peabody's definition, the heat required to raise a pound of water from 62° 
to 6 r J° F. is not generally accepted. (See Thurston, Trans. A. S. M. E., 
xiii. 351.) 

Specific Heat of Saturated Steam.— The specific heat of satu- 
rated steam is .305, that of water being 1; or it is 1.281, if that of air be 1. 
The expression .305 for specific heat is taken in a compound sense, relating 
to changes both of volume and of pressure which takes place in the eleva- 
tion of temperature of saturated steam. (Clark, S. E.) 

This statement by Clark is not strictly accurate. When the temperature 
of saturated steam is elevated, water being present and the steam remain- 
ing saturated, water is evaporated. To raise the temperature of 1 lb. of 
water 1° F. requires 1 thermal unit, and to evaporate it at 1° F. higher would 
require 0.695 less thermal unit, the latent heat of saturated steam decreas- 
ing 0.695 B.T.U. for each increase of temperature of 1° F. Hence 0.305 is 
the specific heat of water and its saturated vapor combined. 

When a unit weight of saturated steam is increased in temperature and in 
pressure, the volume decreasing so as to just keep it saturated, the specific 
heat is negative, and decreases as temperature increases. (See Wood, 
Therm., p. 147; Peahody, Therm., p. 93.) 

Density and Volume of Saturated Steam. — The density of 
steam is expressed by the weight of a given volume, say one cubic foot; and 
the volume is expressed by the number of cubic feet in one pound of steam. 

Mr. Brownlee's expression for the density of saturated steam in terms of 

op-941 

the pressure is D = fl , or log D = .941 p — 2.519, in which D is the den- 

oou.oo 
sity, and p the pressure in pounds per square inch. In this expression, p" 941 
is the equivalent of p raised to the 16/17 power, as employed by Rankine. 
The volume v being the reciprocal of the density, 

.941 log p. 

Relative Volume of Steam.— The relative volume of saturated 
steam is expressed by the number of volumes of steam produced from one 



STEAM. 661 

volume of water, the volume of water being measured at the temperature 
39° F. The relative volume is found by multiplying the volume in cu. ft. of 
one lb. of steam by the weight of a cu. ft. of water at 39° F., or 62.425 lbs. 

Gaseous Steam.— When saturated steam is superheated, or sur- 
charged with heat, it advances from the condition of saturation into that of 
gaseity. The gaseous state is only arrived at by considerably elevating the 
temperature, supposing the pressure remains the same. Steam thus suffi- 
ciently superheated is known as gaseous steam or steam gas. 

Total Heat of Gaseous Steam.— Regnault found that the total 
heat of gaseous steam increased, like that of saturated steam, uniformly 
with the temperature, and at the rate of .475 thermal units per pound for 
each degree of temperature, under a constant pressure. 

The general formula for the total heat of gaseous steam produced from 
1 pound of water at 32° F. is H — 1074.6 -j- A7M. [This formula is for vapor 
generated at 32°. It is not true if generated at 212°, or at any other tempera- 
ture than 32°. (Prof. Wood.)] 

The Specific Heat of Gaseous Steam is .475, under constant 
pressure, as found by Regnault. It is identical with the coefficient of in- 
crease of total heat for each degree of temperature. [This is at atmospheric 
pressure and 212° temperature. He found it not true for any other pressure. 
Theory indicates that it would be less at higher temperatures. (Prof. Wood.)] 

The Specific Density of Gaseous Steam is .622, that of air being 
1. That is to say, the weight of a cubic foot of gaseous steam is about five 
eighths of that of a cubic foot of air, of the same pressure and temperature. 

The density or weight of a cubic foot of gaseous steam is expressible by 
the same formula as that of air, except that the multiplier or coefficient is 
less in proportion to the less specific density. Thus, 

_ 2.7074p X -622 _ 1.684p 
* + 461 ~ «+461' 

in which D' is the weight of a cubic foot of gaseous steam, p the total pres- 
sure per square inch, and t the temperature Fahreuheit. 

Superheated. Steam. —The above remarks concerning gaseous steam 
are taken from Clark's Steam engine. Wood gives for the total heat (above 
32°) of superheated steam H = 1091.7 + 0.48(i - 32°). 

The following is abridged from Peabody (Therm., p. 115, etc.). 

When far removed from the temperature of saturation, superheated steam 
follows the laws of perfect gases very nearly, but near the temperature of 
saturation the departure from those laws is too great to allow of calculations 
by them for engineering purposes. 

The specific heat at constant pressure, Cp, from the mean of three experi- 
ments by Regnault, is 0.4805. 

Values of the ratio of Cp to specific heat at constant volume: 

Pressure p, pounds per square inch.. 5 50 100 200 300 
Ratio Cp -s- Cv = k = 1 .335 1.332 1 .330 1 .324 1 .316 

Zeuner takes fcasa constant = 1.333. 

Specific Heat at Constant Volume, Superheated Steam. 

Pressure, pounds per square inch 5 50 100 200 300 

Specific heat Cv 0.351.348 .346 .344 .341 

It is quite as reasonable to assume that Cv is a constant as to suppose that 
Cp is constant, as has been assumed. If we take Cv to be constant, then Cp 
will appear as a variable. 

If p = pressure in lbs. per sq. ft,, v = volume in cubic feet, and T = 
temperature in degrees Fahrenheit -f- 460.7. then pv = 93.5T — 971pJ. 

Total heat of superheated steam, H = 0.4805(T - 10.38pi) 4- 857.2. 

The Rationalization of Regnault's Experiments on 
Steam. (J. McFarlane Gray, Proc. Inst. M. E., July, 188S>.)— The formulae 

constructed by Regnault are strictly empirical, and were based entirely on 
his experiments. They are therefore not valid beyond the range of temper- 
atures and pressures observed 

Mr. Gray has made a most elaborate calculation, based not on experiments 
but on fundamental principles of thermodynamics, from which he deduces 
formulas for the pressure and total heat of steam, and presents tables calcu- 



662 



lated therefrom which show substantial agreement with Regnault's figures. 
He gives the following examples of steam-pressures calculated for tempera- 
tures beyond the range of Regnault's experiments. 



Temperature. 


Pounds per 
sq. in. 


Tempe 


rature. 


Pounds per 


C. 


Fahr. 


C. 


Fahr 


sq. in. 


230 


446 


406.9 


340 


644 


2156.2 


240 


464 


488.9 


360 


680 


2742.5 


250 


482 


579.9 


380 


716 


3448.1 


260 


500 


691.6 


400 


752 


4300.2 


280 


536 


940.0 


415 


779 


5017.1 


300 


572 


1261.8 


427 


800.6 


5659.9 


320 


608 


1661.9 









These pressures are higher than those obtained by Regnault's formula, 
which gives for 415° C. only 4067.1 lbs. per square inch. 

Table of the Properties of Saturated Steam.— In the table 
of properties of saturated steam on the following pages the figures for tem- 
perature, total heat, and latent heat are taken, up to 210 lbs. absolute pres- 
sure, from the tables in Porter's Steam-engine Indicator, which tables have 
been widely accepted as standard by American engineers. The figures for 
total heat, given in the original as from 0° F., have been changed to heat 
above 32° F. The figures for weight per cubic foot and for cubic feet per 
pound have been taken from Dwelshauvers-Dery's table, Trans. A. S. M. E., 
vol. xi., as being probably more accurate than those of Porter. The figures 
for relative volume are from Buel's table, in Dubois's translation of Weis- 
bach, vol. ii. They agree quite closely with the relative volumes calculated 
from weights as given by Dery. From 211 to 219 lbs. the figures for temper- 
ature, total heat, and latent heat are from Dery's table; and from 220 to 1000 
lbs. all the figures are from Buel's table. The figures have not been carried 
out to as many decimal places as they are in most of the tables given by the 
different authorities: but any figure beyond the fourth significant figure is 
unnecessary in practice, and beyond the limit of error of the observations 
and of the formulae from which the figures were derived. 

Weight of 1 Cubic Foot of Steam in Decimals of a Pound. 
Comparison of Different Authorities. 



_c 


Weight of 1 cubic foot 


a 

a a m 
£2& 


Weight of 1 cubic foot 


SP* 


according to— 


according to— 


£2^ 

JD 


Por- 
ter. 


Clark 


Buel. 


Dery. 


Pea- 
body. 


Por- 
ter. 


Clark 


Buel. 


Dery. 


Pea- 
body 


1 


.0030 


.003 


.00303 


.00299 


.00299 


120 


.27428 


.2738 


.2735 


.2724 


.2695 


14.7 


.03797 


.0380 


.03793 




.0376 


140 


.31386 


.3162 


.3163 


.3147 


.3113 


20 


.0511 


.0507 


.0507 


.0507 


.0502 


160 


.35209 


.3590 


.3589 


.3567 


.3530 


40 


.0994 


.0974 


.0972 


.0972 


.0964 


180 


.38895 


.4009 


.4012 


.3983 


.3945 


60 


.1457 


.1425 


.1424 


.1422 


.1409 


200 


.42496 


.4431 


.4433 


.4400 


.4359 


80 


.19015 


.1865 


.1866 


.1862 


.1843 


220 




.4842 


.4852 




.4772 


100 


.23302 


.2307 


.2303 


.2296 


.2271 


240 




.5248 


.5270 




.5186 



There are considerable differences between the figures of weight and vol- 
ume of steam as given by different authorities. Porter's figures are based 
on the experiments of Fairbairn and Tate. The figures given by the other 
authorities are derived from theoretical formula? which are believed to give 
more reliable results than the experiments. The figures for temperature, 
total heat, and latent heat as given by different authorities show a practical 
agreement, all being derived from Regnault's experiments. See Peabody's 
Tables of Saturated Steam; also Jacobus, Trans. A. S. M. E., vol. xii., 593. 



STEAM. 



663 



Properties of Saturated Steam. 



oTi 


, 




Total 


Heat 




oi , 


fi'S 






2> ®-5 


II 


above 32° F. 


"el j.-£ 


2 Z^ 




3.S 


o=S 


In the 


In the 




S CD . 
SOU 


o aj = 


PUS 


Water 
h 


Steam 
H 


1^1 




<D . 

3 7-1 

c a 


^02 


ac3 


£3(0 


S£ 


Heat- 


Heat- 


%m 


|-a 


!> 


< 


£ 


units. 


units. 


j 


tf 


> 


29.74 


.089 


32 





1091.7 


1091.7 


208080 


3333.3 


.00030 


29.67 


.122 


40 


8. 


1094.1 


1086.1 


154330 


2472.2 


.00040 


29.56 


.176 


50 


18. 


1097.2 


1079.2 


107630 


1724.1 


.00058 


29.40 


.254 


60 


28.01 


1100.2 


1072.2 


76370 


1223.4 


.00082 


29.19 


.359 


70 


38.02 


1103.3 


1065.3 


54660 


875.61 


.00115 


28.90 


.502 


80 


48.04 


1106.3 


1058.3 


39690 


635.80 


.00158 


28.51 


.692 


90 


58.06 


1109.4 


1051.3 


29290 


469.20 


.00213 


28.00 


.943 


100 


68.08 


1112.4 


1044.4 


21830 


349.70 


.00286 


27.88 


1 


102.1 


70.09 


1113.1 


1043.0 


20623 


334.23 


.00299 


25.85 


2 


126.3 


94.44 


1120.5 


1026.0 


10730 


173.23 


.00577 


23.83 


3 


141.6 


109.9 


1125.1 


1015.3 


7325 


117.98 


.00848 


21.78 


4 


153.1 


121.4 


1128.6 


1007.2 


5588 


89.80 


.01112 


19.74 


5 


162.3 


130.7 


1131.4 


1000.7 


4530 


72.50 


.01373 


17.70 


6 


170.1 


138.6 


1133.8 


995.2 


3816 


61.10 


.01631 


15.67 


7 


176.9 


145.4 


1135.9 


990.5 


3302 


53.00 


.01887 


13.63 


8 


182.9 


151.5 


1137.7 


986.2 


2912 


46.60 


.02140 


11.60 


9 


188.3 


156.9 


1139.4 


982.4 


2607 


41.82 


.02391 


9.56 


10 
' 11 


193.2 


161.9 


1140.9 


979.0 


2361 


37.80 


.02641 


7.52 


197.8 


166.5 


1142.3 


975.8 


2159 


34.61 


.02889 


5.49 


12 


202.0 


170.7 


1143.5 


972.8 


1990 


31.90 


.03136 


3.45 


13 


205.9 


174.7 


1144.7 


970.0 


1846 


29.58 


.03381 


1.41 


14 


209.6 


178.4 


1145.9 


967.4 


1721 


27.59 


.03625 


Gauge 


















Pressure 
lbs. per 
sq. in. 


14.7 


212 


180.9 


1146.6 


965.7 


1646 


26.36 


.03794 


















0.304 


15 


213.0 


181.9 


1146.9 


965.0 


1614 


25.87 


.03868 


1.3 


16 


216.3 


185.3 


1147.9 


962.7 


1519 


24.33 


.04110 


2.3 


17 


219.4 


188.4 


1148.9 


960.5 


1434 


22.98 


.04352 


3.3 


18 


222.4 


191.4 


1149 8 


958.3 


1359 


21.78 


.04592 


4.3 


19 


225.2 


194.3 


1150.6 


956.3 


1292 


20.70 


.04831 


5.3 


20 


227.9 


197.0 


1151.5 


954.4 


1231 


19.72 


.05070 


6.3 


21 


230.5 


199.7 


1152.2 


952.6 


1176 


18.84 


.05308 


7.3 


22 


233.0 


202.2 


1153.0 


950.8 


1126 


18.03 


.05545 


8.3 


23 


235.4 


204.7 


.7 


949.1 


1080 


17.30 


.05782 


9.3 


24 


237.8 


207.0 


1154.5 


947.4 


1038 


16.62 


.06018 


10.3 


25 


240.0 


209.3 


1155.1 


945.8 


998.4 


15.99 


.06253 


11.3 


26 


' 242.2 


211.5 


.8 


944.3 


962.3 


15.42 


.06487 


12.3 


27 


244.3 


213.7 


1156.4 


942.8 


928.8 


14.88 


.06721 


13.3 


28 


246.3 


215.7 


1157.1 


941.3 


897.6 


14.38 


.06955 


14.3 


29 


248.3 


217.8 


.7 


939.9 


868.5 


13.91 


.07188 


15.3 


30 


250.2 


219.7 


1158.3 


938.9 


841.3 


13.48 


.07420 


16.3 


31 


252.1 


221.6 


.8 


937.2 


815 8 


13.07 


.07652 


17.3 


32 


254.0 


223.5 


1159.4 


935.9 


791.8 


12.68 


.07884 


18.3 


33 


255.7 


225.3 


.9 


934.6 


769.2 


12.32 


.08115 


19.3 


34 


257.5 


227.1 


1160.5 


933.4 


748.0 


11.98 


.08346 


20.3 


35 


259.2 


228.8 


1161.0 


932.2 


727.9 


11.66 


.08576 


21.3 


36 


260.8 


230.5 


1161.5 


931.0 


708.8 


11.36 


.08806 


22.3 


37 


262.5 


232.1 


1162.0 


929.8 


690.8 


11.07 


.09035 



664 



STEAM. 



Properties of Saturated Steam. 



of = ' 


CO 




Total Heat 
above 32° F. 


^ . 


CD ; 


£ 5 


Z-O 




P- 1 -.2 


CO S 




-us m 




. o 






In the 


In the 


o « 




*2 £ 


U CD 

A* 


Water 
h 


Steam 
H 


a if 


> .OS 

•J3 — 00 
TO ©+-> 


s— 


too . 


§>:2 


w ~ §* 


Heat- 


Heat- 


tgllffl 


1> «s 


o s 


'<££ 


o 


:§ 


H 


units. 


units. 


J 


ti 


> 


£ 


23.3 


38 


264.0 


233.8 


1162.5 


928.7 


673.7 


10.79 


.09264 


20 


39 


265.6 


235.4 


.9 


927.6 


657.5 


10.53 


.09493 


25.3 


40 


267.1 


236.9 


1163.4 


926.5 


642.0 


10.28 


.09721 


26.3 


41 


268.6 


238.5 


.9 


925.4 


627.3 


10.05 


.09949 


27.3 


42 


270.1 


240.0 


1164.3 


924.4 


613.3 


9.83 


.1018 


28.3 


43 


271.5 


241.4 


.7 


923.3 


599.9 


9.61 


.1040 


29.3 


44 


272.9 


242.9 


1165.2 


922.3 


587.0 


9.41 


.1063 


30.3 


45 


274.3 


244.3 


.6 


921.3 


574.7 


9.21 


.1086 


31.3 


46 


275.7 


245.7 


1166.0 


920.4 


563 


9.02 


.1108 


32.3 


47 


277.0 


247.0 


.4 


919.4 


551.7 


8.84 


.1131 


33.3 


48 


278.3 


248.4 


.8 


918.5 


540.9 


8.67 


.1153 


34.3 


49 


279.6 


249.7 


1167.2 


917.5 


530.5 


8.50 


.1176 


35.3 


50 


280.9 


251.0 


.6 


916.6 


520.5 


8.34 


.1198 


36.3 


51 


282.1 


252.2 


1168.0 


915.7 


510.9 


8.19 


.1221 


37.3 


52 


283.3 


253.5 


.4 


914.9 


501.7 


8.04 


.1243 


38.3 


53 


284.5 


254.7 


7 


914.0 


492.8 


7.90 


.1266 


39.3 


54 


285.7 


256.0 


1169.1 


913.1 


484.2 


7.76 


.1288 


40.3 


55 


286.9 


257.2 


.4 


912.3 


475.9 


7.63 


.1311 


41.3 


56 


288.1 


258.3 


.8 


911.5 


467.9 


7.50 


.1333 


42.3 


57 


289.1 


259.5 


1170.1 


910.6 


460.2 


7.38 


.1355 


43.3 


58 


290.3 


260.7 .5 


909.8 


452.7 


7.26 


.1377 


44.3 


59 


291.4 


261.8 


.8 


909.0 


445.5 


7.14 


.1400 


45.3 


60 


292.5 


262.9 


1171.2 


908.2 


438.5 


7.03 


.1422 


46.3 


61 


293.6 


264.0 


.5 


907.5 


431.7 


6.92 


.1444 


47.3 


62 


294.7 


265.1 


.8 


906.7 


425.2 


6.82 


.1466 


48.3 


63 


295.7 


266.2 


1172.1 


905.9 


418.8 


6.72 


.1488 


49.3 


64 


296.8 


267.2 


.4 


905.2 


412.6 


6.62 


.1511 


50.3 


65 


297.8 


268.3 


.8 


904.5 


406.6 


• 6.53 


.1533 


51.3 


66 


298.8 


269.3 


1173.1 


903 7 


400.8 


6.43 


.1555 


52.3 


67 


299.8 


270.4 


.4 


903.0 


395.2 


6.34 


.1577 


53.3 


68 


300.8 


271.4 


.7 


902.3 


389.8 


6.25 


.1599 


54.3 


69 


301.8 


272.4 


1174.0 


901.6 


384.5 


6.17 


.1621 


55.3 


70 


302.7 


273.4 


.3 


900.9 


379.3 


6.09 


.1643 


56.3 


71 


303.7 


274.4 


.6 


900.2 


374.3 


6.01 


.1665 


57.3 


72 


304.6 


275.3 


.8 


899.5 


369.4 


5.93 


.1687 


58.3 


73 


305.6 


276.3 


1175.1 


898.9 


364.6 


5.85 


.1709 


59.3 


74 


306.5 


277.2 


.4 


898.2 


360.0 


5.78 


.1731 


60.3 


75 


307.4 


278.2 


.7 


897.5 


355.5 


5.71 


.1753 


61.3 


76 


308.3 


279.1 


1176.0 


896.9 


351.1 


5.63 


.1775 


62.3 


77 


309.2 


280.0 


.2 


896.2 


346.8 


5.57 


.1797 


63.3 


78 


310.1 


280.9 


.5 


895.6 


342.6 


5.50 


.1819 


64.3 


79 


310.9 


281.8 


1176.8 


895.0 


338.5 


5.43 


.1840 


65.3 


80 


311.8 


282.7 


1177.0 


894.3 


334.5 


5.37 


.1862 


66.3 


81 


312.7 


283.6 


.3 


893.7 


330.6 


5.31 


.1884 


67.3 


82 


313.5 


284.5 


.6 


893.1 


326.8 


5.25 


.1906 


68.3 


83 


314.4 


285.3 


.8 


892.5 


323.1 


5.18 


.1928 


69.3 


84 


315.2 


286.2 


1178.1 


891.9 


319.5 


5.13 


.1950 


70.3 


85 


316.0 


287.0 


.3 


891.3 


315.9 


5.07 


.1971 



665 





Properties of 


Saturated Steam. 






€.5 




is 

a a 

CD ^ 


Total 
above 


Heat 
32° F. 


i4 . 

W | ? 

■gnw 




-e 3 
~ ® 

.O 

3h 

© a 


us 


In the 
Water 

h 
Heat- 


In the 
Steam 

H 
Heat- 


"53 «S 


o 


< 


H 


units. 


units. 


J 


P5 - 


>'" 


£ 


71.3 


86 


316.8 


287.9 


1178.6 


890.7 


312.5 


5 02 


.1993 


72.3 


87 


317.7 


288.7 


.8 


890.1 


309.1 


4.96 


.2015 


73.3 


88 


318.5 


289.5 


1179.1 


889.5 


305.8 


4.91 


.2036 


74.3 


89 


319.3 


290.4 


.3 


888.9 


302.5 


4.86 


.2058 


75.3 


90 


320.0 


291.2 


.6 


888.4 


299.4 


4.81 


.2080 


76.3 


91 


320.8 


292.0 


.8 


887.8 


296.3 


4.76 


.2102 


77.3 


92 


321.6 


292.8 


1180.0 


887.2 


293.2 


4.71 


.2123 


78.3 


93 


322.4 


293.6 


.3 


886.7 


290.2 


4.66 


.2145 


79.3 


94 


323.1 


294.4 


.5 


886.1 


287.3 


4.62 


.2166 


80.3 


95 


323.9 


295.1 


.7 


885.6 


284.5 


4.57 


.2188 


81.3 


96 


324.6 


295.9 


1181.0 


885.0 


281.7 


4.53 


.2210 


82.3 


97 


325.4 


296.7 


.2 


884.5 


279.0 


4.48 


.2231 


83.3 


98 


326.1 


297.4 


.4 


884.0 


276.3 


4.44 


.2253 


84.3 


99 


3:26.8 


298.2 


.6 


883.4 


273.7 


4.40 


.2274 


85.3 


100 


327.6 


298.9 


.8 


882.9 


271.1 


4.36 


.2296 


86.3 


101 


3-28.3 


299 7 


1182.1 


882.4 


268.5 


4.32 


.2317 


87.3 


102 


329.0 


300.4 


.3 


881.9 


266.0 


4.28 


.2339 


88.3 


103 


329.7 


301.1 


.5 


881.4 


263.6 


4.24 


.2360 


89.3 


104 


330.4 


301.9 


.7 


880.8 


261.2 


4.20 


.2382 


90.3 


105 


331.1 


302.6 


.9 


880.3 


258.9 


4.16 


.2403 


91.3 


106 


331.8 


303.3 


1183.1 


879.8 


256.6 


4.12 


.2425 


92.3 


107 


332.5 


304.0 


.4 


879.3 


254.3 


4.09 


.2446 


93.3 


108 


333.2 


304.7 


.6 


878.8 


252.1 


4.05 


.2467 


94.3 


109 


333.9 


305.4 


.8 


878.3 


249.9 


4.02 


.2489 


95.3 


110 


334.5 


306.1 


1184.0 


877.9 


247.8 


3.98 


.2510 


96.3 


111 


335.2 


306.8 


.2 


877.4 


245.7 


3.95 


.2531 


97.3 


112 


335.9 


307.5 


.4 


876.9 


243.6 


3.92 


.2553 


98.3 


113 


336.5 


308.2 


.6 


876.4 


241.6 


3.88 


.2574 


99.3 


114 


337.2 


308.8 


.8 


875.9 


239.6 


3.85 


.2596 


100.3 


115 


337.8 


309.5 


1185.0 


875.5 


237.6 


3.82 


.2617 


101.3 


116 


338.5 


310.2 


.2 


875.0 


235.7 


3.79 


.2638 


102.3 


117 


339.1 


310.8 


'.4 


874.5 


233.8 


3.76 


.2660 


103.3 . 


118 


339.7 


311.5 


.6 


874.1 


231.9 


3 73 


.2681 


104.3 


119 


340.4 


312.1 


.8 


873.6 


230.1 


3>0 


.2703 


105.3 


120 


341.0 


312.8 


.9 


873.2 


228.3 


3.67 


.2724 


106.3 


121 


341.6 


313.4 


1186.1 


872.7 


226.5 


3.64 


.2745 


107.3 


122 


342.2 


314.1 


.3 


872.3 


224.7 


3.62 


.2766 


108.3 


123 


342.9 


314.7 


.5 


871.8 


223.0 


3.59 


.2788 


109.3 


124 


343.5 


315.3 


.7 


871.4 


221.3 


3.56 


.2809 


110.3 


125 


344.1 


316 


.9 


870.9 


219.6 


3.53 


.2830 


111.3 


126 


344.7 


316.6 


1187.1 


870.5 


218.0 


3.51 


.2851 


112.3 


127 


345.3 


317.2 


.3 


870.0 


216.4 


3.48 


.2872 


113.3 


128 


345.9 


317.8 


.4 


869.6 


214.8 


3.46 


.2894 


114.3 


129 


346.5 


318.4 


.6 


869.2 


213.2 


3.43 


.2915 


115.3 


130 


347.1 


319.1 


.8 


868.7 


211.6 


3.41 


.2936 


116.3 


131 


347.6 


319.7 


1188.0 


868.3 


210.1 


3.38 


.2957 


117.3 


132 


348.2 


320.3 


.2 


867.9 


208.6 


3.36 


.2978 


118.3 


133 


34S.8 


320.8 


.3 


867.5 


207.1 


3.33 


.3000 


119,3 


134 


349.4 


321.5 


.5 


867.0 


205.7 


3.31 


.3021 



666 



Properties of Saturated Steam. 









Total Heat 




11 


£*S 




3 — 


fc ..S 

©j§ Q 


Is "a 


above 32° F. 




O02 


&s 


to a" 

© 02 


In the 


In the 


!| 




<D * 


Water 


Steam 




©* ° 


& ft 


ogs 


g-l 


h 


H 


s^l 


1°* 

£ n o 

©►rai 




§02 




CO J- & 

< 




Heat- 
units. 


Heat- 
units. 


Sum 




120.3 


135 


350.0 


322.1 


1188.7 


866.6 


204.2 


3.29 


.3042 


121.3 


136 


350.5 


322.6 


.9 


866.2 


202.8 


3.27 


.3063 


122.3 


137 


351.1 


323.2 


1189.0 


865.8 


201.4 


3.24 


.3084 


123.3 


138 


351.8 


323.8 


.2 


865.4 


200.0 


3.22 


.3105 


124.3 


139 


352.2 


324.4 


.4 


865.0 


198.7 


3.20 


.3126 


125.3 


140 


a52.8 


325.0 


.5 


864.6 


197.3 


3.18 


.3147 


126.3 


141 


353.3 


325.5 


.7 


864.2 


196.0 


3.16 


.3169 


127.3 


142 


353.9 


326.1 


.9 


863.8 


194.7 


3.14 


.3190 


128.3 


143 


354.4 


326.7 


1190.0 


863.4 


193.4 


3.11 


.3211 


129.3 


144 


355.0 


327.2 


.2 


863.0 


192.2 


3.09 


.3232 


130.3 


145 


355.5 


327.8 


.4 


862.6 


190.9 


3.07 


.3253 


131.3 


146 


356.0 


328.4 


.5 


862.2 


189.7 


3.05 


.3274 


132.3 


147 


356.6 


328.9 


.7 


861.8 


185.5 


3.04 


.3295 


133.3 


148 


357.1 


329.5 


.9 


861.4 


187.3 


3.02 


.3316 


134.3 


149 


357.6 


330.0 


1191.0 


861.0 


186.1 


3.00 


.3337 


135.3 


150 


358.2 


330.6 


.2 


860.6 


184.9 


2.98 


.3358 


136.3 


151 


358.7 


331.1 


.3 


860.2 


183.7 


2.96 


.3379 


137.3 


152 


359.2 


331.6 


.5 


859.9 


182.6 


2.94 


.3400 


138.3 


153 


359.7 


332.2 


.7 


859.5 


181.5 


2.92 


.3421 


139.3 


154 


360.2 


332.7 


.8 


859.1 


180.4 


2.91 


.3442 


140.3 


155 


360.7 


333.2 


1192.0 


858.7 


179.2 


2.89 


.3463 


141.3 


156 


361.3 


333.8 


.1 


858.4 


178.1 


2.87 


.3483 


142.3 


157 


361.8 


334.3 


.3 


858.0 


177.0 


2.85 


.3504 


143.3 


158 


362.3 


334.8 


.4 


857.6 


175.0 


2.84 


.3525 


144.3 


159 


362.8 


335.3 


.6 


857.2 


174.9 


2.82 


.3546 


145.3 


160 


363.3 


335.9 


.7 


856.9 


173.9 


2.80 


.3567 


146.3 


161 


363.8 


336.4 


.9 


856.5 


172.9 


2.79 


.3588 


147.3 


162 


364.3 


336.9 


1193.0 


856.1 


171.9 


2.77 


.3609 


14S.3 


163 


364.8 


337.4 


.2 


855.8 


171.0 


2.76 


.3630 


149.3 


164 


365.3 


337.9 


.3 


855.4 


170.0 


2.74 


.3650 


150.3 


165 


365.7 


338.4 


.5 


855.1 


169.0 


2.72 


.3671 


151.3 


166 


366.2 


338.9 


.6 


854.7 


168.1 


2.71 


.3692 


152.3 


167 


366.7 


339.4 


.$ 


854.4 


167.1 


2.69 


.3713 


153.3 


168 


367.2 


339.9 


.9 


854.0 


166.2 


2.68 


.3734 


154.3 


169 


367.7 


340.4 


1194.1 


853.6 


165.3 


2M 


.3754 


155.3 


170 


368.2 


340.9 


.2 


853.3 


164.3 


2.65 


.3775 


156.3 


171 


368.6 


341.4 


.4 


852.9 


163.4 


2.63 


.3796 


157.3 


172 


369.1 


341.9 


.5 


852.6 


162.5 


2.62 


.3817 


158.3 


173 


369.6 


342.4 


.7 


852.3 


161.6 


2.61 


.3838 


159.3 


174 


370.0 


342.9 


.8 


851.9 


160.7 


2.59 


.3858 


160 3 


175 


370.5 


343.4 


.9 


851.6 


159.8 


2.58 


.3879 


161.3 


176 


371.0 


343.9 


1195.1 


851.2 


158.9 


2.56 


.3900 


162.3 


177 


371.4 


344.3 


.2 


850.9 


158.1 


2.55 


.3921 


163.3 


178 


371.9 


344.8 


.4 


850.5 


157.2 


2.54 


.3942 


164.3 


179 


372.4 


345.3 


.5 


850.2 


156.4 


2.52 


.3962 


165.3 


180 


372.8 


345.8 


7 


849.9 


155.6 


2.51 


.3983 


166.3 


181 


373.3 


346.3 


'.8 


849.5 


154.8 


2.50 


.4004 


167.3 


182 


373 7 


346.7 


.9 


S49.2 


154.0 


2.48 


.4025 


168.3 


183 


374^2 


347.2 


1196.1 


848.9 


153.2 


2.47 


.40-46 



STEAM. 



667 



Properties of Saturated Steam. 



- 






Total Heat 




p"£ 


«•§ 




tip 


sa'Z 

s - 

j= fa & 


p '33 
|| 


above 32° F. 


^ . 

gtrj * 
1"W 


s 55 

?« • 
> £~ 
£<« II 
S°fe 


V 

o 

"3 fl 


'§£ 


1 st 

Pi <s 


In the 
Water 

h 
Heat- 


In the 

Steam 

H 
Heat- 




&~ 


Si 3 " 


units. 


units. 


J 


>r 


^<£ 


169.3 


184 


374.6 


347.7 


1196.2 


848.5 


152.4 


2.46 


.4066 


170.3 


185 


375.1 


348.1 


.3 


848.2 


151.6 


2.45 


.4087 


171.3 


186 


375.5 


348.6 


.5 


847.9 


150.8 


2.43 


.4108 


172.3 


187 


375.9 


349.1 


.6 


847.6 


150.0 


2.42 


.4129 


173.3 


188 


376.4 


349.5 


.7 


847.2 


149.2 


2.41 


.4150 


174.3 


189 


376.9 


350.0 


.9 


846.9 


148.5 


2.40 


.4170 


175.3 


190 


377.3 


350.4 


1197.0 


846.6 


147.8 


2.39 


.4191 


176.3 


191 


377.7 


350.9 


.1 


846.3 


147.0 


2.37 


.4212 


177.3 


192 


378.2 


351.3 


.3 


845.9 


146.3 


2.36 


.4233 


178.3 


193 


378.6 


351.8 


.4 


845.6 


145.6 


2.35 


.4254 


179.3 


194 


379.0 


352.2 


.5 


845.3 


144.9 


2.34 


.4275 


180.3 


195 


379.5 


352.7 


7 


845.0 


144.2 


2.33 


.4296 


181.3 


196 


380.0 


353.1 


!8 


844.7 


143.5 


2.32 


.4317 


182.3 


197 


380.3 


353.6 


.9 


844.4 


142.8 


2.31 


.4337 


183.3 


198 


380.7 


354.0 


1198.1 


844.1 


142.1 


2.29 


.4358 


184.3 


199 


381.2 


354.4 


.2 


843.7 


141.4 


2.28 


.4379 


185.3 


200 


381.6 


354.9 


.3 


843.4 


140.8 


2.27 


.4400 


186.3 


201 


382.0 


355.3 


.4 


843.1 


140.1 


2.26 


.4420 


187.3 


202 


382.4 


355.8 


.6 


842.8 


139.5 


2.25 


.4441 


188.3 


203 


382.8 


356.2 


.7 


842.5 


138.8 


2.24 


.4462 


189.3 


204 


383.2 


356.6 


.8 


842.2 


138.1 


2.23 


.4482 


190.3 


205 


383.7 


357.1 


1199.0 


841.9 


137.5 


2.22 


.4503 


191.3 


206 


384.1 


357.5 


.1 


841.6 


136.9 


2.21 


.4523 


192.3 


207 


384.5 


357.9 


.2 


841.3 


136.3 


2.20 


.4544 


193.3 


208 


384.9 


358.3 


.3 


841 


135.7 


2.19 


.4564 


194.3 


209 


385.3 


358.8 


.5 


840.7 


135.1 


2.18 


.4585 


195.3 


210 


385.7 


359.2 


.6 


840.4 


134.5 


2.17 


.4605 


196.3 


211 


386.1 


359.6 


.7 


840.1 


133.9 


2.16 


.4626 


197.3 


212 


386.5 


360.0 


.8 


839.8 


133.3 


2.15 


.4646 


198.3 


213 


386.9 


360.4 


.9 


839.5 


132.7 


2.14 


.4667 


199.3 


214 


387.3 


360.9 


1200.1 


839.2 


132.1 


2.13 


.4687 


200.3 


215 


387.7 


361.3 


.2 


838.9 


131.5 


2.12 


.4707 


201.3 


216 


388.1 


361.7 


.3 


838.6 


130.9 


2.12 


.4728 


202.3 


217 


388.5 


362.1 


.4 


838.3 


130.3 


2.11 


.4748 


203.3 


218 


388.9 


362.5 


.6 


838.1 


129.7 


2.10 


.4768 


204.3 


219 


389.3 


362.9 


.7 


837.8 


129.2 


2.09 


.4788 


205.3 


220 


389.7 


362.2* 


1200.8 


838.6* 


128.7 


2.06 


.4852" 


215.3 


230 


393.6 


366.2 


1202.0 


835.8 


123.3 


1.98 


.5061 


225.3 


240 


397.3 


370.0 


1203.1 


833.1 


118.5 


1.90 


.5270 


235.3 


250 


400.9 


373.8 


1204.2 


830.5 


114.0 


1.83 


.5478 


245.3 


260 


404.4 


377.4 


1205.3 


827.9 


109.8 


1.76 


.5686 


255.3 


270 


407.8 


380.9 


1206.3 


825.4 


105.9 


1.70 


.5894 


265.3 


280 


411.0 


384.3 


1207.3 


823.0 


102.3 


1.64 


.6101 


275.3 


290 


414.2 


387.7 


1208.3 


820.6 


99.0 


1.585 


.6308 


285.3 


300 


417.4 


390.9 


1209.2 


818.3 


95.8 


1.535 


.6515 


335.3 


350 


432.0 


406.3 


1213.7 


807.5 


82.7 


1.325 


.7545 



* The discrepancies at 205.3 lbs. gauge are due to the change from Dery's 
to Buel's figures. 



668 



STEAM. 



Properties of Saturated Steam. 



<B. 






Total Heat 




3 u 


da 
a£ 




3 a 


?%A 


0) . 


above 


32° F. 


^ . 


o£ 


a> a 1 


& .8 


3'v 


In the 
Water 


In the 

Steam 


33 <u 


u* II 




S* 2 - 


© afS 


e as 


h 


H 




tuoW 


§.5 


|si 


Heat- 


Heat- 


% iik 


Xw 


OJ.J 


o~ 


units. 


units. 


j 


w >co 


!>o 


£- 


385.3 


400 


444.9 


419.8 


1217.7 


797.9 


72.8 


1.167 


.8572 


435.3 


450 


456.6 


432.2 


1221.3 


, 789.1 


65.1 


1.042 


.9595 


485.3 


500 


467.4 


443.5 


1224.5 


781.0 


58.8 


.942 


1.062 


535.3 


550 


477.5 


454.1 


1227.6 


773.5 


53.6 


.859 


1.164 


585.3 


600 


486.9 


464.2 


1230.5 


766.3 


49.3 


.7,90 


1.266 


635.3 


650 


495.7 


473.6 


1233.2 


759.6 


45.6 


.731 


1.368 


685.3 


700 


504.1 


482.4 


1235.7 


753.3 


42.4 


.680 


1.470 


735.3 


750 


512.1 


490.9 


1238.0 


747.2 


39.6 


.636 


1.572 


785.3 


800 


519.6 


498.9 


1240.3 


741 .4 


37.1 


.597 


1.674 


835.3 


850 


526.8 


506.7 


1242.5 


735.8 


34.9 


.563 


1.776 


885.3 


900 


533.7 


514.0 


1244.7 


730.6 


33.0 


.532 


1.878 


935.3 


950 


540.3 


521.3 


1246.7 


725.4 


31.4 


.505 


1.980 


985.3 


1000 


546.8 


528.3 


1248.7 


720.3 


30.0 


.480 


2.082 



FLOW OF STEAM. 
Flow of Steam through a Nozzle. (From Clark on the Steam- 
engine.)— The flow of steam ot a greater pressure into an atmosphere of a 
less pressure increases as the difference of pressure is increased, until the 
external pressure becomes only 58$ of the absolute pressure in the boiler. 
The flow of steam is neither increased nor diminished by the fall of the ex- 
ternal pressure below 58$, or about 4/7ths of the inside pressure, even to the 
extent of a perfect vacuum. In flowing through a nozzle of the best form, 
the steam expands to the external pressure, and to the volume due to this 
pressure, so long as it is not less than 58$ of the internal pressure. For an 
external pressure of 58$, and for lower percentages, the ratio of expansion 
is 1 to 1.624. The following table is selected from Mr. Brownlee's data exem- 
plifying the rates of discharge under a constant internal pressure, into 
various external pressures: 

Outflow of Steam ; from a Given Initial Pressure into 
Various Lower Pressures. 

Absolute initial pressure in boiler, 75 lbs. per sq. in. 



Absolute 


External 


Ratio of 


Velocity of 


Actual 


Discharge 


Boiler per 
square 
inch. 


Pressure 

per square 

inch. 


Expansion 

in 

Nozzle. 


Outflow 

at Constant 

Density. 


Velocity of 

Outflow 
Expanded. 


inch of 

Orifice per 

minute. 


lbs. 


lbs. 


ratio. 


feet per sec. 


feet p. sec. 


lbs. 


75 


74 


1.012 


227.5 


230 


16.68 


75 


72 


1.037 


386.7 


401 


28.35 


75 


70 


1.063 


490 


521 


35.93 


75 


65 


1.136 


660 


749 


48.38 


75 


61.62 


1.198 


736 • 


876 


53.97 


75 


60 


1.219 


765 


933 


56.12 


75 


50 


1.434 


873 


1252 


64 


75 


45 


1.575 


890 


1401 


65.24 


75 


( 43.46 J 
1 58 p. cent f 


1.624 


890.6 


1446.5 


65.3 


75 


15 


1.624 


890.6 


1446.5 


65.3 


75 





1.624 


890.6 


1446.5 


65.3 



FLOW OF STEAM, 



When steam of varying initial pressures is discharged into the atmos- 
phere—the atmospheric pressure being not more than 58$ of the initial 
pressure— the velocity of outflow at constant deusity, that is, supposing the 
initial density to be maintained, is given by the formula V — 3.5953 y/i. 

V — the velocity of outflow in feet per minute, as for steam of the initial 
density ; 

h = the height in feet of a column of steam of the given absolute initial 
pressure of uniform density, the weight of which is equal to the pres- 
sure on the unit of base. 

The lowest initial pressure to which the formula applies, when the steam 
is discharged into the atmosphere at 14.7 lbs. per square inch, is (14.7 X 
100/58 =) 25.37 lbs. per square inch. Examples of the application of the 
formula are given in the table below. 

From the contents of this table it appears that the velocity of outflow into 
the atmosphere, of steam above 25 lbs. per square inch absolute pressure, 
or 10 lbs. effective, increases very slowly with the pressure, obviously be- 
cause the density, and the weight to be moved, increase with the pressure. 
An average of 900 feet per second may, for approximate calculations, be 
taken for the velocity of outflow as for constant density, that is, taking the 
volume of the steam at the initial volume. 

Outflow of Steam into the Atmosphere.— External pressure 
per square inch 14.7 lbs. absolute. Ratio of expansion in nozzle, 1.624. 



"3 u . 


?%>> 


>> 






« h . 


a o £■ 


e*> 


o 


. © 




©55 co 


O £t3 

Hi 

cso ft 
tToW 


8.1 S 

©.5 ft 

&£© © 

8 to 


. --co g 


S a 2 

oil 


'3 fe a 
o of? 


1** 

m o <u 

k^a 
cSO ft 

a^ x 


©^ t- 
©•fh » . 

be © © 5 

£ ^A a 




< 


t> 


< 


Q 


w 


< 


> 





ft 


W 


lbs. 


feet 
p. sec. 


feet 
per sec. 


lbs. 


H.P. 


lbs. 


feet 
p. sec. 


feet 
per sec. 


lbs. 


H.P. 


25.37 


863 


1401 


22.81 


45.6 


90 


895 


1454 


77.94 


155.9 


30 


867 


1408 


26.84 


53.7 


100 


898 


1459 


86.34 


172.7 


40 


874 


1419 


35.18 


70.4 


115 


902 


1466 


98.76 


197.5 


50 


880 


1429 


44.06 


88.1 


135 


906 


1472 


115.61 


231.2 


60 


885 


1437 


52.59 


105.2 


155 


910 


1478 


132.21 


264.4 


70 


889 


1444 


61.07 


122.1 


165 


912 


1481 


140.46 


280.9 


75 


891 


1447 


65.30 


130.6 


215 


919 


1493 


181.58 


363.2 



Napier's Approximate Rule.— Flow in pounds per second = ab- 
solute pressure x area in square inches -f- 70. This rule gives results which 
closely correspond with those in the above table, as shown below. 

Abs. press., lbs. p. sq. in. 25.37 40 60 75 100 135 165 215 
Discharge per mm., by 

table, lbs 22.81 35.18 52.59 65.30 86.34 115.61 140.46 181.58 

By Napier's rule 21.74 34.23 51.43 64.29.85.71 115.71 141.43 184.29 

Prof. Peabody, in Trans. A. S. M. E., xi, 187, reports a series of experi- 
ments on flow of steam through tubes J4 inch in diameter, and J4, J^, and \y& 
inch long, with rounded entrances, in which the results agreed closely with 
Napier's formula, the greatest difference being an excess of the experimental 
over the calculated result of 3.2$. An equation derived from the theory of 
thermodynamics is given by Prof. Peabody, but it does not agree with the 
experimental results as well as Napier's rule, the excess of the actual flow 
being 6.6$. 

Flow ot Steam in Pipes.— A formula commonly used for velocity 
of flow of steam in pipes is the same as Downing's for the flow of water in 



smooth cast-iron pipes, viz., V = 50 



i/f a ■ 



in which V = velocity in feet 



per second, L — length and D = diameter of pipe in feet, H = height in 
feet of a column of steam, of the pressure of the steam at the entrance, 



670 STEAM. 

which would produce a pressure equal to the difference of pressures at the 
two ends of the pipe. (For derivation of the coefficient 50, see Briggs on 
"Wanning Buildings by Steam," Proc. Inst. C. E. 1882.) 

If Q = quantity in cubic feet per minute, d = diameter in inches, L and H 
being in feet, the formula reduces to 

<2 = 4.7233|/^, #=.0448^, d = .5374^^. 

(These formulae are applicable to. air and other gases as well as steam.) 

If Pi = pressure in pounds per square inch of the steam (or gas) at the en- 
trance to the pipe, p 2 = the pressure at the exit, then 144(pj — p 2 ) — differ- 
ence in pressure per square foot. Let w = density or weight per cubic foot 
of steam at the pressure p t , then the height of column equivalent to the 
difference in pressures 



= H= Ui ^~P*\ and Q = 60 X .7854 X 5oW ***<£■' ~ P * )D . 

to y %vL 

If W = weight of steam flowing in pounds per minute =. Qw, and d is 
taken in inches, L being in feet, 



S|/ M( P'2 Pa,)d<r ; Q = 56.68i/^: 

Y w(Pi - Pa) y Pi - Pa 



Pi)d\ 



Velocity in feet per minute = V = Q -+- .78! 
For a velocity of 6000 feet per minute, d = 



y«- 



- p?)d 



3(Pi-p a )' 



For a velocity of 6000 feet per minute, a steam-pressure of 100 lbs. gauge, 

or w =.264, and a length of 100 feet, d — : : p^. — Pi = ~r- That is, a 

Pi — Pi d 

pipe 1 inch diameter, 100 feet long, carrying steam of 100 lbs. gauge-pressure 
at 6000 feet velocity per minute, would have a loss of pressure of 8.8 lbs. per 
sq jare inch, while steam travelling at the same velocity in a pipe 8.8 inches 
diameter would lose only 1 lb. pressure. 
G. H. Babcock, in " Steam," gives the formula 



-=v ; 



*o(Pi — P<i)d a 



In earlier editions of " Steam " the coefficient is given as 300.— evidently an 
error,— and this value has been reprinted in Clark's Pocket-Book (1892 edi- 
tion). It is apparently derived from one of the numerous formulae for flow 
of water in pipes, the multiplier of L in the denominator being used for an 
expression of the incre ased resista nce of small pipes. Putting this formula 

in the form W = ca/ ■ ~ P2 ' — , in which c will vary with the diameter 

of the pipe, we have, 

For diameter, inches.... 1 2 3 4 6 9 12 

Value of c 40.7 52.1 58.8 63 68.8 73.7 79.3 

instead of the constant value 56.68, given with the simpler formula. 
One of the most widely accepted formulae for flow of water is D'Arcy's, 

V = ca/ y~p in which c has values ranging from 65 for a ^-inch pipe up to 



FLOW OF STEAM. 



671 



111.5 for 24-inch. Using D'Arcy's coefficients, and modifying his formula to 
make it apply to steam, to the form 



we obtain, 

For diameter, inches 


- p * )d \ or W 
toL 

36.8 45.3 52.7 

9 10 12 
61.2 61.8 62.1 


/w(p 


-Pi)d 5 


V 

3 4 
56.1 57.8 

14 16 
62.3 62.6 


L 

58.4 59.5 60.1 60.7 


For diameter, inches — 
Value of c 


18 20 22 24 
62.7 62.9 63.2 63.2 



In the absence of direct experiments these coefficients are probably as 
accurate as any that may be derived from formulae for flow of water. 



QHvL 
c 2 d 5 " 



Loss of pressure in lbs. per sq. in. = p t — p, 2 = 

Iioss of Pressure due to Radiation as well as Friction.— 

E. A. Rudiger {Mechanics, June 30, 1883) gives the following formulae and 
tables for flow of steam in pipes. He takes into consideration the losses in 
pressure due both to radiation and to friction. 

W 3 fl 
Loss of power, expressed in heat-units due to friction, Hf = _ ^ „ . 

Loss due to radiation, Hr = 0.2Q2rld. 

In which IF is the weight in lbs. of steam delivered per hour, / the coeffi- 
cient of friction of the pipe, I the length of the pipe in feet, p the absolute 
terminal pressure, d the diameter of the pipe in inches, and r the coefficient 
of radiation. / is taken as from .0165 to .0175, and r varies as follows : 

TABLE OF VALUES FOR ?*. 



Pipe Covering. 


Absolute Pressure. 


40 lbs. 


65 lbs. 


90 lbs. 


115 lbs. 


Uncovered pipe :-.•••< - 


437 
146 
157 
150 
100 
61 
48 


555 
178 
192 
185 
122 
76 
58 


620 
193 
202 
197 
145 
85 
66 


684 
209 




222 




210 




151 




93 


2 "- hair felt 


73 



The appended table shows the loss due to friction and radiation in a steam- 
pipe where the quantity of steam to be delivered is 1000 lbs. per hour, I = 
1000 feet, the pipe being so protected that loss by radiation r = 64, and the 
absolute terminal pressure being 90 lbs.: 



Diameter 


Loss by 


Loss by 
Radia- 
tion, 
Hr. 


Total 


Diam. 


Loss by 


Loss by 
Radia- 
tion, 
Hr. 


Total 


of Pipe, 


Friction, 


Loss, 


of Pipe, 


Friction, 


Loss, 


inches. 


Hf. 


L. 


inches. 


Hf. 


L. 


1 


197,531 


16,76^ 


214,300 


3^ 


376 


58,688 


59,064 


M 


64,727 


20,960 


85,687 


4 


193 


67,072 


67,265 


W% 


26,012 


25,152 


51,164 


5 


63 


83,840 


83,903 


Wa 


12,035 


29,344 


41,379 


6 


25 


100,608 


li 0,623 


a 


6,173 


33,536 


39,709 


7 


12 


117,376 


117,388 


2^3 


2,023 


41,920 


43,943 


8 


6 


134,144 


134,150 


3 


813 


50.304 


51.117 











672 



If the pipes are carrying steam with minimum loss, then for same r, f, 
and p, the loss of pressure L for pipes of different diameters varies in- 
versely as the diameters. 

The general equation for the loss of pressure for the minimal loss from 
friction and radiation is 

_ 0.0007023 drip 
L ~ W • 

The loss of pressure for pipes of 1 inch diameter for different absolute 
terminal pressures when steam is flowing with minimal loss is expressed by 

the formula L = Cl^/r 2 , in which the coefficient C has the following values: 

For 65 lbs. abs. term, pressure C = 0.00089337 

" 75 " " " " 0.00093684 

" 90 " " " " 0.00099573 

"100" " " " 0.00103132 

"115" " " " 0.00108051 

In order to find the loss of pressure for any other diameter, divide the loss 
of pressure in a 1-inch pipe for the given terminal pressure by the given 
diameter, and the quotient will be the loss of pressure for that diameter. 

The following is a general summary of the results of Mr. Rudiger's inves- 
tigation : 

The flow of steam in a pipe is determined in the same manner as the flow 
of water, the formula for the flow of steam being modified only by substi- 
tuting the equivalent loss of pressure, divided by the density of the steam, 
for the loss of head. 

The losses in the flow of steam are two in number— the loss due to the 
friction of flow and that due to radiation from the sides of the pipe. The 
sum of these is a minimum when the equivalent of the loss due to fric- 
tion of flow is equal to one fifth of the loss of heat by radiation. For a 
greater or less loss of pressure — i.e., for a less or greater diameter of pipe 
— the total loss increases very rapidly. 

For delivering a given quantity of steam at a given terminal pressure, 
with minimal total loss, the better the non-conducting material employed, 
the larger the diameter of the steam-pipe to be used. 

The most economical loss of pressure for a pipe of given diameter is equal 
to the most economical loss of pressure in a pipe of 1 inch diameter for same 
conditions, divided by the diameter of the given pipe in inches. 

The following table gives the capacity of pipes of different diameters, to 
deliver steam at different terminal pressures through a pipe one half mile 
long for loss of pressure of 10 lbs., and a mean value of / = 0.0175. Let W 
denote the number of pounds of steam delivered per hour : 



Diameter 
of Pipe, 


Abs. Term. Pressure. 


Diameter 
of Pipe, 
inches. 


Abs. Term. Pressure. 


inches. 


65 lbs. 


80 lbs. 


100 lbs. 


65 lbs. 


80 lbs. 


100 lbs. 


1 


W 

102 

179 

282 

415 

579 

1,011 

1,595 

2,346 

3,275 


W 
113 
198 

312 
459 
641 
1,121 
1,768 
2,599 
3.629 


W 
125 
219 
346 

508 
710 
1,240 
1,956 
2,875 
4,042 


4^ 

5 


W 

4,397 

5,721 
9,024 
13,268 
18,526 
24,870 
32,364 
41,081 
51,049 


W 

4,872 
6,339 
10,000 
14,701 
20,528 
27,556 
35,860 
45,507 
56,564 


W 
5,390 


\y^ 


7,013 
11,063 


v/% 


6 


\y± 


7 


16,265 


2 

2V 2 

3 . 


8 

9 


22,711 

30,488 
39,675 
50,349 
62,581 


10 


3 Va 


11 


4 


12 







Resistance to Flow by Bends, Valves, etc. (From Briggs on 
Warming Buildings by Steam.)— The resistance at the entrance to a tube 
when no special bell-mouth is given consists of two parts. The head v*-i-2g 

is expended in giving the velocity of flow; and the head 0.505 —in over' 



FLOW OF STEAM. 673 

coming the resistance of the mouth of the tube. Hence the whole loss of 

head at the entrance is 1.505 — . This resistance is equal to the resistance 

2g 
of a straight tube of a length equal to about 60 times its diameter. 

The loss at each sharp right-angled elbow is the same as in flowing 
through a length of straight tube equal to about 40 times its diameter. For 
a globe steam stop-valve the resistance is taken to be 1J^ times that of the 
right-angled elbow. 

Sizes of Steam-pipes for Stationary Engines.— Authorities 
on the steam-engine generally agree that steam-pipes supplying engines 
should be of such size that the mean velocity of steam in them does not 
exceed 6000 feet per minute, in order that the loss of pressure due to friction 
may not be excessive. The velocity is calculated on the assumption that the 
cylinder is filled at each stroke. In very long pipes, 100 feet and upward, it 
is well to make them larger than this rule would give, and to place a large 
steam receiver on the pipe near the engine, especially when the engine cuts 
off early in the stroke. 

An article in Power, May, 1893, on proper area of supply-pipes for engines 
gives a table showing the practice of leading builders. To facilitate com- 
parison, all the engines have been rated in horse-power at 40 pounds mean 
effective pressure. The table contains all the varieties of simple engines, 
from the slide-valve to the Corliss, and it appears that there is no general 
difference in the sizes of pipe used in the different types. 

The averages selected from this table are as follows: 

Diam. of pipe, in 2 2^ 3 3^ 4 4^ 5 6 7 8 9 10 

Av.H.P.of engines.... 25 39 56 77 100 126 156 225 306 400 506 625 

Calculated,formula (1) 23 36 51 70 91 116 143 206 278 366 463 571 

formula (2) 24 37.5 54 73 96 121 150 216 294 384 486 600 

Formula (1) is: 1 H P. requires .1375 sq. in. of steam-pipe area. 

Formula (2) is: Horse-power — 6d 2 . d = diam. of pipe in inches. 

The factor .1375 in formula (1) is thus derived: Assume that the linear 
velocity of steam in the pipe should not exceed 6000 feet per minute, then 
pipe area = cyl. area X piston-speed -f- 6000 (a). Assume that the av. mean 
effective pressure is 40 lbs. per sq. in., then cyl. area X piston-speed x 40 -=- 
33,000 = horse-power (b). DivMing (a) by (b) and cancelling, we have pipe 
area-=-H.P. = .1375 sq. in. If we use 8000 ft. per min. as the allowable 
velocity, then the factor .1375 becomes .1031; that is, pipe area -f- H.P. = 
.1031, or pipe area X .97 = horse-power. This, however, gives areas of pipe 
smaller than are used in the most recent practice. A formula which gives 
results closely agreeing with practice, as shown in the above table is 

Horse-power = 6cZ 2 , or pipe diameter —if - — ~ = .408 VH.P. 

Diameters of Cylinders corresponding to Various Sizes of Steam- 
pipes BASED ON PlSTON-SPEED OF ENGINE OF 600 FT. PER MlNUTE, AND 

Allowable Mean Velocity of Steam in Pipe of 4000, 6000, and 8000 
ft. per Minute. 

Diam. of pipe, inches ... 2 2^ 3 3^ 4 4)4 5 6 

Vel. 4000 5.2 6.5 7.7 9.0 10.3 11.6 12.9 15.5 

" 6000 6.3 7.9 9.5 11.1 12.6 14.2 15.8 19. 

" 8000 7.3 9.1 10.9 12.8 14.6 16.4 18.3 21. S 

Horse-power, approx 20 31 45 62 80 100 125 180 

Diam. of pipe, inches 7 8 9 10 11 12 13 14 

Vel. 4000 18.1 20.7 23.2 25.8 28.4 31.0 33.6 36.1 

" 6000 22.1 25.3 28.5 31.6 34.8 37.9 41.1 44.3 

" 8000 25.6 29.2 32.9 36.5 40.2 43.8 47.5 51.1 

Horse-power, approx 245 320 406 500 606 718 845 981 

t-, . . . Area of cylinder X piston-speed 

Formula. Area of pipe = ■—. ^-^- : — -. — -. 

mean velocity of steam in pipe 

For piston-speed of 600 ft. per min. and velocity in pipe of 4000, 6000, and 
8000 ft. per min. area of pipe = respectively .15, .10, and .075 x area of cyl- 
inder. Diam. of pipe = respectively .3S73, .3162, and .2739 X diam. of cylin- 
der. Reciprocals of these figures are 2.582, 3.162, and 3.651. 

The first line in the above table may be used for proportioning exhaust- 



674 



pipes, in which a velocity not exceeding 1 4000 ft. per minute is advisable. 
The last line, approx. H.P. of engine, is based on the velocity of 6000 ft. per 
min. in the pipe, using the corresponding diameter of piston, and taking 
H.P. = J^(diam. of piston in inches) • 

Sizes of Steam-pipes for Marine Engines.— In marine-engine 
practice the steam -pipes are generally not as large as in stationary practice 
for the same sizes of cylinder. Seaton gives the following rules: 

Main Steam-pipes should be of such size that the mean velocity of flow 
does not exceed 8000 ft. per min. 

In large engines, 1000 to 2000 H.P., cutting off at less than half stroke, the 
steam-pipe maybe designed for a mean velocity of 9000 ft., and 10.000 ft. 
for still larger engines. 

In small engines and engines cutting later than half stroke, a velocity of 
less than 8000 ft. per minute is desirable. 

Taking 8100 ft. per min. as the mean velocity, S speed of piston in feet per 
min., and D the diameter of the cyl., 



Diam. of main steam-pipe = 



Area of piston x speed of piston in ft. per min. 
6000 



Stop and Throttle Valves should have a greater area of passages than the 
area of the main steam-pipe, on account of the friction through the cir- 
cuitous passages. The shape of the passages should be designed so as to 
avoid abrupt changes of direction and of velocity of flow as far as possible. 

Area of Steam Ports and Passages — 

(Diam.) 2 x speed 

; 7631) ' 

Opening of Port to Steam.— To avoid wire-drawing during admission the 
area of opening to steam should be such that the mean velocity of flow does 
not exceed 10,000 ft. per min. To avoid excessive clearance the width of 
port should be as short as possible, the necessary area being obtained by 
length (measured at right angles to the line of travel of the valve). In 
practice this length is usually 0.6 to 0.8 of the diameter of the cylinder, but 
in long-stroke engines it may equal or even exceed the diameter. 

Exhaust Passages and Pipes. — The area should be such that the mean 
velocity of the steam should not exceed 6000 ft. per min., and the area 
should be greater if the length of the exhaust-pipe is comparatively long. 
The area of passages from cylinders to receivers should be such that the 
velocity will not exceed 5000 ft. per min. 

The following table is computed on the basis of a mean velocity of flow 
of 8000 ft. per min. for the main steam-pipe, 10.000 for opening to steam, 
and 6000 for exhaust. A = area of piston, D its diameter. 

Steam and Exhaust Openings. 



Piston- 


Diam. of 


Area of 


Diam. of 


Area of 


Opening 


speed, 


Steam-pipe 


Steam-pipe 


Exhaust 


Exhaust 


to Steam 


ft. per min. 


H-£>. 


-t-A. 


-H D. 


-*- A. 


-^A. 


300 


0.194 


0.0375 


0.223 


0.0500 


0.03 


400 


0.224 


0.0500 


0.258 


0.0667 


04 


500 


0.250 


0.0625 


0.288 


0.0833 


0.05 


600 


0.274 


0.0750 


0.316 


0.1000 


0.06 


700 


0.296 


0.0875 


0.341 


0.1167 


0.07 


800 


0.316 


0.1000 


0.365 


0.1333 


0.08 


900 


0.335 


0.1125 


0.387 


0.1500 


0.09 


1000 


0.353 


0.1250 


0.400 


0.1667 


0.10 



STEAM PIPES. 

Bursting-tests of Copper Steam-pipes. (From Report of Chief 
Engineer Melville, U. S. N., for 1892.)— Some tests were made at the New 
York Navy Yard which show the unreliability of brazed seams in cop- 
per pipes. Each pipe was 8 in. diameter inside and 3 ft. 1% in . long. 
Both ends were closed by ribbed heads and the pipe was subjected to a hot- 
water pressure, the temperature being maintained constant at 371° F- Three 



STEAM-PIPES. 675 

of the pipes were made of No. 4 sheet copper (" Stubbs " gauge) and the 
fourth was made of No. 3 sheet. 
The following were the results, in lbs. per sq. in., of bursting-pressure: 



Pipe number 

Actual bursting-strength. 
Calculated " " 
Difference 



1 


2 


3 


4 


4' 


835 


785 


950 


1225 


1275 


336 


1336 


1569 


1568 


1568 


501 


551 


619 


343 


293 



The theoretical bursting-pressure of the pipes was calculated by using the 
figures obtained in the tests for the strength of copper sheet with a brazed 
joint at 350° F. Pipes 1 and 2 are considered as having been annealed. 

The tests of specimens cut from the ruptured pipes show the injurious 
action of heat upon copper sheets; and that, while a white heat does not 
change the character of the metal, a heat of only slightly greater degree 
causes it to lose the fibrous nature that it has acquired in rolling, and a 
serious reduction in its tensile strength and ductility results. 

All the brazing was done by expert workmen, and their failure to make a 
pipe-joint without burning the metal at some point makes it probable that, 
with copper of this or greater thickness, it is seldom accomplished. 

That it is possible to make a joint without thus injuring the metal was 
proven in the cases of many of the specimens, both of those cut from the 
pipes and those made separately, which broke with a fibrous fracture. 

Rule for Thickness of Copper Steam-pipes. (U. S. Super- 
vising Inspectors of Steam Vessels.)— Multiply the working steam-pressure 
in lbs. per sq. in. allowed the boiler by the diameter of the pipe in inches, 
then divide the product by the constant whole number 8000, and add .0625 to 
the quotient; the sum will give the thickness of material required. 

Example.— Let 175 lbs. = working steam-pressure per sq. in. allowed the 

boiler, 5 in. = diameter of the pipe; then — ^r— — \- .0625 = .1718 -f- inch, 

thickness required. 

Reinforcing Steam-pipes. (Eng., Aug. 11, 1893.)— In the Italian 
Navy copper pipes above 8 in. diam. are reinforced by wrapping them with 
a close spiral of copper or Delta-metal wire. Two or three independent 
spirals are used for safety in case one wire breaks. They are wound at a 
tension of about lVg tons per sq. in. 

Wire-wound Steam-pipes.— The system instituted by the British 
Admiralty of winding all steam -pipes over 8 in. in diameter with 3/16-in. 
copper wire, thereby about doubling the bursting-pressure, has within re- 
cent years been adopted on many merchant steamers using high-pressure 
steam, says the London Engineer. The results of some of the Admiralty 
tests showed that a wire pipe stood just about the pressure it ought to have 
stood when unwired, had the copper not been injured in the brazing. 

Riveted Steel Steam-pipes have recently been used for high 
pressures. See paper on A Method of Manufacture of Large Steam-pipes, 
by Chas. H. Manning, Trans. A. S. M. E., vol. xv. 

Valves in Steam-pipes.— Should a globe-valve on a steam-pipe have 
the steam -pressure on top or underneath the valve is a disputed question. 
With the steam-pressure on top, the stuffing-box around the valve-stem can- 
not be repacked without shutting off steam from the whole line of pipe; on 
the other hand, if the steam-pressure is on the bottom of the valve it all has 
to be sustained by the screw-thread on the valve-stem, and there is danger 
of stripping the thread. 

A correspondent of the American Machinist, 1892, says that it is a very 
uncommon thing in the ordinary globe-valve to have the thread give out, 
but by water-hammer and merciless screwing the seat will be crushed down 
quite -frequently. Therefore with plants where only one boiler is used he 
advises placing the valve with the boiler-pressure underneath it. On plants 
where several boilers are connected to one main steam-pipe he would re- 
verse the position of the valve, then when one of the valves needs repacking 
the valve can be closed and the pressure in the boiler whose pipe it controls 
can be reduced to atmospheric by lifting the safety-valve. The repacking 
can then be done without interfering with the operation of the other boilers 
of the plant. 

He proposes also the following other rules for locating valves: Place 
valves with the stems horizontal to avoid the formation of a water-pocket. 
Never put the junction-valve close to the boiler if the main pipe is above 
the boiler, but put it on the highest point of the junction-pipe. If the other 



676 STEAM. 

plan is followed, the pipe fills with water whenever this boiler is stopped 
and the others are running, and breakage of the pipe may cause serious re- 
sults. Never let a junction-pipe run into the bottom of the main pipe, but 
into the side or top. Always use an angle-valve where convenient, as there 
is more room in them. Never use a gate valve under high pressure unless a 
by-pass is used with it. Never open a blow-off valve on a boiler a little and 
then shut it; it is sure to catch the sediment and ruin the valve; throw it 
well open before closing. Never use a globe-valve on an indicator-pipe. For 
water, always use gate or angle valves or stop-cocks to obtain a clear pas- 
sage. Buy if possible valves with renewable disks. Lastly, never let a man 
go inside a boiler to work, especially if he is to hammer on it, unless you 
break the joint between the boiler and the valve and put a plate of steel 
between the flanges. 

Flanges for Steam-nozzles and Steam-pipe, used with the 
Gill Water-tube Boiler, Phila., 1892. 

Sizeofpipe 3 4 5 6 7 8 9 

Outside diameter of flange, inches.. 9 10 11 12 13 14 15 

Pitch-circle for bolts, diam., " .. 7 8 9 10 11 12 13 

Outside diam. of gaskets, " .. $% 6% 7% 8)4 9% 10J^ llj^ 

Inside diam. of gaskets, " .. 3^| 4^| 5^ <o% 7^ 8*4 9^ 

Number of bolts 5 6 7 8 9 10 11 

Size of pipe 10 11 12 13 14 15 16 

Outside diameter of flange, inches.. 16 17 18 19 20 21 22 

Pitch-circle for bolts, diam., " .. 14 15 16 17 18 19 20 

Outside diam. of gaskets, " .. 12J^ 13^ 14^ 15^ 16}^ 17^ 18*^ 

Inside diam. of gaskets, " .. 10^ \\X6 12^ 13V^ 14V6 ^ l A ^Y% 

Number of bolts 12 13 14 15 16 17 18 

All holes drilled 15/16 in., with a jig accurately laid out. 

All bolts to be % in. diam. by 3^> in. long under the head. 

All bolts to have square heads and hexagon nuts. 

The "Steam Lioop" is a system of piping by which water of con- 
densation in steam-pipes is automatically returned to the boiler. In its 
simplest form it consists of three pipes, which are called the riser, the hori- 
zontal, and the drop-leg. When the steam-loop is used for returning to the 
boiler the water of condensation and entrainment from the steam-pipe 
through which the steam flows to the cylinder of an engine, the riser is gen- 
erally attached to a separator; this riser empties at a suitable height into 
the horizontal, and from thence the water of condensation is led into the 
drop-leg, which is connected to the boiler, into which the water of condensa- 
tion is fed as soon as the hydrostatic pressure in drop-leg in connection with 
the steam-pressure in the pipes is sufficient to overcome the boiler-pressure. 
The action of the device depends on the following principles: Difference of 
pressure may be balanced by a water-column: vapors or liquids tend to flow 
to the point of lowest pressure; rate of flow depends on difference of pres- 
sure and mass; decrease of static pressure in a steam-pipe or chamber is 
proportional to rate of condensation; in a steam-current water will be car- 
ried or swept along rapidly by friction. (Illustrated in Modern Mechanism, 
p. 807.) 

Lioss from an Uncovered Steam-pipe. (Bjorling on Pumping- 
engines.) — The amount of loss by condensation in a steam-pipe carried down 
a deep mine-shaft has been ascertained by actuai practice at the Clay Cross 
Colliery, near Chesterfield, where there is a pipe 7\4 in. internal diam.. 1 100 
ft. long. The loss of steam by condensation was ascertained by direct 
measurement of the water deposited in a receiver, and was found to be 
equivalent to about 1 lb. of coal per I.H.P. per hour for every 100 ft. of 
steam-pipe; but there is no doubt that if the pipes had been in ihe upcast 
shaft, and well covered with a good non-conducting material, the loss would 
have been less. (For Steam-pipe Coverings, see p. 469, ante.) 



THE HORSE-POWER OF A STEAM-BOILER. 677 



THE STEAM-BOILER. 

The Horse-power of a Steam-boiler.-— The term horsepower 
has two meanings in engineering : First, an absolute unit or measure of the 
rate of work, that is, of the work done in a certain definite period of time, 
by a source of energy, as a steam-boiler, a waterfall, a current of air or 
water, or by a prime mover, as a steam-engine, a water-wheel, or a wind- 
mill. The value of this unit, whenever it can be expressed in foot-pounds 
of energ}^, as in the case of steam-engines, water-wheels, and waterfalls, is 
33,000 foot-pounds per minute. In the case of boilers, where the work done, 
the conversion of water into steam, cannot be expressed in foot-pounds of 
available energy, the usual value given to the term horse-power is the evap- 
oration of 30 .'bs. of water of a temperature of 100° F. into steam at 70 lbs. 
pressure above the atmosphere. Both of these units are arbitrary; the first, 
33,000 foot-pounds per minute, first adopted by James Watt, being considered 
equivalent to the power exerted by a good London draught-horse, and the 
30 lbs. of water evaporated per hour being considered to be the steam re- 
quirement per indicated horse-power of an average engine. 

The second definition of the term horse-power is an approximate measure 
of the size, capacity, value, or " rating " of a boiler, engine, water-wheel, or 
other source or conveyer of energy, by which measure it may be described, 
bought and sold, advertised, etc. No definite value can be given to this 
measure, which varies largely with local custom or individual opinion of 
makers and users of machinery. The nearest approach to uniformity which 
can be arrived at in the term "horse power," used in this sense, is to say 
that a boiler, engine, water-wheel, or other machine, " rated 1 ' at a certain 
horse-power, should be capable of steadily developing that horse-power for 
a long period of time under ordinary conditions of use and practice, leaving 
to local custom, to the judgment of the buyer and seller, to written contracts 
of purchase and sale, or to legal decisions upon such contracts, the interpre- 
tation of what is meant by the term "ordinary conditions of use and 
practice.' 1 (Trans. A. S. M. E., vol. vii. p. 226.) 

The committee of the A. S. M. E. on Trials of Steam-boilers in 1S84 (Trans., 
vol. vi. p. 265) discussed the question of the horse-power of boilers as follows: 

The Committee of Judges of the Centennial Exhibition, to whom the trials 
of competing boilers at that exhibition were intrusted, met with this same 
problem,. and finally agreed to solve it, at least so far as the work of that 
committee was concerned, by the adoption of the unit, 30 lbs. of water evap- 
orated into dry steam per hour from feed-water at 100° F., and under a 
pressure of 70 lbs. per square inch above the atmosphere, these conditions 
being considered by them to represent fairly average practice. The quan- 
tity of heat demanded to evaporate a pound of water under these conditions 
is 1110.2 British thermal units, or 1.1496 units of evaporation. The unit of 
power proposed is thus equivalent to the development of 33,305 heat units 
per hour, or 34 488 units of evaporation. . . . 

Your committee, after due consideration, has determined to accept the 
Centennial Standard, the first above mentioned, and to recommend that in 
all standard trials the commercial horse-power be taken as an evaporation 
of 30 lbs. of water per hour from a feed-water temperature of 100° F. into 
steam at 70 lbs. gauge pressure, which shall be considered to be equal to 34}^ 
units of evaporation, that is, to 34^ lbs. of water evaporated from a feed- 
water temperature of 212° F. into steam at the same temperature. This 
standard is equal to 33.305 thermal units per hour. 

It is the opinion of this committee that a boiler rated at any stated number 
of horse-powers should be capable of developing that power with easy firing, 
moderate draught, and ordinary fuel, while exhibiting good economy ; and 
further, that the boiler should be capable of developing at least one third 
more than its rated power to meet emergencies at times when maximum 
economy is not the most important object to be attained. 

Unit ot Evaporation. — It is the custom to reduce results of boiler- 
tests to the common standard of weight of water evaporated by the unit 
weight of the combustible portion of the fuel, the evaporation being consid- 
ered to have taken place at mean atmospheric pressure, and at the temper- 
ature due that pressure, the feed-water being also assumed to have been 
supplied at that temperature. This is, in technical language, said to be the 
equivalent evaporation from and at the boiling-point at atmospheric pres- 
sure, or "from and at 212° F." This unit of evaporation, or one pound of 



678 THE STEAM-BOILER, 

water evaporated from and at 212°, is equivalent to 965.7 British thermal 
units. 

Measures for Comparing the Buty of Boilers.— The meas- 
ure of the efficieuey of a boiler is the number of pounds of water evaporated 
per pound of combustible, the evaporation being reduced to the standard of 
"from and at 212°;'" that is, the equivalent evaporation from feed-water at a 
temperature of 212° F. into steam at the same temperature. 

The measure of the capacity of a boiler is the amount of "boiler horse- 
power " developed, a horse-power being defined as the evaporation of 30 lbs. 
of water per hour from 100° F. into steam at 70 lbs. pressure, or 34J/£ lbs. per 
hour from and at 212°. 

The measure of relative rapidity of steaming of boilers is the number of 
pounds of water evaporated per hour per square foot of water-heating sur- 
face. 

The measure of relative rapidity of combustion of fuel in boiler-furnaces 
is the number of pounds of coal burned per hour per square foot of grate- 
surface. 

STEAM-BOILER PROPORTIONS. 

Proportions of Grate and Heating Surface required for 

a given Horse-power. — The term horse-power here means capacity 

to evaporate 30 lbs. of water from 100° F., temperature of feed-water, to 

steam of 70 lbs., gauge-pressure = 34.5 lbs. from and at 212° F. 

Average proportions for maximum economy for land boilers fired with 
good anthracite coal: 

Heating surface per horse-power 1 1.5 sq. ft. 

Grate " " " 1/3 " 

Ratio of heating to grate surface 34.5 " 

Water evap'd from and at 212° per sq. ft H.S. per hour 3 lbs. 

Combustible burned per H. P. per hour 3 " 

Coal with 1/6 refuse, lbs. per H.P. per hour 3.6 " 

Combustible burned per sq. ft. grate per hour 9 " 

Coal with 1/6 refuse, lbs. per sq.ft. grate pe 1- hour 10.8 " 

Water evap'd from and at 212° per lb. combustible. .. 11.5 " 
" " " " " coal (1/6 refuse) 9.6 >l 
The rate of evaporation is most conveniently expressed in pounds evapo- 
rated from and at 212° per sq. ft. of water-heating surface per hour, and the 
rate of combustion in pounds of coal per sq. ft. of grate-surface per hour. 

Heating-surface.— For maximum economy with any kind of fuel a 
boiler should be proportioned so that at least one square foot of heating- 
surface should be given for every 3 lbs. of water to be evaporated from and 
at 212° F. per hour. Still more liberal proportions are required if a portion 
of the heating-surface has its efficiency reduced by: 1. Tendency of the 
heated gases to short-circuit, that is, to select passages of least resistance 
and flow through them with high velocity, to the neglect of other passages. 
2. Deposition of soot from smoky fuel. 3. Incrustation. If the heating-sur- 
faces are clean, and the heated gases pass over it uniformly, little if any 
increase in economy can be obtained by increasing the heating-surface be- 
yond the proportion of 1 sq. ft. to every 3 lbs. of water to be evaporated, and 
with all conditions favorable but little decrease of economy will take place 
if the proportion is 1 sq. ft. to every 4 lbs. evaporated; but in order to pro- 
vide for driving of the boiler beyond its rated capacity, and for possible 
decrease of efficiency due to the causes above named, it is better to adopt 1 
sq. ft. to 3 lbs. evaporation per hour as the minimum standard proportion. 

Where economy may be sacrificed to capacity, as where fuel is very cheap, 
it is customary to proportion the heating-surface much less liberally. The 
following table shows approximately the relative results that may be ex- 
pected with different rates of evaporation, with anthracite coal. 
Lbs. water evapor 1 d from and at 212° per sq. ft. heating-surface per hour: 
2 2.5 3 3.5 4 5 6 7 8 9 10 

Sq. ft. heating-surface required per horse-power: 
17.3 13.8 11.5 9.8 8.6 6.8 5.8 4.9 4.3 3.8 3.5 

Ratio of heating to grate surface if 1/3 sq. ft. of G. S. is required per H.P.: 
52 41.4 34.5 29.4 25.8 20.4 17.4 13.7 12.9 11.4 10.5 

Probable relative economy: 
100 100 100 95 90 85 80 75 70 65 60 

Probable temperature of chimney gases, degrees F.: 
450 450 450 518 585 652 720 787 855 922 990 



STEAM-BOILER PROPORTIONS. 679 

The relative economy will vary not only with the amount of heating-sur- 
face per horse-power, but with the efficiency of that heating-surface as 
regards its capacity for transfer of heat from the heated gases to the water, 
which will depend on its freedom from soot and. incrustation, and upon the 
circulation of the water and the heated gases. 

With bituminous coal the efficiency will largely depend upon the thorough- 
ness with which the combustion is effected in the furnace. 

The efficiency with any kind of fuel will greatly depend upon the amount 
of air supplied to the furnace in excess of that required to support com- 
bustion. With strong draught and thin fires this excess may be very great, 
causing a serious loss of economy. 

Measurement of Heating-surface. —Authorities are not agreed 
as to the methods of measuring the heating-surface of steam-boilers. The 
usual rule is to consider as heating-surface all the surfaces that are sur- 
rounded by water on one side and by flame or heated gases on the other, but 
there is a difference of opinion as to' whether tubular heating-surface should 
be figured from the inside or from the outside diameter. Some writers say, 
measure the heating-surface always on the smaller side— the fire side of the 
tube in a horizontal return tubular boiler and the water side in a water-tube 
boiler. Others would deduct from the heating-surface thus measured an 
allowance for portions supposed to be ineffective on account of being cov- 
ered by dust, or being out of the direct current of the gases. 

For the sake of uniformity, however, it would appear to be the best method 
to consider all surfaces as heating-surfaces which transmit heat from the 
flame or gases to the water, making no allowance for different degrees of 
effectiveness; also, to use the external instead of the internal diameter 
of tubes, for greater convenience in calculation, the external diameter of 
boiler-tubes usually being made in even inches or half inches. There would 
seem to be no good reason for considering the smaller surface in a tube as 
the heating-surface, for the transmission of heat through plates that are 
ribbed or corrugated on one side does not appear to be proportional to the 
smaller surface, but rather to the larger. Thus the Serve ribbed tube trans- 
mits more heat to the water per foot of length than a plain tube of same 
external diameter, and a ribbed steam-radiator radiates more heat than a 
plain radiator having the same internal or smaller surface. 

Rule for finding the heating-surface of vertical tubular boilers : Multiply 
the circumference of the fire-box (in inches) by its height above the grate ; 
multiply the combined circumference of all the tubes by their length, and 
to these two products add the area of the lower tube-sheet ; from this sum 
subtract the area of all the tubes, and divide by 144 : the quotient is the 
number of square feet of heating-surface. 

Rule for finding the heating-surface of horizontal tubular boiler« : Multi- 
ply two thirds of the circumference of the shell (in inches) by its length ; 
multiply the combined length of the tubes by their combined circumference, 
to the sum of these products add two thirds of the area of both tube-sheets; 
from this sum subtract the combined area of all the tubes, and divide the 
remainder by 144: the result is the number of square feet of heatiug-surface. 

Rule for finding the square feet of heating -surface in tubes : Multiply the 
number of tubes by the diameter of a tube in inches, by its length in feet, 
and by .2618. 

Horse-power, Builder's Rating. Heating-surface per 
Horse-power.— If is a general practice among builders to furnish about 
12 square feet of heating-surface per horse-power, but as the practice is not 
uniform, bids and contracts should always specify the amount of heating- 
surface to be furnished. Not less than one third square foot of grate-surface 
should be furnished per horse-power. 

Engineering Neivs, July 5, 1894, gives the following rough-and-ready rule 
for finding approximately the commercial horse-power of tubular or water- 
tube boilers : Number of tubes X their length in feet X their nominal 
diameter in inches -s- 50 = nLd -*- 50. The number of square feet of surface 

in the tubes is = -5-0.5, an( * tne horse-power at 12 square feet of surface 

of tubes per horse-power, not counting the shell, = nLd -s- 45.8. If 15 square 
feet of surface of tubes be taken, it is nLd -4- 57.3. Making allowance for 
the heating-surface in the shell will reduce the divisor to about 50. 

Horse-power of Marine and Locomotive Boilers.— The 
term horse-power is not generally used in connection with boilers in marine 
practice, or with locomotives. The boilers are designed to suit the engines, 
and are rated by extent of grate and heating-surface only, 



680 



THE STEAM-BOILER. 



Grate-surface.— The amount of grate-surface required per horse 
power, and the proper ratio of heating-surface to grate-surface are ex- 
tremely variable, depending chiefly upon the character of the coal and upon 
the rate of draught. With good coal, low in ash, approximately equal results 
may be obtained with large grate-surface and light draught and with small 
grate-surface and strong draught, the total amount of coal burned per hour 
being the same in both cases. With good bituminous coal, like Pittsburgh, 
low in ash, the best results apparently are obtained with strong draught 
and high rates of combustion, provided the grate-surfaces are cut down so 
that the total coal burned per hour is not too great for the capacity of the 
heating-surface to absorb the heat produced. 

With coals high in ash, especially if the ash is easily fusible, tending to 
choke the grates, large grate-surface and a slow rate of combustion are 
required, unless means, such as shaking grates, are provided to get rid of 
the ash as fast as it is made. 

The amount of grate-surface required per horse-power under various con- 
ditions may be estimated from the following table : 





Lbs. Water 
from and 
at 212° 
per lb., 
Coal. 




Pounds of Coal burned per square foot 




6ffi2 

rA t-> '-• 

52 <D 0> 
5** 


of Grate per hour. 






8 | 10 | 12 | 15 | 20 | 25 | 30 


35 | 40 




J 10 


3.45 


Sq. Ft. Grate per H. P. 


Good coal 


.48 


35 


28 


.23 


.17 


14 


11 


10 


09 


and boiler, 


1 9 


3.83 


.48 


38 


. : J .2 


. 25 


19 


.15 


13 


.11 


,10 


Fair coal or 
boiler, 


( 8.61 

1 8 


4. 
4.31 


.50 
54 


.40 
.48 


.33 

.36 


.26 
.29 


.20 
.22 


.16 
.17 


.13 
.14 


.12 
.13 


.10 
.11 


I 7 


4.93 


. 62 


.49 


.41 


.88 


.24 


.20 


.17 


.14 


.12 


Poor coal or 
boiler, 


I 6.9 

1 6 


5. 

5.75 


.63 

.72 


.50 

.58 


.42 

.48 


.34 

.38 


.25 
.29 


.20 
.23 


.17 
.19 


.15 

17 


.13 
.14 


1 5 


6.9 


.86 


.69 


58 


.46 


.35 


.28 


.23 


.22 


.17 


Lignite and 
poor boiler, 


[ 3.45 


10. 


1.25 


1.00 


.83 


.67 


.50 


.40 


.33 


.29 


.25 



In designing a boiler for a given set of conditions, the grate-surface should 
be made as liberal as possible, say sufficient for a rate of combustion of 10 
lbs. per square foot of grate for anthracite, and 15 lbs. per square foot for 
bituminous coal, and in practice a portion Of the grate-surface may be 
bricked over if it is found that the draught, fuel, or other conditions render 
it advisable. 

Proportions of Areas of Flues and other Gas-passages. 
— Rules are usually given making the area of gas-passages bear a certain 
ratio to the area of the grate-surface; thus a common rule for horizontal 
tubular boilers is to make the area over the bridge wall 1/7 of the grate- 
surface, the flue area 1/8, and the chimney area 1/9. 

For average conditions with anthracite coal and moderate draught, say a 
rate of combustion of 12 lbs. coal per square foot of grate per hour, and a ratio 
of heating to grate surface of 30 to 1, this rule is as good as any, but it is evi- 
dent that if the draught were increased so as to cause a rate of combustion 
of 24 lbs., requiring the grate-surface to be cut down to a ratio of 60 to 1, the 
areas of gas-passages should not be reduced much, because the grate-sur- 
face is reduced. The coal burned being the same under the changed condi- 
tions, and there being no reason why the gases should travel at a higher 
velocity, the actual areas of the passages should remain as before, but the 
ratio of the area to the grate-surface would in that case be doubled. 

Mr. Barrus states that the highest efficiency with anthracite coal is 
obtained when the tube area is 1/9 to 1/10 of the grate-surface, aud with 
bituminous coal when it is 1/6 to 1/7, for the conditions of medium rates of 
combustion, such as 10 to 12 lbs. per square foot of grate per hour, and 12 
.square feet of heating surface allowed to the horse-power. 

The tube area should be made large enough not to choke the draught, and 
so lessen the capacity of the boiler; if made too large the gases are apt to 
select the passages of least resistance and escape from them at a high 
velocity and high temperature. 

This condition is very commonly found in horizontal tubular boilers where 



PERFORMANCE OF BOILERS. 



681 



the gases go chiefly through the upper rows of tubes; sometimes also in 
vertical tubular boilers, where the gases are apt to pass most rapidly 
through the tubes nearest to the centre. 

Air-passages through Grate-bars.— The usual practice is, air- 
opening = 30$ to 50$ of area of the grate ; the larger the better, to avoid 
stoppage of the air-supply by clinker; but with coal free from clinker much 
smaller air-space may be used without detriment. See paper by F. A. 
Scheffler, Trans. A. S. M. E., vol. xv. p. 503. 

PERFORMANCE OF ISOIL.ERS. 

Clark (Steam-engine, vol. i. p. 327) gives the following formulas for the 
relation of coal and water consumed in steam-boilers per square foot of 
grate-area per hour, and the ratio of the heating-surface to the area of the 
fire-grate. Water taken as evaporated from and at 212° F. 

Stationary boilers to = M22r* + 9.56c 

Marine boilers w = .016r a + 10.25c 

Portable-engine boilers ru = .OOSr 2 -j- 8.6c 

Locomotive boilers icoal-burning) iv — .009r a -j- 9.7c 

Locomotive boilers (coke-burning) to = ,0178r 2 -j- 7.94c 

In which w = weight of water in pounds per square foot of grate per hour; 
c = pounds of fuel per square foot of grate per hour; 
r = ratio of heating to grate surface. 

There are minimum rates of consumption of fuel below which these 
formulas are not applicable. The limit varies for each kind of boiler, and it 
varies with the surface-ratio. It is imposed by the fact that the maximum 
evaporative power of fuel is a fixed quantity, and is naturally at that point 
where the reduction of the rate of combustion for a given ratio procures the 
absorption into the boiler of the whole of the proportion of the heat which 
is available for evaporation. In the combustion of good coal the limit of 
evaporative efficiency may be taken as measured by 12}^ lbs. of water from: 
and at 212° F. ; and in that of good coke by 12 lbs. of water from and at- 
212° F. Based on these formulas Clark gives the following table : 
Evaporative Performance of Steam-boilers for increasing 
Rates of Combustion and different Surface-ratios. 
For best coal; surface-ratio 30. 



Kind of 
Boiler. 



Stationary. 
Marine. 
Portable. 
Locomotive. 



Water from and at 
212° F. per hour. 



Per sq. ft. of grate 
Per lb. of coal 



Per sq. ft. of grate 

Per lb. of coal 

Per sq. ft. of grate 

Per lb. of coal 

Per sq. ft. of grate 
Per lb. of coal.. . 



Fuel per Square Foot of Grate per hour, 



lbs. 

62.5* 

12.5 

62.5* 

12.5 

50 

10 

57 
11.4 



10 



lbs. 

116 
11.56 

117 
11. 6J 

93 

9.3 



15 20 30 40 50 



10.89 
16f 

11.25 
136 
9.01 
154 

10.26 



lbs. 
211 

10.56 
219 

10.95 
179 
8.95 
202 

10.10 



lbs. 
307 
10.23 



lbs. 

402 
10.06 

424 
10.61 

351 

8.77 



lbs. 

498 



10.54 
437 
8.74 
493 
9.86 



Surface-ratio 50. 







5 


10 


15 


20 


30 


40 


50 






lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs 


lbs, 


Stationary. 


Per sq. ft. of grate 


62.5* 


125* 


187.5* 


247 


342 


438 


534 


" 


Per lb. of coal 


12.5 


12.5 


12.5 


12.33 


11.41 


10.95 


10.67 


Marine. 


Per sq. ft. of grate 


62.5* 


125* 


187.5* 


245 


348 


450 


552 


" 


Per lb. of coal 


12.5 


12.5 


12.5 


12.25 


11.58 


11.25 


11.05 


Portable. 


Per sq. ft. of grate 


62.5* 


106 


149 


192 


278 


364 


450 


" 


Per lb. of coal 


12.5 


10.6 


9.93 


9.6 


9.27 


9.10 


9.00 


Locomotive. 


Per sq. ft. of grate 


62.5* 


120 


168 


217 


314 


411 


508 


" 


Per lb. of coal 


12.5 


11.95 


11. zO 


10.85 


10 45 


10.26 


10.15 



* These quantities fall below the scope of the formulas for the water, as 
explained in the text. 



682 



THE STEAM-BOlLim 





Surface ratio 75. 
















30 


40 


50 


60 


75 


90 


100 


Locomotive. 


Pei* sq. ft. of grate. 
Per lb. of coal 


lbs. 
342 
11.39 


lbs. 
439 

10.97 


lbs. 
536 
10.71 


lbs. 
633 
10.65 


lbs. 
778 
10.37 


lbs. 
927 
10.26 


lbs. 
102C 
10.20 



General Conditions which secure Economy of Steam- 
boilers.— In general, the highest results are produced where the tempera- 
ture of the escaping gases is the least. An examination of this question is 
made by Mr. G. H. Barrus in his book on " Boiler Tests, 11 by selecting those 
tests made by him, six in number, in which the temperature exceeds the 
average, that is, 375° F., and comparing with five tests in which the temper- 
ature is less than 375°. The boilers are all of the common horizontal type, 
and all use anthracite coal of either egg or broken size. The average flue 
temperatures in the two series v/as 444° and 343° respectively, and the dif- 
ference was 101°. The average evaporations are 10.40 lbs. and 11.02 lbs. re- 
spectively, and the lowest result corresponds to the case of the highest flue 
temperature. In these tests it appears, therefore, that a reduction of 101° 
in the temperature of the waste gases secured an increase in the evaporation 
of 6%. This result corresponds quite closely to the effect of lowering the 
temperature of the gases by means of a flue-heater where a reduction of 
107° was attended by an increase of 7% in the evaporation per pound of coal. 

A similar comparison was made on horizontal tubular boilers using Cum- 
berland coal. The average flue temperature in four tests is 450° and the 
average evaporation is 11.34 lbs. Six boilers have temperatures below 415°, 
the average of which is 383°, and these give an average evaporation of 11.75 
lbs. With 67° less temperature of the escaping gases the evaporation is 
higher by about 4%. 

The wasteful effect of a high flue temperature is exhibited by other boilers 
than those of the horizontal tubular class. This source of waste was shown 
to be the main cause of the low economy produced in those vertical boilers 
which are deficient in heating-surface. 

Relation between the Heating-surface and Grate-surface to obtain the 
Highest Efficiency. — A comparison of thi*ee tests of horizontal tubular 
boilers with anthracite coal, the ratio of heating- surf ace to grate-surf ace 
being 364 to 1, with three other tests of similar boilers, in which the ratio 
was 48 to 1, showed practically no difference in the results. The evidence 
shows that a ratio of 36 to 1 provides a sufficient quantity of heating-surface 
to secure the full efficiency of anthracite coal where the rate of combustion 
is not more than 12 lbs. per sq. ft. of grate per hour. 

In tests with bituminous coal an increase in the ratio from 36.8 to 42.8 se- 
cured a small improvement in the evaporation per pound of coal, and a high 
temperature of the escaping gases indicated that a still further increase 
would be beneficial. Among the high results produced on common horizon- 
tal tubular boilers using bituminous coal, the highest occurs where the ratio 
is 53.1 to 1. This boiler gave an evaporation of 12.47 lbs. A double-deck 
boiler furnishes another example of high performance, an evaporation of 
12.42 lbs. having been obtained with bituminous coal, and in this. case the 
ratio is 65 to 1. These examples indicate that a much larger amount of 
heating-surface is required for obtaining the full efficiency of bituminous 
coal than for boilers using anthracite coal. The temperature of the escap- 
ing gases in the same 1 oiler is invariably higher when bituminous coal is 
used than when anthracite coal is used. The deposit of soot on the surfaces 
when bituminous coal is used interferes with the full efficiency of the sur- 
face, and an increased area is demanded as an offset to the loss which this 
deposit occasions. It would seem, then, that if a ratio of 36 to 1 is sufficient 
for anthracite coal, from 45 to 50 should be provided when bituminous coal 
is burned, especially in cases where the rate of combustion is above 10 or 12 
lbs. per sq. ft. of grate per hour. 

The number of tubes controls the ratio between the area of grate-surface 
and area of tube opening. A certain minimum amount of tube-opening is 
required for efficient work. 

The best results obtained with anthracite coal in the common hoiizontal 
boiler are in cases where the ratio of area of grate-surface to area of tube- 
opening is larger than 9 lo 1. The conclusion is drawn that the highest effi- 
ciency with anthracite coal is obtained when the tube-opening is from 1/9 to 
1/10 of the grate-surface. 



PERFORMANCE OF BOILERS. 683 

When bituminous coal is burned the requirements appear to be different. 
The effect of a large tube opening does not seem to make the extra tubes 
inefficient when bituminous coal is used. The highest result on any boiler of 
the horizontal tubular class, fired with bituminous coal, was obtained where 
the tube-opening was the largest. This gave an evaporation of 12.47 lbs., the 
ratio of grate-surface to tube-opening being 5.4 to 1. The next highest re- 
sult was 12.42 lbs., the ratio being 5.2 tol. Three high results, averaging 
12.01 lbs., were obtained when the average ratio was 7.1 to 1. Without going 
to extremes, the ratio to be desired when bituminous coal is used is that 
which gives a tube-opening having an area of from 1/6 to 1/7 of the grate- 
surface. This applies to medium rates oi ; combustion of, say, 10 to 12 lbs. per 
sq. ft. of grate per hour, 12 sq. ft. of water-heating surface being allowed per 
horse-power. 

A comparison of results obtained from different types of boilers leads to 
the general conclusion that the economy with which different types of 
boilers operate depends much more upon their proportions and the condi- 
tions under which they work, than upon their type ; and, moreover, that 
when these proportions are suitably carried out, and when the conditions 
are favorable, the various types of boilers give substantially the same eco- 
nomic result. 

Efficiency of a Steam-boiler.— The efficiency of a boiler is the 
percentage of the total heat generated by the combustion of the fuel 
which is utilized in heating the water and in raising steam. With anthracite 
coal the heating-value of the combustible portion is very nearly 14,500 
B. T. U. per lb., equal'to an evaporation from and at 212° of 14,500-^966 
= 15 lbs. of water. A boiler which when tested with anthracite coal shows 
an evaporation of 12 lbs. of water per lb. of combustible, has an efficiency of 
12 -=- 15 = 80$, a figure which is approximated, but scarcely ever quite 
reached, in the best practice. With bituminous coal it is necessary to have 
a determination of its heating-power made by a coal calorimeter before the 
efficiency of the boiler using it can be determined, but a close estimate may 
be made from the chemical analysis of the coal. (See Coal.) 

The difference between the efficiency obtained by test and 100$ is the sum 
of the numerous wastes of heat, the chief of which is the necessary loss due 
to the temperature of the chimney-gases. If we have an analysis and a 
calorimetric determination of the heating-power of the coal (properly sam- 
pled), and an average analysis of the chimney-gases, the amounts of the 
several loses may be determined with approximate accuracy by the method 
described below. 

Data given : 

1. Analysis of the Coal. 2. Analysis of the Dry Chimney- 

Cumberland Semi-bituminous. gases, by Weight. 

Carbon 80.55 

Hydrogen 4.50 C0 2 = 13.6 

Oxygen 2.70 CO = .2 

Nitrogen 1.08 O = 11.2 

Moisture 2.92 N = 75.0 

Ash 8.25 



o. 


N. 


9.89 




.11 




11.20 






75.00 



100.00 



100.0 



Heating-value of the coal by Dulong's formula, 14,243 heat-units. 
The gases being collected over water, the moisture in them is n&t deter- 
mined. 

3. Ash and refuse as determined by boiler-test, 10.25, or 2$ more than that 
found by analysis, the difference representing carbon in the ashes obtained 
in the boiler-test. 

4. Temperature of external atmosphere, 60° F. 

5. Relative humidity of air, 60$, corresponding (see air tables) to .007 lb. of 
vapor in each lb. of air. 

6. Temperature of chimney-gases, 560° F. 
Calculated results : 

The carbon in the chimnej^-gases being 3.8$ of their weight, the total 
weight of dry gases per lb. of carbon burned is 100 -h 3.8 = 26.32 lbs. Since 
the carbon burned is 80.55 — 2 = 78.55$ of the weight of the coal, the weight 
of the dry gases per lb. of coal is 26.32 X 78.55 -h 100 = 20.67 lbs. 

Each pound of coal furnishes to the dry chimney-gases .7855 lb. C, .0108N, 

and (g.70 - ^-) -s- 100 = .0214 lb, O; a total of .8177, say .82 lb. This sub- 



684 THE STEAM-BOILER. 

tracted from 20.6? lbs. leaves 19.85 lbs. as the quantity of dry air (not includ- 
ing moisture) which enters the furnace per pound of coal, not counting the 
air required to burn the available hydrogen, that is, the hydrogen minus one 
eighth of the oxygen chemically combined in the coal. Each lb. of coal 
burned contained .045 lb. H, which requires .045 X 8 = .36 lb. O for its com- 
bustion. Of this, .027 lb. is furnished by the coal itself, leaving .333 lb. 10. 
come from the air. The quantity of air needed to supply this oxygen (air 
containing 23$ by weight of oxygen) is .333 -h .23 — 1.45 lb., which added to 
the 19.85 lbs. already found gives 21.30 lbs. as the quantity of dry air sup- 
plied to the furnace per lb. of coal burned. 

The air carried in as vapor is .0071 lb. for each lb. of dry air, or 21.3 X .0071 
= 0.15 lb. for each lb. of coal. Each lb. of coal contained .029 lb. of mois- 
ture, which was evaporated and carried into the chimney-gases. The .045 lb. 
of H per lb. of coal when burned formed .045 x9= .405 lb. of H 2 0. 

From the analysis of the chimney-gas it appears that .09 -s- 3.80 — 2.37$ of 
the carbon in the coal was burned to CO instead of to C0 2 . 

We now have the data for calculating the various loses of heat as follows, 
for each pound of coal burned : 



Heat- 
units. 



Per cent of 
Heat-value 
of the Coal. 



21.3 lbs. dry air X (560° - 60°) X sp. heat .238 = 2534.7 17.80 
.15 lb. vapor in air X (560° - 60°) x sp. heat .48 = 36.0 0.25 
.029 lb. moisture in coal heated from 60° to 212° — 4.4 0.03 
evaporated from and at 212°; .029 X 966 = 28.0 0.20 
steam (heated from 212° to 560°) X 348 X .48 = 4.8 0.03 
.405 lb. H 2 from H in coal X (560° - 60°) x .48 = 97.2 0.68 
.0237 lb. C burned to CO ; loss by incomplete com- 
bustion, .0237 X (14544 - 4451) = 239.2 1.68 
.02 lb. coal lost in ashes ; .02 x 14544 = 290.9 2.04 
Radiation and unaccounted for, by difference = 712.1 5.00 

3,947.3 27.71 

Utilized in making steam, equivalent evaporation 

10.66 lbs. from and at 212° per lb. of coal = 10,295.7 72.29 

14,243.0 100.00 

The heat lost by radiation from the boiier and furnace is not easily deter- 
mined directly, especially if the boiler is enclosed in brickwork, or is pro- 
tected by non-conducting covering. It is customary to estimate the heat 
lost by radiation by difference, that is, to charge radiation with all the heat 
lost which is not otherwise accounted for. 

One method of determining the loss by radiation is to block off a portion 
of the grate-surface and build a small fire on the remainder, and drive this 
fire with just enough draught to keep up the steam-pressure and supply the 
heat lost by radiation without allowing any steam to be discharged, weigh- 
ing the coal consumed for this purpose during a test of several hours' dura- 
tion. 

Estimates of radiation by difference are apt to be greatly in error, as in 
this difference are accumulated all the errors of the analyses of the coal 
and of the gases. An average value of the heat lost by radiation from a 
boiler set in brickwork is about 4 per cent. When several boilers are in a 
battery and enclosed in a boiler-house the loss by radiation maybe very 
much less, since much of the heat radiated from the boiler is returned to it 
in the air supplied to the furnace, which is taken from the boiler-room. 

An important source of error in making a "heat balance" such as the 
one above given, especially when highly bituminous coal is used, may be 
due to the non-combustion of part of the hydrocarbon gases distilled from 
the coal immediately after firing, when the temperature of the furnace may 
be reduced below the point of ignition of the gases. Each pound of hydro- 
gen which escapes burning is equivalent to a loss of heat in the furnace of 
62,500 heat-units. 

In analyzing the chimney gases by the usual method the percentages of 
the constituent gases are obtained by volume instead of by weight. To 
reduce percentages by volume to percentages by weight, multiply the per- 
centage by volume of each gas by its specific gravity as compared with air, 
and divide each product by the sum of the products. 



TESTS OF STEAM-BOILER. 



685 



The pounds of air required to burn a pound of carbon may be obtained 
directly from the analysis by volume by the following formula: 

Lbs. of air required to burn I _ t j 3(CQ 2 -f O) + CO < + 23- 
one pound of carbon j 3) C0 2 + CO j 
In which O, CO.^, and CO are the per cents, by volume, of the several con- 
stituents of the flue gases. 

Lbs. of air per pound | _ j Lbs. of air per pound \ v \ Per cent of carbon 
of coal f ) of carbon \ ) in coal. 

To reduce to volume at temperature of 32° F. make use of the formula 
V = 12.387 X lbs. of air per pound of coal. 

TESTS OF STEAM-BOILERS. 

Boiler-tests at the Centennial Exhibition, Philadel- 
phia, 18 76. -(See Reports and Awards Group XX, International Exhibi- 
tion, Phila., 1876; also, Clark on the Steam-engine, vol. i, page 253.) 

Competitive tests were made of fourteen boilers, using good anthracite 
coal, one boiler, the Galloway, being tested with both anthracite and semi- 
bituminous coal. Two tests were made with each boiler : one called the 
capacity trial, to determine the economy and capacity at a rapid rate of 
driving; and the other called the economy trial, to determine the economy 
when driven at a rate supposed to be near that of maximum economy and 
rated capacilv. The following table gives the principal results obtained in 
the economy "trial, together with the capacity and economy figures of the 
capacity trial for comparison 





Economy Tests. 


Capacity 
Tests. 




co - 


^ 


6 


o ^ 


§J=CO 


J* 




s 






"g'FI 






















oS O 


Name 

of 
Boiler. 


.5 u 

v i 


"~ - 
■°ta 


-a 

< 

a 
o 


z* 
1- 
ft£ 

aS J. 

V & 


& 

a 


g 


co 
o 


© 

o 
ft 


© 

o 
ft 
© 


41 & 5 




s-j 


* = 


«2% 


H 


o 


ft 


o 


o 


^^^ 




m 


o 


Ph 


£f- 


^ ww 


EH 


S 


X 


M 


K 


£ 






lbs. 


p.Ct 


lbs. 


lbs. 


deg 


* 


deg 


H.P. 


H.P. 


lbs. 




84.6 
64.3 


9.1 

12 


10 4 


•> OS, 


12.094 
11.988 


393 
415 




41.4 


119.8 

57.8 


148.6 
68.4 


10.441 


Firmenich 


10.4 1.68 


11.064 




• J ,ll fi 


6.8 

12.1 


11.31.87 
11.1 2.42 


11.923 
11.906 


333 
411 


l"3 


9.4 


47.0 
99.8 


69.3 
125.0 


11.163 


Smith. 


45 8 


11.925 


Bahcock & Wilcox 


37.7 


10.0 


11.02.43 


11.822 


296 


2.7 




135.6 


186.6 


10.330 


Galloway 


23.7 


9.(5 


ll.l!3.63 


11 583 


303 




1.4 


103.3 


133.8 


11.216 


Do. semi-bit. coal 


23.7 


7.9 


8.8 3.20 


12.125 


825 


0.3 




90.9 


125.1 


11.609 


Andrews 


15.6 


8.0 


10.32.32 


11.039 


420 




71.7 


42.6 


58.7 


9.745 


Harrison 


27.3 


12.4 


8.5 2.75 


10.930 


517 


0.9 




82.4 


108.4 


9.889 


Wiegand 


30. 7 


12.3 


9.5 3.30 


10.834 


524 




20.5 


147.5 


162.8 


9.145 


Anderson 


17.5 


9.7 


9.3 2.64 


10.618 


417 




15.7 


98.0 


132.8 


9.568 


Kellv 


20.9 

33.5 


10.8 
9.3 


9.0 3.82 

11.41.38 


10.312 
10.041 


'430 


5.6 
4.2 




81.0 

72.1 


99.9 
108.0 


8.397 


Exeter 


9.974 




14.0 
19.0 


8.C 
8.6 


11.0 4.44 
9.9 3.43 


10.021 
9.613 


871 
572 


5.2 

2.1 





51.7 
45.7 


67.8 
67.2 


9.865 


Rogers & Black . . . 


9.429 


Averages 








2.77 


11.123 








85.0 


110.8 


10.251 



The comparison of the economy and capacity trials shows that an average 
increase in capacity of 30 per cent was attended by a decrease in economy 
of 8 per cent, but the relation of economy to rate of driving varied greatly 
in the different boilers. In the Kelly boiler an increase in capacity of 22 per 
cent was attended by a decrease in economy of over 18 per cent, while the 
Smith boiler with an increase of 2k- per cent in capacity showed a slight 
increase in economy. 



686 



THE STEAM-BOILER. 



One of the most important lessons gained from the above tests is that 
there is no necessary relation between the type of a boiler and economy. Of 
the five boilers that gave the best results, the total range of variation be- 
tween the highest and lowest of the five being only 2.3%, thi'ee were water- 
tube boilers, one was a horizontal tubular boiler, and the fifth was a com- 
bination of the two types. The next boiler on the list, tbe Galloway, was an 
internally fired boiler, all of the others being externally fired. The following 
is a brief description of the principal constructive features of the fourteen 
boilers: 
!,--,. j 4-in. water-tubes, incliued 20° to horizontal; reversed 

Root 1 draught. 

3 in. water-tubes, nearly vertical; reversed draught. 
Cylindrical shell, multitubular flue. 
( Cylindrical shell, multitubular flue — water-tubes in 
) side flues. 

j 33^-in. water-tubes, inclined 15° to horizontal; re- 
I versed draught. 

Galloway Cylindrical shell, furnace-tubes and water-tubes. 

Andrews Square fire-box and double return multitubular flues. 

tto,...^™ j 8 slabs of cast-iron spheres, 8 in. in diameter; re- 

nai f lson f versed draught. 

wTi a „ a „A j 4-in. water tubes, vertical, with internal circulating 

wiegana 1 tubes. 

Anderson 3-in. flue-tubes, nearly horizontal; return circulation. 

K ,j ) 3-in. water-tubes, slightly inclined; each divided by 

y j internal diaphragm to promote circulation. 

Exeter 27 hollow rectangular cast-iron slabs. 

Pierce Rotating horizontal cylinder, with flue-tubes. 

Rogers & Black Vertical cylindrical boiler, with external water-tubes. 

Tests of Tubulous Boilers.— The following tables are given by S. 
H. Leonard, Asst. Engr. U. 8. N., in Jour. Am. Soc. Naval Engrs. 1890. The 
tests were made at different times by boards of U. S. Naval Engineers, ex- 
cept the test of the locomotive-torpedo boiler, which was made in England. 



Firmenich 

Lowe 

Smith 

Babcock & Wilcox. 



Type. 



Herreshoff 
Towne 



Ward 



Scotch 

Locom'tive 
torpedo, 

Ward 

Thorny- 
croft, (U. 
S.S.Cush- 
ing.) 



■33 

o 

O 



25. 1 
4.3 
24.1 
7 A 
15.1 



j 24 

I 38 



Evaporation 

from and at 

212° F. 



17.1 
•,'0.05 



23.8 
10 



Weights, lbs. 






E40,6 
S 42,7 
E 2,9 
S 3,0 
E 1,3 
S 1,6 



E 1,6 
S 1,9 

E18,9 
S 30,0 

S 34,9 

E26,5 
S 30.4 
E 20.1 

S -J4.C 








. 





£ 


a 


<v 










a> - 




m* 


<D 


ft* 


ft, 


ft 


a 






< 


GO 


Nat'l. 


111 


Jet. 


120 


Jet. 


195 


NatL 


148 


1.14 


152 


Nat'l. 





Jet. 


17 


Jet. 


161 


2.08 




4.01 


78 


3.13 


125 


4.95 


123 


2 


160 


3 


245 



n 



Per cent moistur 
Scotch, 1st, 3,44; 2d, 



* Approximate, 
•e in steam: Belleville, 6.31; Herreshoff (first test), 3.5; 
4,29; Ward, 11.6; others not given. 



TESTS OF STEAM-BOILERS. 



687 



Dimensions of the Boilers. 



No. 



Length, ft. and in.. 
Width, " " " 
Height, " " " 

Space, cu. ft 

Grate- area, sq. ft 
Heating-surface, 

sq. ft.... 

Ratio H.S. -r- G . 



1 


2 


3 


4 


5 


6 


7 


8' 6" 


4' 9" 


2' 6" 


3' 2" 


9' 0" 


16' 8 


10' 3"* 


7 


3 8 


2 6 


1 7 


9 


6 4 


4 6 t 


11 


4 


3 3 


7 2 




7 6 


11 8 


645.5 


69.6 


20 3 


42.7 


572.5 


630.3 


729.3 


34.17 


9 


4.25 


3.68 


31.16 


28- 


66.5 


804 


205 


75 


146 


727 


1116 


2490 


23.5 


22 


17.6 


39.5 


23.3 


39.8 


37.4 



10' o"% 

7 0J 

8 0$ 
560* 
38.3 

2375 



* Diameter, t Diam. of drum. % Approximate. 

The weight per I.H.P. is estimated on a basis of 20 lbs. of water per hour 
for all cases expecting the Scotch boiler, where 25 lbs. have been used, as this 
boiler was limited to 80 lbs. pressure of steam. 

The following approximation is made from the large table, on the assump- 
tion that the evaporation varies directly as the combustion, and 25 lbs. of 
coal per square foot of grate per hour used as the unit. 



Type of Boiler. 


Com 
bustion. 


Evapora- 
tion per 
cu. ft. of 
Space. 


Weight 
per 
T.H.P. 


Weight 
per sq. ft. 
Heating- 
surface. 


Weight 
per lb. 
Water 
Evapo- 
rated. 




0.50 
1.00 
1.00 
1.00 
3.90 
2.20 


0.50 
0.95 
1.20 
0.44 
0.31 
0.58 


2.02 
0.72 
1.12 
2.40 
3.70 
1.27 


2.10 
0.60 
0.87 
1.64 
1.25 
0.50 


2.50 




0.90 


Towne 

Scotch 


1.30 
2.30 




3.50 


Ward 


1.53 







The Belleville boiler has no practical advantage over the Scotch either in 
space occupied or weight. All the other tubulous boilers given greatly 
exceed the Scotch in these advantages of weight and space. 
Some High Rates of Evaporation.— Eng'g, May 9, 1884, p. 415. 

Locomotive. Torpedo-boat. 

Water evap. per sq. ft. H.S. per hour 12.57 13.73 12.54 20.74 

" lb. fuel from and at 212°. 8.22 8.94 8.37 7.04 

Thermal units trausf 'd per sq. ft. of H.S. 12,142 13,263 12,113 20,034 

Efficiency 586 .637 .542 .468 

It is doubtful if these figures were corrected for priming. 
Economy Effected by Heating the Air Supplied to 
Boiler-furnaces. (Clark, S. E.)— Meunier and Scheurer-Kestner ob- 
tained about 7% greater evaporative efficiency in summer than in winter, 
from the same boilers under like conditions,— an excess which had been ex- 
plained by the difference of loss by radiation and conduction. But Mr. 
Poupardin, surmising that the gain might be due in some degree also to the 
greater temperature of the air in summer, made comparative trials with 
two groups of three boilers, each working one week with the heated air, 
and the next week with cold air. The following were the several efficien- 
cies: 

First Trials: Three Boilers; Ronchamp Coal. 

Water per lb. of Water per lb. of 
Coal. Combustible. 

With heated air (128° F.) 7.77 lbs. 8.95 lbs. 

With cold air (69°.8) 7.33 " 8.63 " 

Difference in favor of heated air 0.44 " 0.32 " 

Second Trials: Same Coal; Three Other Boilers. 

With heated air (!20°.4 F.) 8.70 lbs. 10.08 lbs. 

With cold air (75°.^) 8.09 " 9.34 " 

Difference in favor of heated air 0.61 " 0.64 " 



688 



THE STEAM-BOILER. 



These results show economies in favor of heating the air of 6% and 7*^#. 

Mr. Poupardin believes that the gain in efficiency is due chiefly to the 
better combustion of the gases with heated air. It was observed that with 
heated air the flames were much shorter and whiter, and that there was 
notably less smoke from the chimney. 

An extensive series of experiments was made by J. C. Hoadley (Trans. 
A. S. M. E., vol. vi., 676) on a "Warm-blast Apparatus," for utilizing the 
heat of the waste gases in heating the air supplied to the furnace. The ap- 
paratus, as applied to an ordinary horizontal tu ular boiler 60 in. diameter, 
21 feet long, with 65 3J^-inch tubes, consisted of 240 2-inch tubes, 18 feet long, 
through which the hot gases passed while the air circulated around them. 
The net saving of fuel effected by the warm blast was from 10.7$ to 15.5$ of 
the fuel used with cold blast. The comparative temperatures averaged as 
follows, in degrees F. : 

Cold-blast Warm-blast rn«^„^™ 
Boiler. Boiler. Difference. 

In heat of Are 2493 2793 300 

At bridge wall 1340 1600 260 

In smoke box 373 375 2 

Air admitted to furnace 32 332 300 

Steam and water in boiler 300 300 

Gases escaping to chimney 373 162 211 

Externalair 32 32 

With anthracite coal the evaporation from and at 212° per lb. combustible 
was, for the cold-blast boiler, days 10.85 lbs., days and nights 10.51 ; and for 
the warm-blast boiler, days 11.83, days and nights 11.03. 

Results of Tests of Heine Water-tube Boilers with 
Different Coals. 

(Communicated by E. D. Meier, C.E., 1894.) 





1 


. 


3 


4 


5 


6 


7 


8 








Kind of Coal. 


"SIS 

IS 

8" 


2d Pool, 

Youghiogh- 

eny. 


9 


eg 

O 



M 


•~ ft 


O 




5.1 

2900 
54 
53.7 
24.7 

5.03 

10.91 
11.50 

530° 
13,800 

77.0 


4.89 
2040 
448 
45.5 
23.5 

5.14 

9.94 

10.48 

12,936 

74.3 


2040 
44.8 
45.5 

22.7 

5.24 

10.51 

"400 
12,936 
78.5 


11.6 
2300 

50 

46 

35 

5.56 

7.31 
8.27 
567 
10,487 
67.2 


16.1 
1260 

21 

60 
33.7 

4.26 

7.59 
9.05 
571 
11,785 
62.5 


11.5 
3730 
73.3 
50.9 

26.2 

4.28 

8.33 
9.41 

11,610 

69.3 


PI .8 

1168 
27.9 
41.9 

27.7 

4.86 

7.36 
9.41 
609 
9,739 
73.0 


12.8 


Heating-surface, sq. ft.. 

Grate-surface, sq. ft 

Ratio H.S. toG.S 

Coal per sq. ft. G.per hr. 
Water per sq. ft. H.S.per 

hr. from and at 212° 

Water evap. from and at 

212° per lb. coal 

Per lb. combustible.. ... 
Temp, of chimney gases 
Calorific value of fuel. .. 
Efficiency of boiler perc. 


2770 

50 
55.4 

36 

5.08 

7.81 
8.96 
707 
10,359 
72.6 



Tests Nos. 7 and 8 were made with the Hawley Down-draught Furnace, 
the others with ordinary furnaces. 

These tests confirm the statement already made as to the difficulty of 
obtaining, with ordinary grate-furnaces, as high a percentage of the calo- 
rific value of the fuel with the Western as with the Eastern coals. 

Test No 3, 78.5$ efficiency, is remarkably good for Pittsburgh (Youghiogh- 
eny) coal. If the Washington coal had given equal efficiency, the saving of 

fuel would be — — -=x-r— — 20.2$. The results of tests Nos. 7 and 8 indicate 

<o.5 
that the downward-draught furnace is well adapted for burning Illinois 
coals. 



BOILERS USIXG WASTE GASES. 689 

Maximum Boiler Efficiency with Cumberland Coal.— 

About 12.5 lbs. of water per lb. combustible from and at 212° is about the 
highest evaporation that can be obtained from the best steam fuels in the 
United States, such as Cumberland, Pocahontas, and Clearfield. In excep- 
tional cases 13 lbs. has been reached, and one test is on record (F. W. Dean, 
Eng'g Neivs, Feb. 1, 1S94) giving 13.23 lbs. The boiler was internally fired, 
of the Belpaire type, 82 inches diameter, 31 feet long, with 160 3-inch tubes 
12^ feet long. Heating surface, 1998 square feet; grate-surface, 45 square feet, 
reduced during the test to 30^ square feet. Double furnace, with fire-brick 
arches and a long combustion -chamber. Feed-water neater in smoke-box. 
The following are the principal results : 

1st Test. 2d Test. 

Dry coal burned per sq. ft. of grate per hour, lbs 8.85 16.06 

Water evap. per sq. ft. of heating-surface per hour, lbs 1.63 3.00 
Water evap. from and at 212° per lb. combustible, in- 
cluding feed- water heater 13.17 13.23 

Water evaporated, excluding feed-water heater 12.88 12.90 

Temperature of gases after leaving heater, F 360° 463° 

BOILERS USING WASTE GASES. 

Proportioning Boilers for Blast-Furnaces. —(F. W. Gordon, 
Trans. A. I. M. E., vol. xii., 1883.) 

Mr. Gordon's recommendation for proportioning boilers when properly set 
for burning blast-furnace gas is, for coke practice, 30 sq. ft. of heating-sur- 
face per ton of iron per 24 hours, which the furnace is expected to make, 
calculating the heating-surface thus : For double-flued boilers, all shell- 
surface exposed to the gases, and half the flue-surface; for the French type, 
all the exposed surface of the upper boiler and half the lower boiler- 
surface; for cylindrical boilers, not more than 60 ft. long, all the heating- 
surface. 

To the above must be added a battery for relay in case of cleaning, repairs, 
etc., and more than one battery extra in large plants, when the water carries 
much lime. 

For anthracite practice add 50% to above calculations. For charcoal prac- 
tice deduct 20$. 

In a letter to the author in May, 1894, Mr. Gordon says that the blast- 
furnace practice at the time when his article (from which the above extract 
is taken) was written was very different from that existing at the present 
time; besides, more economical engines are being introduced, so that less 
than 30 sq. ft. of boiler-surface per ton of iron made in 24 hours may now be 
adopted. He says further: Blast-furnace gases are seldom used for other 
than furnace requirements, which of course is throwing away good fuel. In 
this case a furnace in an ordinary good condition, and a condition where it 
can take its maximum of blast, which is in the neighborhood of 200 to 225 
cubic ft., atmospheric measurement, per sq. ft. of sectional area of hearth, 
will generate the necessary H.P. with very small heating-surface, owing to 
the high heat of the escaping gases from the boilers, which frequently is 
1000 degrees. 

A furnace making 200 tons of iron a day will consume about 900 H.P. in 
blowing the engine. About a pound of fuel is required in the furnace per 
pound of pig metal. 

In practice it requires 70 cu. ft. of air-piston displacement per lb. of fuel 
consumed, or 22,400 cu. ft. per minute for 200 tons of metal in 1400 working 
minutes per day, at, say, 10 lbs. discharge-pressure. This is equal to 9J4 lbs. 
M.E.P. on the steam-piston of equal area to the blast-piston, or 900I.H.P. To 
this add 20% for hoisting, pumping and other purposes for which steam is em- 
ployed around blast-furnaces, and we have 1100 H.P., or say 5*4 H.P. per 
ton of iron per day. Dividing this into 30 gives approximately 5f& sq. ft. of 
heating-surface of boiler per H.P. 

Water-tube Boilers using Blast-furnace Gases.— D. S. 
Jacobus (Trans. A. I. M. E., xvii. 50) reports a test of a water- tube boiler using 
blast-furnace gas as fuel. The heating-surface was 2535 sq. ft. It developed 
328 H.P. (Centennial standard), or 5.01 lbs. of water from and at 212° per 
sq. ft. of heating-surface per hour. Some of the principal data obtained 
were as follows: Calorific value of 1 lb. of the gas, 1413 B T.U., including 
the effect of its initial temperature, which was 650° F. Amount of air used 
to burn 1 lb. of the gas = 0.9 lb. Chimney draught, \y§ in. of water. Area of 
gas inlet, 300 sq. in.; of air inlet, 100 sq. in. Temperature of the chimney 



690 



THE STEAM-BOILER. 



gases, 775° F. Efficiency of the boiler calculated from the temperatures 
and analyses of the gases at exit and entrance, 61$. The average analyses 
were as follows, hydrocarbons being included in the nitrogen : 





By Weight. 


By Volume. 




At Entrance. 


At Exit. 


At Entrance. 


At Exit. 


co 2 


10.69 
.11 
26.71 
62.48 
2.92 
11.45 
14.37 


26.37 
3.05 

1.78 
68.80 
7.19 
.76 
7.95 


7.08 

.10 

27.80 

65.02 


18 64 


o 

CO 


2.96 
1.98 




76.42 


Cin C0 2 

Cin CO 

Total C 





Steam-boilers Fired witli Waste Gases from Paddling 
and Heating Furnaces.— The Iron Age, April 6, 1893, contains a report 
of a number of tests of steam-boilers utilizing the waste heat from pud- 
dling and heating furnaces in rolling-mills. The following principal data are 
selected: In Nos. 1, 2, and 4 the boiler is a Babcock & Wilcox water-tube 
boiler, and in No. 3 it is a plain cylinder boiler, 42 in. diam. and 26 ft. long. 
No. 4 boiler was connected with a heating-furnace, the others with puddling 
furnaces. 

No.l. No. 2. No. 3. No. 4. 

Heating-surface, sq. ft 1026 1196 143 1380 

Grate-surface, sq. ft 19.9 13 6 13.6 16.7 

Ratio H.S. to G.S 52 87.2 10.5 82.8 

Water evap. per hour, lbs 3358 2159 1812 3055 

persq. ft. H.S. per hr., lbs... 3.3 1.8 12.7 2.2 

per lb. coal from and at 212°. 5.9 6.24 3.76 6.34 

" "comb." " " " .... 7.20 4.31 8.34 

In No. 2, 1 .38 lbs. of iron were puddled per lb. of coal. 
In No." 3, 1 .14 lbs. of iron were puddled per lb. of coal. 
No. 3 shows that an insufficient amount of heating- surface was provided 
for the amount of waste heat available. 



RULES FOR CONDUCTING BOILER-TESTS. 

The Committee of the A. S. M. E. on Boiler-tests, consisting of Wm. Kent 
(chairman), J. C. Hoadley, R. H. Thurston, Chas. E. Emery, and Chas. T. 
Porter, recommended the following code of rules for boiler-tests (Trans., 
vol. vi. p. 256) : 

Preliminaries to a Test. 

I. In preparing for and conducting trials of steam-boilers the specific 
object of the proposed trial should be clearly defined and steadily kept in 
view. 

II. Measure and record the dimensions, position, etc., of grate and heat- 
ing surfaces, flues and chimneys, proportion of air-space in the grate-sur- 
face, kind of draught, natural or forced. 

III. Put the boiler in good condition. Have heating -surface clean inside 
and out, grate-bars and sides of furnace free from clinkers, dust and ashes 
removed from back connections, leaks in masonry stopped, and all obstruc- 
tions to draught removed. See that the damper will open to full extent, and 
that it may be closed when desired. Test for leaks in masonry by firing a 
little smoky fuel and immediately closing damper. The smoke will then 
escape through the leaks. 

IV. Have an understanding with the parties in whose interest the test is 
to be made as to the character of the coal to be used. The coal must be dry, 
or, if wet, a sample must be dried carefully and a determination of the 
amount of moisture in the coal made, and the calculation of the results of 
the test corrected accordingly. Wherever possible, the test should be made 
with standard coal of a known quality. For that portion of the country 
east of the Alleghany Mountains good anthracite egg coal or Cumberland 
semi-bituminous coal may be taken as the standard for making tests. West 



RULES FOR CONDUCTING BOILER-TESTS. 691 

of the Alleghany Mountains and east of the Missouri River, Pittsburgh lump 
coal may be used.* 

V. In all important tests a sample of coal should be selected for chemical 
analysis. 

VI. Establish the correctness of ail apparatus used in the test for weighing 
and measuring. These are: 1. Scales for weighing coal, ashes, and water. 
2. Tanks, or water-meters for measuring water. Water-meters, as a rule, 
should only be used as a check on other measurements. For accurate work 
the water should be weighed or measured in a tank. 3. Thermometers and 
pyrometers for taking temperatures of air, steam, feed-water, waste gases, 
etc. 4. Pressure-gauges, draught-gauges, etc. 

VII. Before beginning a test, the boiler and chimney should be thoroughly 
heated to their usual working temperature. If the boiler is new, it should 
be in continuous use at least a week before testing, so as to dry the mortar 
thoroughly and heat the walls. 

VIII. Before beginning a test, the boiler and connections should be free 
from leaks, and all water connections, including blow and extra feed pipes, 
should be disconnected or stopped with blank flanges, except the particular 
pipe through which water is to be fed to the boiler during the trial. In lo- 
cations where the reliability of the power is so important that an extra feed- 
pipe must be kept in position, and in general when for any other reason 
water-pipes other than the feed-pipes cannot be disconnected, such pipes 
may be drilled so as to leave openings in their lower sides, which should be 
kept open throughout the test as a means of detecting leaks, or accidental 
or unauthorized opening of valves. During the test the blow-off pipe should 
remain exposed. 

If an injector is used it must receive steam directly from the boiler being 
tested, and not from a steam-pipe or from any other boiler. 

See that the steam-pipe is so arranged that water of condensation cannot 
run back into the boiler. If the steam-pipe has such an inclination that the 
water of condensation from any portion of the steam-pipe system may rnn 
back into the boiler, it must be trapped so as to prevent this water getting 
into the boiler without being measured. 

Starting and Stopping a Test. 

A test should last at least ten hours of continuous running, and twenty- 
four hours whenever practicable. The conditions of the boiler and furnace 
in all respects should be, as nearly as possible, the same at the end as at 
the beginning of the test. The steam-pressure should be the same, the 
water-level the same, the fire upon the grates should be the same in quan- 
tity and condition, and the walls, flues, etc., should be of the same tempera- 
ture. To secure as near an approximation to exact uniformity as possible 
in conditions of the fire and in temperatures of the walls and flues, the 
following method of starting' and stopping a test should be adopted : 

X. Standard Method.— Steam being raised to the working pressure, re- 
move rapidly all the fire from the grate, close the damper, clean the ash-pit, . 
and as quickly as possible start a new fire with weighed wood and coal, 
noting the time of starting the test and the height of the water-level while 
the water is in a quiescent state, just before lighting the fire. 

At the end of the test remove the whole Are, clean the grates and ash-pit, 
and note the water-level when the water is in a quiescent state : record the 
time of hauling the fire as the end of the test. The water-level should be 
as nearly as possible the same as at the beginning of the test. If it is not 
the same, a correction should be made by computation, and not by operat- 
ing pump after test is completed. It will generally be necessary to regulate 
the discharge of steam from the boiler tested by means of the stop-valve 
for a time while fires are being hauled at the beginning and at the end of 
the test, in order to keep the steam-pressure in the boiler at those times up 
to the average during the test. 

XL Alternate Method.— Instead of the Standard Method above described, 
the following may be employed where local conditions render it necessary : 

At the regular time for slicing and cleaning fires have them burned rather 
low, as is usual before cleaning, and then thoroughly cleaned ; note the 
amount of coal left ou the grate as nearly as it can be estimated ; note the 

* These coals are selected because they are about the only coals which 
contain the essentials of excellence of quality, adaptability to various kinds 
of furnaces, grates, boilers, and methods of filing, and wide distribution and 
general accessibility in the markets. 



692 THE STEAM-BOILER. 

pressure of steam and the height of the water-level— which should be at the 
medium height to be carried throughout the test— at the same time ; and 
note this time as the time of starting the test. Fresh coal, which has been 
weighed, should now be fired. The ash-pits should be thoroughly cleaned 
at once after starting. Before the end of the test the fires should be burned 
low, just as before the start, and the fires cleaned in such a manner as to 
leave the same amount of fire, and in the same condition, on the grates as at 
the start. The water-level and steam-pressure should be brought to the 
same point as at the start, and the time of the ending of the test should be 
noted just before fresh coal is fired. 

During the Test. 

XII. Keep the Conditions Uniform.— The boiler should be run continu- 
ously, without stopping for meal-times or for rise or fall of pressure of 
steam due to change of demand for steam. The draught being adjusted 10 
the rate of evaporation or combustion desired before the test is begun, it 
should be retained constant during the test by means of the damper. 

If the boiler is not connected to the same steam pipe with other boilers, 
an extra outlet for steam with valve in same should be provided, so that in 
case the pressure should rise to that at which the safety-valve is set it may 
be reduced to the desired point by opening the extra outlet, without check- 
ing the fires. 

If the boiler is connected to a main steam-pipe with other boilers, the 
safety-valve on the boiler being tested should be set a few pounds higher 
than ' those of the other boilers, so that in case of a rise in pressure the 
other boilers may blow off, and the pressure be reduced by closing their 
dampers, allowing the damper of the boiler being tested to remain open, 
and firing as usual. 

All the conditions should be kept as nearly uuiform as possible, such as 
force of draught, pressure of steam, and height of water. The time of 
cleaning the fires will depend upon the character of the fuel, the rapidity of 
combustion, and the kind of grates. When very good coal is used, and the 
combustion not too rapid, a ten-hour test may be run without any cleaning 
of the grates, other than just before the beginning and just before the end 
of the test. But in case the grates have to be cleaned during the test, the 
intervals between one cleaning and another should be uniform. 

XIII. Keeping the Records. — The coal should be weighed and delivered to 
the firemen in equal portions, each sufficient for about one hour's run, and 
a fresh portion should not be delivered until the previous one has all been 
fired. The time required to consume each portion should be noted, the 
time being recorded at the instant of firing the first of each new portion. It 
is desirable that at the same time the amount of water fed into the boiler 
should be accurately noted and recorded, including the height of the water 
in the boiler and the average pressure of steam and temperature of feed 
during the time. By thus recording the amount of water evaporated by 
successive portions of coal, the record of the test may be divided into 
several divisions, if desired, at the end of the test, to discover the degree 
of uniformity of combustion, evaporation, and economy at different stages 
of the test. 

XIV. Priming Tests.— In all tests in which accuracy of results is impor- 
tant, calorimeter tests should be made of the percentage of moisture in the 
steam, or of the degree of superheating. At least ten such tests should be 
made during the trial of the boiler, or so many as to reduce the probable 
average error to less than one per cent, and the final records of the boiled- 
test corrected according to the average results of the calorimeter tests. 

On account of the difficulty of securing accuracy in these tests, the great- 
est care should be taken in the measurements of weights and temperatures. 
The thermometers should be accurate within a tenth of a degree, and the 
scales on which the water is weighed to within one hundredth of a pound. 

Analyses of Gases.— Measurement of Air-supply, etc. 

XV. In tests for purposes of scientific research, in which the determina- 
tion of all the variables entering into the test is desired, certain observations 
should be made which are in general not necessary in tests for commercial 
purposes. These are the measurement of the air-supply, the determination 
of its contained moisture, the measurement and analysis of the flue gases, 
the determination of the amount of heat lost by radiation, of the amount of 
infiltration of air through the setting, the direct determination by calorim- 
eter experiments of the absolute heating value of the fuel, and (by condeu- 



KULES FOR CONDUCTING BOILER-TESTS. 



693 



sation of all the steam made by the boiler) of the total heat imparted to the 
water. 

The analysis of the flue-gases is an especially valuable method of deter- 
mining the relative value of different methods of firing, or of different kinds 
of furnaces. In making these analyses great care should be taken to pro- 
cure average samples— since the composition is apt to vary at different 
points of the flue, and the analyses should be intrusted only to a thoroughly 
competent chemist, who is provided with complete and accurate appai'atus. 

As the determinations of the other variables mentioned above are not 
likely to be undertaken except by engineers of high scientific attainments, 
and as apparatus for making them is likely to be improved in the course of 
scientific research, it is not deemed advisable to include in this code any 
specific directions for making them. 

Record op the Test. 

XVI. A "log" of the test should be kept on properly prepared blanks, 
containing headings as follows: 





Pressures. 




Temperatures. 




Fuel. 


Feed- 
water. 


Time. 


i 

£ 
p . . 

03 3 


£ 6/0 

0) o3 


2f ox 

?* o3 


"3 

0J .; 






^ el 
1* 


1 

w 


6 

£ 




6 

£ 


5° 























Reporting the Trial. 
XVII. The final results should be recorded upon a properly prepared 
blank, and should include as many of the following items as are adapted for 
the specific object for which the trial is made. The items marked with a * 
may be omitted for ordinary trials, but are desirable for comparison with 
similar data from other sources. 



Results of the trials of a. . 

Boiler at 

To determine 



1 . Date of trial 

2. Duration of trial . 



DIMENSIONS AND PROPORTIONS. 

Leave space for complete description. 

3. Grate-surface wide long area 

4. Water-heating surface 

5. Superheating surface 

6. Ratio of water-heating surface to grate sur- 

face 



AVERAGE PRESSURES. 

7. Steam-pressure in boiler, by gauge. . . 

*8. Absolute steam-pressure 

*9. Atmospheric pressure, per bai'ometer.. 
10. Force of draught in inches of water . . 

AVERAGE TEMPERATURES. 

* 11. Of external air 

*12. Of fire- room.., 

*13. Of steam 

14. Of escaping gast s 

15. Of feed-water 



sq. ft. 
sq. ft. 
sq. ft, 



lbs. 
lbs. 



deg. 
deg. 
deg. 
deg. 
deg. 



* See reference in paragraph preceding table. 



694 



THE STEAM-BOILER. 



FUEL. 

16. Total amount of coal consumed t 

1 7. Moisture in coal 

18. Dry coal consumed 

19. Total refuse, dry pounds =.. 

20. Total combustible (dry weight of coal, Item 

18; less refuse. Item 19) 

*21 . Dry coal consumed per hour 

*22. Combustible consumed per hour 



RESULTS OP CALORIMKTRIC TESTS. 

23. Quality of steam, dry steam being taken as 

unity 

24. Percentage of moisture in steam 

25. Number of degrees superheated 



26. Total weight of water pumped into boiler and 

apparently evaporated^ 

27. Water actually evaporated, corrected for 

quality of steam § 

28. Equivalent water evaporated into dry steam 

from and at 212° F. § 

*29. Equivalent total heat derived from fuel in 

British thermal units §.. 

30. Equivalent water evaporated into dry steam 
from and at 212° F. per hour 



ECONOMIC EVAPORATION. 

, Water actually evaporated per pound of dry 
coal, from actual pressure and tempera- 
ture § . . 



lbs. 
per cent. 

lbs. 
per cent, 

lbs. 
lbs. 
lbs. 



per cent, 
deg. 



lbs. 

lbs. 

B.T.U. 

lbs. 



lbs. 



* See reference in paragraph preceding table. 

t Including equivalent of wood used in lighting fire. 1 pound of wood 
equals 0.4 pound coal. Not including unburnt coal withdrawn from fire at 
end of test. 

X Corrected for inequality of water-level and of steam-pressure at begin- 
ning and end of test. 

§ The following shows how some of the items in the above table are 
derived from others: 

Item 27 = Item 26 X Item 23. 

Item 28 = Item 27 X Factor of evaporation. 



Factor of evaporation = 



, H and h being respectively the total heat- 



units in steam of the average observed pressure and in water of the average 
observed temperature of feed, as obtained from tables of the properties of 
steam and water. 

Item 29 = Item 27 X Off - h). 

Item 31 = Item 27 -=- Ttem 18. 

Item 32 = Item 28 -h- Item 18, or = Item 31 X factor of evaporation. 

Item 33 = Item 28 -h Item 20, or = Item 32 -=- (per cent 100 - Item 19). 

Items 36 to 38. First term = Item 22 x 6/5. 

Items 40 to 42. First term = Item 30 x 0.8698. 



Item :■: 
Difference of Items 43 and 44 



Item 43 = Item 29 X 0.00003, or = 
Item 45 = 



-, or 



Item 29 
"33,305 " 



Item. 44 



RULES EOR CONDUCTING BOILER-TESTS. 



695 



3-2. Equivalent water evaporated per pound of 
dry coal from and at 212° F, § 

33. Equivalent water evaporated per pound of 
combustible from and at 212° F. § 



COMMERCIAL EVAPORATION. 

*34. Equivalent water evaporated per pound of 
dry coal with one sixth refuse, at 70 pounds 
gauge-pressure, from temperature of 100° 
F. = Item 33 X 0.7249 



RATE OF COMBUSTION. 

35. Dry coal actually burned per square foot of 

grate-surface per hour 

f 1 Per sq. ft. of grate- 

.j. OR I Consumption of dry | surface 

t o" J coal per hour. Coal ! Per sq. ft. of watei - 

Jjo* | assumed with one f heating surface.. 

6ii - I sixth refuse. § I Per sq. ft. of least 

[ J area for draught. 

RATE OF EVAPORATION. 

39. Water evaporated from and at 212° F. per 
sq. ft. of heating-surface per hour. 



*40. 
*41. 
*42. 



Water evaporated 
per hour from tem- 

] perature of 100° F. 

I into steam of 70 lbs. 
gauge -pressure. § 



Per sq. ft. of grate- 
surface 

Per sq. ft. of water 
heating surface. 

Per sq. ft. of least 
J area for draught. 



COMMERCIAL HORSE-POWER. 

43. On basis of thirty pounds of water per hour 

evaporated from temperature of 100° F. 
into steam of 70 pounds gauge-pressure 
(= 34^2 lbs. from and at 212°) § 

44. Horse-power, builders' rating, at square 

feet per horse-power 

45. Per cent developed above, or below, rating§. 



lbs. 
lbs. 
lbs. 



lbs. 
lbs. 
lbs. 
lbs. 



H.P. 

per cent 



Factors of Evaporation.— The table on the following pages was 
originally published by the author in Trans. A. S. M. E., vol. vi., 1884, under 
the' title, Tables for Facilitating Calculations of Boiler-tests. The table 
gives the factors for every 3° of temperature of feed-water from 32° to 212° 
F., and for every two pounds pressure of steam within the limits of ordinary 
working steam-pressures. 

The difference in the factor corresponding to a difference of 3° tempera- 
ture of feed is always either .0031 or .0032. For interpolation to find a factor 
for a feed-water temperature between 32° and 212°, not given in the table, 
take the factor for the nearest temperature and add or subtract, as the case 
may be, .0010 if the difference is .0031, and .0011 if the difference is .0032. As 
in nearly all cases a factor of evaporation to three decimal places is accu- 
rate enough, any error which may be made in the fourth decimal place by 
interpolation is of no practical importance. 

The tables used in calculating these factors of evaporation are those given 
in Charles T. Porter's Treatise on the Richards' Steam-engine Indicator. 

The formula is Factor = -^^-=- , in which H is the total heat of steam at the 
965. < 

observed pressure, and h the total heat of feed-water of the observed 
temperature. 



THE STEAM-BOILER. 



CSaage- pressures. 
Absolute pressur 


Lbs. 
.0 -f 
s 15 


10 + 
25 


20 + 
35 


30 + ! 40 + 
15 1 55 


45 + 
60 


50 + 
65 


52 + 
67 


54 + 
69 


56 + 
71 


Feed-water 1 
temperature.! 








Factors of Evaporation. 









1.0003 1.0088 


1.0149 


1.0197 


1.0237 


1.0254 


1.0271 


1.0277 


1.0283 


35 


1.0120 


80 


1.0228 


681 86 1.0302 1.0309 


1.0315 


66 


51 


1.0212 


60 


99 1.0317 


34 


40 


46 


98 


83 


43 


91 


1.0331 


49 


65 


72 


78 


1.0129 


1.0214 


75 


1.0323 


62 


80 


97 


1.0403 


1.0409 


60 


46 


1.0306 


54 


94 


1.0412 


1.0428 


34 


41 


92 


77 


38 


85 


1.0425 


43 


60 


66 


72 


1.0223 


1.0308 


69 


1.0417 


57 


74 


91 


97 


1.0503 


55 


40 


1.0400 


48 


88 


1.0506 


1.0522 


1.0528 


35 


86 


71 


32 


80 


1.0519 


37 


54 


60 


66 


1.0317 


1.0403 


63 


1.0511 


51 


68 


85 


91 


98 


49 


34 


95 


42 


82 


1.0600 


1.0616 


1.0623 


1.0629 


80 


65 


1.0526 


74 


1.0613 


31 


48 


54 


60 


1.0411 


97 


57 


1.0605 


45 


63 


79 


85 


92 


43 


1.0528 


89 


36 


76 


94 


1.0710 


1.0717 


1.0723 


74 


59 


1.0620 


68 


1.0707 


1.0725 


42 


48 


54 


1.0505 


91 


51 


99 


39 


56 


73 


80 


86 


37 


1.0622 


82 


1.0730 


70 


88 


1.0804 


1.0811 


1.0817 


68 


53 


1.0714 


62 


1.0801 


1.0819 


36 


42 


48 


99 


84 


45 


93 


33 


50 


67 


73 


80 


1.0631 


1.0716 


76 


1.0824 


64 


82 


98 


1.0905 


1.0911 


62 


47 


1.0808 


55 


95 


1.0913 


1.0930 


36 


42 


93 


78 


39 


87 


1.0926 


44 


61 


67 


73 


1.0724 


1.0810 


70 


1.0918 


58 


75 


92 


98 


1.1005 


56 


41 


1.0901 


49 


89 


1.1007 


1.1023 


1.1030 


36 


87 


72 


33 


80 


1 . 1020 


38 


55 


61 


67 


1.0818 


1.0903 


64 


1.1012 


51 


69 


86 


92 


98 


49 


34 


95 


43 


83 


1.1100 


1.1117 


1.1123 


1.1130 


81 


66 


1.1026 


74 


1.1114 


32 


48 


55 


61 


1.0912 


97 


57 


1.1105 


45 


63 


79 


86 


92 


43 


1.1028 


89 


36 


76 


94 


1.1211 


1.1217 


1.1223 


74 


59 


1.1120 


68 


1.1207 


1.1225 


42 


48 


54 


1.1005 


90 


51 


99 


39 


56 


73 


79 


86 


36 


1.1122 


82 


1.1230 


70 


88 


1.1304 


1.1310 


1.1317 


68 


. 53 


1.1213 


61 


1.1301 


1.1319 


35 


42 


48 


99 


84 


45 


92 


32 


50 


66 


73 


79 


1.1130 


1.1215 


76 


1.1323 


63 


81 


98 


1.1404 


1.1410 


61 


46 


1.1307 


55 


94 


1.1412 


1.1429 


35 


41 


92 


77 


38 


86 


1.1426 


43 


60 


66 


73 


1.1223 


1.1309 


69 


1.1417 


57 


75 


91 


97 


1.1504 


55 


40 


1.1400 


48 


88 


1.1506 


1.1522 


1.1529 


35 


86 


71 


31 


79 


1.1519 


37 


53 


60 


66 


1.1317 


1.1402 


63 


1.1510 


50 


68 


84 


91 


97 


48 


33 


94 


41 


81 


99 


1.1616 


1.1622 


1.1628 


79 


64 


1.1525 


73 


1.1612 


1.1630 


47 


53 


59 


1.1410 


95 


56 


1.1604 


44 


61 


78 


84 


90 


41 


1.1526 


87 


35 


75 


92 


1.1709 


1.1715 


1.1722 


72 


58 


1.1618 


66 


1.1706 


1.1723 


40 


46 


53 


1.1504 


89 


49 


97 


37 


55 


71 


78 


84' 


35 


1.1620 


80 


1.1728 


68 


86 


1.1802 


1.1809 


1.1815 


66 


51 


1.1711 


59 


99 


1.1817 


33 


40 


46 


97 


82 


43 


90 


1.1830 


48 


64 


71 


77 


1.1628 


1.1713 


74 


1.1821 


61 


79 


96 


1.1902 


1.1908 


59 


44 


1 . 1805 


52 


92 


1.1910 


1.1927 


33 


39 


90 


75 


36 


84 


1.1923 


41 


58 


64 


70 


1.1721 


1.1806 


67 


1.1915 


54 


72 


89 


95 


1.2001 


52 


37 


98 


46 


86 


1.2003 


1.2020 


1.2026 


32 


83 


68 


1.1929 


■ 77 


1.2017 


34 


51 


57 


64 


1.1814 


1.1900 


60 


1.2008 


48 


65 


82 


88 


95 


45 


31 


91 


39 


79 


96 


1.2113 1.2119 


1.2126 


76 


62 


1.2022 


70 


1.2110 


1.2128 


44 


51 


57 



1.0290 
1.0321 

84 
1.0415 

47 



1.0604 
35 



1.0729 
60 
92 



1.0917 
48 



'3 
1.1104 



1.1229 
60 



72 
1.1603 
34 



1.1821 
52 



76 
1.2007 

39 

70 
1.2101 

32 



FACTORS OF EVAPORATION. 



697 



Gaugre-press., 


lbs. 58 + 


60+ 1 


62 + 


64 + 


66 + 1 


68 + I 


70 + I 


72 + 1 


74 + 1 


76 + 


Absolute Pres 


sures..73. 


75 | 77 | 79 | 81 | 83 | 85 I 87 1 89 | 91 


Feed-water 
Temp. 


Factors of Evaporation. 


212° F. 


1.0295 


1.0301 


1.0307 


1.0312 


1.0318 


1.0323 


1.0329 


1.0334 1.0339 


1.0344 


209 


1.0327 


33 


38 


44 


49 


55 


60 


65 


70 


75 


206 


58 


64 


70 


75 


81 


86 


41 


97 


1.0402 


1.0407 


203 


90 


96 


1.0401 


1.0407 


1.0412 


1.0418 


1.0423 


1.0428 


33 


38 


200 


1.0421 


1.0427 


33 


38 


44 


49 


54 


59 


65 


69 


197 


53 


58 


64 


70 


75 


80 


86 


91 


96 


1.0501 


194 


84 


90 


96 


1.0501 


1.0507 


1.0512 


1.0517 


1.0522 


1.0527 


32 


191 


1.0515 


1.0521 


1.0527 


33 


38 


43 


49 


54 


59 


64 


188 


47 


53 


58 


64 


69 


"75 


80 


85 


90 


95 


185 


78 


84 


90 


95 


1.0601 


1.0006 


1.0611 


1.0616 


1.0622 


1.0626 


• 182 


1.0610 


1.0615 


1.0621 


1.0627 


32 


37 


43 


48 


53 


58 


179 


41 


47 


52 


58 


63 


69 


74 


79 


84 


89 


176 


72 


78 


84 


89 


95 


1.0700 


1.0705 


1.0711 


1.0716 


1.0721 


173 


1.0704 


1.0709 


1.0715 


1.0721 


1.0726 


32 


37 


42 


47 


52 


170 


35 


41 


46 


52 


57 


63 


68 


73 


78 


83 


167 


65 


72 


78| 83 


89 


94 


99 


1.0805 1.0810 


1.0815 


164 


m 


1.0803 


1.0809 1.08!5 


1.0820 


1.0825 


1.0831 


36 41 


46 


161 


1.0829 


35 


40 


46 


51 


57 


62 


67 72 


77 


158 


60 


66 


72 


77 


83 


88 


93 


981.0904 


1.0908 


155 


92 


97 


1.0903 


1.0909 


1.0914 


1.0919 


1.0925 


1.0930, 35 


40 


152 


1.0923 


1.0929 


34 


40 


45 


51 


56 


61 


66 


71 


149 


54 


60 


66 


71 


77 


82 


87 


92 


97 


1.1002 


146 


85 


91 


97 


1.1002 


1.1008 


1.1013 


1.1018 


1.1024 1.1029 


34 


143 


1.1017 


1.1022 


1.1028 


34 


39 


44 


50 


55 


60 


65 


140 


48 


54 


59 


65 


70 


76 


81 


86 


91 


96 


137 


79 


85 


91 


96 


1.1102 


1.1107 


1.1112 


1.1117 


1.1122 


1.1127 


134 


1.1110 


1.1116 


1.1122 


1.1127 


33 


38 


43 


49 


54 


59 


131 


42 


47 


53 


59 


64 


69 


75 


80 


85 


90 


128 


73 


79 


84 


90 


95 


1.1201 


1.1206 


1.1211 


1.1216 


1.1221 


125 


1.1204 


1.1210 


1.1215 


1.1221 


1.1226 


32 


37 


42 


47 


52 


122 


35 


41 


47 


52 


58 


63 


6S 


73 


78 


83 


119 


66 


72 


78 


83 


89 


94 


99 


1.1305 


1.1310 


1.1315 


116 


98 


1.1303 


1.1309 


1.1315 


1.1320 


1.1325 


1.1331 


36 


41 


46 


113 


1.1329 


34 


40 


46 


51 


57 


62 


67 


72 


77 


110 


60 


66 


71 


77 


82 


88 


93 


98 


1.1403 


1.1408 


107 


91 


97 


1.1403 


1 . 1408 


1.1414 


1.1419 


1.1424 


1.1489 


34 


39 


104 


1.1422 


1.1428 


34 


39 


45 


50 


55 


60 


65 


70 


101 


53 


59 


65 


70 


76 


81 


86 


92 


97 


1.1502 


98 


85 


90 


96 


1.1502 


1.1507 


1.1512 


1.1518 


1.1523 


1.1528 


33 


95 


1.1516 


1.1521 


1.1527 


33 


38 


43 


49 


54 


59 


64 


92 


47 


53 


58 


64 


69 


75 


80 


85 


90 


95 


89 


78 


84 


89 


95 


1.1600 


1.1606 


1.1611 


1.1616 


1.1621 


1.1626 


86 


1.1609 


1.1615 


1.1621 


1.1626 


32 


37 


42 


47 


52 


57 


83 


40 


46 


52 


57 


63 


68 


73 


78 


83 


88 


80 


71 


77 


83 


88 


94 


99 


1.1704 


1.1710 


1.1715 


1.1720 


77 


1.1702 


1 1708 


1.1714 


1.1719 


1.1725 


1.1730 


35 


41 


46 


51 


74 


34 


39 


45 


51 


56 


61 


67 


72 


77 


82 


71 


65 


70 


76 


82 


87 


92 


98 


1.1803 


1.1808 


1.1813 


68 


96 


1 lf02 


1.1807 


1.1813 


1.1818 


1.1824 


1.1829 


34 


39 


44 


65 


1.1827 


33 


38 


4-1 


49 


55 


60 


65 


70 


75 


62 


58 


64 


69 


75 


80 


86 


91 


96 


1.1901 


1.1906 


59 


89 


95 


1.1901 


1.1906 


1.1912 


1.1917 


1.1922 


1.1927 


32 


37 


56 


1.1920 


1.1926 


32 


37 


43 


48 


53 


58 


63 


68 


53 


51 


57 


63 


68 


74 


79 


84 


89 


94 


P9 


50 


82 


88 


94 


99 


1.2005 


1.2010 


1.2015 


1.2021 


1.2026 


1.2031 


47 


1.2013 


1.2019 


1.2025 


1.2030 


36 


41 


46 


52 


57 


62 


44 


44 


50 


56 


61 


67 


72 


78 


83 


88 


93 


41 


76 


81 


87 


93 


98 


1.2103 


1.2109 


1.2114 


1.2119 


1.2124 


38 


1.2107 


1.2112 


1.2118 


1.2124 


1.2129 


34 


40 


45 


50 


55 


35 


38 


43 


49 


55 


60 


65 


71 


76 


81 


86 


32 


09 


75 


80 


86 


91 


97 


1.2202 


1.2207 


1.2212 


1.221, 



698 



THE STEAM-BOILER. 



o+ I i 



92+ I 94+ I ' 
107 1 109 I 1 



Temp 


*| 






Factors 


of Evaporation. 








212 


1.0349 


1.0353 


1.0358 


1.0363 


1.0367 


1.0372 


1.0376 


1.0381 1.0385 


1.0389 1 1.0393 


209 


80 


85 


90 


94 


99 


1.0403 


1.0408 


1.0412 


1.0416 


1.0421 '1.0425 


206 


1.0411 


1.0416 


i.0421 


1.0426 


1.0430 


35 


39 


43 


48 


52 


56 


203 


43 


48 


52 


57 


62 


66 


71 


75 


79 


83 


88 


200 


74 


79 


84 


89 


93 


98 


1.0502 


1.0506 


1.0511 


1.0515 


1.0519 


197 


1.0506 


1.0511 


1.0515 


1.0520 


1.0525 


1.0521) 


33 


38 


42 


46 


50 


194 


37 


42 


47 


51 


56 


60 


65 


69 


73 


78 


82 


191 


69 


73 


78 


83 


87 


92 


96 


1.0601 


1.0605 


1.0609 


1.0613 


188 


1.0600 


1.0605 


1.0610 


1.0614 


1.0619 


1.0623 


1.0628 


32 


36 


40 


45 


185 


31 


36 


41 


46 


50 


55 


59 


63 


68 


72 


76 


182 


63 


68 


72 


77 


81 


86 


90 


95 


99 


1.0703 


1.0707 


179 


94 


99 


1.0704 


1.0708 


1.0713 


1.0717 


1.0722 


1.0726 


1.0730 


35 


39 


176 


1.0725 


1.0730 


35 


40 


44 


49 


53 


57 


62 


66 


70 


173 


57 


62 


66 


71 


75 


80 


84 


89 


93 


97 


1.0801 


■ 170 


88 


93 


98 


1.C802 


1.0807 


1.0811 


1.0816 


1.0820 


1.0824 


1.0829 


33 


167 


1.0819 


1.0884 


1.0829 


34 


38 


43 


47 


51 


56 


60 


64 


164 


51 


56 


60 


65 


69 


74 


78 


83 


87 


91 


95 


161 


82 


87 


92 


96 


1.0901 


1.0905 


1.0910 


1.0914 


1.0918 


1.0923 


1.0927 


158 


1.0913 


1.0918 


1.0923 


1.0927 


32 


37 


41 


45 


50 


54 


58 


155 


45 


49 


54 


59 


63 


68 


72 


77 


81 


85 


89 


152 


76 


81 


85 


90 


95 


99 


1.1004 


1.1008 


1.1012 


1.1016 


1.1021 


149 


1.1007 


1.1012 


1.1017 


1.1021 


1.1026 


1.1030 


35 


39 


43 


48 


52 


146 


38 


43 


48 


53 


57 


62 


66 


70 


75 


79 


83 


143 


70 


74 


79 


84 


88 


93 


97 


1.1102 


1.1106 


1.1110 


1.1114 


140 


1.1101 


1.1106 


1.1110 


1.1115 


1.1120 


1.1124 


1.1129 


33 


37 


41 


46 


137 


32 


37 


42 


46 


51 


55 


60 


64 


68 


73 


77 


134 


63 


68 


73 


78 


82 


87 


91 


95 


1.1200 


1.1204 


1.1208 


131 


95 


99 


1.1204 


1.1209 


1.1213 


1.1218 


1.1222 


1.1227 


31 


35 


39 


128 


1.1226 


1.1231 


35 


40 


45 


49 


53 


58 


62 


66 


71 


125 


57 


62 


67 


71 


76 


80 


85 


89 


93 


98 


1.1302 


122 


88 


93 


98 


1.1302 


1 . 1307 


1.1311 


1.1316 


1.1320 


1.1325 


1.1329 


33 


119 


1.1320 


1 . 1324 


1.1329 


34 


38 


43 


47 


51 


56 


60 


64 


116 


51 


55 


60 


65 


69 


74 


78 


83 


87 


91 


95 


113 


82 


87 


91 


96 


1.1401 


1.1405 


1.1409 


1.1414 


1.1418 


1.1422 


1.1426 


110 


1.1413 


1.1418 


1.1422 


1.1427 


32 


36 


41 


45 


49 


53 


58 


107 


44 


49 


54 


58 


63 


67 


72 


76 


80 


85 


89 


104 


75 


80 


85 


89 


94 


99 


1.1503 


1 . 1507 


1.1512 


1.1516 


1.1520 


101 


1.1506 


1.1511 


1.1516 


1.1521 


1.1525 


1.1530 


34 


38 


43 


47 


51 


98 


38 


42 


47 


52 


56 


61 


65 


70 


74 


78 


82 


95 


69 


74 


78 


83 


87 


92 


96 


1.1601 


1.1605 


1.1609 


1.1613 


92 


1.1600 


1.1605 


1.1609 


1.1614 


1.1619 


1.1623 


1.1628 


32 


36 


40 


45 


89 


31 


36 


41 


45 


50 


54 


59 


63 


67 


72 


76 


86 


62 


67 


72 


76 


81 


85 


90 


94 


98 


1.1703 


1.1707 


83 


93 


98 


1.1703 


1.1707 


1.1712 


1.1717 


1.1721 


1.1725 


1 .1730 


34 


38 


80 


1.1724 


1.1729 


34 


39 


43 


48 


52 


56 


61 


65 


69 


77 


56 


60 


65 


70 


74 


79 


83 


88j 92 


96 


1.1800 


74 


87 


91 


96 


1.1801 


1.1805 


1.1810 


1.1814 


1.1819 1.1823 


1.1827 


31 


71 


1.1818 


1.1823 


1.1827 


32 


36 


41 


45 


50 


54 


58 


62 


68 


49 


54 


58 


63 


68 


72 


77 


81 


85 


89 


94 


65 


80 


85 


89 


94 


99 1.1903 


1.1908 


1.1912 


1.1916 


1 1920 


1.1925 


62 


1.1911 


1.1916 


1.19*1 


1.1925 


1.1930 


34 


39 


43 


47 


52 


56 


59 


42 


4? 


52 


56 


61 


65 


70 


74 


78 


83 


87 


56 


73 


78 


83 


87 


92 


96 


1.2001 


1.2005 


1.2010 


1.2014 


1.2018 


53 


1 2004 


1 :?009 


1.2014 


1.2018 


1 2023 


1.2028 


32 


36 


41 


45 


49 


50 


35 


40 


45 


50 


54 


59 


63 


67 


72 


76 


80 


47 


66 


71 


70 


81 


1.2116 


90 


94 


98 


1.2103 


1.2107 


1.2111 


44 


98 


1.2102 


1.2107 


1.2112 


1.2121 


1.2125 


1.2130 


34 


38 


42 


41 


1.2129 


33 


38 


43 


47 


52 


56 


61 


65 


69 


73 


38 


60 


64 


69 


74 


78 


83 


87 


92 


96 


1.2200 


1.2204 


35 


91 


96 


1.2200 


1.2205 


1.2209 


1.2214 


1.2218 


1.2223 


1.2227 


31 


35 


32 


1.2222 


1.2227 


31 


36 


41 


45 


49 


54 


58 


62 


67 



FACTORS OF EVAPORATION". 



699 



Gauge-pressures 1 1 I 

lbs. 100 + 1 105 + 1 110 + 1 115 + 
Absolute Press. j 

lbs. 115. | 120 | 125 | 130 


120 + 
135 


125 + 
140 


130 + 
145 


135 f 
150 


110 + 
| 155 


145 + 
160 


150 + 
165 


Fe Temp ter | Factors of Evaporation. 



212° 


1.0397 


1.0407 


1.0417 


1.0427 


1.0436 


1.0445 


1.045: 


I1.046S 


1.0470:1.0478 1.0486 


209 


1.0429 


39 


49 


58 


67 


76 


85 9311.0501 


1.050C 


1.0517 


206 


60 


70 


80 


89 


9fa 


1.0508 


1.0516 


1.052" 


31 


4 


48 


203 


92 


1.0502 


1.0511 


1.0521 


1.053C 


39 


4£ 


56 


64 


I 7S 


80 


200 


1.0523 


33 


43 


52 


62 


70 


7t 


87 


96 


1.060^ 


1.0611 


19? 


55 


65 


74 


84 


93 


1.0602 


1.061C 


1.061S 


1 . 062" 


3f 


43 


194 


86 


9b 


1.0606 


1.0615 


1.0624 


33 


4£ 


5C 


5b 


66 


74 


191 


1.0617 


1.0627 


37 


47 


56 


65 


73 


82 


9C 


9< 


1.0706 


188 


49 


59 


69 


78 


8? 


96 


1.070c 


1.0713 


1.0721 


1.072f 


37 


185 


80 


90 


1.0700 


1.0709 


1.0719 


1.072? 


36 


44 


5c 


61 


68 


182 


1.0712 


1.0722 


31 


41 


50 


59 


67 


76 


84 


9; 


1.0800 


179 


43 


53 


63 


72 


81 


90 


9Q 


1.080? 


1.081? 


1 . 0S2: 


31 


176 


74 


84 


94 


1.0803 


1.0813 


1.0821 


1.0830 


39 


47 


5c 


62 


173 


1.0806 


1.0816 


1.0825 


35 


44 


53 


61 


70 


78 


86 


94 


170 


37 


47 


57 


66 


75 


84 


93 


1.0901 


1.090S 


1.0917 


1.0925 


167 


68 


78 


88 


97 


1.0907 


1.0915 


1.0924 


32 


41 


4£ 


56 


164 


1.0900 


1.0910 


1.0919 


1.0929 


38 


47 


55 


64 


72 


86 


88 


161 


31 


41 


51 


60 


69 


78 


8? 


95 


1 . 1003 


1.1011 


1.1019 


158 


62 


72 


82 


91 


1.1000 


1.1009 


1.1018 


1.1026 


35 


43 


50 


155 


03 


1.1003 


1.1013 


1.1023 


32 


41 


49 


58 


66 


74 


82 


152 


1.1025 


35 


44 


54 


63 


72 


81 


89 


9? 


1.1105 


1.1113 


149 


56 


66 


76 


85 


94 


1.1103 


1.1112 


1.1120 


1.1128 


36 


44 


146 


87 


97 


1.1107 


1.1116 1.1126 


34 


43 


51 


60 


68 


75 


143 


1.1118 


1.1129 


38 


48 


57 


66 


74 


83 


91 


99 


1 . 1207 


140 


50 


60 


70 


79 


88 


97 


1.1206 


1.1214 


1.1222 


1.1230 


38 


137 


81 


91 


1.1201 


1.1210 


1.1219 


1.1228 


37 


45 


53 


61 


69 


134 


1.1212 


1 . 1222 


32 


41 


51 


59 


68 


76 


85 


93 


1.1300 


131 


43 


53 


63 


73 


82 


91 


99 


1.1308 


1.1316 


1.1324 


32 


128 


75 


85 


94 


1.1304 


1.1313 


1.1322 


1.1331 


39 


47 


55 


63 


125 


1.1306 


1.1316 


1.1326 


35 


44 


53 


62 


70 


78 


86 


94 


122 


37 


47 


57 


66 


75 


84 


93 


1.1401 


1.1409 


1.1417 


1.1425 


119 


68 


78 


88 


97 


1.1407 


1.1415 


1.1424 


32 


41 


49 


56 


116 


99 


1.1409 


1.1419 


1 . 1429 


38 


47 


55 


64 


72 


80 


88 


113 


1.1431 


41 


50 


60 


69 


78 


86 


95 


1.1503 


1.1511 


1.1519 


110 


62 


72 


82 


91 


1.1500 


1.1509 


1.1518 


1.1516 


34 


42 


50 


107 


93 


1.1503 


1.1513 


1.1522 


31 


40 


49 


57 


65 


73 


81 


104 


1.1524 


34 


44 


53 


62 


71 


80 


88 


97 


1.1605 


1.1612 


101 


55 


65 


75 


■ 84 


94 


1.1602 


1.1611 


1.1620 


1.1628 


36 


43 


98 


86 


96 


1.1606 


1.1616 


1.1625 


34 


42 


51 


59 


67 




95 


1.1618 


1.1628 


37 


47 


56 


65 


73 


82 


90 


98 


1.1706 


92 


49 


50 


68 


78 


87 


96 


1.1705 


1.1713 


1.1721 


1.1729 


37 


89 


80 


90 


1.1700 


1.1709 


1.1718 


1.1727 


36 


44 


52 


60 


68 


86 


1.1711 


1.1721 


31 


40 


49 


58 


67 


75 


83 


91 


99 


83 


42 


52 


62 


71 


80 


89 


98 


1.1806 


1.1815 


1.1823 


1.1830 


80 


73 


83 


93 


1.1802 


1.1812 


1.1820 


1.1829 


3? 


46 


54 


61 


77 


1.1804 


1.1814 


1.1824 


34 


43 


52 


60 


69 


77 


85 


93 


74 


35 


45 


55 


65 


74 


83 


91 


1.1900 


1.1908 


1.1916 


1.1924 


71 


67 


77 


86 


96 


1.1905 


1.1914 


1.1922 


31 


39 


4? 


55 


68 


98 


1.1908 


1.1917 


1.192? 


36 


45 


54 


62 


70 


78 


86 


65 


1.1929 


39 


49 


58 


% 


76 


85 


93 


1.2001 


1.2009 


1.2017 


62 


60 


70 


80 


89 


98 


1.200? 


1.2016 


1.2024 


32 


40 


48 


59 


91 


1.2001 


1.2011 


1.2020 


1.2029 


38 


47 


55 


63 


71 


79 


56 


1.2022 


32 


42 


51 


60 


69 


78 


86 


94 


1.2102 


1.2110 


53 


53 


63 


73 


82 


91 


1.2100 


1 . 1209 


1.2117 1.2126 


34 


41 


50 


84 


94 


1.2104 


1.2113 


1 2123 


31 


40 


48 


5? 


65 


72 


47 


1.2115 


1.2125 


35 


44 


54 


63 


71 


80 


88 


96 


1.2203 


44 


. 46 


56 


66 


76 


85 


94 


1.2202 


1.2211 


1.2219 


1.2227 


35 


41 


77 


87 


97 


1.2207,1.2216 


1.2225 


33 


42 


50 


58 


66 


38 


1.2208 


1.2219 


1.2228 


38 47 


56 


64 


73 


81 


89 


97 


35 


40 


50 


59 


69 78 


8? 


95 


1. 2304 i 1.2312 


1.2320 


1.2328 


32 


71 


81 


90 


1.2300 1.2309 


1.2318 1.2326 


351 43 


51 


59 



700 THE STEAM-BOILER. 

STRENGTH OF STEAM-BOILEBS, VARIOUS RULES 
FOR CONSTRUCTION. 

There is a great lack of uniformity in the rules prescribed by differ- 
ent writers and by legislation governing the construction of steam-boilers 
In the United States, boilers for merchant vessels must be constructed ac- 
cording to the rules and regulations prescribed by the Board of Supervising 
Inspectors of Steam Vessels; in the U. S. Navy, according to rules of the 
Navy Department, and in some cases according to special acts of Congress. 
On land, in some places, as in Philadelphia, the construction of boilers is 
governed by local laws; but generally there are no laws upon the subject, 
and boilers are constructed according to the idea of individual engineers and 
boiler-makers. In Europe the construction is generally regulated by strin- 
gent inspection laws. The rules of the U. 8. Supervising Inspectors of 
Steam- vessels, the British Lloyd's and Board of Trade, the French Bureau 
Veritas, and the German Lloyd's are ably reviewed in a paper by Nelson 
Foley, M. Inst. Naval Architects, etc., read at the Chicago Engineering Con- 
gress, Division of Marine and Naval Engineering. From this paper the fol- 
lowing notes are taken, chiefly with reference to the U. S. and British rules: 

(Abbreviations.— T. S., for tensile strength; El., elongation; Con tr., con- 
traction of area.) 

Hydraulic Tests.— Board of Trade, Lloyd's, and Bureau Veritas — 
Twice the working pressure. 

United States Statutes.— One and a half times the working pressure. 

Mr. Foley proposes that the proof pressure should be 1J^ times the work- 
ing pressure + one atmosphere. 

Established Nominal Factors of Safety. —Boat -d of Trade.— 
4.5 for a boiler of moderate length and of the best construction and work- 
manship. 

Lloyd's.— Not very apparent, but appears to lie between 4 and 5. 

United States Statutes.— Indefinite, because the strength of the joint is 
not considered, except by the broad distinction between single and double 
riveting. 

Bureau Veritas: 4.4. 

German Lloyd's: 5 to 4.65, according to the thickness of the plates. 

Material tor Riveting.— Board of Trade.— Tensile strength of 
rivet bars between 26 and 30 tons, el. in 10" not less than 25$, and contr. of 
area not less than 50$. 

Lloyd's.— T. S., 26 to 80 tons; el. not less than 20$ in 8". The material 
must stand bending to a curve, the inner radius of which is not greater than 
\y^ times the thickness of the plate, after having been uniformly heated to 
a low cherry- red, and quenched in water at 82° F. 

United States Statutes. — No special provision. 

Rules Connected with Riveting.— Board of Trade.— The shear- 
ing resistance of the rivet steel to be taken at 23 tons per square inch, 5 to 
be used for the factor of safety independently of any addition to this factor 
for the plating. Rivets in double shear to have only 1.75 times the single 
section taken in the calculation instead of 2. The diameter must not be less 
than the thickness of the plate and the pitch never greater than 8}^". The 
thickness of double butt-straps (each) not to be less than % the thickness of 
the plate; single butt-straps not less than 9/8. 

Distance from centre of rivet to edge of hole = diameter of rivet x V/%- 

Distance between rows of rivets 

= 2 X diam. of rivet or = [(diam. x 4) + 1] -=- 2, if chain, and 



V[(pitch x 11) + (diam. x 4)] x (pitch -f diam. X 4) . 



10 



if zigzag. 



Diagonal pitch = (pitch x 6 -f- diam. x 4) -+- 10. 

Lloyd's.— Rivets in double shear to have only 1.75 times the single section 
taken in the calculation instead of 2. The shearing strength of rivet steel to 
be taken at 85$ of the T. S. of the material of shell plates. In any case 
where the strength of the longitudinal joint is satisfactorily shown by ex- 
periment to be greater than given by the formula, the actual strength may 
be taken in the calculation. 

United States Statutes. — No rules. 

Material for Cyindrical Shells Subject to Internal Pres- 
sure.— Board of Trade.—?. S l erween 27 and 32 tons. In the normal con- 
dition, el. not less than 18$ in 10", but should be about 25$ ; if annealed, not 



STRENGTH OF STEAM-BOILERS. 701 

less than 20$. Strips 2" wide should stand bending: until the sides are 
parallel at a distance from each other of not more than three times the 
plate's thickness. 

Lloyd's— T. S. between the limits of 26 and 30 tons per square inch. El. 
not less than 20$ in 8". Test strips heated to a low cherry-red and plunged 
into water at 82° F. must stand bending to a curve, the inner radius of 
which is not greater than \% times the plate's thickness. 

U. S. Statutes. — Plates of %" thick and under shall show a contr. of not 
less than 50$; when over y%' and up to %", not less than 45$ ; when over 
%", not less than 40$. 

Mr. Foley's comments : The Board of Trade rules seem to indicate a steel 
of too high T. S. when a lower and more ductile one can be got : the lower 
tensile limit should be reduced, and the bending test might with advantage 
be made after tempering, and made to a smaller radius. Lloyd's rule for 
quality seems more satisfactory, but the temper test is not severe. The 
United States Statutes are not sufficiently stringent to insure an entirely 
satisfactory material. 

Mr. Foley suggests a material which wou'd meet the following : 25 tons 
lower limit in tension ; 25$ in 8" minimum elongation ; radius for bending 
test after tempering = the plate's thickness. 

Shell-plate Formulae.- Board of Trade: P = T * B *J , * 2 . 

U X -ft X 100 
D = diameter of boiler in inches ; 
P — working-pressure in lbs. per square inch ; 
t = thickness in inches ; 

B = percentage of strength of joint compared to solid plate ; 
T — tensile strength allowed for the material in lbs. per square inch ; 
F = a factor of safety, being 4.5, with certain additions depending on 
method of construction. 

n ^ . t> CX(t -2)XB 
Lloyd's : P = — . 

t = thickness of plate in sixteenths ; B and D as before ; C = a constant 
depending on the kind of joint. 

When longitudinal seams have double butt-straps, C = 20. When longi- 
tudinal seams have double butt-straps of unequal width, only covering on 
one side the reduced section of plate at the outer line of rivets, C — 19.5. 

When the longitudinal seams are lap-jointed, C = 18.5. 

U. S. Statutes.— Using same notation as for Board of Trade, 

t X 2 X T 

P — -— — for single-riveting ; add 20$ for double-riveting ; 

D X o 
where 7 is the lowest T. S. stamped on any plate. 

Mr. Foley criticises the rule of the United States Statutes as follows : The 
rule ignores the riveting, except that it distinguishes between single and 
double, giving the latter 20$ advantage; the circumferential riveting or 
class of seam is altogether ignored. The rule takes no account of workman- 
ship or method adopted of constructing the joints. The factor, one sixth, 
simply covers the actual nominal factor of safety as well as the loss of 
strength at the joint, no matter what its percentage ; we may therefore 
dismiss it as unsatisfactory. 

C(t + 1 )2 

Rules for Flat Plates.— Board of Trade ; P= ^ _ ^ . 

P — working- pressure in lbs. per square inch; 
S = surface supported in square inches; 
t = thickness in sixteenths of an inch; 
C = a constant as per following table: 
C = 125 for plates not exposed to heat or flame, the stays fitted with nuts 
and washers, the latter at least three times the diameter of the stay 
and % the thickness of the plate; 
C = 187.5 for the same condition, but the washers % the pitch of stays in 

diameter, and thickness not less than plate; 
C — 200 for the same condition, but doubling plates in place of washers, the 

width of which is % the pitch and thickness the same as the plate; 
C = 112.5 for the same condition, but the stays with nuts only; 
C = 75 when exposed to impact of heat or flame and steam in contact with 
the plates, and the stays fitted with nuts and washers three times the 
diameter of the slay and % the plate's thickness; 



702 THE STEAM-BOILER. 

C — 67.5 for the same condition, but stays fitted with nuts only; 

C — 100 when exposed to heat or flame, and water in contact with the plates, 

and stays screwed into the plates and fitted with nuts; 
C = 66 for the same condition, but stays with riveted heads. 

C X 
U. S. Statutes.— U sing same notation as for Board of Trade. P — — > 

where p = greatest pitch in inches, P and t as above; 

C — 112 for plates 7/16" thick and under, fitted with screw stay-bolts 
and nuts, or plain bolt fitted with single nut and socket, or 
riveted head and socket; 

C — 120 for plates above 7/16", under the same conditions; 

C — 140 for flat surfaces where the stays are fitted with nuts inside 
and outside; 

C = 200 for flat surfaces under the same condition, but with the addi- 
tion of a washer riveted to the plate at least J^ plate's thick- 
ness, and of a diameter equal to 2/5 pitch. 

N.B.— Plates fitted with double angle-irons and riveted to plate, with leaf 
at leasts the thickness of plate and depth at least J4 of pitch, would be 
allowed the same pressure as determined by formula fur plate with washer 
riveted on. 

N.B.— No brace or stay-bolt used in marine boilers to have a greater pitch 
than 10^" on fire-boxes and back connections. 

Certain experiments were carried out by the Board of Trade which showed 
that the resistance to bulging does not vary as the square of the plate's 
thickness. There seems also good reason to believe that it is not inversely 
as the square of the greatest pitch. Bearing in mind, says Mr. Foley, that 
mathematicians have signally failed to give us true theoretical foundations 
for calculating the resistance of bodies subject to the simplest forms of 
stresses, we therefore cannot expect much from their assistance in the 
matter of flat plates. 

The Board of Trade rules for flat surfaces, being based on actual experi- 
ment, are especially worthy of respect; sound judgment appears also to 
have been used in framing them. 

Furnace Formulae.— Board of Trade.— Long Furnaces.— 

P = -=— ■, -, but not where L is shorter than (1 1.5* — 1), at which length 

the rule for short furnaces comes into play. 

P = working-pressure in pounds per square inch; t — thickness in inches; 
D = outside diameter in inches; L — length of furnace in feet up to 10 ft.; 
C — a constant, as per following table, for drilled holes : 

C — 99,000 for welded or butt-jointed with single straps, double- 
riveted; 
C = 88,000 for butts with single straps, single-riveted; 
C = 99,000 for butts with double straps, single-riveted. 

Provided always that the pressure so found does not exceed that given by 
the following formulas, which apply also to short furnaces : 



_ OX*/. tXl2\ . ... '•"/. 

P — o w r, ( 5 — <^~7 j. ) when with Adamson rings. 

o X D\ 67.5 X */ 
C— 8,800 for plain furnaces; 
C = 14,000 for Fox; minimum thickness 5/16", greatest %"; plain part 

not to exceed 6" in length; 
C = 13,500 for Morison; minimum thickness 5/16", greatest %''; plain 

part not to exceed 6" in length; 
C = 14,000 for Purves-Brown ; limits of thickness 7/16" and %" ; plain 

part 9" in length; 
C — 8,800 for Adamson rings; radius of flange next fire 1%" '• 

U. S. Statutes.— Long Furnaces.— Same notation. 

89 600 X * 2 

P = — '— — , but L not to exceed 8 ft. 

IXD 
N.B.— If rings of wrought iron are fitted and riveted on properly arour.d 
and to the flue in such a manner that the tensile stress on the rivets shall 



STRENGTH OF STEAM-BOILERS. 703 

not exceed 6000 lbs. per sq. in., the distance between the rings shall be taken 
as the length of the flue in the formulas. 
Short Furnaces, Plain and Patent.— P, as before, when not 8 ft. 
89.600 X < 2 
l ° ng = LXD > 

„ txo . 

P — — - — when 

C = 1-1,000 for Fox corrugations where D = mean diameter; 
C — 1-1,000 for Purves-Brown where D = diameter of flue; 
C ~ 5677 for plain flues over 16" diameter and less than 40", when 
not over 3 ft. lengths. 

Mr. Foley comments on the rules for long furnac s as follows: The Board 
of Trade general formula, where the length is a factor, has a very limited 
range indeed, viz., 10 ft. as the extreme length, and 135 thicknesses — 12", 

Cx t 2 
as the short limit. The original formula, P — j -, is that of Sir W. 

Fairbairn, and was, I believe, never intended by him to apply to short fur- 
naces. On the very face of it, it is apparent, on the other hand, that if it is 
true for moderately long furnaces, it cannot be so for very long ones. We 
are therefore driven to the conclusion that any formula which includes 
simple L as a factor must be founded on a wrong basis. 

With Mr. Traill's form of the formula, namely, substituting (L + 1) for L, 
the results appear sufficiently satisfactory for practical purposes, and in- 
deed, as far as can be judged, tally with the results obtained from experi- 
ment as nearly as could be expected. The experiments to which I refer 
were six in number, and of great variety of length to diameter; the actual 
factors of safety ranged from 4.4 to 6.2, the mean being 4.78, or practically 
5. It seems tome, therefore, that, within the limits prescribed, the Board of 
Trade formula may be accepted as suitable for our requirements. 

The United States Statutes give Fairbairn's rule pure and simple, except 
that the extreme limit of length to which it applies is fixed at 8 feet. As 
far as can be seen, no limit for the shortest length is prescribed, but the 
rules to me are by no means clear, flues and furnaces being mixed or not 
well distinguished. 

Material for Stays.— The qualities of material prescribed are as 
follows: 

Board of Trade.— The tensile strength to lie between the limits of 27 and 
32 tons per square inch, and to have an elongation of not less than 20$ in 
10". Steel stays which have been welded or worked in the fire should not 
be used. 

Lloyd's.— 26 to 30 ton steel, with elongation not less than 20% in 8". 

U. S. Statutes.— The only condition is that the reduction of area must not 
be less than 40% if the test bar is over %" diameter. 

Loads allowed on Stays.— Board of Trade.— 9000 lbs. per square 
inch is allowed on the net section, provided the tensile strength ranges from 
27 to 32 tons. Steel stays are not to be welded or worked in the fire. 

Lloyd's.— For screwed and other stays, not exceeding 1J^" diameter effec- 
tive, 8000 lbs. per square inch is allowed; for stays above 1J^", 9000 lbs. No 
stays are to be welded. 

U. S. Statutes.— Braces and stays shall not be subjected to a greater stress 
than 6000 lbs. per square inch. 

[Raukine, S. E., p. 459, says: " The iron of the stays ought not to be ex- 
posed to a greater working tension than 3000 lbs. on the square inch, in 
order to provide against their being weakened by corrosion. This amounts 
to making the factor of safety for the working pressure about 20." It is 
evident, however, that an allowance in the factor of safety for corrosion may 
reasonably be decreased with increase of diameter. W. K.] 
C X d 2 X t 

I Girders.— Board of Trade. P == — rr- -, P = working pres- 

( *y — P)ls X j-i 
sure in lbs. per sq. in.; W — width of flame-box in inches; L = length of 
girder in inches; p — pitch of bolts in inches; D = distance between girders 
from centre to centre in inches; d = depth of girder in inches; t = thick- 
ness of sum of same in inches; C — a constant = 6600 for 1 bolt, 9900 for 2 
or 3 bolts, and 11,220 for 4 bolts. 

Lloyd's.— The same formula and constants, except that C = 11,000 for 4 or 
5 bolt's, 11,550 for 6 or 7, and 11,880 for 8 or more. 

L 7 . S. Statutes.— The matter appears to he left to the designers, 



704 THE STEAM-BOILER. 

t(D- 



Tube-riates.— Board of Trade. P - 

horizontal distance between centres of tubes in inches; d — inside diameter 
of ordinary tubes; t = thickness of tube-plate in inches; W — extreme 
width of combustion-box in inches from front tube-plate to back of fire- 
box, or distance between combustion-box tube plates when the boiler is 
double-ended and the box common to both ends. 

The crushing stress on tube-plates caused by the pressure on the flame- 
box top is to be limited to 10,000 lbs. per square inch. 

Material for Tubes.— Mr. Foley proposes the following: If iron, the ,' 
quality to be such as to give at least 22 tons per square inch as the minimum 
tensile strength, with an elongation of not less than 15$ in 8". If steel, the 
elongation to be not less than 26$ in 8" for the material before being rolled 
into strips; and after tempering, the test bar to stand completely closing 
together. Provided the steel welds well, there does not seem to be any ob- 
ject in providing tensile limits. 

The ends should be annealed after manufacture, and stay-tube ends should 
be annealed before screwing. 

Holding-power of Boiler-tubes.— Experiments made in Wash- 
ington Navy Yard show that wit h 2}/ 2 in . brass tubes in no case was the holding- 
power less, roughly speaking, than 6000 lbs., while the average was upwards 
of 20,000 lbs. It was further shown that with these tubes nuts were super- 
fluous, quite as good results being obtained with tubes simply expanded into 
the tube-plate and fitted with a ferrule. When nuts were fitted it was shown 
that they drew off without injuring the threads. 

In Messrs. Yarrow's experiments on iron and steel tubes of 2" to 2%" 
diameter the first 5 tubes gave way on an average of 23,740 lbs., which would 
appear to be about % the ultimate strength of the tubes themselves. In all 
these cases the hole through the tube-plate was parallel with a sharp edge 
to it, and a ferrule was driven into the tube. 

Tests of the next 5 tubes were made under the same conditions as the first 
5, with the exception that in this case the ferrule was omitted, the tubes be- 
ing simply expanded into the plates. The mean pull required was 15,270 lbs., 
or considerably less than half the ultimate strength of the tubes. 

Effect of beading the tubes, the holes through the plate being parallel and 
ferrules omitted. The mean of the first 3, which are tubes of the same 
kind, gives 26,876 lbs. as their holding-power, under these conditions, as com- 
pared with 23,740 lbs. for the tubes fitted with ferrules only. This high 
figure is, however, mainly due to an exceptional case where'the holding- 
power is greater than the average strength of the tubes themselves. 

It is disadvantageous to cone the hole through the tube-plate unless its 
sharp edge is removed, as the results are much worse than those obtained, 
with parallel holes, the mean pull being but 16.031 lbs., the experiments be- 
ing made with tubes expanded and ferruled but not beaded over. 

In experiments on tubes expanded into tapered holes, beaded over and 
fitted with ferrules, the net result is that the holding-power is, for the size 
experimented on, about % of the tensile strength of the tube, the mean pull 
being 28,797 lbs. 

With tubes expanded into tapered holes and simply beaded over, better 
results were obtained than with ferrules; in these cases, however, the sharp 
edge of the hole was rounded off, which appears in general to have a good 
effect. 

In one particular the experiments are incomplete, as it is impossible to 
reproduce on a machine the racking the tubes get by the expansion of a 
boiler as it is heated up and cooled down again, and it is quite possible, 
therefore, that the fastening giving the best results on the testing-machine 
may not prove so efficient in practice. 

N.B.— It should be noted that the experiments were all made under the 
cold condition, so that reference should be made with caution, the circum- 
stances in practice being very different, especially when there is scale on 
the tube-plates, or when the tube-plates are thick and subject to intense 
heat. 

Iron versus Steel Boiler-tubes. (Foley.) — Mr. Blechynden 
prefers iron tubes to those of steel, but how far he would go in attributing 
the leaky-tube defect to the use of steel tubes we are not aware. It appears, 
however, that the results of his experiments would warrant him in going a 
considerable distance in this direction. The test consisted of heating and 
cooling two tubes, one of wrought iron and the other of steel. Both tubes 
were 2^4 in. in diameter and .16 in. thickness of metal. The tubes were 



Steel. 


Iron. 


55.495 in. 


55.495 in. 


0.52 " 


0.48 " 


.0000067 


.0000062 


.00? in. 


.003 in. 


.031 in. 


.004 in. 


.017 in. 


.006 in. 


.055 in. 


.013 in. 



STRENGTH OF STEAM-BOILERS. 705 

put in the same furnace, made red-hot, and then dipped in water. The 
length was gauged at a temperature of 46° F. 
This operation was twice repeated, with results as follows : 

Steel. 

Original length 55.495 in. 

Heated to 186° F. ; increase 

Coefficient of expansion per degree F 

Heated red-hot and dipped in water; decrease 

Second heating and cooling, decrease 

Third heating and cooling, decrease 

Total contraction 

Mr. A. C. Kirk writes : That overheating of tube ends is the cause of the 
leakage of the tubes in boilers is proved by the fact that the ferrules at 
present used by the Admiralty prevent it. These act by shielding the tube 
ends from the action of the flame, and consequently reducing evaporation, 
and so allowing free access of the water to keep them cool. 

Although many causes contribute, there seems no doubt that thick tube- 
plates must bear a share of causing the mischief. 

Rules for Construction of Boilers in Merchant Vessels 
in tne United States. 

(Extracts from General Rules and Regulations of the Board of Supervising 
Inspectors of Steam-vessels (as amended 1893 and 1894).) 
Tensile Strength of Plate. (Section 3.)— To ascertain the tensile 
strength and other qualities of iron plate there shall be taken from each 
sheet to be used in shell or other 
parts of boiler which are subject to 
tensile strain a test piece prepared 
in form according to the following 
diagram, viz.: 10 inches in length, 2 
inches in width, cut out in the 
centre in the manner indicated. 
To ascertain the tensile strength 
and other qualities of steel plate, there small be taken from each sheet to be 
used in shell or other parts of boiler which are subject to tensile strain, a test- 
piece prepared in form according 

to the following diagram, the length "T~| fs. /[ 

of straight part in centre varying as 
called for by different thickness of 
material, as follows: 

The straight portion shall be in 
length at least eight times the width multiplied by the thickness of said pa' t, 
and have a reduction of area as called for by the present rules of the Boaio.. 
and an elongation of at least 2".#. The stra'ght part shall be of a width of 1 
inch. This rule to take effect on and after July 1, 1894. 

Provided, that where contracts for boilers for ocean-going steamers re- 
quire a test of material in compliance with the British Board of Trade, 
British Lloyd's, or Bureau Veritas rules for testing, the inspectors shall 
make the tests in compliance with the following rules: 

Steel plates shall in all cases to have an ultimate elongation not less than 20# 
in a length of 8 inches. It is to be capable of being bent to a curve of which 
the inner radius is not greater than one and a half times the thickness of 
the plates after having been heated uniformly to a low cherry-red, and 
quenched in water of 82° F. 

[Prior to 1894 the shape of test-piece for steel was the same as that for iron, 
viz., the grooved shape. This shape has been condemned by authorities on 
strength of materials for over twenty years. It always gives results which 
are too high, the error sometimes amounting to 25 per cent. See pages 242, 
243, ante; also. Strength of Materials, W. Kent. Van N. Science Series No. 41, 
and Beardslee on Wrought-iron and Chain Cables.] 

Ductility. (Section 6.)— To ascertain the ductility and other lawful 
qualities, iron of 45.000 lbs. tensile strength shall show a contraction of area 
of 15 per cent, and each additional 1000 lbs. tensile strength shall show 1 
per cent additional contraction of area, up to and including 55,000 tensile 
strength. Iron of 55,000 tensile strength and upwards, showing 25 per cent 
reduction of. area, shall be deemed to have the lawful ductility. All steel 
plate of % inch thickness and under shall show a contraction of area of not 
less than 50 per cent. Steel plate over y % inch in thickness, up to % inch in 




-6-inches- "-*T??4*j ° 'rp r^rJP* 1 6-iirches— 



jy xl 



706 



THE STEAM-BOILER. 



thickness, shall show a reduction of not less than 45 per cent. All steel plate 
over % inch thickness shall show a reduction of not less than 40 per cent. 

Bumped Heads of Boilers. (Section 17 as amended 1894.) — 
Pressure Allowed on Bumped Heads.— Multiply the thickness of the plate 
by one sixth of the tensile strength, and divide by six tenths of the radius to 
which head is bumped, which will give the pressure per square inch of 
steam allowed. 

Pressure Allowable for Concaved Heads of Boilers.— Multiply the pressure 
per square inch allowable for bumped heads attached to boilers or drums 
convexly, by the constant .6, and the product will give the pressure per 
square inch allowable in concaved heads. 

Tlie pressure on unstayed flat-heads on steam-drums or shells 
of boilers, when flanged and made of wrought iron or steel or of cast steel, 
shall be determined by the following rule: 

The thickness of plate in inches multiplied by one sixth of its tensile 
strength in pounds, which product divided by the area of the head in square 
inches multiplied by .09 will give pressure per square inch allowed. The 
material used in the construction of flat-heads when tensile strength has 
not been officially determined shall be deemed to have a tensile strength of 
45,000 lbs. 

Table of Pressures allowable on Steam-boilers made of 
Riveted Iron or Steel Plates. 

(Abstract from a table published in Rules and Regulations of the U. S, 
Board of Supervising Inspectors of Steam-vessels.) 
Plates 14 i ncn thick. For other thicknesses, multiply by the ratio of the 
thickness to J4 inch. 





50,000 Tensile 


55,000 Tensile 


60,000 Tensile 


65,000 Tensile 


70,000 Tensile 


O .£ 


Strength. 


Strength. 


Strength. 


Strength. 


Strength. 


|.s 


£ 


■ "^ 


s 


, "^ 


s 


a 


ai 


. "3 


33 


■ "3 




3 


•a 


3 


ts 9 




•ag 


3 


^•g 


3 


-ig 


■So 

on 






GO 






I'' 3 




<§ 

©^ 




«j.2 


36 


115.74 


138.88 


127.31 


152.77 


138.88 


166.65 


150.46 


180.55 


162.03 


191.43 


38 


109.64 


131.56 


120.61 


144.73 


131.57 


157.88 


142.54 


171.04 


153.5 


184.20 


40 


104.16 


124.99 


114.58 


137.49 


125 


150 


135.41 


162.49 


145.83 


174.99 


42 


99.2 


119.04 


109.12 


130.94 


119.04 


142 81 


128.96 


154.75 


138.88 


166.65 


44 


94.69 


113.62 


104.16 


124.99 


113.63 


136.35 


123.1 


147.72 


132.56 


159.07 


46 


90.57 


108.68 


99.63 


119.55 


108.69 


130.42 


117.75 


141.3 


126.8 


153.16 


48 


86.8 


104.16 


95.48 


114.57 


104.16 


124.99 


112.84 


135.4 


121.52 


145.82 


54 


77.16 


92.59 


84.87 


101.84 


92.59 


111.10 


100.3 


120.36 


108.02 


129.62 


60 


69.44 


83.32 


76.38 


91.65 


83.33 


99.99 


90.27 


108.32 


97.22 


116.66 


66 


63.13 


75.75 


69.44 


83.32 


75.75 


90.90 


8.2.07 


98.48 


88.37 


106.04 


72 


57.87 


69.44 


63.65 


76.38 


69 44 


83.32 


75.22 


90.26 


81.01 


97.21 


78 


53.41 


64.09 


58.76 


70.5 


64.4 


76.92 


69.44 


83.32 


74.78 


89.73 


84 


49.6 


59.52 


54.56 


65.47 


59.52 


71.42 


64.48 


77.37 


69.44 


83.32 


90 


46.29 


55.44 


50.92 


61.1 


55.55 


66.66 


60.18 


72.21 


64.81 


77.77 


96 


43.4 


52.08 


47.74 


57.28 


52.08 


62.49 


56.42 


67.67 


60.76 


72.91 



The figures under the columns headed "pressure" are for single-riveted 
boilers. Those under the columns headed " 20$ Additional 1 ' are for double- 
riveted. 

U. S. Rule for Allowable Pressures. 

The pressure of any dimension of boilers not found in the table annexed 
to these rules must be ascertained by the following rule: 

Multiply one sixth of the lowest tensile strength found stamped on any 
plate in the cylindrical shell by the thickness (expressed in inches or parts 
of an inch) of the thinnest plate in the same cylindrical shell, and divide by 
the radius or half diameter (also expressed in inches), and the sum will be 
the pressure allowable per square inch of surface for single-riveting, to 
which add twenty per centum for double-riveting. 

The author desires to express his condemnation of the above rule, and of 
the tables derived from it, as giving too low a factor of safety. (See also 
criticism by Mr. Foley, page 701, ante.) 



STRENGTH OF STEAM-BOILERS. 



70? 



If Pb = bursting-pressure, t = thickness, T — tensile strength, c = coef- 
ficient of strength of riveted joint, that is, ratio of strength of the joint to 

that of the solid plate, d = diameter, Pb = —p, or if c be taken for double- 

1 4tT 
riveting at 0.7, then Pb — - L -r--. 

■\/QtT 4tT 

By the U. S. rule the allowable pressure Pa = -jy-r X 1.20 = - 1 — - ; whence 

Pb = Z.bPa\ that is, the factor of safety is only 3.5, provided the "tensile 
strength found stamped in the plate 11 is the real tensile strength of the 
material. But in the case of iron plates, since the stamped T.S. is obtained 
from a grooved specimen, it may be greatly in excess of the real T.S., which 
would make the factor of safety still lower. According to the table, a boiler 
40 in. diam., % in - thick, made of iron stamped 60,000 T.S., would be licensed 
to carry 150 lbs. pressure if double-riveted. If the real T.S. is only 50,000 lbs. 
the calculated bursting-strength would be 

p= 2JTC = L xa»mxMXM = 43M lbs _ 

and the factor of safety only 437.5 -=- 150 = 2.91 ! 
The author's formula for safe working-pressure of extern ally -fired boilers 

with longitudinal seams double-riveted, is P-— - — ; t = -—-;P = gauge- 
pressure in lbs. per sq. in. ; t = thickness and d — diam. in inches. 
This is derived from the formula P = -j— , taking c at 0.7 and / = 5 for 

steel of 50,000 lbs. T.S., or 6 for 60,000 lbs. T.S.; the factor of safety being 
increased* in the ratio of the T.S., since with the higher T.S. there is greater 
danger of cracking at the rivet-holes from the effect of punching and rivet- 
ing and of expansion and contraction caused by variations of temperature. 
For external shells of internally -fired boilers, these shells not being exposed 
to the fire, with rivet-holes drilled or reamed after punching, a lower factor 
of safety and steel of a higher T.S. may be allowable. 

If the T.S. is 60,000, a working pressure P = would give a factor of 

safety of 5.25. 

The following table gives safe working pressures for different diameters 
of shell and thicknesses of plate calculated from the author's formula. 

Safe Working Pressures in Cylindrical Shells of Boilers, 
Tanks, Pipes, etc., in Pounds per Square Inch. 

Longitudinal seams double'-riveted. 
(Calculated from formula P = 14,000 x thickness -*- diameter.) 













Diameter in Inches 










50 1 — 1 


24 

36.5 


30 

29.2 


36 


38 


40 


42 


44 


46 


48 


50 


52 


l 


24.3 


23.0 


21.9 


20.8 


19.9 


19.0 


18.2 


17.5 


16 8 


2 


72.9 


58.3 


48.6 


46.1 


43.8 


41.7 


39.8 


38.0 


36.5 


25.0 


33.7 


3 


109.4 


87.5 


72.9 


69.1 


65.6 


62.5 


59.7 


57.1 


54.7 


52.5 


50.5 


4 


145.8 


116.7 


97.2 


92.1 


87.5 


83.3 


79.5 


76.1 


72.9 


70.0 


67.3 


5 


182.3 


145.8 


121.5 


115.1 


109.4 


104.2 


99.4 


95.1 


91.1 


87.5 


84.1 


6 


218.7 


175.0 


145.8 


138.2 


131.3 


125.0 


119.3 


114.1 


109.4 


105.0 


101.0 


7 


255.2 


204.1 


170.1 


161.2 


153.1 


145.9 


139.2 


133.2 


127.6 


122.5 


117.8 


8 


291.7 


233.3 


194.4 


184.2 


175.0 


166.7 


159.1 


152.2 


145.8 


140.0 


134.6 


9 


328.1 


262.5 


218.8 


207.2 


196.9 


187.5 


179.0 


171.2 


164.1 


157.5 


151.4 


10 


364.6 


291.7 


243.1 


230.3 


218.8 


208.3 


198.9 


190.2 


182.3 


175.0 


168.3 


11 


401.0 


320.8 


267.4 


253.3 


240.6 


229.2 


218.7 


209.2 


200.5 


192.5 


185.1 


12 


437.5 


350.0 


291.7 


276.3 


262.5 


250.0 


238.6 


228.3 


218.7 


210.0 


201.9 


13 


473.9 


379.2 


316.0 


299.3 


284.4 


270.9 


258.5 


247.3 


337.0 


227.5 


218.8 


14 


410.4 


408.3 


340.3 


322.4 


306.3 


291.7 


278.4 


266.3 


255.2 


245.0 


235.6 


15 


546.9 


437.5 


364.6 


345.4 


328.1 


312.5 


298 3 


285.3 


273.4 


266.5 


252.4 


16 


583.3 


466.7 


388.9 


368.4 


350.0 


333.3 


318.2 


304.4 


291.7 


280.0 


269.2 



708 



THE STEAM-BOILER. 





Diameter in Inches. 


Eh.2 e8 


54 


60 


66 


72 


78 


84 
10.4 


90 


96 


102 


108 


114 

7 I' 


120 


1 


16.2 


14.6 


13.3 


12.2 


11.2 


9.7 


9.1 


8.6 


8 1 


7 3 


2 


32.4 


29.2 


26.5 


24.3 


22.4 


20.8 


19.4 


18.2 


17.2 


16.2 


15 4 


14 fi 


3 


48.6 


43.7 


39.8 


36.5 


33.7 


31 .3 


29.2 


27 3 


25 7 


24 3 


23 


21.9 


4 


64.8 


58.3 


53.0 


48.6 


44.9 


41.7 


38.9 


36 5 


34 3 


32.4 


3(1 7 


29.2 


5 


81.0 


72.9 


66.3 


60.8 


56.1 


52.1 


48.6 


45.6 


42 9 


40 5 


aft 4 


36.5 


6 


97.2 


87.5 


79.5 


72.9 


67.3 


62.5 


58.3 


54.7 


51 5 


48 6 


46 1 


43.8 


7 


113.4 


102.1 


92.8 


85.1 


78.5 


72.9 


C8.1 


63.8 


60.0 


56 7 


S3 , 7 


51.0 


8 


129.6 


116.7 


106.1 


97.2 


89.7 


83.3 


77.8 


72.9 


68 6 


64 8 


61.4 


58.3 


9 


145.8 


131.2 


119.3 


109.4 


101.0 


93.8 


87.5 


82.0 


77 2 


72.9 


69.1 


65.6 


10 


162.0 


145.8 


132.6 


121.5 


112.2 


104.2 


97.2 


91 1 


85 8 


81,0 


76 8 


72.9 


11 


178.2 


160.4 


145.8 


133.7 


123.4 


114.6 


106 9 


100 3 


94.4 


89 1 


84.4 


80.2 


12 


194.4 


175.0 


159.1 


145.8 


134.6 


125.0 


116.7 


109 4 


102.9 


97 2 


92 1 


87 5 


13 


210.7 


189.6 


172.4 


158.0 


145.8 


135.4 


126.4 


118 5 


111 5 


105 3 


99 8 


94.8 


14 


226. 9 


204.2 


185.6 


170.1 


157.1 


145.8 


136.1 


127 6 


120 1 


113 4 


107 5 


102.1 


15 


243.1 


218.7 


198.9 


182.3 


168.3 


156.3 


145.8 


136.7 


128 7 


121 5 


115.1 


109 4 


16 


259.3 


233.3 


212.1 


194.4 


179.5 


166.7 


155.6 


145.8 137.3 


129.6 


122.8 


116.7 



Rules governing Inspection of Boilers in Philadelphia. 

In estimating the strength of the longitudinal seams in the cylindrical 
shells of boilers the inspector shall apply two formulae, A and B : 

j Pitch of rivets — diameter of holes punched to receive the rivets 
' ' pitch of rivets 

percentage of strength of the sheet at the seam. 



B, 



( Area of hole filled by rivet x No. of rows of rivets in seam x shear- 
< ipg strength of rivet 



pitch of rivets X thickness of sheet x tensile strength of sheet 

percentage of strength of the rivets in the seam. 

Take the lowest of the percentages as found by formulae A and B and 
apply that percentage as the " strength of the seam " in the following 
formula C, which determines the strength of the longitudinal seams: 

( Thickness of sheet in parts of inch X strength of seam as obtained 
p -< by formula A or B X ultimate strength of iron stamped on plates _ 
' internal radius of boiler in inches X 5 as a factor of safety 

safe working pressure. 

Table of Proportions and Safe Working Pressures with Formula A 

AND C, @ 50,000 LBS., T.S. 



Diameter of rivet 

Diameter of rivet-hole. 

Pitch of rivets 

Strength of seam, %.. .. 
Thickness of plate. . . . 



11/16" 
2" 
.656 
Y4," 



11/16 
2 1/16 



13/16 



2 3/16 
.60 
7/16 



15/16 

.58 



Diameter of boiler, in. 



Safe Working Pressure with Longitudinal £ 
Single-riveted. 



24 


137 


165 


193 


220 


242 


30 


109 


132 


154 


176 


194 


32 


102 


124 


144 


165 


182 


34 


96 


117 


136 


155 


171 


36 


91 


110 


129 


147 


161 


38 


86 


104 


122 


139 


153 


40 


82 


99 


116 


132 


145 


44 


74 


91 


105 


120 


132 


48 


68 


83 


96 


110 


121 


54 


60 


73 


86 


98 


107 


60 


55 


66 


77 


88 


97 



STRENGTH OF STEAM-BOILERS. 



709 



Diameter of rivet 


%" 


11/16 


% 


13/16 


% 


Diameter of rivet-hole. . . 


11/16" 


u 


13/16 


% 


15/16 




3" 


.76 


.75 


m 

.74 


3^o 
.73 


Strength of seam, % 




Thickness of plate 


ji" 


5/16 


% 


7/16 


X 




Safe Working Pressure with Longitudinal Seams, 






Double- riveted. 




24 


160 


198 


235 


269 


305 


30 


127 


158 


188 


215 


243 


32 


119 


148 


176 


202 


228 


34 


112 


140 


166 


190 


215 


36 


106 


132 


156 


179 


203 . 


38 


101 


125 


148 


170 


192 


40 


96 


119 


141 


161 


183 


44 


87 


108 


128 


147 


166 


48 


79 


99 


118 


135 


152 


54 


70 


88 


104 


120 


135 


60 


64 


79 


94 


108 


122 



Flues and. Tubes for Steam-boilers.— (From Rules of U. S- 
Supervising Inspectors. Steam-pressures per square inch allowable on 
riveted and lap-welded flues made in sections. Extract from table in Rules 
of U. S. Supervising Inspectors.) 

T = least thickness of material allowable, D = greatest diameter in inches, 
P — allowable pressure. For thickness greater than T with same diameter 
P is increased in the ratio of the thickness. 

D=in. 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 

T=h\. .18 .20.21 .21 .22 .22 .23 .24 .25 .26 .27.28.29 30.31 .32 .33 

P = lbs. 189 184 179 174 172 158 152 147 143 139 136 134 131129 126 125 122 

D = in. 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 

T = in. .34 .35 .36 .37 .38 .39 .40 .41 .42 .43 .44 .45 .46 .47 .48 .49 .50 

P= lbs. 121 120 119 117 116 115 115 114 112 112 110 110 109 109 108 108 107 

For diameters not over 10 inches the greatest length of section allowable 
is 5 feet; for diameters 10 to 23 inches, 3 feet; for diameters 23 to 40 inches, 30 
inches. If lengths of sections are greater than these lengths, the allowable 
pressure is reduced proportionately. 

TheU. S. rule for corrugated flues, as amended in 1894, is as follows: Rule 
II, Section 14. The strength of all corrugated flues, when used for furnaces 
or steam chimneys (corrugation not less than \% inches deep and notexceed- 
ing 8 inches from centres of corrugation), and provided that the plain parts 
at the ends do not exceed 6 inches in length, and the plates are not less than 
5/16 inch thick, when new, corrugated, and practically true circles, to be 
calculated from the following formula: 



14,000 



XT— pressure. 



T — thickness, in inches; D — mean diameter in inches. 

Ribbed Flues. — The same formula is given for ribbed flues, with rib 
projections not less than 1% inches deep and not more than 9 inches 
apart. 

Flat Stayed Surfaces in Steam-boilers.— Rule II., Section 6, of 
the rules of the U. S. Supervising Inspectors provides as follows: 

No braces or stays hereafter employed in the construction of boilers 
shall be allowed a greater strain than 6000 lbs. per square inch of 
section. 

Clark, in his treatise on the Steam-engine, also in his Pocket-book, gives 
the following formula: p = 40?fo -=- d, in which p is the internal pressure in 
pounds per square inch that will strain the plates to their elastic limit, t is 
the thickness of the plate in inches, d is the distance between two rows of 
stay-bolts in the clear, aud s is the tensile stress in the plate in tons of 
2240 lbs. per square inch, at the elastic limit. Substituting values of s 
for iron, steel, and copper, 12, 14, and 8 tons respectively, we have the 
following : 



710 



THE STEAM-BOILER. 



FORMULAE FOR ULTIMATE ELASTIC STRENGTH OF FLAT STAYED SURFACES. 





Iron. 


Steel. 


Copper. 




p = 5000-, 
d 

. pXd 

5000 

5000* 

~ p 


p= 5700 j 

. p X d 
1 ~ 5700 
d _ 5700 t 
P 


p = 3300-* 

PXd 

~ 3300 

3300f 


Thickness of plate 

Pitch of bolts 




P 



For Diameter of the Stay-bolts, Clark gives d' = .0024i, 



' PP'p 



in which d' = diameter of screwed bolt at bottom of thread, P = longitudi- 
nal and P' transverse pitch of stay-bolts between centres, p = internal 
pressure in lbs. per sq. in. that will strain the plate to its elastic limit, s = 
elastic strength of the stay-bolts in lbs. per sq. j n . Taking s = 12, 14, and 8 
tons, respectively for iron, steel, and copper, we have 

For iron, d' = .00069 ^PP'^'or if P = P, d' = .00069P Vp; 

For steel, d' = .00064 VPP'p , " " d' = .00064 P Vp; 

For copper, d' = .00084 VPP'p, " " d' = .00084P Vp. 

In using these formulas a large factor of safety should be taken to allow 

for i-eduction of size by corrosion. Thurston's Manual of Steam-boilers, p. 

144, recommends that the factor be as large as 15 or 20. The Hartford 

Steam Boiler Insp. & Ins. Co. recommends not less than 10. 

Strength of Stays.— A. F. Yarrow (Engr., March 20, 1891) gives the 
following results of experiments to ascertain the strength of water-space 
stays : 



Description. 



Length 
between 
Plates. 



Diameter of Stay over 
Threads. 



Ulti- 
mate 

Stress. 



Hollow stays screwed into j 
plates and hole expanded ( 

Solid stays screwed into j 
plates and riveted over. ) 



1 in. (hole 7/16 in. and 5/16 in. 
lin.(hole 9/16 in. and 7/16 in. 



lbs. 

25,457 
20,992 
22,008 
22,070 



The above are taken as a fair average of numerous tests. 

Stay-bolts in Curved Surfaces, as In Water-legs of Verti- 
cal Boilers.— The rules of the U. S. Supervising Inspectors provide as 
follows: All vertical boiler-furnaces constructed of wrought iron or steel 
plates, and having a diameter of over 42 in. or a height of over 40 in. shall be 
stayed with bolts as provided by § 6 of Rule II, for flat surfaces; and the 
thickness of material required for the shells of such furnaces shall be de- 
termined by the distance between the centres of the stay-bolts in the fur- 
nace and not in the shell of the boiler; and the steam-pressure allowable 
shall be determined by the distance from centre of stay-bolts in the furnace 
and the diameter of such stay-bolts at the bottom of the thread. 

The Hartford Steam-boiler Insp. & Ins. Co. approves the above rule {The 
Locomotive, March, 1892) as far as it states that curved surfaces are to be 
computed the same as flat ones, but prefers Clark's formulae for flat 
stayed surfaces to the rules of the U. S. Supervising Inspectors. 

Fusible-plugs.— Fusible-plugs should be put in that portion of the 
heating-surface which first becomes exposed from lack of water. The rules 
of the U. S. Supervising Inspectors specify Banca tin for the purpose. Its 
melting-point is about 445° F. The rule says: All steamers shall have 
inserted in their boilers plugs of Banca tin, at least \i> in. in diameter at the 
smallest end of the internal opening, in the following manner, to wit: 
Cylinder- boilers with flues shall have one plug inserted in one flue of each 
boiler; and also one plug inserted in the shell of each boiler from the inside, 
immediately before the fire line and not less than 4 ft. from the forward 
end of the boiler. All fire-box boilers shall have one plug inserted in the 
crown of the back connection, or in the highest fire-surface of the boiler. 



IMPROVED METHODS OF FEEDING COAL. 711 

All upright tubular boilers used for marine purposes shall have a fusible 
plug inserted in one of the tubes at a point at least 2 in. below the lower 
gauge-cock, and said plug may be placed in the upper head sheet when 
deemed advisable by theHocal inspectors. 

Steam-domes.— Steam domes or drums were formerly almost univer- 
sally used on horizontal boilers, but their use is now generally discontinued, 
as they are considered a useless appendage to a steam-boiler, and unless 
properly designed and constructed are an element of weakness. 

Height of Furnace.— Recent practice in the United States makes 
the height of furnace much greater than it was formerly. With large sizes 
of anthracite there is no serious objection to having the furnace as low as 12 
to 18 in., measured from the surface of the grate to the nearest portion of 
the heating surface of the boiler, but with coal containing much volatile 
matter and moisture a much greater distance is desirable. With very vola- 
tile coals the distance may be as great as 4 or 5 ft. Rankine (S. E., p. 457) 
says: The clear height of the " crown " or roof of the furnace above the grate- 
bars is seldom less than about 18 in., and often considerably more. In the 
fire-boxes of locomotives it is on an average about 4 ft. The height of 18 in. 
is suitable where the crown of the furnace is a brick arch. Where the crown 
of the furnace, on the other hand, forms part of the heating-surface of the- 
boiler, a greater height is desirable in every case in which it can be 
obtained; for the temperature of the boiler-plates, being much lower than 
that of the flame, tends to check the combustion of the inflammable gases 
which rise from the fuel. Asa general principle a high furnace is favorable 
to complete combustion. 

IMPROVED METHODS OF FEEDING COAIi, 

Mechanical Stokers. (William R. Roney, Trans. A. S. M. E., vol. 
xii.)— Mechanical stokers have been used in England to a limited extent 
since 1785. In that year one was patented by James Watt. It was a simple 
device to push the coal, after it was coked at the front end of the grate, 
back towards the bridge. It was worked intermittently by levers, and was 
designed primarily to prevent smoke from bituminous coal. (See D. K. 
Clark's Treatise on the Steam-engine.) 

After the year 1840 many styles of mechanical stokers w r ere patented in 
England, but nearly all were variations and modifications of the two forms 
of stokers patented by John Jukes in 1841, and by E. Henderson in 1843. 

The Jukes stoker consisted of longitudinal fire-bars, connected by links, 
so as to form an endless chain, similar to the familiar treadmill horse-power. 
The small coal was delivered from a hopper on the front of the boiler, on to 
the grate, which slowly moving from front to rear, gradually advanced the 
fuel into the furnace and discharged the ash and clinker at the back. 

The Henderson stoker consists primarily of two horizontal fans revolving 
on vertical spindles, which scatter the coal over the fire. 

Numerous faults in mechanical construction and in operation have limited 
the use of these and other mechanical stokers. The first American stoker 
was the Murphy stoker, brought out in 1878. It consists of tw r o coal maga- 
zines placed in the side walls of the boiler furnace, and extending back from 
the boiler front 6 or 7 feet. In the bottom of these magazines are rectangu- 
lar iron boxes, which are moved from side to side by means of a rack and 
pinion, and serve to push the coal upon the grates, which incline at an angle 
of about 35° from the inner edge of the coal magazines, forming a V-shaped 
receptacle for the burning coal. The grates are composed of narrow parallel 
bars, so arranged that each alternate bar lifts about an inch at the lower 
end, while at the bottom of the V, and filling the space between the ends of 
the grate-bars, is placed a cast-iron toothed bar, arranged to be turned by a 
crank. The purpose of this bar is to grind the clinker coming in contact 
with it. Over this V-shaped receptacle is sprung a fire-brick arch. 

In the Roney mechanical stoker the fuel to be burned is dumped into a 
hopper on the boiler front. Set in the lower part of the hopper is a "pusher" 
to which is attached the u feed-plate " forming the bottom of the hopper. 
The " pusher, 1 ' by a vibratory motion, carrying with it the "feed-plate," 
gradually forces the fuel over the " dead-plate " and on the grate. The 
grate-bars, in their normal condition form a series of steps, to the top step 
of which coal is fed from the " dead-plate." Each bar rests in a concave 
seat in the bearer, and is capable of a rocking motion through an adjustable 
angle. All the grate-bars are coupled together by a "rocker- bar." A vari- 
able bagk-and-forth motion being given to the «' rocker-bar," through a corj- 



712 ! THE STEAM-BOILEK. 

\ 

necting-rod, the grate-bars rock in unison, now forming a series of steps, 
and now approximating to an inclined plane, with the grates partly over- 
lapping, like shingles on a roof. When the grate-bars rock forward the fire 
will tend to work down in a body. But before the coal can move too far 
the bars rock back to the stepped position, checking the downward motion, 
breaking up the cake over the whole surface, and admitting a free volume 
of air through the fire. The rocking motion is slow, being from 7 to 10, 
strokes per minute, according to the kind of coal. This alternate starting j 
and checking motion is continuous, and finally lands the cinder and ash on ' 
the dumping-grate below. 

Mr. Roney gives the following record of six tests to determine the com- 
parative economy of the Roney mechanical stoker and hand-firing on return 
tubular boilers. 60 inches X 20 feet, burning Cumberland coal with natural 
draught. Rating of boiler at 12.5 square feet, 105 H. P. 

Three tests, hand-firing. Three tests, Stoker. 

E 3Sora^ r aS" C !bs r ^ >«•« >«■« »■«» "•» *» !»« 

H.P. developed above rating, % 5.8 13.5 68 54.6 66.7 84.3 

Results of comparative tests like the above should be used with caution 
in drawing generalizations. It by no means follows from these results that 
a stoker will always show such comparative excellence, for in this case the 
results of hand-firing are much below what may be obtained under favor- 
able circumstances from hand-firing with good Cumberland coal. 

Tlie Mawley Down-draught Furnace.— A foot or more above 
the ordinary grate there is carried a second grate composed of a series of 
water-tubes, opening at both ends into steel drums or headers, through which 
water is circulated. The coal is fed on this second grate, and as it is par- 
tially consumed falls through it upon the lower grate, where the combustion 
is completed in the ordinary manner. The draught through the coal on the 
upper grate is downward through the coal and the grate. The volatile gases 
are therefore carried down through the bed of coal, where they are thor- 
oughly heated, and are burned in the space beneath, where they meet the 
excess of hot air drawn through the fire on the lower grate. In tests in 
Chicago, from 30 to 45 lbs. of coal were burned per square foot of grate upon 
this system, with good economical results. (See catalogue of the Hawley 
Down Draught Furnace Co., Chicago, 1894.) 

Under-feed. Stokers.— Results similar to those that may be obtained 
with downward draught are obtained by feeding the coal at the bottom of the 
bed. pushing upward the coal already on the bed which has had its volatile 
matter distilled from it. The volatile matter of the freshly fired coal then 
has to pass through a body of ignited coke. (See circular of the Jones Un- 
der-feed Stoker, Fraser & Chalmers, Chicago, 1894.) 

SMOKE PREVENTION. 

A committee of experts was appointed in St. Louis in 1891 to report on the 
smoke problem. A summary of its report is given in the Iron Age of April 
7, 1892. It describes the different means that have been tried to prevent 
smoke, such as gas-fuel, steam-jets, fire-brick arches and checker-work, 
hollow walls for preheating air, coking arches or chambers, double combus- 
tion furnaces, and automatic stokers. All of these means have been more or 
less effective in diminishing smoke, their effectiveness depending largely 
upon the skill with which they are operated ; but none is entirely satisfac- 
tory. Fuel-gas is objectionable chiefly on account of its expense. The 
average quality of fuel-gas made from a trial run of several car-loads of 
Illinois coal, in a well-designed fuel-gas plant, showed a calorific value of 
243,391 heat-units per 1000 cubic feet. This is equivalent to 5052.8 heat-units 
per lb. of coal, whereas by direct calorimeter test an average sample of the 
coal gave 11,172 heat-units. One lb. of the coal showed a theoretical evap- 
oration of 11.56 lbs. water, while the gas from 1 lb. showed a theoretical 
evaporation of 5.23 lbs. 48.17 lbs. of coal were required to furnish 1000 cubic 
feet of the gas. In 39 tests the smoke-preventing furnaces showed only 74% 
of the capacity of the common furnaces, reduced the work of the boilers 
28$, and required about 2% more fuel to do the same work. In one case with 
steam-jets the fuel consumption was increased 12% for the same work. 

Prof. O. H. Landreth, in a report to the State Board of Health of Tennes- 
see {Engineering News, June 8, 1S93), writes as follows on the subject of 
smoke prevention; 



SMOKE PREVENTION. 713 

As pertaius to steam-boilers, the object must be attained by one or more 
of the following: agencies : 

1. Proper design and setting of the boiler-plant. This implies proper grate 
area, sufficient draught, the necessary air-space between grate-oars and 
through furnace, and ample combustion-room under boilers. 

2. That system of firing that is best adapted to each particular furnace to 
secure the perfect combustion of bituminous coal. This may be either: (a) 
••coke-firing," or charging all coal into the front of the furnace until par- 
tially coked, then pushing back and spreading; or (b) "alternate side-fir- 
ing"; or (c) "spreading," by which the coal is spread over the whole grate 
area in thin, uniform layers at each charging. 

3. The admission of air through the furnace-door, bridge- wall, or side walls. 

4. Steam-jets and other artificial means for thoroughly mixingjthe air and 
combustible gases. 

5. Preventing the cooling of the furnace and boilers by the inrush of cold 
air when the furnace-doors are opened for charging coal and handling the 
fire. 

6. Establishing a gradation of the several steps of combustion so that the 
coal may be charged, dried, and warmed at the coolest part of the furnace, 
and then moved by successive steps to the hottest place, where the final 
combustion of the coked coal is completed, and compelling the distilled 
combustible gases to pass through this hottest part of the fire. 

7. Preventing the cooling by radiation of the unburned combustible gases 
until perfect mixing and combustion have been accomplished. 

8. Varying the supply of air to suit the periodic variation in demand. 

9. The substitution of a continuous uniform feeding of coal instead of 
intermittent charging. 

10. Down-draught burning or causing the air to enter above the grate and 
pass down through the coal, carrying the distilled products down to the high 
temperature plane at the bottom of the fire. 

The number of smoke-prevention devices which have been invented is 
legion. A brief classification is : 

(a) Mechanical stokers. They effect a material saving in the labor of 
firing, and are efficient smoke-preventers when not pushed above their 
capacity, and when the coal does not cake badly. They are rarely suscepti- 
ble to the sudden changes in the rate of firing frequently demanded in 
service. 

(b) Air-flues in side walls, bridge-wall, and grate-bars, through which air 
when passing is heated. The results are always beneficial, but the flues aie 
difficult to keep clean and in order. 

(c) Coking arches, or spaces in front of the furnace arched over, in which 
the fresh coal is coked, both to prevent cooling of the distilled gases, and to 
force them to pass through the hottest part of the furnace just beyond the 
arch. The results are good for normal conditions, but ineffective when the 
fires are forced. The arches also are easily burned out and injured by 
working the fire. 

(d) Dead-plates, or a portion of the grate next the furnace-doors, reserved 
for warming and coking the coal before it is spread over the grate. These 
give good results when the furnace is not forced above its normal capacity. 
This embodies the method of "coke-firing" mentioned before. 

(e) Down-draught furnaces, or furnaces in which the air is supplied to the 
coal above the grate, and the products of combustion are taken away from 
beneath the grate, thus causing a downward draught through the coal, carry- 
ing the distilled gases down to the highly heated incandescent coal at the 
bottom of the layer of coal on the grate. This is the most perfect manner 
of producing combustion, and is absolutely smokeless. 

(/) Steam- jets to draw air in or inject air into the furnace above the grate, 
and also to mix the air and the combustible gases together. A very efficient 
smoke-preventer, but one liable to be wasteful of fuel by inducing too rapid 
a draught. 

(g) Baffle-plates placed in the furnace above the fire to aid in mixing the 
combustible gases with the air. 

(h) Double furnaces, of which there are two different styles; the first of 
which places the second grate below the first grate; the coal is coked on the 
first grate, during which process the distilled gases are made to pass over 
the second grate, where they are ignited and burned ; the coke from the first 
grate is dropped onto the second grate: a very efficient and economical 
smoke-preventer, but rather complicated to construct and maintain. In the 
second form the products of combustion from the first furnace pass through 



^14 THE STEAM-BOiLER\ 

the grate and fire of the second, each fttrnace being charged with fresh fuel 
when needed, the latter generally with a smokeless coal or coke : an irra- 
tional and unpromising method. 

Mr. C. F. White, Consulting Engineer to the Chicago Society for the Pre- 
vention of Smoke, writes under date of May 4, 1893 : 

The experience had in Chicago has shown plainly that it is perfectly easy 
to equip steam-boilers with furnaces which shall burn ordinary soft coal iu 
such a manner that the making of smoke dense enough to obstruct the vision 
shall be confined to one or two intervals of perhaps a couple of minutes' 
duration in the ordinary day of 10 hours. 

Gas-fired Steam-boilers,— Converting coal into gas in a separate 
producer, before burning it under the steam-boiler, is an ideal method of 
smoke-prevention, but its expense has hitherto prevented its general intro- 
duction. A series of articles on the subject, illustrating a great number of 
devices, by F. J. Rowan, is published in the Colliery Engineer, 1889-90. See 
also Clark on the Steam-engine. 

FORCED COMBUSTION IN STEAM-BOILERS. 

For the purpose of increasing the amount of steam that can be generated 
by a boiler of a given size, forced draught is of great importance. It is 
universally used in the locomotive, the draught being obtained by a steam- 
jet in the smoke-stack. It is now largely used in ocean steamers, especially 
in ships of war, and to a small extent in stationary boilers. Economy of fuel 
is generally not attained by its use, its advantages being confined to the 
securing of increased capacity from a boiler of a given bulk, weight, or cost. 
The subject of forced draught is well treated in a paper by James Howden, 
entitled, "Forced Combustion in Steam-boilers" (Section G, Engineering 
Congress at Chicago, in 1893), from which we abstract the following: 

Edwin A. Stevens at Bordentown, N. J., in 1827, in the steamer "North 
America," fitted the boilers with closed ash-pits, into which the air of com- 
bustion was forced by a fan. In 1828 Ericsson fitted in a similar manner the 
steamer "Victory," commanded by Sir John Ross. 

Messrs. E. A. and R. L. Stevens continued the use of forced draught for 
a considerable period, during which they tried three different modes of using 
the fan for promoting combustion: 1, blowing direct into a closed ash-pit; 
2, exhausting the base of the funnel by the suction of the fan; 3, forcing air 
into an air-tight boiler-room or stoke-hold. Each of these three methods 
was attended with serious difficulties. 

In the use of the closed ash-pit the blast-pressure would frequently force 
the gases of combustion, in the shape of a serrated flame, from thie joint 
around the furnace doors in so great a quantity as to affect both the effi- 
ciency and health of the firemen. 

The chief defect of the second plan was the great size of the fan required 
to produce the necessary exhaustion. The size of fan required grows in a 
rapidly increasing ratio as the combustion increases, both on account of the 
greater air-supply and the higher exit temperature enlarging the volume of 
the waste gases. 

The third method, that of forcing cold air by the fan into an air-tight 
boiler-room— the present closed stoke-hold system— though it overcame the 
difficulties in working belonging to the two forms first tried, has serious 
defects of its own, as it cannot be worked, even with modern high-class 
boiler-construction, much, if at all, above the power of a good chimney 
draught, in most boilers, without damaging them. 

In 1875 John I. Thorn ycroft & Co., of London, began the construction of 
torpedo-boats with boilers of the locomotive type, in which a high rate of 
combustion was attained by means of the air-tight boiler-room, into which 
air was forced by -means of a fan. 

In 1882 H.B.M. ships "Satellite" and "Conqueror" were fitted with this 
system, the former being a small ship of 1500 I.H.P., and the latter an iron- 
clad of 4500 I.H.P. On the trials with forced draught, which lasted from two 
to three hours each, the highest rates of combustion gave 16.9 I.H.P. per 
square foot of fire-grate in the " Satellite," and 13.41 I.H.P. in the " Con- 
queror." 

None of the short trials at these rates of combustion were made without 
injury to the seams and tubes of the boilers, but the system was adopted, 
and it has been contiuued in the British Navy to this day (1893). 

In Mr. Howden's opinion no advantage arising from increased combustion 
over natural-draught rates is derived from using forced draught in a closed 
ash-pit sufficient to compensate the disadvantages arising from difficulties 



FUEL ECONOMIZERS. 715 

in working, there being either excessive smoke from bituminous coal or 
reduced evaporative economy. 

In 1880 Mr. Howden designed an arrangement intended to overcome the 
defects of both the closed ash-pit and closed stokehold systems. 

An air-tight reservoir or chamber is placed on the front end of the boiler 
and surrounding the furnaces. This reservoir, which projects from 8 to 10 
inches from the end of the boiler, receives the air under pressure, which is 
passed by the valves into the ash-pits and over the fires in proportions 
suited to 'the kind of fuel used and the rate of combustion required. The 
air nsed above the fires is admitted to a space between the outer and inner 
furnace-doors, the inner having perforations and an air-distributing box 
through which the air passes under pressure. 

By means of the balance of air-pressure above and below the fires all 
tendency for the fire to blow out at the furnace-door is removed. 

By regulating the admission of the air by the valves above and below the 
fires, the highest rate of combustion possible by the air-pressure used can 
be effected, and in same manner the rate of combustion can be reduced to 
far below that of natural draught, while complete and economical combus- 
tion at all rates is secured. 

A feature of the system is the combination of the heating of the air of 
combustion by the waste gases with the controlled and regulated admission 
of air to the furnaces. This arrangement is effected most conveniently by 
passing the hot fire- gases after they leave the boiler through stacks of 
vertical tubes enclosed in the uptake, their lower ends being immediately 
above the smoke-box doors. 

Installations on Howden's system have hitherto been arranged for a rate 
of combustion to give at full sea-power an average of from 18 to 22 I.H.P. 
per square foot of fire-grate with fire-bars from 5' 0" to 5' 6" in length. 

It is believed that with suitable arrangement of proportions even SO 
I.H.P. per square foot can be obtained. 

For an account of recent uses of exhaust-fans for increasing draught, see 
paper by W. R. Roney, Trans. A. S. M. E., vol. xv. 

FUEL ECONOMIZERS. 

Green's Fuel Economizer.— Clark gives the following average re- 
sults of comparative trials of three boilers at Wigan used with and without 
economizers : 

Without With 

Economizers. Economizers. 

Coal per square foot of grate per hour 21.6 21.4 

Water at 100° evaporated per hour 73 . 55 79 . 32 

Water at 212° per pound of coal 9.60 10.56 

Showing that in burning equal quantities of coal per hour the rapidity of 
evaporation is increased 9.3$ and the efficiency of evaporation 10$ by the 
addition of the economizer. 

The average temperatures of the gases and of the feed-water before and 
after passing the economizer were as follows: 

With 6-f t. grate. With 4-ft. grate. 
Before. After. Before. After. 

Average temperature of gases 649 340 501 312 

Average temperature of feed- water. 47 157 41 137 - 

Taking averages of the two grates, to raise the temperature of the feed- 
water 100° the gases were cooled down 250°. 
Performance of a Green Economizer with a Smoky Coal. 

—The action of Green's Economizer was tested by M. W. Grosseteste for a 
period of three weeks. The apparatus consists of four ranges of vertical 
pipes, Q}4 feet high, 3% inches in diameter outside, nine pipes in each range, 
connected at top and bottom by horizontal pipes. The water enters all the 
tubes from below, and leaves them from above. The system of pipes is en- 
veloped in a brick casing, into which the gaseous products of combustion 
are introduced from above, and which they leave from below. The pipes 
are cleared of soot externally by automatic scrapers. The capacity for 
water is 24 cubic feet, and the total external heating-surface is 290 square 
feet. The apparatus is placed in connection with a boiler having 355 square 
feet of surface. 

This apparatus had been at work for seven weeks continuously without 
having been cleaned, and had accumulated a J^-inch coating of soot and 



716 



THE STEAM-BOILER. 



ash, when its performance, in the same condition, was observed for one 
week. During the second week it was cleaned twice every day; but during 
the third week, after having been cleaned on Monday morning, it was 
worked continuously without further cleaning. A smoke-making coal was 
used. The consumption was maintained sensibly constant from day to day. 

Green's Economizer.— Results of Experiments on its Efficiency as 

AFFECTED BY THE STATE OF THE SURFACE. (W, GrOSSeteSte.) 





Temperature of Feed-, 
water. 


Temperature of Gas- 
eous Products. 


Time 
(February and March). 


Enter- 
ing 
Feed- 
heater. 


Leav- 
ing 
Feed- 
heater. 


Differ- 
ence. 


Enter- 
ing 
Feed- 
heater 


Leav- 
ing 
Feed- 
heater. 


Differ- 
ence. 


1st Week 

2d Week 


Fahr. 
73.5° 
77.0 
73.4 
73.4 
79.0 
80.6 
80.6 
79.0 


Fahr. 

161.5° 
230 
196.0 
181.4 
178.0 
170.6 
169.0 
172.4 


Fahr. 

88.0° 
153.0 
122.6 
108.0 
99.0 
90.0 
88.4 
93.4 


Fahr. 

849° 
882 
831 

871 

952 
889 
901 


Fahr. 

261° 
297 
284 
309 

329 
338 
351 


Fahr. 

588° 
585 


3d Week— Monday 

Tuesday 

Wednesday 

Thursday 

Friday 

Saturday 


547 
562 

623 

551 
550 



Coal consumed per hour 

Water evaporated from 32° F. per hour 
Water per pound of coal 



1st Week. 2d Week. 3d Week. 
. 214 lbs. 216 lbs. 213 lbs. 
. 1424 1525 1428 

. 6.65 7.06 6.70 

It is apparent that there is a great advantage in cleaning the pipes daily 
—the elevation of temperature having been increased by it from 88° to 153°. 
In the third week, without cleaning, the elevation of temperature relapsed 
in three days to the level of the first week; even on the first day it was 
quickly reduced by as much as half the extent of relapse. By cleaning the 
pipes daily an increased elevation of temperature of 65° F., was obtained, 
whilst a gain of 6% was effected in the evaporative efficiency. 

INCRUSTATION AND CORROSION. 

Incrustation and Scale.— Incrustation (as distinguished from 
mere sediments due to dirty water, which are easily blown out, or gathered 
up, by means of sediment-collectors) is due to the presence of salts in the 
feed-water (carbonates and sulphates of lime and magnesia for the most 
part), which are precipitated when the water is heated, and form hard de- 
posits upon the boiler-plates. (See Impurities in Water, p. 551, ante.) 

Where the quantity of these salts is not very large (12 grains per gallon, 
say) scale preventives may be found effective. The chemical preventives 
either form with the salts other salts soluble in hot water; or precipitate 
them in the form of soft mud, which does not adhere to the plates, and can 
be washed out from time to time. The selection of the chemical must de- 
pend upon the composition of the water, and it should be introduced regu- 
larly with the feed. 

Examples.— Sulphate-of -lime scale prevented by carbonate of soda: The 
sulphate of soda produced is soluble in water; and the carbonate of lime 
falls down in grains, does not adhere to the plates, and may therefore be 
blown out or gathered into sediment- collectors. The chemical reaction is: 



Sulphate of lime H 
CaSOj 



Carbonate of soda = Sulphate of soda -f- Carbonate of lime 
NA 9 COa NA 2 S0 4 CaC0 3 



Sodium phosphate will decompose the sulphates of lime and magnesia: 
Sulphate of lime -4- Sodium phosphate = Calcium phos. + Sulphate of soda. 

CaS0 4 Na 2 HP0 4 CaHP0 4 Na 2 S0 4 

Sul. of magnesia 4- Sodium phosphate = Phosphate of magnesia + Sul. of soda. 
MgS0 4 Na 2 HP0 4 MgHP0 4 Na 2 S0 4 



INCRUSTATION" AND CORROSION. 717 

Where the quantity of salts is large, scale preventives are not of much 
use. Some other source of supply must be sought, or the bad water purified 
before it is allowed to enter the boilers. The damage done to boilers by un- 
suitable water is enormous. 

Pure water may be obtained by collecting rain, or condensing steam by 
means of surface condensers. The water thus obtained should be mixed 
with a little bad water, or treated with a little alkali, as undiluted, pure 
water corrodes iron ; or, after each periodic cleaning, the bad may be used 
for a day or two to put a skin upon the plates. 

Carbonate of lime and magnesia may be precipitated either by heating the 
water or by mixing milk of lime (Porter Clark process) with it, the water 
being then filtered. 

Corrosion may be produced by the use of pure water, or by the presence 
of acids in the water, caused perhaps in the engine-cylinder by the action of 
high- pressure steam upon the grease, resulting in the production of fatty 
acids. Acid water may be neutralized by the addition of lime. 

Amount of Sediment which may collect in a 100-H.P. steam-boiler, 
evaporating 3000 lbs. of water per hour, the water containing different 
amounts of impurity in solution, provided that no water is blown off: 

Grains of solid impurities per gallon: 

5 10 20 30 40 50 60 70 80 90 100 

Equivalent parts per 100,000: 

8.57 17.14 34.28 51.42 68.56 85.71 102.85 120 137.1 154.3 171.4 
Sediment deposited in 1 hour, pounds: 

2.57 5.14 10.28 15.42 20.56 25.71 30.85 36 41.1 46.3 51.4 
In one day of 10 hours, pounds: 

25.7 51.4 102.8 154.2 205.6 257.1 308.5 360 411 463 514 
In one week of 6 days, pounds: 
154.3 3u8.5 617.0 925.5 1234 1543 1851 2160 2468 2776 3085 

If a 100-H.P. boiler has 1200 sq. ft. heating-surface, one week's running 
without blowing off, wish water containing 100 grains of solid matter per 
gallon in solution, would make a scale nearly 1/5 in. thick, if evenly depos- 
ited all over the heating-surface, assuming the scale to have a sp. gr. of 
2.5 = 156 lbs. percu. ft.; 1/5 x 1200 X 156 X 1/12 - 3120 lbs. 

fSoiler-scale Compounds.— The Bavarian Steam-boiler Inspection 
Assn. in 1885 reported as follows: 

Generally the unusual substances in water can be retained in soluble form 
or precipitated as mud by adding caustic soda or lime. This is especially 
desirable when the boilers have small interior spaces. 

It is necessary to have a chemical analysis of the water in order to fully 
determine the kind and quantity of the preparation to be used for the 
above purpose. 

All secret compounds for removing boiler-scale should be avoided. (A list 
of 27 such compounds manufactured and sold by German firms is then given 
which have been analyzed by the association.) 

Such secret preparations are either nonsensical or fraudulent, or contain 
either one of the two substances recommended by the association for re- 
moving scale, generally soda, which is colored to conceal its presence, and 
sometimes adulterated with useless or even injurious matter. 

These additions as well as giving the compound some strange, fanciful 
name, are meant simply to deceive the boiler owner and conceal from him 
the fact that he is buying colored soda or similar substances, for which he is 
paying an exorbitant price. 

The Chicago, Milwaukee & St. P. R. R. uses for the prevention of scale in 
locomotive-boilers an alkaline compound consisting of 3750 gals, of water, 
2G00 lbs. of 70£ caustic soda, and 1600 lbs. of 58$ soda-ash. Between Milwau- 
kee and Madison the water-supply contains from 1 to 4]4 lbs. of incrusting 
solids per 1000 gals., principally calcium carbonate and sulphate and mag- 
nesium sulphate. The amount of compound necessary to prevent the in- 
crustation is V& to 7 pints per 1000 gals, of water. This is really only one 
fourth of the quantity needed for chemical combination, but the action of 
the compound is regenerative. The soda-ash (sodium carbonate) extracts 
carbonic acid from the carbonates of lime and magnesia and precipitates 
them in a granular form. The bicarbonate of soda thus formed, however, 
loses its carbonic acid by the heat, and is again changed to the active car- 
bonate form. Theoretically this action might continue indefinitely; but on 



718 THE STEAM-BOILER. 

account of the loss by blowing off and the presence of other impurities in 
the water, it is found that the soda-ash will precipitate only about four 
times the theoretical quantity. Scaling is entirely prevented. One engine 
made 122,000 miles, and inspection of the boiler showed that it was as clean 
as when new. This compound precipitates the impurities in a granular 
form, and careful attention must be paid to washing out the precipitate. 
The practice is to change the water every 600 miles and wash out the boiler 
every 1200 miles, using the blow-off cocks also whenever there is any indica- 
tion of foaming, winch seems to be caused by the precipitate in the water, 
but not by the alkali itself. (Eng'g News, Dec. 5, 1891.) 

Kerosene and otner Petroleum Oils ; Foaming.— Kerosene 
has recently been highly recommended- as a scale preventive. See paper 
by L. F. Lyne (Trans. A. S. M. E., ix. 247). The Am. Mach., May 22, 1890,. 
says: Kerosene used in moderate quantities will not make the boiler foam;, 
it is recommended and used for loosening the scale and for preventing the 
formation of scale. Neither will a small quantity of common oil always 
cause foaming; it is sometimes injected into small vertical boilers to pre- 
vent priming, and is supposed to have the same effect on the disturbed sur- 
face of the water that oil has when poured on the rough sea. Yet oil in boilers 
will not have the same effect, and give the desired results in all cases. The 
presence of oil in combination with other impurities increases the tendency 
of many boilers to foam,as the oil with the impurities impedes the free escape 
of steam from the water surface. The use of common oil not only tends to 
cause foaming, but is dangerous otherwise. The grease appears to combine 
with the impurities of the water, and when the boiler is at rest this com- 
pound sinks to the plates and clings to them in a loose, spongy mass, pre- 
venting the water from coming in contact with the plates, and thereby pro- 
ducing overheating, which may lead to an explosion. Foaming may also 
be caused by forcing the fire, or by taking the steam from a point over the 
furnace or where the ebullition is violent; the greasy and dirty state of new 
boilers is another good cause for foaming. Kerosene should be used at first 
in small quantities, the effect carefully noted, and the quantity increased if 
necessary for obtaining the desired results. 

R. 0. Carpenter (Trans. A. S. M. E., vol. xi.) says: The boilers of the State 
Agricultural College at Lansing, Mich., were badly incrusted with a hard 
scale. It was fully three eighths of an inch thick in many places. The first 
application of the oil was made while the boilers were being but little used, 
by inserting a gallon of oil, filling with water, heating to the boiling-point 
and allowing the water to stand in the boiler two or three weeks before 
removal. By this method fully one half the scale was removed during the 
warm season and before the boilers were needed for heavy firing. The oil 
was then added in small quantities when the boiler was in actual use. For 
boilers 4 ft. in diam. and 12 ft. long the best results were obtained by the 
use of 2 qts. for each boiler per week, and for each boiler 5 ft. in diam. 3 qts. 
per week. The water used in the boilers has the following analysis: 

CaC0 3 (carbonate calcium) 20G parts in 1,000,000. 

Mg0O 3 (carbonate magnesium) 78 " " " 

1< 2 C0 3 (carbonate iron) 22 " " " 

Traces of sulphates and chlorides of potash and soda. 

Total solid parts, 325 to 1,000,000. 

Tannate of Soda Compound.— T. T. Parker writes to Am. Mach.: 
Should you find kerosene not doing any good, try this recipe: 50 lbs. sal-soda, 
35 lbs. japonica; put the ingredients in a 50-gal. barrel, fill half full of Avater, 
and run a steam hose into it until it dissolves and boils. Remove the hose, 
fill up with water, and allow to settle. Use one quart per day of ten hours 
for a40-H.P. boiler, and, if possible, introduce it as you do cylinder- oil to 
your engine. Barr recommends tannate of soda as a remedy for scale com- 
posed of sulphate and carbonate of lime. As the japonica yields the tannic 
acid, I think the resultant equivalent to the tannate of soda. 

Petroleum Oils heavier than kerosene have been used with good re- 
sults. Crude oil should never be used. The more volatile oils it contains 
make explosive gases, and its tarry constituents are apt to form a spongy 
incrustation. 

Removal of Hard Scale.— When boilers are coated with a hard 
scale difficult to remove the addition of J4 lb. caustic soda per horse-power, 
and steaming for some hours, according to the thickness of" the scale, just 
before cleaning, will greatly facilitate that operation, rendering the scale 



INCRUSTATIOK AKD CORROSION. 719 

soft and loose. This should be done, if possible, when the boilers are not 
otherwise in use. {Steam.) 

Corrosion in Marine Boilers. (Proc. Inst. M. E., Aug. 1884).— The 
investigations of the Committee on Boilers served to show that the internal 
corrosion of boilers is greatly due to the combined action of air and sea- 
water when under steam, and when not under steam to the combined action 
of air and moisture upon the unprotected surfaces of the metal. There are 
other deleterious influences at work, such as the corrosive action of fatty- 
acids, the galvanic action of copper and brass, and the inequalities of tem- 
perature; these latter, however, are considered to be of minor importance. 

Of the several methods recommended for protecting the internal surfaces 
of boilers, the three found most effectual are: First, the formation of a 
thin layer of hard scale, deposited by working the boiler with sea-water; 
second, the coating of the surfaces with a thin wash of Portland cement, 
partially wherever there are signs of decay; third, the use of zinc slabs 
suspended in the water and steam spaces. 

As to general treatment for the preservation of boilers in store or when 
laid up in the reserve, either of the two following methods is adopted, as 
may be found most suitable in particular cases. First, the boilers are 
dried as much as possible by airing-stoves, after which 2 to 3 cwt. of quick- 
lime, according to the size of the boiler, is placed on suitable trays at the 
bottom of the boiler and on the tubes. The boiler is then closed and made 
as air-tight as possible. Periodical inspection is made every six months, 
when if the lime be found slacked it is renewed. Second, the other 
method is to fill the boilers up with sea or fresh water, having added soda 
to it in the proportion of 1 lb. of soda to every 100 or 120 lbs. of water. The 
sufficiency of the saturation can be tested by introducing a piece of clean 
new iron and leaving it in the boiler for ten or twelve hours; if it shows 
signs of rusting, more soda should be added. It is essential that the boilers 
be entirely filled, to the complete exclusion of air. 

Great care is taken to prevent sudden changes of temperature in boilers. 
Directions are given that steam shall not be raised rapidly, and that care 
shall be taken to prevent a rush of cold air through the tubes by too sud- 
denly opening the smoke-box doors. The practice of emptying boilers by 
blowing out is also prohibited, except in cases of extreme urgency. As a 
rule the water is allowed to remain until it becomes cool before the boilers 
are emptied. 

Mineral oil has for many years been exclusively used for internal lubrica- 
tion of engines, with the view of avoiding the effects of fatty acid, as this oil 
does not readily decompose and possesses no acid properties. 

Of all the preservative methods adopted in the British service, the use of 
zinc properly distributed and fixed has been found the most effectual in 
saving the iron and steel surfaces from corrosion, and also in neutralizing 
by its own deterioration the hurtful influences met with in water as ordina- 
rily supplied to boilers. The zinc slabs now used in the navy boilers are 12 
in. long, 6 in. wide, and y% inch thick; this size being found convenient for 
general application. The amount of zinc used in new boilers at present is 
one slab of the above size for every 20 I.H.P., or about one square foot of 
zinc surface to two square feet of grate surface. Rolled zinc is found the 
most suitable for the purpose. To make the zinc properly efficient as a 
protector especial care must betaken to insure perfect metallic contact 
between the slabs and the stays or plates to which they are attached. The 
slabs should be placed in such positions that all the surfaces in the boiler 
shall be protected. Each slab should be periodically examined to see that 
its connection remains perfect, and to renew any that may have decayed ; 
this examination is usually made at intervals not exceeding three months. 
Under ordinary circumstances of working these zinc slabs may be expected 
to last in fit condition from sixty to ninety days, immersed in hot sea-water; 
but in new boilers they at first decay more rapidly. The slabs are generally 
secured by means of iron straps 2 in. wide and % inch thick, and long 
enough to reach the nearest stay, to which the strap is firmly attached by 
screw-bolts. 

To promote the proper care of boilers when not in use the following order 
has been issued to the French Navy by the Government: On board all ships 
in the reserve, as well as those which are laid up, the boilers will be com- 
pletely filled with fresh water. In the case of large boilers with large tubes 
there will be added to the water a certain amounts of milk of lime, or a 
solution of soda may be used instead. In the case of tubulous boilers with 
small tubes milk of lime or soda may be added, but the solution will not be 



no 



THE STEAM-BOILER. 



so strong as in the case of the larger tube, so as to avoid any danger of 
contracting the effective area by deposit from the solution ; but the strength 
of the solution will be just sufficient to neutralize any acidity of the water. 
{Iron Age, Nov. 2, 1893.) 

Use of Zinc— Zinc is often used in boilers to prevent the corrosive 
action of water on the metal. The action appears to be an electrical one, 
the iron being one pole of the battery and the zinc being the other. The 
hydrogen goes to the iron shell and escapes as a gas into the steam. The 
oxygen goes to the zinc. 

On account of this action it is generally believed that zinc will always 
prevent corrosion, and that it cannot be harmful to the boiler or tank. 
Some experiences go to disprove this belief, and in numerous cases zinc has 
not only been of no use, but has even been harmful. In one case a tubular 
boiler had beeu troubled with a deposit of scale consisting chiefly of or- 
ganic matter and lime, and zinc was tried as a preventive. The beneficial 
action of the zinc was so obvious that its continued use was advised, with 
frequent opening of the boiler and cleaning out of detached scale until all 
the old scale should be removed and the boiler become clean. Eight or ten 
months later the water supply was changed, it being now obtained from 
another stream supposed to be free from lime and to contain only organic 
matter. Two or three months after its introduction the tubes and shell 
were found to be coated with an obstinate adhesive scale, and composed of 
zinc oxide and the organic matter or sediment of the water used. The 
deposit had become so heavy in places as to cause overheating and bulging 
of the plates over the fire. {The Locomotive.) 

Effect of Deposit on Flues. (Rankine.) — An external crust of a 
carbonaceous kind is often deposited from the flame and smoke of the fur- 
naces in the flues and tubes, and if allowed to accumulate seriously impairs 
the economy of fuel. It is removed from time to time by means of scrapers 
and wire brushes. The accumulation of this crust is the probable cause of 
the fact that in some steamships the consumption of coal per indicated 
horse-power per hour goes on gradually increasing until it reaches one and 
a half times its original amount, and sometimes more. 

Dangerous Steam-boilers discovered by Inspection.— 
The Hartford Steam-boiler Inspection and Insurance Co. reports that its 
inspectors during 1893 examined 163,328 boilers, inspected 66,698 boilers, 
both internally and externally, subjected 7861 to hydrostatic pressure, and 
found 597 unsafe for further use. The whole number of defects reported 
was 122,893, of which 12,390 were considered dangerous. A summary is 
given below. (The Locomotive, Feb. 1894.) 



Summary, by Defects, for the Year 18! 



Nature of Defects. 



Whole Dan- 
No. geroos 



Nature of Defects. 



Whole Dan- 
No. gerous. 
Leakage around tubes. . . 21 ,211 2,90S 

Leakage at seams 5,424 

Water-gauges defective. 3,670 

Blow outs defective 1,620 

Deficiency of water 204 

Safety-valves overloaded 723 
Safety-valves defective.. 942 
Pressure-gauges def'tive 5,953 
Boilers without pressure- 
gauges 115 

Unclassified defects 755 



482 



Deposit of sediment. . 
Incrustation and scale. . .18,369 

Internal grooving 1,249 

Internal corrosion 6,252 

External corrosion 8,600 

Deftive braces and stays 1,966 

Settings defective 3,094 

Furnaces out of shape. . . 4,575 

Fractured plates 3,532 

Burned plates 2,762 

Blistered plates 3,331 

Defective rivets 17,415 

Defective heads 1,357 

The above-named company publishes annually a classified list of boiler- 
explosions, compiled chiefly from newspaper reports, showing that from 
200 to 300 explosions take place in the United States every year, killing from 
200 to 300 persons, and injuring from 300 to 450. The lists are not pretended 
to be complete, and may include only a fraction of the actual number of 
explosions. 

Steam-boilers as Magazines of Explosive Energy.— Prof. 
R. H. Thurston (Trans. A. 8. M. E., vol. vi.), in a paper with the above 
title, presents calculations showing the stored energy in the hot water and 
s'eam of various boilers. Concerning the plain tubular boiler of the 
form and dimensions adopted as a standard by the Hartford Steam-boiler 



Total. . 



.122,893 12,3 



SAFETY-VALVES. 721 

Insurance Co., he says: It is 60 inches in diameter, containing 66 3-inch 
tubes, and is 15 feet long. It has 850 feet of heating and 30 feet of grate 
surface; is rated at 60 horse-power, but isoftener driven up to 75; weighs 
9500 pounds, and contains nearly its own weight of water, but only 21 
pounds of steam when under a pressure of 75 pounds per square inch, 
which is below its safe allowance. It stores 52,000,000 foot-pounds of en- 
ergy, of which but 4 per cent is in the steam, and this is enough to drive 
the' boiler just about one mile into the air, with an initial velocity of nearly 
600 feet per second. 

SAFETY-VALVES. 

Calculation of Weight, etc., for Lever Safety-valves. 

Let W = weight of ball at end of lever, in pounds; 
to — weight of lever itself, in pounds; 
V = weight of valve and spindle, in pounds; 
L = distance between fulcrum and centre of ball, in inches; 
I — " " " " " " valve, in inches; 

g — " " " " " " gravity of lever, in in. ; 

A = area of valve, in square inches; 
,P — pressure of steam, in lbs. per sq. in., at which valve will open. 



Then PA x I = W x L + w X g + V x l; 
whence P = 



WL -f tog + VI . 
Al ' 

w ^ PAl-wg- Vl m 

T PAl - wg - VI 

L = w • 

Example.— Diameter of valve, 4"; distance from fulcrum to centre of ball, 
36"; to centre of valve, 4"; to centre of gravity of lever, \5%"\ weight of 
valve and spindle, 3 lbs.; weight of lever, 7 lbs. ; required the weight of ball 
to make the bio wing-off pressure 80 lbs. per sq. in.; area of 4" valve = 12.566 
sq. in. Then 

w _ PAl - wg - VI _ 80 X 12.566 X 4 - 7 X 15^2 - 4 X 4 _ 10g Jbg 
L 36 

The following rules governing the proportions of lever- valves are given by 
the U. S. Supervisors. The distance from the fulcrum to the valve-stem 
must in no case be less than the diameter of the valve-opening; the length 
of the lever must not be more than ten times the distance from the fulcrum 
to the valve-stem; the width of the bearings of the fulcrum must not be 
less than three quarters of an inch; the length of the fulcrum-link must not 
be less than four inches; the lever and fulcrum-link must be made of 
wrought iron or steel, and the knife-edged fulcrum points and the beatings 
for these points must be made of steel and hardened; the valve must be 
guided by its spindle, both above and below the ground seat and above the 
lever, through supports either made of composition (gun-metal) or bushed 
with it; and the spindle must fit loosely in the bearings or supports. 

Rules for Area of Safety-valves. 

(Rule of U. S. Supervising Inspectors of Steam-vessels (as amended 1891).) 

Lever safety-valves to be attached to marine boilers shall have an area of 

not less than 1 sq. in. to 2 sq. ft. of the grate surface in the boiler, and the 

seats of all such safety-valves shall have an angle of inclination of 45° to the 

centre line of their axes. 

Spring-loaded safety-valves shall be required to have an area of not less 
than 1 sq. in. to 3 sq. ft. of grate surface of the boiler, except as hereinafter 
otherwise provided for water-tube or coil and sectional boilers, and each 
spring-loaded valve shall be supplied with a lever that will raise the valve 
from its seat a distance of not less than that equal to one eighth the diam- 
eter of the valve-opening, and the seats of all such safety-valves shall have 
an angle of inclination to the centre line of their axes of 45°. All spring- 
loaded safety-valves for water-tube or coil and sectional boilers required to 



722 THE STEAM-BOILER. 

carry a steam -pressure exceeding 175 lbs. per square inch shall be required 
to have an area of not less than 1 sq. in. to 6 sq. ft. of the grate surface of 
the boiler. Nothing herein shall be construed so as to prohibit rhe use of 
two safety-valves on one water- tube or coil and sectional boiler, provided 
the combined area of such valves is equal to that required by rule for one 
such valve. 

Rule In Philadelphia Ordinances : Bureau of Steam- 
engine and Boiler Inspection.— Every boiler when fired sepa- 
rately, and every set or series of boilers when placed over one fire, shall 
have attached thereto, without the interposition of any other valve, two or 
more safety-valves, the aggregate area of which shall have such relations to 
the area of the grate and the pressure within the boiler as is expressed in 
schedule A. 

Schedule A. — Least aggregate area of safety-valve (being the least sec- 
tional area for the discharge of steam) to be placed upon all stationary boil- 
ers with natural or chimney draught [see note a]. 

a - 22 - hG 
P+8.62' 

in which A is area of combined safety-valves in inches; G is area of grate in 
square feet; P is pressure of steam in pounds per square inch to be carried 
in the boiler above the atmosphere. 

The following table gives the results of the formula for one square foot of 
grate, as applied to boilers used at different pressures: 

Pressures per square inch: 

10 20 30 40 50 60 70 80 90 </v > 110 120 

Area corresponding to one square foot of grate: 
1.21 0.79 0.58 0.46 0.38 0.33 0.29 0.25 0.23 0.21 0.19 0.17 

[Note a.] Where boilers have a forced or artificial draught, the inspector 
must estimate the area of grate at the rate of one square foot of grate-sur- 
face for each 10 lbs. of fuel burned on the average per hour. 

Comparison of Various Bules for Area of Lever Safety- 
valves. (From an article by the author in American- Machinist, May z4, 
1894, with some alterations and additions.) — Assume the case of a boiler 
rated at 100 horse-power; 40 sq. ft. grate; 1200 sq. ft. heating-surface; using 
400 lbs. of coal per hour, or 10 lbs. per sq. ft. of grate per hour, and evapora- 
ting 3600 lbs. of water, or 3 lbs. per sq. ft. of heating-surface per hour; 
steam-pressure by gauge, 100 lbs. What size of safety-valve, of the lever 
type, should be required ? 

A compilation of various rules for finding the area of the safety-vale disk, 
from The Locomotive of July, 1892, is given in abridged form below, to- 
gether with the area calculated by each rule for the above example. 

Disk Area in sq. in. 

U. S. Supervisors, heating-surface in sq. f t. -h 25 * 48 

English Board of Trade, grate-surface in sq. ft. -s- 2 20 

Molesworth, four fifths of grate-surface in sq. f t 32 

Thurston, 4 times coal burned per hour X (gauge pressure + 10) 14.5 

„, 1 (5 X heating-surface) 

Thurston,- — . 273 

2 gauge pressure -4- 10 

Rankine, .006 X water evaporated per hour 21 .6 

Committee of U. S. Supervisors, .005 X water evaporated per hour 18 

Suppose that, other data remaining the same, the draught were increased- 
so as to burn 13^, lbs. coal per square foot of grate per hour, and the grate- 
surface cut down to 30 sq. ft. to correspond, making the coal burned per 
hour 400 lbs., and the water evaporated 3600 lbs., the same as before; then 
the English Board of Trade rule and Molesworth's rule would give an area 
of disk of only 15 and 24 sq. in., respectively, showing the absurdity of mak- 
ing the area of grate the basis of the calculation of disk area. 

Another rule by Prof. Thurston is given in American Machinist, Dec. 1877, 
viz.: 

Disk area - ^ max. wt. of water evap. per hour 
gauge pressure + 10 
This gives for the example considered 16.4 sq. in. 

* The edition of 1893 of the Rules of the Supervisors does not contain this 
rule, but gives the rule grate-surface -5- 2. 



SAFETY-VALVES. 723 

One rule by Rankine is 1/150 to 1/180 of the number of pounds of water 
evaporated per hour, equals for the above case 27 to 20 sq. in. A communi- 
tion in Power, July, 1890. gives two other rules: 

1st. 1 sq. in. disk area for 3 sq. ft. grate, which would give 13.3 sq. in. 

2d. % sq. in. disk area for 1 sq. ft. grate, which would give 30 sq. in.; but 
if the grate-surface were reduced to 30 sq. ft. on account of increased 
draught, these rules would make the disk area only 10 and 22.5 sq. in., 
respectively. 

The Philadelphia rule for 100 lbs. gauge pressure gives a disk area of 0.21 
sq. in. for each sq. ft. of grate area, which would give an area of 8.4 sq. in. 
for 40 sq. ft. grate, and only 6.3 sq. in. if the grate is reduced to 30 sq. ft. 

According to the rule this aggregate area would have to be divided between 
two valves. But if the boiler was driven by forced draught, then the in- 
spector " must estimate the area of grate at 1 sq. ft. for each 16 lbs. of fuel 
burned per hour." 

Under this condition the actual grate-surface might be cut down to 400 -=- 
16 = 25 sq. ft., and by the rule the combined area of the two safety-valves 
would be only 25 X 0.21 = .25 sq. in. 

Nystrom's Pocket-book, edition of 1891, gives % sq. in. for 1 sq. ft. grate; 
also quoting from Weisbach, vol. ii, 1/3000 of the heating-surface^ This in 
the case considered is 1200/3000 = .4 sq. ft. or 57.6 sq. in. 

We thus have rules w r hich give for the area of safety-valve of the same 100- 
horse-power boiler results ranging all the way from 5.25 to 57.6 sq. in. 

All of the rules above quoted give the area of the disk of the valve as the 
thing to be ascertained, and it is this area which is supposed to bear some 
direct ratio to the grate-surface, to the heating-surface, to the water evap- 
orated, etc. It is difficult to see why this area has been considered even 
approximately proportional to these quantities., for with small lifts the area, 
of actual opening bears a direct ratio, not to tne area of disk, but to the 
circumference. 

Thus for various diameters of valve : 

Diameter 1 2 3 4'' " 6 7 

Area 785 3.14 7.07 12.57 19.64 28.27 38.48 

Circumference 3.14 6.28 9.42 12.57 15.71 18.85 21.99 

Ciicum. X lift of 0.1 in 31 .63 .94 1.26 1.57 1.89 2.20 

Ratio to area 4 .2 .13 .1 .08 .067 .057 

The apertures, therefore, are therefore directly proportional to the diam- 
eter or to the circumference, but their relation to the area is a varying one. 

If the lift = J4 diameter, then the opening would be equal to the area of 
the disk, for circumference X f4 diameter = area, but such a lift is far 
beyond the actual lift of an ordinary safety-valve. 

A correct rule for size of safety-valves should make the product of the 
diameter and the lift proportional" to the weight of steam to be discharged. 

A " logical " method for calculating the size of safety-valve is given in 
The Locomotive, July, 1892, based on the assumption that the actual opening 
should be sufficient to discharge all the steam generated by the boiler. 
Napier's rule for flow of steam is taken, viz., flow through aperture of one 
sq. in. in lbs. per second = absolute pressure -^- 70, or in lbs. per hour — 51.43 
X absolute pressure. 

If the angle of the seat is 45°, as specified in the rules of the U. S. Super- 
visors, the area of opening in sq. in. = circumference of the disk X the lift 
X .71, .71 beiug the cosine of 45°; or diameter of disk X lift X 2.23. 

A. G. Brown in his book on The Indicator and -its Practical Working 
(London, 1894) gives the following as the lift of the ordinary lever safetj : - 
valve for 100 lbs. gauge-pressure: 

Diam. of valve.. 2 2^ 3 3}& 4 4% 5 6 inches. 

Rise of valve 0583 .0523 .0507 .049~2 .0478 .0462 .0446 .0430 inch. 

The lift decreases with increase of steam -pressure; thus for a 4-inch valve: 
Abs. pressure, lbs. 45 65 85 105 115 135 155 175 195 215 
Gauge-press., lbs.. 30 50 70 90 100 120 140 160 180 200 
Rise, inch 1034 .0775 .0620 .0517 .0478 .0413 .0365 .0327 .0296 .0270 

The effective area of opening Mr. Brown takes at 70$£ of the rise multiolied 
by the circumference. 

An approximate formula corresponding to Mr. Brown's figures for diam- 
eters between 2J^ and 6 in. and gauge-pressures between 70 and 200 lbs. is 

Lift = (.0603 - 0031d) x -* — , in which d = diam. of valve in in, 

abs. pressure 



724 



THE STEAM-BOILER. 



If we combine this formula with the formulae 

Flow in lbs. per hour = area of opening in sq. in. x 51.43X abs. pressure, and 

Area = diameter of valve X lift X 2.23, we obtain the following, which the 
author suggests as probably a more correct formula for the discharging 
capacity of the ordinary lever safety-valve than either of those above given. 

Flow 'in lbs. per hour = d(.0603 - .0031cZ) X 115 X 2.23 X 51.43 = d(7% — Aid). 

From which we obtain : 
Diameter, inches ... . 1 1*4 2 2\i 3 3^ 4 5 6 7 
Flow, lbs. per hour.. 754 1100 1426 1733 2016 2282 2524 2950 3294 3556 

Horse-power 25 37 47 58 67 76 84 98 110 119 

the horse-power being taken as an evaporation of 30 lbs. of water per hour. 

If we solve the example, above given, of the boiler evaporating 3600 lbs. of 
water per hour by this table, we find it requires one 7-inch valve, or a 2)4- 
and a 3-inch valve combined. The 7-inch valve has an area of 38.5 sq. in., 
and the two smaller valves taken together have an area of only 12 sq. in.; 
another evidence of the absurdity of considering the area of disk as the 
factor which determined the capacity of the valve. 

It is customary in practice not to use safety-valves of greater diameter 
than 4 in. If a greater diameter is called for by the rule that is adopted, 
then two or more valves are used instead of one. 

Spring-loaded Safety-valves.— Instead of weights, springs are 
sometimes employed to hold down safety-valves. The calculations are 
similar to those for lever safety-valves, the tension of the spring correspond- 
ing to a given rise being first found by experiment (see Springs, page 347). 

The rules of the U. S. Supervisors allow an area of 1 sq. in. of the valve 
to 3 sq. ft. of grate, in the case of spring-loaded valves, except in water-tube, 
coil, or sectional boilers, in which 1 sq. in. to 6 sq. ft. of grate is allowed. 

Spriug-loaded safety-valves are usually of the reactionary or "pop " type, 
in which the escape of the steam is opposed by a lip above the valve-seat, 
against which the escaping steam reacts, causing the valve to lift higher 
than the ordinary valve. 

A. G. Brown gives the following for the rise, effective area, and quantity 
of steam discharged per hour by valves of the "pop " or Richardson type. 
The effective is taken at only 50% of the actual area due to the rise, on account 
of the obstruction which the lip of the valve offers to the escape of steam. 



Dia. value, in 


1 


Wo 


2 


2Yo, 


3 


3% 


4 


4Vo, 


5 


6 


Lift, inches. 


.125 


.150 


.175 


.200 


.225 


.250 


.275 


.300 


.325 


.375 


Area, sq. in. 


.196 


.354 


.550 


.785 


1.061 


1.375 


1.728 


2.121 


2.553 


3.535 


Gauge-pres., 


Steam discharged per hour, lbs. 


30 lbs. 


474 


856 


1330 


1897 


2563 


3325 


4178 


5128 


6173 


8578 


50 


669 


1209 


1878 


2680 


3620 


4695 


5901 


7242 


8718 


12070 


70 


861 


1556 


2417 


3450 


4660 


6144 


7596 


9324 


11220 


15535 


90 


1050 


1897 


2947 


4207 


5680 


7370 


9260 


11365 


13685 


18945 


100 


1144 


2065 


3208 


4580 


6185 


8322 


10080 


12375 


14895 


20625 


120 


1332 


2405 


3736 


5332 


7202 


9342 


11735 


14410 


173-10 


24015 


140 


1516 


2738 


4254 


6070 


8200 


10635 


13365 


16405 


19745 


27340 


160 


1696 


3064 


4760 


6794 


9175 


11900 


14955 


18355 


22095 


30595 


180 


1883 


3400 


5283 


7540 


10180 


13250 


16595 


20370 


24520 


33950 


200 


2062 


3724 


5786 


8258 


11150 


14165 18175 


22310 




37185 



If we take 30 lbs. of steam per hour, at 100 lbs. gauge-pressure = 1 H.P., 
we have from the above table: 

Diameter, inches... 1 1]4 2 2% 3 3^ 4 4J^ 5 6 

Horse-power 38 69 107 153 206 277 336 412 496 687 

A safety-valve should be capable of discharging a much greater quantity 
of steam than that corresponding to the rated horse-power of a boiler, since 
a. boiler having ample grate surface and strong draught may generate more 
than double the quantity of steam its rating calls for. 

The Consolidated Safety-valve Co.'s circular gives the following rated 
capacity of its nickel-seat " pop " safety-valves: 

Size, in .... 1 1J4 Vy 2 2 2\& 3 3}^ 4 4)4 5 5^£ 

Boiler t from 8 10 20 35 60 75 100 125 150 175 200 

H.P. 1 to 10 15 30 50 75 100 125 150 175 200 275 

The figures in the lower line from 2 inch to 5 inch, inclusive, correspond to 

the formula H.P. — 50(diameter — 1 inch). 



THE INJECTOR. 



725 



Lets 

W 



J 

L 

778 
Then 



THE INJECTOR. 
Equation of the Injector. 

be the number of pounds of steam used ; 

the number of pounds of water lifted and forced into the boiler; 

the height in feet of a column of water, equivalent to the absolute 
pressure in the boiler; 

the height in feet the water is lifted to the injector; 

the temperature of the water before it enters the injector; 

the temperature of the water after leaving the injector; 

the total heat above 32° F. in one pound of steam in the boiler, in 
heat-units; 

the lost work in friction and the equivalent lost work due to radia- 
tion and lost heat; 

the mechanical equivalent of heat. 



8[H - (t 2 - 32°)] = W(t. z - t,) + 



(W+ S)h -f- irft + Z, 



An equivalent formula, neglecting Wh -f Las small, is 



= [w 2 



■*i) + 



W+S 



144T 



1 



d ' * '778J.H" - (t 2 - 32°)' 
_ W[(t 2 - tjd + .1851p] 
H-{t 2 - 32°)cZ - .185123' 
in which d = weight of 1 cu. ft. of water at temperature £ 2 ; p — absolute 
pressure of steam, lbs. per sq. in. 

The rule for finding the proper sectional area for the narrowest part of 
the- nozzles is given as follows by Rankine, S. E. p. 477: 

. , cubic feet per hour gross feed-water 

Area in square inches = • 

800 ^/pressure in atmospheres 
An important condition which must be fulfilled in order that the injector 
will work is that the supply of water must be sufficient to condense the 
steam. As the temperature of the supply or feed -water is higher, the 
amount of water required for condensing purposes will be greater. 

The table below gives the calculated value of the maximum ratio of water 
to the steam, and the values obtained on actual trial, also the highest admis- 
sible temperature of the feed-water as shown by theory and the highest 
actually found hy trial with several injectors. 





Maximum Ratio Water 
to Steam. 


Gauge- 
pres- 
sure, 

pounds 

per 
sq. in. 


Maximum Temperature of 
Feed -Water. 


Gauge- 
pres- 


Calculated 
from 
Theory. 


Actual Expe- 
riment. 


Theoretical. 


Experrtal Results. 


pounds 
per 


111 

H.| 
T3 




H. 


P. 


M. 




sq. in. 


H. 


P. 


M. 


S. 


10 


36.5 
25.6 
20.9 

17.87 

16.2 

14.7 

13.7 

12.9 

12.1 

11.5 


30.9 
2^2.5 
19.0 
15.8 
13.3 
11.2 
12.3 
11.4 


19^9 
17.2 
15.0 
14.0 
11.2 
11.7 
11.2 


21 ^5 
19.0 

15.86 
13.3 
12.6 
12.9 


10 
20 
30 
40 
50 
60 
70 
80 
90 
100 
120 
150 






135° 
140* 

141* 

141* 


120° 

113 

lis 

118 


130° 

125 

123 
123 
122 


132° 


20 
30 
40 

50 
60 
70 
80 
90 


142° 
132 
126 
120 
114 
109 
105 

99 

95 

87 

77 


173° 
162 
156 
150 
143 
139 
134 
129 
125 
117 
107 


134 
134 
132 
131 
130 
130 
131 
132* 


100 








132* 
134* 
121* 



* Temperature of delivery above 212°. Waste-valve closed. 
H, Hancock inspirator; P, Park injector; M, Metropolitan injector; S, 
lers 1876 injector. 



Sel- 



72b THE STEAM-BOILER. 

Efficiency of the Injector.— Experiments at Cornell University, 
described by Prof. R. C. Carpenter, in Cassier , s Magazine, Feb. 1892, show- 
that the injector, when considered merely as a pump, has an exceedingly 
low efficiency, the duty ranging from 161^000 to 2,752.000 under different cir- 
cumstances of steam and delivery pressure. Small direct-acting pumps, 
such as are used for feeding boilers, show a duty of from 4 to 8 
million lbs , and the best pumping-engines from 100 to 140 million. When 
used for feeding- water into a boiler, however, the injector has a thermal 
efficiency of 100$, less the trifling loss due to radiation, since all the heat re- 
jected passes into the water which is carried into the boiler. 

The loss of work in the injector due to friction reappears as heat which is 
carried into the boiler, and the heat which is converted into useful work in 
the injector appears in the boiler as stored-up energy. 

Although the injector thus has a perfect efficiency as a boiler-feeder, it is 
nevertheless not the most economical means for feeding a boiler, since it 
can draw only cold or moderately warm water, while a pump can feed 
water which has been heated by exhaust steam which would otherwise be 
wasted. 

Performance of Injectors.— In Am. Mach., April 13, 1893, are a 
number of letters from different manufacturers of injectors in reply to the 
question: "What is the best performance of the injector in l-aising or lifting 
water to any height ?" Some of the replies are tabulated below. 

W. Sellers & Co.— 25.51 lbs. water delivered to boiler per lb. of steam; tem- 
perature of water, 64°; steam pressure, 65 lbs. 

Schaeffer & Budenberg— 1 gal. water delivered to boile- for 0.4 to 0.8 lb. 
steam. 

Injector will lift by suction water of 

140° F. 136° to 133° 122° to 180° 113° to 107° 
If boiler pressure is. 30 to 60 lbs. 60 to 90 lbs. 90 to 120 lbs. 120 to 150 lbs. 

If the water is not over 80° F., the injector will force against a pressure 75 
lbs. higher than that of the steam. 
Hancock Inspirator Co.: 

Lift in feet 22 22 22 11 

Boiler pressure, absolute, lbs 75.8 54.1 95.5 . 75.4 

Temperature of suction 34.9° 35.4° 47.3° 53.2° 

Temperature of delivery 134° 117.4° 173.7° 131.1 

Water fed per lb. of steam, lbs... 11.02 13.67 8.18 13.3 

The theory of the injector is discussed in Wood's. Peabody's, and Ront- 
gen's treatises on Thermodynamics. See also "Theory and Practice of the 
Injector,"" by Strickland L. Kueass, New York, 1895. 

Boiler-feeding Pumps.- Since the direct-acting pump, commonly 
used for feeding boilers, has a very low efficiency, or less than one tenth 
that of a good engine, it is generally better to use a pump driven by belt 
from the main engine or driving shaft. The mechanical work needed to feed 
a boiler may be estimated as follows: If the combination of boiler and en- 
gine is such that half a cubic foot, say 32 lbs. of water, is needed per horse- 
power, and the boiler-pressure is 100 lbs. per sq. in., then the work of feed- 
ing the quantity of water is 100 lbs. X 144 sq. in. x J4 ft. -lbs. per hour = 120 
ft.-lbs. per min. = 120/33.000 = .0036 H.P., or less than 4/10 of H of the 
power exerted by the engine. If a direct-acting pump, which discharges its 
exhaust steam into the atmosphere, is used for feeding, and it has only 1/10 
the efficiency of the main engine, then the steam used by the pump will be 
equal to nearly 4% of that generated by the boiler. 

The following table by Prof. D. S. Jacobus gives the relative efficiency of 
steam and power pumps and injector, with and without heater, as used 
upon a boiler with 80 lbs. gauge-pressure, the pump having a duty of 
10,000,000 ft.-lbs. per 100 lbs. of coal when no heater is used ; the injector 
heating the water from 60° to 150° F. 

Direct-acting pump feeding water at 60°, without a heater 1 .000 

Injector feeding water at 150°, without a heater 985 

Injector feeding water through a heater in which it is heated from 

150° to 200° 938 

Direct-acting pump feeding water through a heater, in which it is 

heated from 60° to 200° 879 

Geared pump, run from the engine, feeding water through a heater, 

in which it is heated from 60° to 200° .868 



PEED-WATER HEATERS. 



m 



FEED-WATER HEATERS. 

Percentage of Saving for Each Degree of Increase in Tem- 
perature of Feed-water Heated by Waste Steam. 





Pressure of Steam in Boiler 


lbs. 


per sq 


. iu. above 






Initial 

Temp. 

of 


Atmosphere. 


Initial 
Temp. 


















Feed. 





20 


40 


60 


80 


100 


120 


140 


160 


180 


200 




32° 


.0872 


.0861 


.0855 


.0851 


.0847 


.0844 


.0841 


.0839 


.0837 


.0835 


.0833 


32 


40 


.0878 


.0867 


.0861 


.0856 


.0853 


.0850 .0847 


.0845 


.0813 


.0841 


.0839 


40 


50 


.0886 


.0875 


.0868 


.0864 


.0860 


.08571.0854 


.0852 


.0850 


.0848 


.0846 


50 


60 


.0894 


.0883 


.0876 


.0872 


.0867 


.0864 |.0862 


.0859 


.0850 


.0855 


.osr,3 


60 


70 


.0902 


.0890 


.0884 


.0879 


.0875 


.08721.0869 


.0867 


.0801 


.0802 


.0800 


70 


80 


.0910 


.0898 


.0891 


.0887 


.oss:-; 


.0879 .0877 


.0874 


. 0872 


.0870 


.0808 


80 


90 


.0919 


.0907 


.0900 


.0895 


.osss 


.08871.0884 


.0883 


.0879 


.087; 


.0875 


90 


100 


.0927 


.0915 


.0908 


.0903 


.0899 


.08951.089.2 


0890 


.0887 


.0885 


.OS 83 


100 


110 


.0936 


.0923 


0916 


.0911 


.0907 


.0903; .0900 


.0898 


.0895 


.0^93 


.0891 


110 


120 


.0945 


.0932 


.0925 


.0919 


0915 


.09111.0908 


.0906 


.0903 


.0901 


.0899 


120 


130 


.0954 


.0941 


.0934 


.0928 


.0924 


.09201.0917 


.0914 


.0912 


.0909 


.0907 


130 


140 


.0963 


.0950 


.0943 


.0937 


.0932 


.0929 


.0925 


.0923 


.0920 


.0918 


.0916 


140 


150 


.0973 


.0959 


.0951 


.0946 


.0941 


.0937 


.0934 


.0931 


.11929 


.0926 


.0921 


150 . 


160- 


.0982 


.0968 


.0961 


.0955 


.0950 


.0946 


.0943 


.0940 


.0937 


.0935 


.0933 


160 


170 


.0992 


.0978 


.0970 


.0964 


.0959 


.0955 


.0952 


.0949 


.0946 


.0944 


.0941 


170 


180 


.1002 


.0988 


.0981 


.0973 


.0969 


.0965 


.0961 


.0958 


0955 


.0953 


.0951 


180 


190 


.1012 


.0998 


.0989 


.0983 


.0978 


.0974 


.0971 


.0968 


.0964 


.0962 


0960 


190 


200 


.1022 


.1008 


.0999 


.09'.):; 


.0988 


.0984 


.0980 


.0977 


.0974 


.0972 


.0969 


200 


210 


.1033 


.1018 


.1009 


.1003 


.0998 


.0994 


.0990 


.0987 


.0984 


.0981 


.0979 


210 


220 




.1029 


.1019 


.1013 


.1008 


.1001 


.1000 


.0997 


.0994 


.0991 


.0989 


220 


230 




.1039 


.1031 


.1024 


.1018 


.1012 


.1010 


.1007 


.1003 


.1001 


.0999 


230 


240 




.1050 


.1041 


.1034 


.1029 


.1021 


.1020 


.1017 


.1014 


.1011 


.1009 


210 


250 




.1062 


.1052 


.1045 


.1040 


.1035 


.1031 


.1027 


.1025 


.1022 


.1019 


250 



An approximate rule for the conditions of ordinary practice is a saving 
of 1$ is made by each increase of 11° in the temperature of the feed-water. 
This corresponds to .0909$ per degree. 

The calculation of saving is made as follows: Boiler-pressure, 100 lbs. 
gauge; total heat in steam above 32° = 1185 B.T.U. Feed-water, original 
temperature 60°, final temperature 209° F. Increase in heat-units, 150. 
Heat-units above 32° in feed -water of original temperature = 28. Heat- 
units in steam above that in cold feed-water, 1185 - 28 = 1157. Saving by the 
feed- water heater = 150/1157 = 12.96$. The same result is obtained by the 
use of the table. Increase in temperature 150° X tabular figure .0864 = 
12.96$. Let total heat of 1 lb. of steam at the boiler-pressure = H\ total 
heat of 1 lb. of feed-water before entering the heater = h 1 , and after pass- 
ing through the heater = ft 2 ! t^ en the saving made by the heater is -| - 1 . 

Strains Caused by Cold Feed-water.— A calculation is made 
in The Locomotive of March, 1893, of the possible strains caused in the sec- 
tion of the shell of a boiler by cooling it by the injection of cold feed-water. 
Assuming the plate to be cooled 200° F., and the coefficient of expansion of 
steel to be .0000067 per degree, a strip 10 in. long would contract .013 in., if it 
were free to contract. To resist this contraction, assuming that the strip is 
firmly held at the ends and that the modulus of elasticity is 29,000,000, would 
require a force of 37,700 lbs. per sq. in. Of course this amount of strain can- 
not actually take place, since the strip is not firmly held at the ends, but is 
allowed to contract to some extent by the elasticity of the surrounding 
metal. But, says The Locomotive, we may feel pretty confident that in the 
case considered a longitudinal strain of somewhere in the neighborhood of 
8000 or 10,000 lbs. per sq. in. may be produced by the feed-water striking 
directly upon the plates; and this, in addition to the normal strain pro- 
duced by the steam-pressure, is quite enough to tax the girth-seams beyond 
their elastic limit, if the feed-pipe discharges anywhere near them. Hence 
it is not surprising that the girth-seams develop leaks and cracks in 99 
cases out of every 100 in which the feed discharges directly upon the fire- 
sheets. 



m 



THE STEAM-BOILEK. 



STEAM SEPARATORS, 

If moist steam flowing at a high velocity in a pipe has its direction sud- 
denly changed, the particles of water are by their momentum projected in 
Cheir original direction against the bend in the pipe or wall of the chamber 
in which the change of direction takes place. By making proper provision 
for drawing off the water thus separated the steam may be dried to a 
greater or less extent. 

For long steam-pipes a large drum should be provided near the engine 
for trapping the water condensed in the pipe. A drum 3 feet diameter, 15 
feet high, has given good results in separating the water of condensation of 
a steam-pipe 10 inches diameter and 800 feet long. 

Efficiency of Steam Separators.— Prof. R. C. Carpenter, in 1S91, 
made a series of tests of six steam separators, furnishing them with steam 
containing different percentages of moisture, and testing the quality of 
steam before entering and after passing the separator. A condensed table 
of the prin cipal results is given below. 



o| 


Test with Steam of about 10$ of 
Moisture. 


Tests with Varying Moisture. 


1* 


Quality of 
Steam 
before. 


Quality of 
Steam 
after. 


Efficiency 
per cent. 


Quality of 
Steam 
before. 


Quality of 
Steam 
after. 


Av'ge 
Effi- 
ciency. 


B 

A 
D 
C 
E 
F 


87.0$ 
90.1 
89.6 
90.6 

88.4 
88.9 


98.8$ 

98.0 

95.8 

93.7 

90.2 

93.1 


90.8 
80.0 
59.6 
33.0 
15.5 
28.8 


66.1 to 97.5$ 
51.9 " 98 

72.2 " 96.1 
67.1 " 96.8 
68.6 " 98.1 
70.4 " 97.7 


97.8 to 99$ 

97.9 " 99.1 
95.5 " 98.2 
93.7 " 98.4 
79.3 " 98.5 
84.1 " 97.9 


87.6 
76.4 
71.7 
63.4 
36.9 
28.4 



Conclusions from the tests were: 1. That no relation existed between the 
eolume of the several separators and their efficiency. 

2. No marked decrease in pressure was shown by any of the separators, 
the most being 1.7 lbs. in E. 

3. Although changed direction, reduced velocity, and perhaps centrifugal 
force are necessary for good separation, still some means must be provided 
to lead the water out of the current of the steam. 

The high efficiency obtained from B and A was largely due to this feature, 
[n B the interior surfaces are corrugated and thus catch the water thrown 
out of the steam and readily lead it to the bottom. 

In A, as soon as the water falls or is precipitated from the steam, it comes 
in contact with the perforated diaphragm through which it runs into the 
space below, where it is not subjected to the action of the steam. 

In D, the next in efficiency, this is accomplished by means of a >-shaped 
diaphragm which throws the water back into the corners out of the current 
of steam. 

DETERMINATION OF THE MOISTURE IN STEAM- 
STEAM CALORIMETERS. 

In all boiler tests it is important to ascertain the quality of the steam, 
i.e., 1st, whether the steam is "saturated" or contains the quantity 
of heat due to the pressure according to standard experiments; 2d, whether 
the quantitv of heat is deficient, so that the steam is wet; and 3d. whether 
the heat is in excess and the steam superheated. The best method of ascer- 
taining the quality of the steam is undoubtedly that employed by a com- 
mittee which tested the boilers at the American Institute Exhibition of 
1871-2, of which Prof. Thurston was chairman, i.e., condensing all the water 
evaporated by the boiler by means of a surface condenser, weighing the 
condensing water, and taking its temperature as it enters and as it leaves 
the condenser; but this plan cannot always be adopted. 

A substitute for this method is the barrel calorimeter, which with careful 
operation and fairly accurate instruments may generally be relied on to 
give results within Iwo per cent of accuracy (that is, a sample of steam 
which gives the apparent result of 2$ of moisture may contain anywhere be 
tween and 4$). This calorimeter is described as follows: A sample of the 
steam is taken by inserting a perforated J^-inch pipe into and through the 
main pipe near the boiler, and led by a hose, thoroughly felted, to a barrel, 
holding preferably 400 lbs. of water, which is set upon a platform scale and 



DETERMINATION OF THE MOISTURE IN STEAM. 729 

provided with a cock or valve for allowing the water to flow to waste, and 
with a small propeller for stirring the water. 

To operate the calorimeter the barrel is filled with water, the weight and 
temperature ascertained, steam blown through the hose outside the barrel 
until the pipe is thoroughly warmed, when the hose is suddenly thrust into 
the water, and the propeller operated until the temperature of the water is 
increased to the desired point, say about 110° usually. The hose is then 
withdrawn quickly, the temperature noted, and the weight again taken. 

An error of 1/10 of a pound in weighing the condensed steam, or an error 
of y 2 degree in the temperature, will cause an error of over \% in the calcu- 
lated percentage of moisture. See Trans. A. S. M. E., vi. 293. 

When all the steam generated is not condensed, the method of making the 
connection for the purpose of taking out a sample is of the utmost impor- 
tance. Unless great care be exercised, the results will frequently show that 
the steam is superheated when the boiler has no superheating surface. 

The samples should be taken from the main steam-pipe, but not from the 
bottom, as this would take all the water draining to that point. 

The calculation of the percentage of moisture is made as below: 

Q = =4— [-Uh - h) ~ (T - >h)l 
H - T I- iv J 

Q = quality of the steam, dry saturated steam being unity. 

H = total heat of 1 lb. of steam at the observed pressure. 

T "== " " " " " water at the temperature of steam of the ob- 
served pressure. 

h = " " " " " condensing water, original. 

/ij = " " " " " " " final. 

W — weight of condensing w r ater, corrected for water-equivalent of the 
apparatus. 

w = weight of the steam condensed. 

Percentage of moisture = 1 — Q. 

If Q is greater than unity, the steam is superheated, and the degrees of 
superheating = 2.0833 (H - T) (Q - 1). 

Coil Calorimeters.— Instead of the open barrel in which the steam 
is condensed, a coil acting as a surface-condenser may be used, which is 
placed in the barrel, the water in coil and barrel being weighed separately. 
For description of an apparatus of this kind designed by the author, which 
he lias found to give results w r ith a probable error not exceeding ^ per cent 
of moisture, see Trans. A. S. M. E., vi. 294. This calorimeter may be used 
continuously, if desired, instead of intermittently. In this case a continu- 
ous flow of condensing water into and out of the barrel must be established, 
and the temperature of inflow and outflow and of the condensed steam 
read at short intervals of time. 

Throttling Calorimeter.— For percentages of moisture not ex- 
ceeding 3 per cent the throttling calorimeter is most useful and convenient 
and remarkably accurate. In this instrument the steam which reaches it 
in a J^-inch pipe is throttled by an orifice 1/16 inch diameter, opening into a 
chamber which has an outlet to the atmosphere. The steam in this cham- 
ber has its pressure reduced nearly or quite to the pressure of the atmos- 
phere, but the total heat in the steam before throttling causes the steam in 
the chamber to be superheated more or less according to whether the 
steam before throttling was dry or contained moisture. The only observa- 
tions required are those of the temperature and pressure of the steam on 
each side of the orifice. 

The author's formula for reducing the observations of the throttling 
calorimeter is as follows (Experiments on Throttling Calorimeters, Am. 

Mach., Aug. 4, 1892) : w = 100 X j , in which iv — percent- 
age of moisture in the steam ; H = total heat, and L = latent heat of steam 
in the main pipe; h = total heat due the pressure in the discharge side of 
the calorimeter, — 1146 6 at atmospheric pressure: K— specific heat of su- 
perheated steam; T = temperature of the throttled and superheated steam 
in the calorimeter; t = temperature due the pressure in the calorimeter, 
= 212° at atmospheric pressure. 

Taking K a,t 0.48 and the pressure in the discharge side of the calorimeter 
as atmospheric pressure, the formula becomes 

« = ioox H "" 46 - fl - £ °- 48(r - 218B \ 

From this formula the fojlowipg table is calculated ; 






730 



THE STEAM-BOILER. 



Moisture in Steam— Determinations by Throttling Calorimeter. 



£ 










Gauge-pressures. 










& tuO°? 
© <D ' 

Q 


5 


10 


20 


30 


40 


50 


60 


70 


75 


80 85 


90 


Per Cent of Moisture in Steam. 


0° 
10° 
20° 


0.51 
0.01 


90 

0.39 


1.54 
1.02 

.51 
.00 


2.06 

1.54 

1.02 

.50 


2.50 
1.97 
1.45 
.92 

.39 


2.90 
2.36 
1.83 
1.30 

.77 
.24 


3.24 

2.71 
2.17 
1.64 
1.10 
.57 
.03 


3.56 
3.02 
2.48 
1.94 
1.40 
.87 
.33 


3.71 

3.17 
2.63 
2.09 
1.55 
1.01 
.47 


3.86 
3 32 
2.77 
2.23 
1.69 
1.15 
.60 
.06 


3.99 
3.45 
2.90 
2.35 
1.80 
1.26 
.72 
.17 


4.13 
3.58 
3.03 


30° 






2 49 


40° 






1.94 


50° 










1 40 


60° 
70° 












.85 
.31 
























D if. p. de- 


.0503 


.0507 


.0515 


.0521 


.0526 


.0531 


.0535 


.0539 


.0541 


.0542 


.0544 


.0546 


ft 


Gauge-pressures. 


|| * 

to 
ft 


100 


110 


120 


130 


140 


150 


160 


170 


180 


190 


200 


250 


Per Cent of Moisture in Steam. 


0° 
10° 
20° 
30° 
40° 
50° 
60° 
70° 
80° 
90° 


3^84 
3.29 
2.74 
2.19 
1.64 
1.09 
.55 
.00 


4.63 
4.08 
3.52 
2.97 
2.42 
1.87 
1.32 
.77 
.22 


4.85 
4.29 
3.74 
3.18 
2.63 
2.08 
1.52 
.97 
.42 


5.08 
4.52 
3.96 
3.41 
2.85 
2.29 
1.74 
1.18 
.63 
.07 


5 29 
4.73 
4.17 
3.61 
3.05 
2.49 
1.93 
1.38 
.82 
.26 


5.49 
4.93 

4.37 
3.80 
3.24 
2.68 
2.12 
1.56 
1.00 
.44 


5.68 
5.12 
4.56 
3.99 
3.43 
2.87 
2.30 
1.74 
1.18 
.61 
.05 


5.87 
5.30 
4.74 
4.17 
3.61 
3.04 
2.48 
1.91 
1.34 
.78 
,21 


6.05 
5.48 
4.91 
4.34 
3.78 
3.21 
2.64 
2.07 
1.50 
.94 
.37 


5.65 
5.08 
4.51 
3.94 
3.37 
2.80 
2.23 
1.66 
1.09 
.52 


6.39 
5.82 
5.25 
4.67 
4.10 
3.53 
2.96 
2.38 
1.81 
1.24 
.67 
.10 

.0572 


7.16 
6.58 
6.00 
5.41 
4.83 
4.25 
3.67 
3.09 
2.51 
1.93 


100° 








1.34 


110° 














.76 






.0551 


.0554 








.0564 


.0566 








Dif.p.deg 


.0549 


.0556 


.0559 


.0561 


.0568 


.0570 


.0581 



Separating Calorimeters.— For percentages of moisture beyond 
the range of the throttling calorimeter the separating calorimeter is used, 
which is simply a steam separator on a small scale. An improved form of 
this calorimeter is described by Prof. Carpenter in Poiver, Feb. 1893. 

For fuller information on various kinds of calorimeters, see papers by 
Prof. Peabody, Prof. Carpenter, and Mr. Barrus in Trans. A. S. M. E., vols, 
x, xi, xii, 1889 to 1891; Appendix to Report of Com. on Boiler Tests, 
A. S. M. E., vol. vi, 1884; Circular of Schaeffer & Budenberg, N. Y., "Calo- 
rimeters, Throttling and Separating," 1894. 

Identification of Dry Steam by Appearance of a Jet.— 
Prof. Denton (Trans. A. S. M. E., vol. x.) found that jets of steam show un- 
mistakable change of appearance to the eye when steam varies less than \% 
from the condition of saturation either in the direction of wetness or super- 
heating. 

If a jet of steam flow from a boiler into the atmosphere under circumstances 
such that very little loss of heat occurs through radiation, etc., and the jet 
be transparent close to the orifice, or be even a grayish-white color, the 
steam may be assumed to be so nearly dry that no portable condensiug 
calorimeter will be capable of measuring the amount of water in the steam. 
If the jet be strongly white, the amount of water may be roughly judged up 
to about 2%, but beyond this a calorimeter only can determine the exact 
amount of moisture. 



CHIMNEYS. 



731 



A common brass pet-cock may be used as an orifice, but it should, if possi- 
ble, be set into the steam-drum of the boiler and never be placed further 
away from the latter than 4 feet, and then only when the intermediate reser- 
voir or pipe is well covered. 

Usual Amount of Moisture in Steam Escaping from a 
Boiler.— In the common forms of horizontal tubular land boilers and 
water-tube boilers with ample horizontal drums, and supplied with water 
free from substances likely to cause foaming, the moisture in the steam 
does not generally exceed 2% unless the boiler is overdriven or the water- 
level is carried too high. 



CHIMNEYS. 

Chimney Draught Theory.— The commonly accepted theory of 
chimney draught, based on Peelet's and Rankine's hypotheses (see Rankine, 
S. E.), is discussed by Prof. De Volson Wood in Trans. A. S. M. E., vol. xi. 

Peclet represented the law of draught by the formula 

h ~(i + 0+ S), 

in which h is the " head, 1 ' defined as such a height of hot gases as, if added 
to the column of gases in the chimney, would produce the 
same pressure at the furnace as a column of outside air, of the 
same area of base, and a height equal to that of the chimney ; 

u is the required velocity of gases in the chimney; 

G a constant to represent the resistance to the passage of air 
through the coal ; 

I the length of the flues and chimney; 

m the mean hydraulic depth or the area of a cross-section divi- 
ded by the perimeter; 

/ a constant depending upon the nature of the surfaces over which 
the gases pass, whether smooth, or sooty and rough. 

Rankine"^ formula (Steam Engine, p. 288), derived by giving certain values 
to the constants (so-called) in Peelet's formula, is 



^(u.0807) 

. T 2 

&( 0.084) 



H-H= (o.96^-l)iJ; 



in which H = the height of the chimney in feet; 

t = 493° F., absolute (temperature of melting ice); 

t 2 = absolute temperature of the gases in the chimney; 

t 2 = absolute temperature of the external air. 
Prof. Wood derives from this a still more complex formula which gives 
the height of chimney required for burning a given quantity of coal per 
second, and from it he calculates the following table, showing the height of 
chimney required to burn respectively 24, 20, and 16 lbs. of coal per square 
foot of grate per hour, for the several temperatures of the chimney gases 
given. 





Chimney Gas. 


Coal per sq. ft. of grate per hour, lbs. 


Outside Air. 






24 1 20 


16 


T 2 


T x 


Temp. 


1 






Absolute. 


Fahr. 


Height H, feet 




520° 


700 


239 


250.9 


157.6 


67.8 




800 


339 


172.4 


115.8 


55.7 


59° F. 


1000 


539 


149.1 


100.0 


48.7 




1100 


639 


148.8 


98.9 


48.2 




1200 


739 


152.0 


100.9 


49.1 




1400 


939 


159.9 


105.7 


51.2 




1600 


1139 


168.8 


111.0 


53.5 




2000 


1539 


206.5 


132.2 


63.0 



T32 



CHIMNEYS. 



Rankine's formula gives a maximum draught when t — 2 1/12t 2 , or 622° F., 
when the outside temperature is 00°. Prof. Wood says: " This result is not 
a fixed value, but departures from theory in practice do not affect the result 
largely. There is, then, in a properly constructed chimney, properly work- 
ing, a temperature giving a maximum draught,* and that temperature is not 
far from the value given by Rankine, although in special cases it may be 50° 
or 75° more or less. 1 ' 

All attempts to base a practical formula for chimneys upon the theoret- 
ical formula of Peclet and Rankine have failed on account of the impos- 
sibility of assigning correct values to the so-called "constants" G and /. 
(See Trans. A. S. M. E., xi. 984.) 

Force or Intensity of Draught.— The force of the draught is equal 
to the difference between the weight of the column of hot gases inside of the 
chimney and the weight of a column of the external air of the same height. 
It is measured by a draught-gauge, usually a U-tube partly filled with water, 
one leg connected by a pipe to the interior of the flue, and the other open to 
the external air. 

If D is the density of the air outside, d the density of the hot gas inside, 
in lbs. per cubic foot, h the height of the chimney in feet, and .192 the factor 
for converting pressure in lbs. per sq. ft. into inches of water column, then 
the formula for the force of draught expressed in inches of water is, 
F= A92h(D - d). 

The density varies with the absolute temperature (see Rankine). 

d = ^0.084; Z> = 0.0S07 — , 

where t is the absolute temperature at 32° F., = 493., t x the absolute tem- 
perature of the chimney gases and t 2 that of the external air. Substituting 
these values the formula for force of draught becomes 

41 - 41 -) = /,(^-^Y 

y v t 2 t x y 



.192/1 



/ 39.79 
^ t 2 



.5 ^ 
























• ® a 


Temperature of the Ex 


ternal Air— 


Barometer, 14.7 lbs 


per sq. in. 


1*1 














































E-i O 


0° 


10° 


20° 


30° 


40° 


50° 


60° 


70° 


80° 


90° 


100° 


200 


.453 


.419 


-.384 


.353 


.321 


.292 


.263 


.234 


.209 


.182 


.157 


220 


.488 


.453 


.419 


.388 


.355 


.326 


.298 


.269 


.244 


.217 


.192 


240 


.520 


.488 


.451 


.421 


.388 


.359 


.330 


.301 


.276 


.250 


.225 


260 


.555 


.528 


.484 


.453 


.420 


.392 


.363 


.331 


.309 


.282 


.257 


280 


.584 


.549 


.515 


.482 


.451 


.422 


.394 


.365 


.310 


.313 


.288 


300 


.611 


.576 


.541 


.511 


.478 


.449 


.420 


.392 


.367 


.340 


.315 


320 


.637 


.603 


.568 


.538 


.505 


.476 


.447 


.419 


.394 


.367 


.342 


340 


.662 


.638 


.593 


.563 


.530 


.501 


.472 


.443 


.419 


.392 


.367 


360 


.687 


.653 


.618 


.588 


.555 


.526 


.497 


.468 


.444 


.117 


.392 


380 


.710 


.676 


.641 


.611 


.578 


.549 


.520 


.492 


.467 


.440 


.415 


400 


.732 


.697 


.662 


.632 


.598 


.570 


.541 


.513 


.488 


.461 


.436 


420 


.753 


.718 


.684 


.653 


.620 


.591 


.563 


.534 


.509 


.482 


.457 


440 


.774 


.739 


.705 


.674 


.641 


.612 


.584 


.555 


.530 


.503 


.478 


460 


.793 


.758 


,724 


.694 


.660 


.632 


.603 


.574 


.549 


.522 


.497 


480 


.810 


.776 


.741 


.710 


.678 


.649 


.620 


.£91 


.566 


.540 


.515 


500 


.829 


.791 


.760 


.730 


.697 


.669 


.689 


.610 


.586 


.559 


.534 



To find the maximum intensity of draught for any given chimney, the 

heated column being 600° F., and the external air 60°, multiply the height 

above grate in feet by .0073, and the product is the draught in inches of water. 

Height of Water Column Due to Unbalanced Pressure in 

Chimney 100 Feet High. {The Locomotive, 1884.) 



* Much confusion to students of the theory of chimneys has resulted from 
their understanding the words maximum draught to mean maximum inten- 
sity or pressure of draught, as measui'ed by a draught-gauge. It here means 
maximum quantity oj- weight of gases passed up the chimney. The maxi- 
mum intensity is found only with maximum temperature, but after the 
temperature reaches about 622° F. the density of the gas decreases more 
rapidly than its velocity increases, so that the weight is a maximum about 
622° F., as shown by Rankine. — W. K. 



CHIMKEYS. 



738 



For any other height of chimney than 100 ft. the height of water column 
is found by simple proportion, the height of water column being directly 
proportioned to the height of chimney. 

The calculations have been made for a chimney 100 ft. high, with various 
temperatures outside and inside of the flue, and on the supposition that the 
temperature of the chimney is uniform from top to bottom. This is the 
basis on which all calculations respecting the draught-power of chimneys 
have been made by Rankine and other writers, but it is very far from the 
truth in most cases. The difference will be shown by comparing the read- 
ing of the draught-gauge with the table given. In one case a chimney 122 ft. 
high showed a temperature at the base of 320°, and at the top of 230°. 

Box, in his " Treatise on Heat," gives the following table : 

Draught Powers op Chimneys, etc., with the Internal Air at 552°, and 
the External Air at 02°, and with the Damper nearly Closed. 



<t-l s 


w 


Theoretical Velocity 


sw = 


w 


Theoretical Velocity 


°>, . 


S"" 1 fD 


in feet pe 


• second. 




S'=53 


in feet per second. 




cs Z £ 

(SI'S 
Pi 






£PPs£ 


£*3 

eg Z ? 

a Is 
p-i 








Cold Air 


Hot Air 


Cold Aii- 


Hot Air 




Entering. 


at Exit. 


MS 


Entering. 


at Exit. 


10 


.073 


17.8 


35.6 


80 


.585 


50.6 


101.2 


20 


.146 


25.3 


50.6 


90 


.657 


53.7 


107.4 


30 


.219 


31.0 


62.0 


100 


.730 


56,5 


113.0 


40 


.292 


35.7 


71.4 


120 


.876 


62.0 


124.0 


50 


.365 


40.0 


80.0 


150 


1.095 


69.3 


138.6 


60 


.438 


43.8 


87.6 


175 


1.277 


74.3 


149 6 


70 


.511 


47.3 


94.6 


200 


1.460 


80.0 


160.0 



Rate of Combustion Due to Height of Chimney. — 

Trowbridge's "Heat and Heat Engines 1 ' gives tne following table showing 
the heights of chimney for producing certain rates of combustion per sq. 
ft. of section of the chimney. It may be approximately true for anthracite 
in moderate and large sizes, but greater heights than are given in the table 
are needed to secure the given rates of combustion with small sizes of 
anthracite, and for bituminous coal smaller heights will suffice if the coal 
is reasonably free from ash— 5% or less. 







Lbs. of Coal 






Lbs. of Coal 




Lbs. of Coal 


Burned per 




Lbs. of Coal 


Burned per 




Burned per 


Sq. Ft. of 




Burned per 


Sq. Ft. of 


Heights 


Hour per 


Grate, the 


Heights 


Hour per 
Sq. Ft. 


Grate, the 


in 


Sq. Ft. 


Ratio of 


in 


Ratio of 


feet. 


of Section 


Grate to Sec- 


feet. 


of Section 


Grate to Sec- 




of 


tion of 




of 


tion of 




Chimney. 


Chimney be- 
ing 8 to 1. 




Chimney. 


Chimney be- 
ing 8 to 1. 


20 


60 


7.5 


70 


126 


15.8 


25 


68 


8.5 


75 


131 


16.4 


30 


76 


9.5 


80 


135 


16.9 


35 


84 


10.5 


85 


139 


17.4 


40 


93 


11.6 


90 


144 


18.0 


45 


99 


12.4 


95 


148 


18.5 


50 


105 


13.1 


100 


152 


19 


55 


111 


13.8 


105 


156 


19.5 


60 


116 


14.5 


110 


160 


20 


65 


121 


15.1 









Thurston's rule for rate of combustion effected by a given fieight of chim- 
ney (Trans. A. S. M. E., xi. 991) is: Subtract 1 from twice the square root of 
the height, and the result is the rate of combustion in pounds per square foot 
of grate per hour, for anthracite. Or rate ~ 2 yTi — \, in which h is the 
height in feet. This rule gives the following: 

_ h = .50 60 70 80 90 100 110 125 150 175 2G0 
2 tyJi- 1 = 13.14 14.49 15.73 16.89 17.97 19 19.97 21.36 23.49 25.45 27.28 

The results agree closely with Trowbridge's table given above. In prac- 



m 



CHIMNEYS. 



tice the high rates of combustion for high chimneys given by the formula 
are not generally obtained, for the reason that with high chimneys there are 
usually long horizontal flues, serving many boilers, and the friction and the 
interference of currents from the several boilers are apt to cause the inten- 
sity of draught in the branch flues leading to each boiler to be much less 
than that at the base of the chimney. The draught of each boiler is also 
usually restricted by a damper and by bends in the gas-passages. In a bat- 
tery of several boilers connected to a chimney 150 ft. high, the author found 
a draught of %-inch water-column at the boiler nearest the chimney, and 
only 14-inch at the boiler farthest away. The first boiler was wasting fuel 
from too high temperature of the chimney-gases, 900°, having too large a 
grate-surface for the draught, and the last boiler was working below its 
rated capacity and with poor economy, on account of insufficient draught. 

The effect of changing the length of the flue leading into a chimney 60 ft. 
high and 2 ft. 9 in. square is given in the following table, from Box on 
" Heat" : 



Length of Flue in 

feet. 


Horse-power. 


Length of Flue in 
feet. 


Horse-power. 


50 

100 
200 
400 
600 


107.6 
100.0 
85.3 
70.8 
62.5 


800 
1,000 
1,500 
2,000 
3,000 


56.1 
51.4 
43.3 
38.2 
31.7 



The temperature of the gases in this chimney was assumed to be 552° F., 
and that of the atmosphere 62°. 

High Chimneys not Necessary.— Chimneys above 150 ft. in height 
are very costly, and their increased cost is rarely justified by increased ef- 
ficiency. In recent practice it has become somewhat common to build two or 
more smaller chimneys instead of one large one. A notable example is the 
Speckels Sugar Refinery in Philadelphia, where five separate chimneys are 
used for one boiler-plant of 7500 H.P. The five chimneys are said to have 
cost several thousand dollars less than a single chimney of their combined 
capacity would have cost. Very tall chimneys have been characterized by 
one writer as " monuments to the folly of their builders. 1 '' 

Heights of Chimney required for Different Fuels.— The 
minimum height necessary varies with the fuel, wood requiring the least, 
then good bituminous coal, and fine sizes of anthracite the greatest. It 
also varies with the character of the boiler — the smaller and more circuitous 
the gas-passages the higher the stack required; also with the number of 
boilers, a single boiler requiring less height than several that discharge 
into a horizontal flue. No general rule can be given. 

SIZE OF CHIMNEYS. 

The formula given below, and the table calculated therefrom for chimneys 
up to 96 in. diameter and 200 ft. high, were first published by the author 
in 1884 (Trans. A. S. M. E. vi , 81). They have met with much approval 
since that date by engineers who have used them, and have been frequently 
published in boiler-makers' catalogues and elsewhere. The table is now 
extended to cover chimneys up to 12 ft. diameter and 300 ft. high. The sizes 
corresponding to the given commercial horse-powers are believed to be 
ample for all cases in which the draught areas through the boiler-flues and 
connections are sufficient, say not less than 20$ greater than the area of the 
chimney, and in which the draught between the boilers and chimney is not 
checked by long horizontal passages and right-angled bends. 

Note that the figures in the table correspond to a coal consumption of 5 lbs. 
of coal per horse-power per hour. This liberal allowance is made to cover 
the contingencies of poor coal being used, and of the boilers being driven 
beyond their rated capacity. In large plants, with economical boilers and 
engines, good fuel and other favorable conditions, which will reduce the 
maximum rate of coal consumption at any one time to less than 5 lbs. per 
H. P. per hour, the figures in the table may be multiplied by the ratio of 5 to 
the maximum expected coal consumption per H.P. per hour. Thus, with 
conditions which make the maximum coal consumption only 2.5 lbs. per 
hour, the chimney 300 ft. high X 12 ft. diameter should be sufficient for 6155 
X 2 = 12,310 horse-power. The formula is based on the following data : 



SIZE OF CHIMNEYS. 



735 



I'.Ji 



(OCOOCO COiOtOOO 

- - _■ z 77 .<• "i -^ 









:-t^coc© 00 00 t^ t- CO -*" iH 






38S 333§ s?l 



736 CHIMNEYS. 

1. The draught power of the chimney varies as the square root of the 
height. 

2. The retarding of the ascending gases by friction may be considered as 
equivalent to a diminution of the area of the chimney, or to a lining of the 
chimney by a layer of gas which has no velocity. The thickness of this 
lining is assumed to be 2 inches for all chimneys, or the diminution of area 
equal to the perimeter x 2 inches (neglecting the overlapping of the corners 
of the lining). Let D = diameter in feet, A = area, and E — effective area 
in square feet. 

For square chimneys, E — D? — — = A — ■= VA. 

For round chimeys, E = ^(l> 2 - ~) = A - 0.591 ]/A. 

For simplifying calculations, the coefficient of YA may be taken as 0.6 
for both square and round chimneys, and the formula becomes 

E = A - 0.6 YA. 

3. The power varies directly as this effective area E. 

4. A chimney should be proportioned so as to be capable of giving sufficient 
draught to cause the boiler to develop much more than its rated power, in 
case of emergencies, or to cause the combustion of 5 lbs. of fuel per rated 
horse-power of boiler per hour. 

5. The power of the chimney varying directly as the effective area, E, and 
as the square root of the height, H, the formula for horse-power of boiler for 
a given size of chimney will take the form H.P. = CE YH, in which C is a 
constant, the average value of which, obtained by plotting the results 
obtained from numerous examples in practice, the author finds to be 3.33. 

The formula for horse-power then is 

H.P. = 3.33.EJ \/H, or H.P. = 3.33(4 - .6 YA) Y~H. 

If the horse-power of boiler is given, to find the size of chimney, the height 
being assumed, 

For round chimneys, diameter of chimney = diam. of E -\- 4". 
For square chimneys, side of chimney = * ' E + 4". 

If effective area E is taken in square feet, the diameter in inches_is d — 
13.54 YE 4-4", and the side of a square chimney in inches is s — 12 YE-\- 4". 

(nou p \ 2 
— ^ 'J . 

In proportioning chimneys the height is generally first assumed, with due 
consideration to the heights of surrounding buildings or hills near to the 
proposed chimney, the length of horizontal flues, the character of coal to be 
used, etc., and then the diameter required for the assumed height and 
horse-power is calculated by the formula or taken from the table. 

The Protection of Tall Chimney-shafts from Lightning. 
— C. Molyneux and J. M. Wood (Industries, March 28, 1890) recommend for 
tall chimneys the use of a coronal or hea\y band at the top of the chimney, 
with copper points 1 ft. in height at intervals of 2 ft. throughout the circum- 
ference. The points should be gilded to prevent oxidation. The most ap- 
proved form of conductor is a copper tape about % in. by % in. thick, 
weighing 6 ozs. per ft. If iron is used it should weigh not less than 2J4 lbs. 
per ft. There must be no insulation, and the copper tape should be fastened 
to the chimney with holdfasts of the same material, to prevent voltaic 
action. An allowance for expansion and contraction should be made, say 1 
in. in 40 ft. Slight bends in the tape, not too abrupt, answer the purpose. 
For an earth terminal a plate of metal at least 3 ft. sq. and 1/16 in. thick 
should be buried as deep as possible iu a damp spot. The plate should be of 
the same metal as the conductor, to which it should be soldered. The best 
earth terminal is water, and when a deep well or other large body of water 
is at hand, the conductor should be carried down into it. Pught-angled 
beucla in the conductor should be avoided. No bend in it should be over 30*, 



SIZE OF CHIMNEYS. 



737 



Some Tall Brick Chimneys. 



1. Hallsbriickner Hiitte, Sax. 

2. Townsend's. Glasgow 

3. Tennant's, Glasgow 

4. Dobson & Barlow, Bolton. 

Eng 

5. Fall River Iron Co., Boston 

6. Clark Thread Co., Newark, 

N.J 

7. Merrimac Mills, Low'l,Mass 

8. Washington Mills, Law- 

rence, Mass . 

9. Amoskeag Mills, Manches- 

ter, N. H 

10. Narragansett E. L. Co., 
Providence, R. I 

11. Lower Pacific Mills, Law- 
rence, Mass . ... 

12. Passaic Print Works, Pas 
saic, N.J 

13. Edison Sta,B'kIyn,Two e'ch 





a 

5 

S 
D 

a 


Outside 
Diameter. 


Capacity by the 
Author's 
Formula. 


be 


6 

M 


o 


H. P. 


Pounds 
Coal 
per 
hour. 


460 
454 
435 


15.7' 


33' 

32 
40 


16' 


13^221 
9,795 


66,105 
48,975 


367^ 
350 


13' 2" 
11 


33'10" 
30 


21 


8,245 
5,558 


41,225 
27, 790 


335 

282'9" 


11 
12 


28' 6" 


14 


5,435 

5,980 


27,175 
29,900 


250 


10 






3,839 


19,195 


250 


10 






3,839 


19,195 


238 


14 






7,515 


37,575 


214 


8 






2,248 


11,240 


200 
150 


9 

50" x 120" 




each 


2,771 
1,541 


13,855 
7,705 



Notes on the Above Chimneys. — 1. This chimney is situated near 
Freiberg, on the right bank of the Mulde, at an elevation of 219 feet above 
that of the foundry works, so that its total height above the sea will be 711% 
feet. The works are situated on the bank of the river, and the furnace- 
gases are conveyed across the river to the chimney on a bridge, through a 
pipe 3227 feet in length. It is built throughout of brick, and will cost about 
$40,000.— Mfr. and Bldr. 

2. Owing to the fact that it was struck by lightning, and somewhat 
damaged, as a precautionary measure a copper extension subsequently was 
added to it, making its entire height 488 feet. 

1, 2, 3, and 4 were built of these great heights to remove deleterious 
gases from the neighborhood, as well as for draught for boilers. 

5. The structure rests on a solid granite foundation, 55 X 30 feet, and 
16 feet deep. In its construction there were used 1,700,000 bricks, 2000 tons 
of stone, 2000 barrels of mortar, 1000 loads of sand, 1000 barrels of Portland 
cement, and the estimated cost is $40,000. It is arranged for two flues, 9 
feet 6 inches by 6 feet, connecting with 40 boilers, which are to be run in 
connection with four triple-expansion engines of 1350 horse-power each. 

6. It has a uniform batter of 2.85 inches to every 10 feet. Designed 
for 21 boilers of 200 H. P. each. It is surmounted by a cast-iron cop- 
ing which weighs six tons, and is composed of thirty-two sections, 
which are bolted together by inside flanges, so as to present a smooth 
exterior. The foundation is in concrete, composed of crushed lime- 
stone 6 parts, sand 3 parts, and Portland cement 1 part. It is 40 feet 
square and 5 feet deep. Two qualities of brick were used; the outer 
portions were of the first quality North River, and the backing up was of 
good quality New Jersey brick. Every twenty feet in vertical measurement 
an iron ring, 4 inches wide and % to y% inch thick, placed edgewise, was 
built into the walls about 8 inches from the outer circle. As the chimney 
starts from the base it is double. The outer wall is 5 feet 2 inches in thick- 
ness, and inside of this is a second wall 20 inches thick and spaced off about 
20 inches from main wall. From the interior surface of the main wall eight 
buttresses are carried, nearly touching this inner or main fluB wall in 
order to keep it in line should it tend to sag. The interior wall, starting 
with the thickness described, is gradually reduced until a height of about 
90 feet is reached, when it js diminished to 8 inches. At 165 feet it ceases, 



738 CHIMNEYS. 

and the rest of the chimney is without lining. The total weight of the chim- 
ney and foundation is 5000 tons. It w r as completed in .September, 1888. 

7. Connected to 12 boilers, with 1200 square feet of grate-surface. Draught- 
Degauge 1 9/16 inches. 

8. Connected to 8 boilers, 6' 8" diameter X 18 feet. Grate-surface 448 
square feet. 

9. Connected to 64 Manning vertical boilers, total grate surface 1810 sq. ft. 
Designed to burn 18,000 lbs. anthracite per hour. 

10. Designed for 12,000 H. P. of engines; (compound condensing). 

11. Grate-surface 431 square feet; H.P. of boilers (Galloway) about 2500. 
13. Eight boilers (water-tube) each 450 H.P. ; 12 engines, each 300 H.P. Plant 

designed for 36,000 incandescent lights. For the first 60 feet the exterior 
wall is 28 inches thick, then 24 inches for 20 feet, 20 inches for 30 feet, 16 
inches for 20 feet, and 12 inches for 20 feet. The interior wall is 9 inches 
thick of fire-brick for 50 feet, and then 8 inches thick of red brick for the 
next 30 feet. Illustrated in Iron Age, January 2, 1890. 
A number of the above chimneys are illustrated in Power, Dec, 1890. 
Chimney at Knoxville, Tenn., illustrated in Eng'g Neivs, Nov. 2, 1893. 
6 feet diameter, 120 feet high, double wall: 

Exterior wall, height 20 feet, 30 feet, 30 feet. 40 feet; 

" " thickness 211^ in., 17in., 13 in,, 8*^ in. ; 

Interior wall, height 35 ft., 35 ft., 29 ft., 21ft.; 
" " thickness 13J^ in., 8}^ in., 4 in., 0. 

Exterior diameter, 15' 6" at bottom ; batter, 7/16 inch in 12 inches from bot- 
tom to 8 feet from top. Interior diameter of inside wall, 6 feet uniform to 
top of interior wall. Space between walls, 16 inches at bottom, diminishing 
to at top of interior wall. The interior wall is of red brick except a lining 
of 4 inches of fire-brick for 20 feet from bottom. 

Stability of Chimneys.— Chimneys must be designed to resist the 
maximum force of the wind in the locality in which they are built, (see 
Weak Chimneys, below). A general rule for diameter of base, of brick 
chimneys, approved by many years of practice in England and the United 
States, is to make the diameter of the base one tenth of the height. If the 
chimney is square or rectangular, make the diameter of the inscribed circle 
of the base one tenth of the height. The " batter " or taper of a chimney 
should be from 1/16 to 14 inch to the foot on each side. The brickwork 
should be one brick (8 or 9 inches) thick for the first 25 feet from the top, in- 
creasing }4 brick (4 or 4}^ inches) for each 25 feet from the top downwards. 
If the inside diameter exceed 5 feet, the top length should be 1J^ bricks; and 
if under 3 feet, it may be \y% brick for ten feet. 

(From The Locomotive, 1884 and 1886.) For chimneys of four feet in diam- 
eter and one hundred feet high, and upwards, the best form is circular, with 
a straight batter on the outside. A circular chimney of this size, in addition 
to being cheaper than any other form, is lighter, stronger, and looks much 
better and more shapely. 

Chimneys of any considerable height are not built up of uniform thickness 
from top to bottom, nor with a uniformly varying thickness of wall, but the 
wall, heaviest of course at the base, is reduced by a series of steps. 

Where practicable the load on a chimney foundation should not exceed two 
tons per square foot in compact sand, gravel, or loam. Where a solid rock- 
bottom is available for foundation, the load may be greatly increased. If 
the rock is sloping, all unsound portions should be removed, and the face 
dressed to a series of horizontal steps, so that there shall be no tendency to 
slide after the structure is finished. 

All boiler-chimneys of any considerable size should consist of an outer 
stack of sufficient strength to give stability to the structure, and an inner 
stack or core independent of the outer one. This core is by many engineers 
extended up to a height of but 50 or 60 feet from the base of the chimney, 
but the better practice is to run it up the whole height of the chimney; it 
may be stopped off, say, a couple feet below the top, and the outer shell con- 
tracted to the area of the core, but the better way is to run it up to about 8 
or 12 inches of the top and not contract the outer shell. But under no cir- 
cumstances should the core at its upper end be built into or connected with 
the outer stack. This has been done in several instances by bricklayers, and 
the result has been the expansion of the inner core wiiich lifted the top of 
the outer stack squarely up and crpcked the brickwork. 

For a height of 100 feet we would make the outer shell in three steps, the 
first 20 feet high, 16 inches thick, the second 30 feet high, 12 inches thick, the 



SIZE OF CHIMNEYS. 739 

third 50 feet high and 8 inches thick. These are the minimum thicknesses 
admissible for chimneys of this height, and the batter should be not less 
than 1 in 36 to give stability. The core should also be built in three steps 
each of which maybe about one third the height of the chimney, the lowest 
12 inches, the middle 8 inches, and the upper step 4 inches thick. This will 
insure a good sound core. The top of a chimney may be protected by a 
cast-iron eap; or perhaps a cheaper and equally good plan is to lay the 
ornamental part in some good cement, and plaster the top with the same 
material. 

Weak Chimneys.— James B. Francis, in a report to the Lawrence 
Mfg. Co. in 1873 (Emfg News, Aug. 28, 1880), gives some calculations con- 
cerning the probable effects of wind on that company's chimney as then 
constructed. Its outer shell is octagonal. The inner shell is cylindrical, 
with an air-space between it and the outer shell; the two shells not being 
bonded together, except at the openings at the base, but with projections in 
the brickwork, at intervals of about 20 ft. in height, to afford lateral sup- 
port by contact of the two shells. The principal dimensions of the chimney 
are as follows : 

Height above the surface of the ground 211 ft. 

Diameter of the inscribed circle of the octagon near the ground . 15 " 

Diameter of the inscribed circle of the octagon near the top 10 ft. 1% in. 

Thickness of the outer shell near the base, 6 bricks, or 23J^ in. 

Thickness of the outer shell near the top, 3 bricks, or 11^ " 

Thickness of the inner shell near the base, 4 bricks, or .15 " 

Thickness of the inner shell near the top, 1 brick, or 3% " 

One tenth of the height for the diameter of the base is the rule commonly 
adopted. The diameter of the inscribed circle of the base of the Lawrence 
Manufacturing Company's chimney being 15 ft., it is evidently much less 
than is usual in a chimney of that height. 

Soon after the chimney was built, and before the mortar had hardened, it ■ 
was found that the top had swayed over about 29 in. toward the east. This 
was evidently due to a strong westerly wind which occurred at that time. 
It was soon brought back to the perpendicular by sawing into some of the 
joints, and other means. 

The stability of the chimney to resist the force of the wind depends mainly 
on the weight of its outer shell, and the width of its base. The cohesion of 
the mortar may add considerably to its strength; but it is too uncertain to 
be relied upon. The inner shell will add a little to the stability, but it may 
be cracked by the heat, and its beneficial effect, if any, is too uncertain to 
be taken into account. 

The effect of the joint action of the vertical pressure due to the weight of 
the chimney, and the horizontal pressure due to the force of the wind is to 
shift the centre of pressure at the base of the chimney, from the axis to- 
ward one side, the extent of the shifting depending on the relative magni- 
tude of the two forces. If the centre of pressure it brought too near the 
side of the chimne}^, it will crush the brickwork on that side,, and the chim- 
ney will fall. A line drawn through the centre of pressure, perpendicular to 
the direction of the wind, must leave an area of brickwork between it and 
the side of the chimney, sufficient to support half the weight of the chim- 
ney; the other half of the weight being supported by the brickwork on the 
windward side of the line. 

Different experimenters on the strength of brickwork give very different 
results. Kirkaldy found the weights which caused several kinds of bricks, 
laid in hydraulic lime mortar and in Roman and Portland cements, to fail 
slightly, to vary from 19 to 60 tons (of 2000 lbs.) per sq. ft. If we take in this 
case 25 tons per sq. ft., as the weight that would cause it to begin to fail, we 
shall not err greatly. To support half the weight of the outer shell of the 
chimney, or 322 tons, at this rate, requires an area of 12.88 sq. ft. of brick- 
work. From these data and the drawings of the chimney, Mr. Francis cal- 
culates that the area of 12.88 sq. ft. is contained in a portion of the chimney 
extending 2.428 ft. from one of its octagonal sides, and that the limit to 
which the centre of pressure may be shifted is therefore 5.072 ft. from the 
axis. If shifted beyond this, he says, on the assumption of the strength 
of the brickwork, it will crush and the chimney will fall. 

Calculating that the wind-pressure can affect only the upper 141 ft. of the 
chimney, the lower 70 ft. being protected by buildings, he calculates that a 
wind-pressure of 44.02 lbs. per sq. ft. would blow the chimney down. 

Rankine, in a paper printed in the transactions of the Institution of Engi- 



740 



CHIMNEYS. 



neers, in Scotland, for 1867-68, says: " It had previously been ascertained 
by observation of the success and failure of actual chimneys, and especially 
of those which respectively stood and fell during the violent storms of 1856, 
that, in order that a round chimney may be sufficiently stable, its weight 
should be such that a pressure of wind, of about 55 lbs. per sq. ft. of a plane 
surface, directly facing the wind, or 27^ lbs. per sq. ft. of the plane projec- 
tion of a cylindrical surface, . . . shall not cause the resultant pressure 
at any bed-joint to deviate from the axis of the chimney by more than one 
quarter of the outside diameter at that joint," 

According to Eankine's rule, the Lawrence Mfg. Co.'s chimney is adapted 
to a maximum pressure of wind on a plane acting on the whole height of 
18.80 lbs. per sq. ft., or of a pressure of 21.70 lbs. per sq. ft. acting on the 
uppermost 141 ft. of the chimney. 

Steel Chimneys are largely coming into use, especially for tall chim- 
neys of iron-works, from 150 to 300 feet in height. The advantages claimed 
are: greater strength and safety; smaller space required; smaller cost, by 
30 to 50 per cent, as compared with brick chimneys; avoidance of infiltra- 
tion of air and consequent checking of the draught, common in brick chim- 
neys. They are usually made cylindrical in shape, with a wide curved flare 
for 10 to 25 feet at the bottom. A heavy cast-iron base-plate is provided, to 
which the chimney is riveted, and the plate is secured to a massive founda- 
tion by holding-down bolts. No guys are used. F. W. Gordon, of the Phila. 
Engineering Works, gives the following method of calculating their resist- 
ance to wind pressure (Poirer, Oct. 1893) : 

In tests by Sir William Fairbairn we find four experiments to determine 
the strength of thin hollow tubes. In the table will be found their elements, 
with their breaking strain. These tubes were placed upon hollow blocks, 
and the weights suspended at the centre from a block fitted to the inside of 
the tube. 



Clear 
Span, 
ft. in. 



Thick- 
ness Iron, 



Outside 
Diame- 
ter, in. 



Sectional 
Area, 



Breaking 

Weight, 

lbs. 



Breaking W't, 

lbs, by Clarke's 

Formula, 

Constant 1.2. 



I. 
II. 
III. 
IV. 



15 7% 
23 5 
23 5 



.037 
.113 
.0631 
.119 



12.4 
17.68 

18.18 



6.74 



2,704 
11,440 

6,400 
14,240 



9,184 
7,302 
13,910 



Edwin Clarke has formulated a rule from experiments conducted by him 
during his investigations into the use of iron ana steel for hollow tube 
bridges, which is as follows : 

Area of material in sq.in. X Mean depth in in. X Constant 



Clear span in feet. 



Center break- | 
ing load,in tons. j 

When the constant used is 1.2, the calculation for the tubes experimented 
upon by Mr. Fairbairn are given in the last column of the table. D. K. 
Clark's "Rules, Tables, and Data," page 513, gives a rule for hollow tubes 
as follows : W= 3A4D 2 TS -=-i. W= breaking weight in pounds in centre; 
D— extreme diameter in inches; T= thickness in inches; L = length be- 
tween supports in inches; S — ultimate tensile strength in pounds per sq. in. 

Taking S, the strength of a square inch of a riveted joint, at 35,000 lbs. 
per. sq. in., this rule figures as follows for the different examples experi- 
mented upon by Mr. Fairbairn : I, 2870; II, 10,190; III, 7700; IV, 15,320. . 

This shows a close approximation to the breaking weight obtained by 
experiments and that derived from Edwin Clarke's and D. K. Clark's rides. 
We therefore assume that this system of calculation is practically correct, 
and that it is eminently safe when a large factor of safety is provided, and 
from the fact that a chimney may be standing for many years without 
receiving anything like the strain taken as the basis of the calculation, viz., 
fifty pounds per square foot. Wind pressure at fifty pounds per square foot 
may be assumed to be travelling in a horizontal direction, and be of the 
same velocity from the top to the bottom of the stack. This is the extreme 
assumption. If, however, the chimney is round, its effective area would be 
only half of its diameter plane. We assume that the entire force may be 
concentrated in the centre of the height of the section of the chimney 
under consideration. 



SIZE OE CHIMNEYS. 



741 



Taking as an example a 125-foot iron chimney at Poughkeepsie, N. Y., the 
average diameter of which is 90 inches, the effective surface in square feet 
upon which the force of the wind may play will therefore be 7% times 125 
divided by 2, which multiplied by 50 gives a total wind force of 23,43? 
pounds. The resistance of the chimney to breaking across the top of the 
foundation would be 3-14 X 168 2 (that is, diameter of base) X .25 x 35,000 -*- 
(750 X 4) = 258,486, or 10.6 times the entire force of the wind. We multiply 
the half height above the joint in inches, 750, by 4, because the chimney is 
considered a fixed beam with a load suspended on one end. In calculating 
its strength half way up, we have a beam of the same character. It is a 
fixed beam at a line half way up the chimney, where it is 90 inches in diam- 
eter and .187 inch thick. Taking the diametrical section above this line, 
and the force as concentrated in the centre of it, or half way up from the 
point under consideration, its breaking strength is: 3.14 X 90 2 x .187 X 35,000 
-5- (381 X 4) = 109,220; and the force of the wind tc tear it apart through its 
cross-section, 7J4 x 62^ x 50-5- 2 = 11,352, or a little more than one tenth of 
the strength of the stack. 

The Babcock & Wilcox Co/s book "Steam 1 ' illustrates a steel chimney 
at the works of the Maryland Steel Co., Sparrow's Point, Md. It is 225 ft. 
in height above the base, with internal brick lining: 13' 9" uniform inside 
diameter. The shell is 25 ft. diam. at the base, tapering in a curve to 1? ft. 
25 ft. above the base, thence tapering almost imperceptibly to 14' 8" at the 
top. The upper 40 feet is of J4 _mcn plates, the next four sections of 40 ft. 
each are respectively 9/32, 5/16, 11/32, and % inch. 

Sizes of Foundations for Steel Chimneys. 

(Selected from circular of Phila. Engineering Works.) 
Half-Lined Chimneys. 

Diameter, clear, feet 3 

Height, feet 100 

Least diameter foundation.. 15'9" 

Least depth foundation 6' 

Height, feet 

Least diameter foundation 

Least depth foundation 

"Weight of Sheet-iron Smoke-stacks per Foot. 

(Porter Mfg. Co.) 



4 


5 


6 


7 


91 


11 


100 


150 


150 


150 


150' 


150 


16'4" 


20'4' 


21'10" 


22'7" 


23'8" 


24'H' 


6' 


9' 


8' 


9' 


10' 


10' 


125 


200 


200 


250 


275 


300 


18'5" 


23'8" 


25' 


29'8" 


33'6" 


36' 


7' 


10' 


10' 


12' 


12' 


14' 



Diam., 


Thick- 


Weight 


Diam., 


Thick- 


Weight 


Diam. 


Thick- 


Weight 


inches. 


w. g: 


per ft. 


inches. 


W. G. 


per ft. 


inches. 


W. G. 


per ft. 


10 


No. 16 


7.20 


26 


No. 16 


17.50 


20 


No. 14 


18.33 


12 


" 


8.66 


28 


" 


18.75 


22 


" 


20.00 


14 


" 


9.58 


30 


" 


20.00 


24 


" 


21.66 


16 


<( 


11.68 


10 


No. 14 


9.40 


26 


" 


23.33 


20 


•' 


13.75 


12 


" 


11.11 


28 


" 


25.00 


22 


" 


15.00 


14 


" 


13.69 


30 


" 


26.66 


24 


" 


16.25 


16 


" 


15.00 









Sheet-iron Chimneys. (Columbus Machine Co.) 



Diameter 
Chimney, 
inches. 


Length 

Chimney, 

feet. 


Thick- 
ness 
Iron, 
B. W. G. 


Weight, 
lbs. 


Diameter 

Chimney, 

inches. 


Length 

Chimney, 

feet. 


Thick- 
ness 
Iron, 
B. W. G 


Weight, 
lbs. 


10 


20 


No. 16 


160 


30 


40 


No. 15 


960 


15 


20 


". 16 


240 


32 


40 


" 15 


1,020 


20 


20 


" 16 


320 


34 


40 


" 14 


1,170 


22 


20 


" 16 


350» 


36 


40 


Y 14 


1,240 


24 


40 


" 16 


760 


38 


40 


" 12 


1,800 


26 


40 


" 16 


826 


40 


40 


" 12 


1,890 


28 


40 


•' 15 


900 











742 



THE STEAM-ENGINE. 



THE STEAM-ENGINE. 

Expansion of Steam. Isothermal and Adlabatic— Accord- 
ing to Mariotte's law, the volume of a perfect gas, the temperature being 



kept constant, varies inversely as its pressure, or p > 



1 



pv = a constant. 



The curve constructed from this formula is called the isothermal curve, or 
curve of equal temperatures, and is a common or rectangular hyperbola. 
The relation of the pressure and volume of saturated steam, as deduced 
from Regnault's experiments, and as given in Steam tables, is approxi- 
mately, according to Rankine (S. E., p. 403), for pressures not exceeding 1^0 

lbs., p oc — , orp qc v~ ib, or pv ~\i — pv~ ' = a constant. Zeuner has 

1516 

found that the exponent 1.0646 gives a closer approximation. 
When steam expands in a closed cylinder, as in an engine, according to 

Rankine (S. E., p. 385), the approximate law of the expansion is p <*- — — , or 

pccv~ 5, ovpv~ = a constant. The curve constructed from this for- 
mula is called the adiabatic curve, or curve of no transmission of heat. 

Peabody (Therm., p. 112) says : " It is probable that this equation was 
obtained by comparing the expansion lines on a large number of indicator- 
diagrams. . . . There does not appear to be any good reason for using an 
exponential equation in this connection, . . . and the action of a lagged steam- 
engine cylinder is far from being adiabatic. . . . For general purposes the 
hyperbola is the best curve for comparison with the expansion curve of an 
indicator-card. . . ." Wolff and Denton, Trans. A. S. M. E., ii. 175, say : 
" From a number of cards examined from a variety of steam-engines in cur- 
rent use, we find that the actual expansion line varies between the 10/9 
adiabatic curve and the Mariotte curve. 1 ' 

Prof. Thurston (A. S. M. E , ii. 203), says he doubts if the exponent ever 
becomes the same in any two engines, or even in the same engines at dif- 
ferent times of the day and under varying conditions of the clay. 

Expansion of Steam according* to Mariotte's Law and 
to the Adiabatic Law, (Trans. A. S. M. E., ii. 156.)— Mariotte's law: 



pv- 



Pm 



1 



= Pi^i ; values calculated from formula — = — (1 + hyp log R), in which 

R — Vj-t-Vi, Pi = absolute initial pressure, Pm — absolute mean pressure, 
Vj = initial volume of steam incylinder at pressure p x , v 2 = final volume of 
steam at final pressure. Adiabatic law: pv$ = PiV 1 x $\ values calculated 

from formula-— = \0R ~ 1 - 9R ~ s - 

Pi 





Ratio of Mean 




Ratio of Mean 




Ratio of Mean 




to Initial 


Ratio 


to Initial 


Ratio 


to Initial 


Ratio of 


Pressure. 


of 


Pressure. 


of 


Pressure. 


Expan- 




Expan- 
sion R. 




Expan- 




sion R. 


Mar. 


Adiab. 


Mar. 


Adiab. 


sion R. 


Mar. 


Adiab. 


1.00 


1.000 


1.000 


3.7 


.624 


.600 


6. 


.465 


.438 


1.25 


.978 


.976 


3.8 


.614 


.590 


6.25 


.453 


.425 


1.50 


.937 


.931 


3.9 


.605 


.580 


6.5 


.442 


.413 


1.75 


.891 


.881 


4. 


.597 


.571 


6.75 


.431 


.403 


2. 


.847 


.834 


4.1 


.588 


.562 


7 


.421 


.393 


2.2 


.813 


.798 


4.2 


.580 


.554 


7~25 


.411 


.383 


2.4 


.781 


.765 


4.3 


.572 


.546 


7.5 


.402 


.374 


2.5 


.766 


.748 


4.4 


.564 


.538 


7.75 


.393 


.365 


2.6 


.752 


.733 


4.5 


.556 


.530 


8. 


.385 


.357 


2.8 


.725 


.704 


4.6 


.549 


.523 


8.25 


.377 


.349 


3. 


.700 


.678 


4.7 


.542 


.516 


8.5 


.369 


.342 


3.1 


.688 


.666 


4.8 


.535 


.509 


8.75 


.362 


.335 


3.2 


.676 


.654 


4.9 


.528 


.502 


9. 


.355 


.328 


3.3 


.665 


.642 


5.05 


.562 


.495 


9.25 


.349 


.321 


3.4 


.654 


.630 


5 2 


.506 


.479 


9.5 


.342 


.315 


3.5 


.644 


.620 


5.5 


.492 


.464 


9.75 


.336 


.309 


3.6 


.634 


.610 


5.75 


.478 


.450 


10. 


.330 


.303 



MEAN" AND TERMINAL ABSOLUTE PRESSURES. 743 



Mean Pressure of Expanded Steam.— For calculations of 
engines it is generally assumed that steam expands according to Mariotte's 
law, the curve of the expansion line being a hyperbola. The mean pressure, 
measured above vacuum, is then obtained from the formula 



Pm = p 



1 + hyp log R 



in which Pm is the absolute mean pressure, p, the absolute initial pressure 
taken as uniform up to the point of cut-off, and R the ratio of expansion. If 
I — length of stroke to the cut-off, L = total stroke, 
L 



Pm = - 



Pjl+Pilhyiplog- 



and if R = 



Pm = p 



1 + hyp log R 



Mean and Terminal Abso lute Pressures.— Mariotte's 
Law. — The values in the following table are based on Mariotte's law, 
except those in the last column, which give the mean pressure of superheated 
steam, which, according to Rankine, expands in a cylinder according to 
the law p <x v~Vk. These latter values are calculated from the formula 



Pm 17 - 1612 - A 



J may be found by extracting the square root of — 



four times. From the mean absolute pressures given deduct the mean back 
pressure (absolute) to obtain the mean effective pressure. 



Rate 
of 
Expan- 
sion. 


Cut- 
off. 


Ratio of 

Mean to 

Initial 

Pressure. 


Ratio of 
Mean to 
Terminal 
Pressure. 


Ratio of 
Terminal 

to Mean 
Pressure. 


Ratio of 

Initial 

to Mean 

Pressure. 


Ratio of 

Mean to 

Initial 

Dry Steam. 


30 

28 


0.033 

0.036 
0.038 
0.042 
0.045 
0.050 
0.055 
0.062 
0.066 
0.071 
0.075 
0.077 
0.083 
0.091 
0.100 
0.111 
0.125 
0.143 
0.150 
0.166 
0.175 
0.200 
0.225 
0.250 
0.275 
0.300 
0.333 
0.350 
0.375 
0.400 
0.450 
0.500 
0.550 
600 
0.625 
0.650 
0.675 


0.1467 
0.1547 
0.1638 
0.1741 
0.1860 
0.1998 
0.2161 
0.2358 
0.2472 
0.2599 
0.2690 
0.2742 
0.2904 
0.3089 
0.3303 
0.3552 
0.3849 
0.4210 
0.4347 
0.4653 
0.4807 
0.5218 
0.5608 
0.5965 
0.6308 
0.6615 
0.6995 
0.7171 
0.7440 
. 0.7664 
0.8095 
0.8465 
0.8786 
0.9066 
0.9187 
0.9292 
0.9405 


4.40 
4.33 
4.26 
4.18 
4.09 
4.00 
3.89 
3.77 
3.71 
3.64 
3.59 
3.56 
3.48 
3.40 
3.30 
3.20 
3.08 
2.95 
2.90 
2.79 
2.74 
2.61 
2.50 
2.39 
2.29 
2.20 
2.10 
2.05 
1.98 
1.91 
1.80 
1.69 
1.60 
1.51 
1.47 
1.43 
1.39 


0.227 

0.231 
0.235 
0.239 
0.244 
0.250 
0.256 
0.265 
0.269 
0.275 
0.279 
0.280 
0.287 
0.294 
0.303 
0.312 
0.321 
0.339 
0.345 
0.360 
0.364 
0.383 
0.400 
0.419 
0.437 
0.454 
0.476 
0.488 
0.505 
0.523 
0.556 
0.591 
0.626 
0.662 
0.680 
0.699 
0.718 


6.82 

6.46 

6.11 

5.75 

5.38 

5.00 

4.63 

4.24 

4.05 

3.85 

3.72 

3.65 • 

3.44 

3.24 

3.03 . 

2.81 

2.60 

2.37 

2. SO 

2.15 

2.08 

1.92 

1.78 

1.68 

1.58 

1.51 

1.43 

1.39 

1.34 

1.31 

1.24 

1.18 

1.14 

1.10 

1.09 

1.07 

1.06 


0.136 


26 




24 
22 




20 
18 


0.186 


16 




15 




14 




13.33 
13 


0.254 


12 




11 




10 
9 


0.314 


8 

6.66 
6.00 


0.370 
6*417 '" 


5.71 




5.00 
4.44 


0.506 


4.00 
3.63 


0.582 


3.33 
3.00 


0.648 


2.86 
2.66 


0.707 


2.50 
2.22 
2.00 
1.82 
1.66 
1.60 


0.756 
0.800 
0.840 

8.874 
0.900 


1.54 

1.48 


0.926 







744 



THE STEAM-ENGINE. 



Calculation of Mean Effective Pressure, Clearance and 
Compression Considered.— In the above tables no account is taken 

LcoU-- 7 J °f clearance, which in actual 

steam-engines modifies the ratio 
of expansion and the mean pres- 
sure ; nor of compression and 
back-pressure, which diminish 
the mean effective pressure. In 
the following calculation these 
elements are considered. 

L — length of stroke, I = length 
before cut-off, x — length of com- 
pression part of stroke, c = clear- 
ance, p x = initial pressure, p b = 
back pressure, pc = pressure of 
clearance steam at end of com- 
w pression. All pressures are abso- 
L b lute, that is, measured from a 
perfect vacuum. 



t 




I 


k 


D 


V 


jr 




\ A ^\^ 


14 




V ^^i 


U 


C 


> V 


-c~ 


—^-—l B 



Area of ABCD = Pl (l + c)(l + hyp log - 



B = p b (L-x); 

C = pcc(l -f byp log ^±^) = p b (x + c)(l + hyp log £±£) 

D = (p, - pc)c = p x c - p h (x + c). 



D = (Pi - pc)c = p x c - p b (x + c), 
Area of A = ABCD -(B + C-fD) 

L + c 
l + c 

x -\- c 



= pS + c)(l + hyp log 4^f) 
[p b (L - x) 4- jo b (z + c)(l + hyp log ^~^J + Pic - p 5 (x + c)J 



= Pi(i+c)(l+hyplog^±^) 



■Pb 



[>- 



») + (* + c) hyp log 



x-\- c~ 



Mean effective pressure = - 



Example.— Let L = 1, Z = 0.25, « = 0.25, c = 0.1, p x = 1 
Area A = 60(.25 4 .l)(l +hyp log -H-). 

- 2 [(I - .25) + .35 hyp log ■ 



>lbs., p & = 2 lbs. 



f] 



= 21(1 + 1.145) -2[.75- 
= 45.045 - 2.377 - 6 = 'c 



5 X 1.253] - 6 

68 = mean effective pressure. 



The actual indicator-diagram generally shows a mean pressure consider- 
ably less than that due to the initial pressure and the rate of expansion. The 
causes of loss of pressure are: 1. Friction in the stop-valves and steam- 
pipes. 2. Friction or wire-drawing of the steam during admission and cut- 
off, due chiefly to defective valve-gear and contracted steam-passages. 
3. Liquefaction during expansion. 4. Exhausting before the engine has 
completed its stroke. 5. Compression due to early closure of exhaust. 
6. Friction in the exhaust-ports, passages, and pipes. 

Re-evaporation during expansion of the steam condensed during admis- 
sion, and valve-leakage after cut off, tend to elevate the expansion line of 
the diagram and increase the mean pressure. 

If the theoretical mean pressure be calculated from the initial pressure 
a-nd the rate of expansion on the supposition that the expansion curve fol- 



EXPANSION OF STEAM, 745 

lows Mariotte's law, pv — a constant, and the necessary corrections are 
made for clearance and compression, the expected mean pressure in practice 
may be found by multiplying the calculated results by the factor in the 
following table, according to Seaton. 

Particulars of Engine. Factor. 

Expansive engine, special valve-gear, or with a separate 
cut-off valve, cylinder jacketed 0.94 

Expansive engine having large ports, etc., and good or- 
dinary valves, cylinders jacketed 0.9 to 0.92 

Expansive engines with the ordinary valves and gear as 
in general practice, and unjacketed 0.8 to 0.85 

Compound engines, with expansion valve to h.p. cylin- 
der; cylinders jacketed, and with large ports, etc 0.9 to 0.92 

Compound engines, with ordinary slide-valves, cylinders 
jacketed, and good ports, etc 0.8 to 0.85 

Compound engines as in general practice in the merchant 
service, with early cut-off in both cylinders, without 
jackets and expansion-valves 0.7 to 0.8 

Fast-running engines of the type and design usually fitted 
in war-ships 0.6 to 0.8 

If no correction be made for clearance and compression, and the engine 
is in accordance with general modern practice, the theoretical mean pres* 
sure may be multiplied by 0.96, and the product by the proper factor in the 
table, to obtain the expected mean pressure. 

Given the Initial Pressure and the Average Pressure, to 
Find the Ratio of Expansion and the Period of Admis- 
sion. 

P = initial absolute pressure in lbs. per sq. in. ; 

p = average total pressure during stroke in lbs. per sq. in.; 

L = length of stroke in inches; 

I — period of admission measured from beginning of stroke; 

c = clearance in inches; 

R = actual ratio of expansion = , T" (1) 

n _ P(l + hyp log R) 
p ~ R 

To find average pressure p, taking account of clearance, 
n _ P(Z + c) + P{1 + c) hyp lug R-Pc 

p ~ L ' {4) 

whence pL + Pc = P(l + c)(l + hyp log R) ; 

—L + c 

Given p and P, to find R and I (by trial and error). — There being two un- 
known quantities R and I, assume one of them, viz., the period of admission 
I, substitute it in equation (3) and solve for R. Substitute this value of R in 

the formula (1), or I = — ^ c, obtained from formula (1), and find I. If 

the result is greated than the assumed value of /, then the assumed value of 
the period of admission is too long; if less, the assumed value is too short. 
Assume a new value of /, substitute it in formula (3) as before, and continue 
by this method of trial and error till the required values of R and I are 
obtained. 
Example.— P = 70, p = 42.78, L = 60", c = 3'-, to find I. Assume I = 21 in 

hyp log B = -j^ - 1 = -4^-3 1 = 1.653 - 1 = .653; 

hyp log R = .653, whence R = 1.92, 



HQ THE STEAM-ENGINE. 



which is greater than the assumed value, 21 inches. 
Now assume I — 15 inches : 



hyp log R = - 1 = 1.204, whence £ = 3.5; 

15 -f- d 



Z = — ^ c = -— — 3 = 18 — 3 = 15 inches, the value assumed. 

K 6.0 

Therefore R = 3.5, and I — 15 inches. 

Period of Admission Required for a Given Actual Ratio of Expansion: 
I = — =- c, in inches . (4) 

In percentage of stroke, I = — ^- — '— — p. ct. clearance. . (5) 



P 

L+c ~ R' 



Pressure at any other' a Point of the Expansion.— Let L x = length of stroke 
up to the given point. 

Pressure at the given point = (7) 

L x -f c 

WORE OF STEAM IN A SINGLE CYLINDER. 

To facilitate calculations of steam expanded in cylinders the table on the 
next page is abridged from Clark on the Steam-engine. The actual ratios 
of expansion, column 1, range from 1.0 to 8.0, for which the hyperbolic 
logarithms are given in column 2. The 3d column contains the periods of 
admission i*elative to the actual ratios of expansion, as percentages of the 
stroke, calculated by formula (5) above. The 4th column gives the values 
of the mean pressures relative to the initial pressures, the latter being taken 
as 1, calculated by formula (2). In the calculation of columns 3 and 4, clear- 
ance is taken into account, and its amount is assumed at 7% of the stroke. 
The final pressures, in the 5th column, are such as would be arrived at by 
the continued expansion of the whole of the steam to the end of the stroke, 
the initial pressure being equal to 1. They are the reciprocals of the ratios 
of expansion, column 1. The 6th column contains the relative total per- 
formances of equal weights of steam worked with the several actual ratios 
of expansion; the total performance, when steam is admitted for the whole 
of the stroke, without expansion, being equal to 1. They are obtained by 
dividing the figures in column 4 by those in column 5. 

The pressui-es have been calculated on the supposition that the pressure of 
steam, during its admission into the cylinder, is uniform up to the point of 
cutting off, and that the expansion is continued regularly to the end of the 
stroke. The relative performances have been calculated without any allow- 
ance for the effect of compressive action. 

The calculations have been made for periods of admission ranging from 
100$, or the whole of the stroke, to 6.4$, or 1/16 of the stroke. And though, 
nominally, the expansion is 16 times in the last instance, it is actually only 
8 times, as given in the first column. The great difference between the 
nominal and the actual ratios of expansion is caused by the clearance, 
which is equal to 7% of the stroke, and causes the nominal volume of steam 
admitted, namely, 6.4$, to be augmented to 6.4 -J- 7 = 13.4$ of the stroke, or, 
say, double, for expansion. When the steam is cut off at 1/9, the actual 
expansion is only 6 times; when cut off at 1/5, the expansion is 4 times; 
when cut off at %, the expansion is 2% times; and to effect an actual expan- 
sion to twice the initial volume, the steam is cut off at 46J^$ of the stroke, 
not at half-stroke. 



WORK OF STEAM IN A SINGLE CYLINDER. 



747 



Expansive Working; of Steam— Actual Ratios of Expan- 
sion, with the Relative Periods ot Admission, Press- 
ures, and Performance. 

Steam-pressure 100 lbs. absolute. Clearance atjeach end of the cylinder 7% 
of the stroke. 

(Single Cylinder.) 



1 


2 


3 


4 


5 


6 


7 


8 


9 


ctual Ratio of Ex- 
pansion, or No. of 
Volumes to which 
the Initial Volume 
is Expanded. 


vperbolic Loga- 
rithm of Actual 
Ratio of Expan- 
sion. 


±riod of Admis- 
sion or Cut-off, 
7% Clearance. 

veras-e Total Press- 


11 


1 1 

•-, CD 
^ 1 

a & 

. eg 

o Sa 


Ratio of Total Per- 
formance of Equal 
Weights of Steam. 
(Col. 4.-*- Col 5.) 


® Cfi" 

= >>£ 


uantity of Steam 
Consumed per 
H.P. of Actual 
Work done per houi 


et Capacity of Cyl- 
inder per lb. of 100 
lbs. Steam ad- 
mitted in 1 stroke. 
Cubic feet. 


< 


W 


cu <t 




H M 


< 


G> 


£ 


1 


.0000 


100 1 


000 


1.000 


1.000 


58,273 


34.0 


4.05 


1.1 


.0953 


90.3 


996 


.909 


1.096 


63,850 


31.0 


4.45 


1.18 


.1698 


83.3 


986 


.847 


1.164 


67,836 


29.2 


4.78 


1.23 


.2070 


80 


980 


.813 


1.206 


70,246 


28.2 


4.98 


1.3 


.2624 


75.3 


969 


.769 


1.261 


73,513 


26.9 


5.26 


1.39 


.3293 


70 


953 


.719 


1.325 


77,242 


25.6 


5.63 


1.45 


.3716 


66.8 


942 


.690 


1.365 


79,555 


24.9 


5.87 


1 54 


.4317 


62.5 


925 


.649 


1.425 


83,055 


23.8 


6.23 


1.6 


.4700 


59.9 


913 


.625 


1.461 


85,125 


23.3 


6.47 


1.75 


.5595 


54.1 


883 


.571 


1.546 


90,115 


22.0 


7.08 


1.88 


.6314 


50 


860 


.532 


1.616 


94,200 


21.0 


7.61 


2 


.3931 


46.5 


836 


.5 


1.672 


97,432 


20.3 


8.09 


2.28 


.8241 


40 


787 


.439 


1.793 


104,466 


19.0 


9.23 


2.4 


.8755 


37.6 


766 


.417 


1.837 


107,050 


18.5 


9.71 


2.65 


.9745 


33.3 


726 


.377 


1.925 


112,220 


17.7 


10.72 


2.9 


1.065 


29.9 


692 


.345 


2.006 


116,88? 


16.9 


11.74 


3.2 


1.163 


26.4 


652 


.313 


2.083 


121,386 


16.3 


12.95 


3.35 


1.209 


25 


637 


.298 


2.129 


124,066 


16.0 


13.56 


3.6 


1.281 


22.7 


608 


.278 


2.187 


127,450 


15.5 


14.57 


3.8 


1.335 


21.2 


589 


.263 


2.240 


130,533 


15.2 


15.38 


4 


1.386 


19.7 


569 


.250 


2.278 


132.770 


14.9 


16.19 


4.2 


1.435 


18.5 


551 


.238 


2.315 


134.900 


14.7 


17.00 


4.5 


1.504 


16.8 


526 


.222 


2.370 


138,130 


14.34 


18.21 


4.8 


1.569 


15.3 


503 


.208 


2.418 


140,920 


14.05 


19.43 


5 


1.609 


14.4 


488 


.200 


2.440 


142,180 


13.92 


20.23 


5.2 


1.649 


13.6 


476 


.193 


2.466 


143,720 


13.78 


21.04 


5.5 


1.705 


12.5 


457 


.182 


2.511 


146.325 


13.53 


22.25 


5.8 


1.758 


11.4 


438 


.172 


2.547 


148,390 


13.34 


23.47 


5.9 


1.775 


11.1 


432 


.169 


2.556 


148,940 


13.29 


23.87 


6.2 


1.825 


10.3 


419 


.161 


2.585 


150,630 


13.14 


25.09 


6.3 


1.841 


10 


413 


.159 


2.597 


151,370 


13.08 


25.49 


6.6 


1.887 


9.2 


398 


.152 


2.619 


152,595 


12.98 


26.71 


7 


1.946 


8.3 


381 


.143 


2.664 


155,200 


12.75 


28.33 


7.3 


',1.988 


7.7 


369 


.137 


2.693 


156,960 


12.61 


29.54 


7.6 


2.028 


7.1 


357 


.132 


2.711 


157,975 


12.53 


30.76 


7.8 


2.054 


6.7 


348 


.128 


2.719 


158,414 


12.50 


31.57 


8 


2.079 


6.4 


342 


.125 


2.736 


159,433 


11.83 


32.38 



Assumptions op the Table.— That the initial pressure is uniform; that 
the expansion is complete to the end of the stroke; that the pressure in ex- 
pansion varies inversely as the volume; that there is no back-pressure of 
exhaust or of compression, and that clearance is 7% of the stroke at each 
end of the cylinder. No allowance has been made for loss of steam by cyl- 
inder-condensation or leakage. 

Volume of 1 lb. of steam of 100 lbs. pressure per sq. in., or 14,400 

lbs. per sq, ft 4.33 cu. ft. 

Product of initial pressure and volume 62,352 f t.-lbs. 



748 



THE STEAM-EKGIKE. 



Though a uniform clearance of 7% at each end of the stroke has been 
assumed as an average proportion for the purpose of compiling the table, 
the clearance of cylinders with ordinary slides varies considerably— say 
from 5% to 10$. (With Corliss engines it is sometimes as low as 2%.) With 
the clearance, 7%, that has been assumed, the table gives approximate re- 
sults sufficient for most practical purposes, and more trustworthy than re- 
sults deduced by calculations based on simple tables of hyperbolic loga- 
rithms, where clearance is neglected. 

Weight of steam of 100 lbs. total initial pressure admitted for one stroke, 
per cubic foot of net capacity of the cylinder, in decimals of a pound = 
reciprocal of figures in column 9. 

Total actual work done by steam of 100 lbs. total initial pressure in one 
stroke per cubic foot of net capacity of cylinder, in foot-pounds = figures 
in column 7 -j- figures in column 9. 

Rule 1: To find the net capacity of cylinder for a given weight of steam 
admitted for one stroke, and a given actual ratio of expansion. (Column 9 
of table.)— Multiply the volume of 1 lb. of steam of the given pressure by the 
given weight in pounds, and by the actual ratio of expansion. Multiply the 
product by 100, and divide by 100 plus the percentage of clearance. The 
quotient is the net capacity of the cylinder. 

Rule 2: To find the net capacity of cylinder for the performance of a 
given amount of total actual work in one stroke, with a given initial press- 
ure and actual ratio of expansion — Divide the given work by the total 
actual work done by 1 lb. of steam of the same pressure, and with the same 
actual ratio of expansion; the quotient is the weight of steam necessary to 
do the given work, for which the net capacity is found by Rule 1 preceding. 

Note.— 1. Conversely, the weight of steam admitted per cubic foot of net 
capacity for one stroke is the reciprocal of the cylinder-capacity per pound 
of steam, as obtained by Rule 1. 

2. The total actual work done per cubic foot of net capacity for one stroke 
is the reciprocal of the cylinder-capacity per foot-pound of work done, as 
obtained by Rule 2. 

3. The total actual work done per square inch of piston per foot of the 
stroke is 1 /144th part of the work done per cubic foot. 

4. The l esistance of back pressure of exhaust and of compression are to 
be added to the net work required to be done, to find the total actual work. 

Appendix to above Table— Multipliers for Net Cylinder-capacity, and 
Total Actual Work done. 

(For steam of other pressures than 100 lbs. per square inch.) 





Multipliers. 


Total Pres- 
sures per 
square inch. 


Multipliers. 


Total Pres- 
sures per 
square inch. 


For Col. 7. 

Total Work 

by 1 lb. of 

Steam. 


For Col. 9. 
Capacity 

of 
Cylinder. 


For Col. 7. 

Total Work 

by 1 lb. of 

Steam. 


For Col. 9. 
Capacity 

of 
Cylinder. 


lbs. 
65 
70 
75 
80 
85 
90 
95 


.975 
.981 
.986 
.988 
.991 
.995 
.998 


1.50 
1.40 
1.31 
1.24 
1.17 
1.11 
1.05 


lbs. 

100 
110 
120 
130 
140 
150 
160 


1.000 
1.009 
1.011 
1.015 
1.022 
1.025 
1.031 


1.00 
.917 
.843 
.781 
.730 
.683 
.644 



The figures in the second column of this table are derived by multiplying 
the total pressure per square foot of any given steam by the volume in 
cubic feet of 1 lb. of such steam, and dividing the product by 62,352, which 
is the product in foot-pounds for steam of 100 lbs. pressure. The quotient 
is the multiplier for the given pressure. 

The figures in the third column are the quotients of the figures in the 
second column diyided by the ratio of the pressure of the given steam to 100 
lbs. 

Measures tor Comparing the Duty of Engines.— Capacity is 
measured in horsepowers, expressed by the initials, H.P.: 1 H.P. = 33.000 
ft. -lbs. per minute, = 550 ft.-lbs. per second, = 1,980,000 ft.-lbs. per hour. 



WOKE OF STEAM IN A SINGLE CYLINDER. 749 

1 ft. -lb. = a pressure of 1 lb. exerted through a space of 1 ft. Economy is 
measured, 1, in pounds of coal per horse-power per hour; 2, in pounds of 
steam per horse-power per hour. The second of these measures is the more 
accurate and scientific, since the engine u»3es steam and not coal, and it is 
indepndent of the economy of the boiler. 

In gas-engine tests the common measure is the number of cubic feet 
of gas (measured at atmospheric pressure) per horse-power, but as all gas 
is not of the same quality, it is necessary for comparison of tests to give the 
analysis of the gas. When the gas for one engine is made in one gas-pro- 
ducer, then the number of pounds of coal used in the producer per hour per 
horse-power of the engine is the proper measure of economy. 

Economy, or duty of an engine, is also measured in the number of foot- 
pounds of work done per pound of fuel. As 1 horse-power is equal to 1,980,- 
000 ft. -lbs. of work in an hour, a duty of 1 lb. of coal per H.P. per hour 
would be equal to 1,980,000 ft. -lbs. per lb. of fuel; 2 lbs. per H.P. per hour 
equals 990,000 ft. -lbs. per lb. of fuel, etc. 

The duty of pumping-engines is commonly expressed by the number of 
foot-pounds of work done per 100 lbs. of coal. 

When the duty of a pumping-engine is thus given, the equivalent number 
of pounds of fuel consumed per horse-power per hour is found by dividing 
198 by the number of millions of foot-pounds of duty. Thus a pumping- 
engine giving a duty of 99 millions is equivalent to 198/99 = 2 lbs. of fuel per 
horse-power per hour. 

Efficiency Measured in Thermal Units per Minute.— 
Some writers express the efficiency of an engine in terms of the number of 
thermal units used by the engine per minute for each indicated horse-power, 
instead of by the number of pounds of steam used per hour. 

The heat chargeable to an engine per pound of steam is the difference be- 
tween the total heat in a pound of steam at the boiler-pressure and that in 
a pound of the feed-water entering the boiler. In the case of condensing 
engines, suppose we have a temperature in the hot-well of 101° F., corre- 
sponding to a vacuum of 28 in. of mercury, or an absolute pressure of 1 lb. 
per sq. in. above a perfect vacuum : we may feed the water into the boiler 
at that temperature. In the case of a non-condensing-engine, by using a por- 
tion of the exhaust steam in a good feed-water heater, at a pressure a trifle 
above the atmosphere (due to the resistance of the exhaust passages 
through the heater), we may obtain feed-water at 212°. One pound of steam 
used by the engine then would be equivalent to thermal units as follows : 
Pressure of steam by gauge: 

50 75 100 125 150 175 200 

Total heat in steam above 32° : 

1172.8 1179.6 1185.0 1189.5 1193.5 1197.0 1200.2 

Subtracting 69.1 and 180.9 heat-units, respectively, the heat above 32° in 
feed -water of 101° and 212° F., we have- 
Heat given by boiler: 

Feed at 101° 1103.7 1110.5 1115. a 1120.4 1124.4 1127.9 1131.1 

Feed at 212° 991.9 998.7 1004.1 1008.6 1012.6 1016.1 1019.3 

Thermal units per minute used by an engine for each pound of steam used 
per indicated horse-power per hour: 

Feedatl0l° 18.40 18.51 18.60 18.67 la. 74 IS. 80 18.85 

Feed at 212° 16.53 16.65 16.74 16.81 16.88 16.94 16.99 

Examples. — A triple-expansion engine, condensing, with steam at 1751bs., 
gauge and vacuum 28 in., uses 13 lbs. of water per I.H.P. per hour, and a 
high-speed non-condensing engine, with steam at 100 lbs. gauge, uses 30 
lbs. How many thermal units per minute does each consume ? 

Ans— 13 X 18.80 = 244.4, and 30 X 16.74 = 502.2 thermal units per minute. 

A perfect engine converting ail the heat -energy of the steam into work 
would require 33,000 ft. -lbs. -f- 778 = 42.4164 thermal units per minute per 
indicated horse-power. This figure, 42.4164, therefore, divided by the num- 
ber of thermal units per minute per I.H.P. consumed by an engine, gives its 
efficiency as compared with an ideally perfect engine. In the examples 
above, 42.4164 divided by 244.4 and by 502.2 gives 17.35^ and 8Ao% efficiency, 
respectively. 

Total Work Done by One Pound of Steam Expanded in 
a Single Cylinder. (Column 7 of table.)— If 1 pound of water be con- 
verted into steam of atmospheric pressure = 2116.8 lbs. per sq. ft., it occu- 
pies a volume equal to 26.36 cu. ft. The work done is equal to 2116.8 lbs. 



750 



THE STEAM-EKGINE. 



X 26.36 ft. = 55,788 ft. -lbs. The heat equivalent of this work is (55,788 -f- 778 
=) 71.7 units. This is the work of 1 lb. of steam of one atmosphere acting 
on a piston without expansion. 

The gross work thus done on a piston by 1 lb. of steam generated at total 
pressures varying from 15 lbs. to 100 lbs. per sq. in. varies in round numbers 
from 56,000 to 62,000 ft.-lbs., equivalent to from 72 to 80 units of heat. 

This work of 1 lb. of steam without expansion is reduced by clearance 
according to the proportion it bears to the net capacity of the cylinder. If 
the clearance be 7$ of the stroke, the work of a given weight of steam with- 
out expansion, admitted for the whole of the stroke, is reduced in the ratio 
of 107 to 100. 

Having determined by this ratio the quantity of work of 1 lb. of steam with- 
out expansion, as reduced by clearance, the work of the same weight of steam 
for various ratios of expansion may be found by multiplying it by the relative 
performance of equal weights of steam, given in the 6th column of the table. 

Quantity of Steam Consumed per Horse-power of Total 
Work, per Hour. (Column 8 of table.)— The measure of a horse-power 
is the performance of 33,000 ft.-lbs. per minute, or 1,980,000 ft.-lbs. per hour. 
This work, divided by the w r ork of 1 lb of steam, gives the weight of steam 
required per horse-power per hour. For example, the total actual work 
done in the cylinder by 1 lb. of 100 lbs. steam, without expansion and with 
1% of clearance, is 58,273 ft.-lbs. ; and ' 5g g ' 73 = 34 lbs. of steam, is the weight 
of steam consumed for the total work done in the cylinder per horse-power 
per hour. For any shorter period of admission with expansion the weight 
of steam per horse-pow'er is less, as the total work of 1 lb. of steam is more, 
and may be found by dividing 1,980,000 ft.-lbs. by the respective total work 
done; or by dividing 34 lbs. by the ratio of performance, column 6 in the 
table. 
Real Ratios of Expansion with Clearances from Q to K%. 

' g S Points of Cut-off. 



5 a> 


.10 





10 


01 


9.111 


0125 


9 


0150 


8.826 


.0175 


8.659 


.02 


8.5 


.0225 


8.346 


.0250 


8.2 


.0275 


8.088 


.03 


7.933 


03-25 


7.792 


0350 


7.666 


.0375 


7.545 


.04 


7.428 


.0425 


7.315 


.0450 


7.206 


.0475 


7.102 


.05 


7 


0525 


6.901 


.0550 


6.806 


.0575 


6.714 


06 


6.625 


.0625 


0.538 


.065C 


6.454 


.0675 


6.373 


.07 


6.294 



7.481 
7.363 
7.25 
7.133 

7.034 
6.932 
6.833 
6.738 

6.645 
6.555 
6.468 
6.390 



5.605 
5.545 



4.677 

4. 
4.595 

4.555 
4.516 

4.41' 
4.440 
4.404 
4.484 

4.333 



4.130 
4.106 

4.076 
4.017 
4.045 
3 



.25 


.30 

3.333 
3.258 
3.24 
3.222 
3.204 


.333 


4 

3.884 
3.875 
3.830 
3.803 


3 
2.944 

2.930 
2.916 
2.902 


3.777 
3.752 
3.727 
3.702 


3.187 
3.170 
3.153 
3.137 


2.889 
2.876 
2.863 
2.850 


3.678 
3.654 
3.631 
3.608 


3.121 
3.105 
3.089 
3.074 


2.837 
2.824 
2.812 
2.800 


3.58 
3.564 
3.542 
3.521 


3.058 
3.043 
3 028 
3.014 


2.788 
2.776 
2.764 
2.752 


3.5 

3.478 
3.459 
3.439 


3 

2.986 
2.971 
2.957 


2.741 

2.730 
2.719 
2. 70S 


3.418 
3 407 
3.380 
3.362 


2.944 
2.931 
2.917 
2.904 


2.697 
2.686 
2.675 
2.665 


3.342 


2.892 


2.655 



2.667 
2.623 
2.612 
2.602 



2.574 
2.562 
2.552 

2 543 
2.533 
2.524 
2.515 

2.506 
2.49' 



2.470 
2.461 
2.453 
2.445 



2.428 
2.420 

2.412 



2.5 

2.463 
2.454 
2.445 



2.428 
2.420 
2.411 
2.403 

2.395 
2.: 

2.379 
2.371 



50 .60 .625 



2 

1.983 
1.975 
1 970 
1.1 



1.952 
1.947 

1.943 

1.938 
1.934 
1.930 



2.363 1.925 

2.355 1.921 

2.348 1.917 

2.340 1.913 



1.1 

1.904 

1.900 



2.325 

2.318 
2.311 

2.304 
2 

2.290 

2.283 

2.276 



1.884 
1.881 



1.43 
1.42 
1.42 
1.42 
1.42 

1.42 

1.41 

1.41 

41 

1.41 
1.41 
1.41 
1.41 

1.40 
1.40 
1.40 

1.40 

1.40 
1.40 



1 39 
1.39 
1.39 
1.39 



WORK OF STEAM IN A SINGLE CYLINDER. 751 

Relative Efficiency of 1 lb. of Steam with and without 
Clearance; back pressure and compression not considered. 

Mean total pressure = p = Jg + + Pff + e^hyp. tog, g - ft 

LetP=l; I, = 100; Z = 25; c = 7. 
107 _ 

32 ~ ' _ 32-f 32X l.i 
*~ 100 ~ 100 

If the clearance be added to the stroke, so that clearance becomes zero, 
the same quantity of steam being used, admission I being then = I -f c = 
32, and stroke L + c = 107. 



: .707. 



107 107 



That is, if the clearance be reduced to 0, the amount of the clearance 7 
being added to both the admission and the stroke, the same quantity of 
steam will do more work than when the clearance is 7 in the ratio 707 : 637, 
or 11$ more. 

Back Pressure Considered.— If back pressure = .10 of P, this 
amount has to be subtracted from p andp x giving p = .537, p x = .607, the 
work of a given quantity of steam used without clearance being greater 
than when clearance is 7 per cent in the ratio of 607 : 537, or 13% more. 

Effect of Compression. —By early closure of the exhaust, so that a 
portion of the exhaust-steam is compressed into the clearance-space, much 
of the loss due to clearance may be avoided. If expansion is continued 
down to the back pressure, if the back pressure is uniform throughout the 
exhaust-stroke, and if compression begins at such point that the exhaust- 
steam remaining in the cylinder is compressed to the initial pressure at the 
end of the back stroke, then the work of compression of the exhaust-steam 
equals the work done during expansion by the clearance-steam. The clear- 
ance-space being filled by the exhaust-steam thus compressed, no new steam 
is required to fill the clearance-space for the next forward stroke, and the 
work and efficiency of the steam used in the cylinder are just the same as if 
there were no clearance and no compression. When, however, there is a 
drop in pressure from the final pressure of the expansion, or the terminal 
pressure, to the exhaust or back pressure (the usual case), the work of com- 
pression to the initial pressure is greater than the work done by the expan- 
sion of the clearance-steam, so that a loss of efficiency results. In this 
case a greater efficiency can be attained by inclosing for compression a less 
quantity of steam than that needed to fill the clearance-space with steam of 
the initial pressure. (See Clark, S. E., p. 399, et seq.; also F. H. Ball, Trans. 
A. S. M. E., xiv. 1067.) It is shown by Clark that a somewhat greater effi- 
ciency is thus attained whether or not the pressure of the steam be carried 
down by expansion to the back exhaust-pressure. As a result of calcula- 
tions to determine the most efficient periods of compression for various 
percentages of back pressure, and for various periods of admission, he gives 
the table on the next page : 

Clearance in Low- and High-speed Engines. (Harris 
Tabor, Am. Mach., April 17, 1891.) — The consrruction of the high-speed 
engine is such, with its i - elatively short stroke, that the clearance must be 
much larger than in the releasing-valve type. The short-stroke engine is, 
of necessity, an engine with large clearance, which is aggravated when a 
variable compression is a feature. Conversely, the releasing-valve gear is, 
from necessity, an engine of slow rotative speed, where great power is 
obtainable from long stroke, and small clearance is a feature in its construc- 
tion. In one case the clearance will vary from 8% to 12% of the piston-dis- 
placement, and in the other from 2% to 3%. In the case of an engino with a 
clearance equalling 10% of the piston-displacement the waste room becomes 
enormous when considered in connection with an early cut-off. The system of 
compounding reduces the waste due to clearance in proportion as the steam 
is expanded to a lower pressure. The farther expansion is carried through 
a train of cylinders the greater will be the reduction of waste due to clear- 
ance. This is shown from the fact that the high-speed engine, expanding 



752 



THE STEAM-ENGINE. 



steam much less than the Corliss, will show a greater gain when changed 
from simple to compound than its rival under similar conditions. 
Compression of Steam in the Cylinder. 
Best Periods of Compression; Clearance 7 per cent. 





Total Back Pressure, in percentages of the total initial pressure. 


Cut-off in 








Percent- 


















ages of 


% 


5 


10 


15 


20 


25 


30 


35 


the 


















Stroke. 




Periods of Compression, in parts of the stroke. 




10$ 


65$ 


57$ 


44$ 


32$ 










15 


58 


52 


40 


29 


23$ 








20 


52 


47 


37 


27 


22 








25 


47 


42 


34 


26 


21 


17$ 






30 


42 


39 


32 


25 


20 


16 


14$ 


12$ 


35 


39 


35 


29 


23 


19 


15 


13 


11 


40 


36 


32 


27 


21 


18 


14 


13 


11 


45 


33 


30 


25 


20 


17 


14 


12 


10 


50 


30 


27 


23 


18 


16 


13 


12 


10 


55 


27 


24 


21 


17 


15 


13 


11 


9 


60 


24 


22 


19 


15 


14 


12 


11 


9 


65 


22 


20 


17 


15 


14 


12 


10 


8 


70 


19 


17 


16 


14 


14 


12 


10 


8 


75 


17 


16 


14 


13 


12 


11 


9 


8 



Notes to Table.— 1. For periods of admission, or percentages of back 
pressure, other than those given, the periods of compression may be readily 
found by interpolation. 

2. For any other clearance, the values of the tabulated periods of com- 
pression are to be altered in the ratio of 7 to the given percentage of 
clearance. 

Cylinder-condensation may have considerable effect upon the best point 
of compression, but it has not yet (1893) been determined by experiment. 
(Trans. A. S. M. E.. xiv. 1078.) 

Cylinder- condensation.— Rankine, S. E., p. 421, says : Conduction 
of heat to and from the metal of the cylinder, or to and from liquid water 
contained in the cylinder, has the effect of lowering the pressure at the be- 
ginning and raising it at the end of the stroke, the lowering effect being on 
the whole greater than the raising effect. In some experiments the quantity 
of steam wasted through alternate liquefaction and evaporation in the 
cylinder has been found to be greater than the quantity wnich performed 
the work. 

Percentage of Loss l>y Cylinder-condensation, taken at 
Cut-off. (From circular of the Ashcroft Mfg. Co. on the Tabor 
Indicator, 1889.) 



13 

§J2 

<D O ^ 
GO 


Percent, of Feed -water accounted 
for by the Indicator diagram. 


Percent, of Feed-water Consump- 
tion due to Cylinder-condensat'n. 


Simple 
Engines. 


Compound 
Engines, 
h.p. cyl. 


Triple-ex- 
pansion 
Engines, 
h.p. cyl. 


Simple 
Engines. 


Compound 
Engines, 
h.p. cyl. 


Triple-ex- 
pansion 
Engines, 
h.p. cyl. 


5 


58 
66 

71 
74 
78 
82 
86 






42 
34 

29 
26 
22 
18 
14 






10 


74 
76 
78 
82 
85 
88 




26 
24 
22 
18 
15 
12 




15 
20 
30 
40 
50 


78 
80 
84 
87 
90 


22 
20 
16 
13 
10 



WORK OF STEAM IK A SIHGLE CYLINDER. 



753 



Theoretical Compared with Actual Water-consump- 
tion, Single-cylinder Automatic Cut-off Engines. (From 
the catalogue of the Buckeye Engine Co.)— The following table has been 
prepared on the basis of the pressures that result in practice with a con- 
stant boiler- pressure of 80 lbs. and different points of cut-off, with Buckeye 
engines and others with similar clearance. Fractions are omitted, except 
in the percentage column, as the degree of accuracy their use would seem 
to imply is not attained or aimed at. 













Cut-off Part 


Mean 
Effective 
Pressure. 


Total 
Terminal 
Pressure. 


Rate, 
lbs. Water, 
per I. H. P. 
per hour. 


Assumed. 


of Stroke. 


Act'l Rate. 


Per ct. Loss. 


.10 


18 


11 


20 


32 


58 


.15 


27 


15 


19 


27 


41 


.20 


35 


20 


19 


25 


31.5 


.25 


42 


25 


20 


25 


25 


.30 


48 


30 


20 


24 


21.8 


.35 


53 


35 


21 


25 


19 


.40 


57 


38 


22 


26 


16.7 


.45 


61 


43 


23 


27 


15 


.50 


64 


48 


24 


27 


13.6 



It will be seen that while the best indicated economy is when the cut-off 
is about at .15 or .20 of the stroke, giving about 30 lbs. M.E.P., and a termi- 
nal 3 or 4 lbs. above atmosphere, when we come to add the percentages due 
to a constant amount of unindicated loss, as per sixth column, the most eco- 
nomical point of cut-off is found to be about .30 of the stroke, giving 48 lbs. 
M.E.P. and 30 lbs. terminal pressure. This showing agrees substantially 
with modern experience under automatic cut-off regulation. 

Experiments on Cylinder-condensation.— Experiments by 
Major Thos. English {Eng\j, Oct. 7, 1887, p. 386) with an engine 10 x 14 in., 
jacketed in the sides but not on the ends, indicate that the net initial con- 
densation (or excess of condensation over re-evaporation) by the clearance 
surface varies directly as the initial density of the steam, and inversely as 
the square root of the number of revolutions per unit of time. The mean 
results gave for the net initial condensation by clearance-space per sq. ft. of 
surface at one rev. per second 6.06 thermal units in the engine when run 
non-condensing and 5.75 units when condensing. 

O. R. Bodmer {Eng'g, March 4, 1892, p. 299) says : Within the ordinary 
limits of expansion desirable in one cylinder the expansion ratio has prac- 
tically no influence on the amount of condensation per stroke, which for 
simple engines can be expressed by the following formula for the weight 
of water condensed [per minute, probably; the original does not state] : 

S(T- t) 
W — O 3/—, where T denotes the mean admission temperature, t the 

mean exhaust temperature, S clearance-surface (square feet), N the num- 
ber of revolutions per second, L latent heat of steam at the mean admission 
temperature, and (7 a constant for any given type of engine. 

Mr. Bodmer found from experimental data that for high-pressure non- 
jacketed engines C = about 0.11, for condensing non-jacketed engines 0.085 
to 0.11, for condensing jacketed engines 0.085 to 0.053. The figures for jack- 
eted engines apply to those jacketed in the usual way, and not at the ends. 

C varies for different engines of the same class, but is practically con- 
stant for any given engine. For simple high-pressure non -jacketed engines 
it was found to range from 0.1 to 0.112. 

Applying Mr. Bodmer's formula to the case of a Corliss non-jacketed non- 
condensing engine, 4-ft. stroke, 24 in. diam., 60 revs, per min., initial pres- 
sure 90 lbs. gauge, exhaust pressure 2 lbs., we have T - t — 112°, N = 1, 
L = 880, S = 7 sq. ft.; and, taking C ~ .112 and W = lbs. water condensed 

112 X 112 X 7 
per minute, W = ' — „ = .09 lb. per minute, or 5.4 lbs. per hour. If 

the steam used per I.H.P. per hour according to the diagram is 20 lbs., the 
actual water consumption is 25.4 lbs., corresponding to a cylinder condensa- 
tion of 27#. 



754 



THE STEAM-ENGINE. 



INDICATOR-DIAGRAM OF A SINGLE-CYLINDER 

ENGINE. 

Definitions.— The Atmospheric Line, AB, is a line drawn by the pencil 
of the indicator when the connections with the engine are closed and both 
sides of the piston are open to the atmosphere. 




Fig. 138. 



The Vacuum Line, OX, is a reference line usually drawn about 14 7/10 
pounds by scale below the atmospheric line. 

The Clearance Line, OY, is a reference line drawn at a distance from the 
end of the diagram equal to the same per cent of its length as the clearance 
and waste room is of the piston-displacement. 

The Line of Boiler-pressure, JK, is drawn parallel to the atmospheric 
line, and at a distance from it by scale equal to the boiler-pressure shown 
by the gauge. 

The Admission Line, CD, shows the rise of pressure due to the admission 
of steam to the cylinder by opening the steam-valve. 

The Steam Line, DE, is drawn when the steam-valve is open and steam is 
being admitted to the cylinder. 

The Point of Cut-off, E, is the point where the admission of steam is 
stopped by the closing of the valve. It is often difficult to determine the 
exact point at which the cut-off takes place. It is usually located where the 
outline of the diagram changes its curvature from convex to concave. 

The Expansion Curve, EF, shows the fall in pressure as the steam in the 
cylinder expands doing work. 

The Point of Release, F, shows when the exhaust-valve opens. 

The Exhaust Line, FG, represents the change in pressure that takes 
place when the exhaust- valve opens. 

The Bach-pressure Line, GH, shows the pressure against which the piston 
acts during its return stroke. 

The Point of Exhaust Closure, H, is the point where the exhaust-valve 
closes. It cannot be located definitely, as the change in pressure is at first 
due to the gradual closing of the valve. 

The Compression Curve, HC, shows the rise in pressure due to the com- 
pression of the steam remaining in the cylinder after the exhaust- valve has 
closed. 

The Mean Height of the Diagram equals its area divided by its length. 

The Mean Effective Pressure is the mean net pressure urging the piston 
forward = the mean height X the scale of the indicator-spring. 

To find the Mean Effective Pressure from the Diagram.— Divide the 
length, LB, into a number, say 10, equal parts, setting off half a part at L, 
half a part at B, and nine other parts between; erect ordinates perpendicu- 
lar to the atmospheric line at the points of division of LB, cutting the dia- 
gram; add together the lengths of these ordinates intercepted between the 
upper and lower lines of the diagram and divide by their number. This 



INDICATED HORSE-POWER OF ENGINES. 755 

gives the mean height, which multiplied by the scale of the indicator-spring 
gives the M.E.P. Or find the area by a planimeter, or other means (see 
Mensuration, p. 55), and divide by the length LB to obtain the mean height. 

TJie Initial Pressure is the pressure acting on the piston at the beginning 
of the stroke. 

The Terminal Pressure is the pressure above the line of perfect vacuum 
that would exist at the end of the stroke if the steam had not been released 
earlier. It is found by continuing the expansion-curve to the end of the 
diagram. 

INDICATED HORSE-POWER OF ENGINES, SINGLE- 

CYLINDER. 

= 33,000' 

in which P= mean effective pressure in lbs. per sq. in. ; L = length of stroke 
in feet; a = area of piston in square inches. For accuracy, one half of the 
sectional area of the piston-rod must be subtracted from the area of the 
piston if the rod passes through one head, or the whole area of the rod if it 
passes through both heads; n = No. of single strokes per min. = 2x No. of 
revolutions. 

PaS 
I.H.P. = , , in which S= piston speed in feet per minute. 

I.H.P. = -J^y = -p^L = .0000238PLd 2 n = .0000238Pd 2 S, 

in which d = diam. of cyl. in inches. (The figures 238 are exa:t, since 
7854 -=- 33 = 23.8 exactly.) If product of piston-speed X mean effective 
pressure = 42,017, then the horse-power would equal the square of the 
diameter in inches. 

Handy Rule for Estimating the Horse-power of a 
Single-cylinder Engine. — Square the diameter and divide by 2. This is 
correct whenever the product of the mean effective pressure and the piston- 
speed = \4, of 42,017, or. say, 21,000, viz., when M.E.P. = 30 and S= 700; 
when M.E.P. = 35 and S - 600; when M.E.P. = 38.2 and S = 550; and when 
M.E.P. — 42 and S = 500. These conditions correspond to those of ordinary 
practice with both Corliss engines and shaft-governor high-speed engines. 

Given Horse-power, Mean Effective Pressure, and 
Piston-speed, to find Size of Cylinder.— 

w = 33,000 x I.H.P Dia ' meter = 205 /L^P, (Exact.) 
PLn \f PS 

Brake Horse-power is the actual horse-power of the engine as 
measured at the fly-wheel by a friction-brake or dynamometer. It is the 
indicated horse-oower minus the friction of the engine. 

Table for Roughly Approximating the Horse-power of 
a Compound Engine from the Diameter of its Iiow- 
pressure Cylinder.— The indicated horse-power of an engine being 
Psd 2 
——-—., in which P = mean effective pressure per sq. in., s = piston-speed in 

ft. per min., and d = diam. of cylinder in inches; if s — 600 ft. per min., 
which is approximately the speed of modern stationary engines, and P = 35 
lbs., which is an approximately average figure for the M.E.P. of single- 
cylinder engines, and of compound engines referred to the low-pressure 
cylinder, then I.H.P. = J^d 2 ; hence the rough-and-ready rule for horse-power 
given above: Square the diameter in inches and divide by 2. This applies to 
triple and quadruple expansion engines as well as to single cylinder and 
compound. For most economical loading, the M.E.P. referred to the low- 
pressure cylinder of compound engines is usually not greater than that of 
simple engines; for the greater economy is obtained by a greater number of 
expansions of steam of higher pressures, and the greater the number of 
expansions for a given initial pressure the lower the mean effective pressure. 
The following table gives approximately the figures of meau total and effec- 



756 



THE STEAM-ENGINE. 



tive pressures for the different types of engines, together with the factor by 
which the square of the diameter 'is to be multiplied to obtain the horse- 
power at most economical loading, for a piston-speed of COO ft. per minute. 



Type of Engine. 






3 - S 3 

3 a ^-3 .? 






.2 £?& 






Non-condensing. 



Single Cylinder. 

Compound 

Triple 

Quadruple 



100 


5. 


20 


.522 


52.2 


15.5 


36.7 


600 


120 


7.5 


16 


.402 


48.2 


15.5 


32.7 




160 


10. 


16 


.330 


52.8 


15.5 


37.3 


" 


•200 


12.5 


16 


.282 


56.4 


15.5 


40.9 


" 



.524 
.467 
.533 

.584 



Condensing Engines. 



Single Cylinder. 

Compound 

Triple. 

Quadruple 



100 


10. 


10 


.330 


33.0 


2 


31.0 


600 


120 


15. 


8 


.247 


29.6 


2 


27 6 




160 


20. 


8 


.200 


32.0 


2 


30.0 


" 


200 


-25. 


8 


.169 


33.8 


2 


1 31.8 


" 



For any other piston-speed than 600 ft. per min., multiply the figures in 
the last column by the ratio of the piston-speed to 600 ft. 

Nominal Horse-power.— The term "nominal horse-power" origi- 
nated in the time of Watt, and was used to express approximately the power 
of an engine as calculated from its diameter, estimating the mean pressure 
in the cylinder at 7 lbs. above the atmosphere. It has long been obsolete in 
America, and is nearly obsolete in England. 

Horse-power Constant of a given Engine for a Fixed 
Speed. = product of its area of piston in square inches, length of stroke in 

feet, and number of single strokes per minute divided by 33,000, or ■ " l 

oo,000 
= C. The product of the mean effective pressure as found by the diagram 
and this constant is the indicated horse-power. - 

Horse-power Constant of a given Engine for Varying 
Speeds = product of its area of piston and length of stroke divided by 
33.000. This multiplied by the mean effective pressure and by the number 
of single strokes per minute is the indicated horse-power. 

Horse-power Constant of any Engine of a given Diam- 
eter of Cylinder, whatever the length of stroke = area of piston -5- 33,000 
= square of the diameter of piston in inches X .0000238. A table of constants 
derived from this formula is given below. 

The constant multiplied by the piston-speed in feet per minute and by 
the M.E.P. gives the I.H.P. 

Errors of Indicators.— The most common error is that of the spring, 
which may vary from its normal rating; the error may be determined by 
proper testing apparatus and allowed for. But after making this correction, 
even with the best work, the results are liable to variable errors which may 
amount to 2 or 3 per cent. See Barrus, Trans. A. S. M. E., v. 310; Denton, 
A. S. M. E., xi. 329; David Smith, U. S. N., Proc. Eng'g Congress, 1893, 
Marine Division. 

Indicator "Rigs," or Reducing-motions ; Interpretation of Diagrams for 
Errors of Steam-disti'ibution, etc. For these see circulars of manufacturers 
of Indicators; also works on the Indicator. 

Table of Engine Constants for Use in Figuring Horse- 
power.—" Horse-power constant " for cylinders from 1 inch to 60 inches in 
diameter, advancing by 8ths, for one foot of piston-speed per minute and one 
pound of M.E.P. Find the diameter of the cylinder in the column at the 
side. If the diameter contains no fraction the constant will be found in the 
column headed Even Inches. If the diameter is not in even inches, follow 
the line horizontally to the column corresponding to the required fraction. 





INDICATED 


HORSE-POWER OF 


ENGINES. 


757 


The constants multiplied by the piston-speed and by the M.E.P. give the 


horse-power. 


Diameter 




+ y s 


+ M 


+ % 


+ H 


+ % 


+ U 


\-% 


of 


Inches . 


or 


or 


or 


or 


or 


or 


or 


\Cylinder. 


.125. 


.25. 


.375. 


.5. 


.625. 


.75. 


.875. 


1 


.0000238 


.0000301 


.0000372 


.0000450 


.0000535 


.0000628 


.0000729 


.0000837 


2 


.0000953 


.0001074 


.0001205 


.0001342 .0001487 


.0001640 


.0001800 


0001967 


3 


. 0002143 


.0002324 


.0002514 


.00027111.0002915 


.0003127 


.0003347 


.0003574 


4 


.0003808 


.0004050 


.0004299 


.0004554 .0004819 


.0005091 


.0005370 


.0005656 


5 


.0005950 


.0006251 


.0006560 


.0006876 .0007199 


.0007530 


.0007869 


.0008215 


6 


.0008568 


.0008929 


.0009297 


.0009672 .0010055 


.0010445 


.0010844 


.0011249 


7 


.0011662 


.0012082 


.0012510 


.0012944 .0013387 


.0013837 


.0014295 


.0014759 


8 


.0015232 


.0015711 


.0016198 


.0016693 .0017195 


.0017705 


.0018222 


.0018746 


9 


.0019278 


.0019817 


.0020363 


.0020916 .0021479 


.0022048 


.0022625 


.0023209 


10 


.0023800 


.0024398 


.0025004 


.0025618 .0026239 


.0026867 


.0027502 


.0028147 


11 


.0028798 


.0029456 


.0030121 


.0030794 .0031475 


.0032163 


.0032859 


.0033561 


12 


.0034272 


.0034990 


.0035714 


.0036447! .0037187 


.0037934 


.0038690 


.0039452 


13 


.0040222 


.0010999 


.0041783 


.0042576 .0043375 


.0044182 


.0044997 


.0045819 


14 


.0046648 


.0047484 


.0048328 


.0049181 -.0050039 


.0050906 


.0051780 


.0052661 


15 


.0053550 


.0054446 


.0055349 


.0056261 .00571 79 


.0058105 


.0059039 


.0059979 


16 


.0060928 


.0061884 


.0062847 


.0063817! .0064795 


.0065780 


.0066774 


.0067774 


17 


.0068782 


.0069797 


.0070819 


.0071850 .0072887 


.0073932 


.0074985 


.0076044 


18 


.0077112 


.0078187 


.0079268 


.0080360i .0081452 


.0082560 


.0083672 


.0084791 


19 


.0085918 


.0087052 


.0088193 .0089343! .0090499 


.0091663 


.0092835 


.0094013 


20 


.0095200 


.0096393 


.0097594 .0098803 .0100019 


.0101243 


.0102474 


.0103712 


21 


.0104958 


.0106211 


.0107472 


.01087391 .0110015 


.0111299 


.0112589 


.0113886 


22 


.0115192 


.0116505 


.0117825 


.0119152! .0120487 


.0121830 


.0123179 


.0124537 


23 


.0125902 


.0127274 


.0128654 


.0130040! .0131435 


.0132837 


.0134247 


.0135664 


24 


.0137088 


.0138519 


.0139959 


.0141405J .0142859 


.0144321 


.0145789 


.0147266 


25 


.0148750 


.0150241 


.0151739 


.0153246 .0154759 


.0156280 


.0157809 


.0159345 


26 


.01608S8 


.0162439 


.0163997 


.0165563 .0167135 


.0168716 


.0170304 


.0171899 


27 


.0173502 


.0175112 


.0176729 .0178355 


.0179988 


.0181627 


.0183275 


.0184929 


28 


.0186592 


.0188262 


.0189939 


.0191624 


.0193316 


.0195015 


.0196722 


.0198436 


29 


.0200158 


.0201887 


.0203624 


.0205368 


.0207119 


.0208879 


.0210645 


.0212418 


30 


.0214200 


.0215988 


.0217785 


.0219588 


.0221399 


.0223218 


.0225044 


.0226877 


31 


.0228718 


.0230566 


.0232422 


.0234285 


.0236155 


.0238033 


.0239919 


.0241812 


32 


.0243712 


.0245019 


.0247535 


.0249457 


.0251387 


.0253325 


.0255269 


.0257222 


33 


.0259182 


.0261149 


.0263124 


.0265106 


.0267095 


.0269092 


.0271097 


.0273109 


34 


.0275128 


.0277155 


.0279189 


.0281231 


.0283279 


.02853S6 


.0287399 


.0289471 


35 


.0291550 


.0293636 


.0295729 .0297831 


.0299939 


. 0302056 


.0304179 


.0306309 


36 


.0308448 


.0310594 


.0312747 .0314908 


.0317075 


.0319251 


.0321434 


.0323624 


37 


.0325822 


.0328027 


239 .0332460 


.0334687 


.0330922 


.0339165 


.0341415 


38 


.0343672 


.0345937 


.0348209|. 0350489 


.0352775 


. 0355070 


.0357372 


.0359681 


39 


.0361998 


.0364322 


.0366654 .0368993 


.0371339 


.0373694 


.0376055 


.0378424 


40 


.0380800 


.0383184 


.0385575 .0387973 


.0390379 


.0392793 


.0395214 


.0397642 


41 


.0400078 


.0402521 


.0404972 


.0407430 


.0409895 


.0412368 


.0414849 


.0417337 


42 


.0419832 


.0422335 


.0424845 


.0427362 


.0429887 


.0432420 


.0434959 


.0437507 


43 


.0440062 


.0442624 


.0445194 


.0447771 


.0450355 


.0452947 


.0455547 


.0458154 


44 


.0460768 


.0463389 


.0466019 


.0468655 


.0471299 


.0473951 


.0476609 


.0479276 


45 


.0481950 


.0484631 


.0487320 


.0490016 


.0492719 


.0495430 


.0498149 


.0500875 


46 


.0503608 


.0506349 


.0509097 


.0511853 


.0514615 


.0517386 


.0520164 


.U522949 


47 


.0525742 


.0528542 


.0531349 


.0534165 


.0536988 


.0539818 


.0542655 


.0545499 


48 


.0548352 


.0551212 


.0554079 


.0556953 


.0559835 


.0562725 


.0565622 


.0568526 


49 


.0571438 


.0574357 


.0577284 


.0580218 


.0583159 


.0586109 


.0589085 


.0592029 


50 


.0595000 


.0597979 


.0600965 


.0603959' .0606959 


.0609969 


.0612984 


.0616007 


51 


.0619038 


.0622076 


.0625122 .0628175 .0632235 


.0634304 


.0637379 


.0640462 


52 


.0643552 


.0646649 


.0619753 .0652867! .0655987 


.0659115 


.0662250 


.0665392 


53 


.0668542 


.0671699 


.0674864 .0678036 .0681215 


.0684402 


.0687597 


.0690799 


54 


.0694008 


.0697225 


.0700449 .0703681 .0705293 


.0710166 


.0713419 


0716681 


55 


.0719950 


.0724226 


.0726510 .0729801;. 0733099 


.0736406 


.0739719 


.0743039 


56 


.0746368 


.0749704 


.0753047 .0756398 .0759755 


.0763120 


.0766494 


.0769874 


57 


.0773262 


.0776657 


.0780060 .0783476 .0786887 


.0790312 


.0793745 


.0797185 


58 


.0800632 ,.0804087 


.0807549 .0811019 .0814495 


.08179SO 


.0821472 


.0824971 


59 


.0828478I.0831992 


.0835514 .0839043' 0842579 


.0846123 


.0849675 


.0853234 


60 


.0856800 


.0860374 


.0863955 


.0867543 


.0871139 


.0874743 


.0878354 


.0881973 



758 THE STEAM-ENGINE. 

Horse-power per Pound Mean Effective Pressure. 

_, . Area in sq. in. X piston-speed 
Formula, ^-^ H— . 



Diam. of 
Cylinder, 


Speed of Piston in feet per minute. 


inches. 


100 


240 


300 


400 


450 


500 


550 


600 


650 


750 


4 


.038 


.091 


.114 


.152 


.171 


.19 


.209 


.228 


.247 


.285 


4Va 


.048 


.115 


.144 


.192 


.216 


.24 


.264 


.288 


.312 


.360 


5 


.06 


.144 


.18 


.■J 4 


.27 


.30 


.33 


.36 


.39 


.450 


Wz 


.072 


.173 


.216 


.288 


.324 


.36 


.396 


.432 


.468 


.540 


6 


.086 


.205 


.256 


.342 


.385 


.428 


.471 


.513 


.555 


.641 


6)4 


.102 


.245 


.307 


.409 


.464 


.512 


.563 


.614 


.698 


.800 




.116 


.279 


.348 


.466 


.524 


.583 


.641 


.699 


.756 


.874 


Wz 


.134 


.321 


.401 


.534 


.602 


.669 


.735 


.802 


.869 


1.002 


8 


.152 


.365 


.456 


.608 


.685 


.761 


.837 


.912 


.989 


1.121 


m 


.172 


.413 


.516 


.688 


.774 


.86 


.946 


1.032 


1.118 


1.290 


9 


.192 


.46* 


.577 


.770 


.866 


.963 


1.059 


1.154 


1.251 


1.444 


9^ 


.215 


.515 


.644 


.859 


.966 


1.074 


1.181 


1.288 


1.395 


1.610 


10 


.238 


.571 


.714 


.952 


1.071 


1.190 


1.309 


1.428 


1.547 


1.785 


11 


.288 


.691 


.864 


1.152 


1.296 


1.44 


1.584 


1.728 


1.872 


2.160 


12 


.342 


.820 


1.025 


1.366 


1.540 


1.708 


1.880 


2.050 


2.222 


2.564 


13 


.402 


.964 


1.206 


1.608 


1.809 


2.01 


2.211 


2.412 


2.613 


3.015 


14 


.466 


1.119 


1.398 


1.864 


2.097 


2.331 


2.564 


2.797 


3.029 


3.495 


15 


.535 


1.285 


1.606 


2.131 


2.409 


2.677 


2.945 


3.212 


3.479 


4.004 


16 


.609 


1.461 


1.827 


2.436 


2.741 


3.045 


3.349 


3.654 


3.958 


4.567 


17 


.685 


1.643 


2.054 


2.739 


3.081 


3.424 


3.766 


4.108 


4.450 


5.135 


18 


.771 


1.849 


2.312 


3.083 


3.468 


3.854 


4.239 


4.624 


5.009 


5.780 


19 


.859 


2.061 


2.577 


3.436 


3.865 


4.295 


4.724 


5.154 


5.583 


6.442 


20 


.952 


2.292 


2.855 


3.807 


4.285 


4.759 


5.234 


5.731 


6.186 


7.138 


21 


1.049 


2.518 


3.148 


4.197 


4.722 


5.247 


5.771 


6.296 


6.820 


7.869 


22 


1.152 


2.764 


3.455 


4.607 


5.183 


5.759 


6.334 


6.911 


7.486 


8.638 


23 


1.259 


3.021 


3.776 


5.035 


5.664 


6.294 


6.923 


7.552 


8.181 


9.44 


24 


1.370 


3.289 


4.111 


5.482 


6.167 


6.853 


7.538 


8.223 


8.908 


10.279 


25 


1.487 


3.569 


4.461 


5.948 


6.692 


7.436 


8.179 


8.923 


9.566 


11.053 


26 


1.609 


3.861 


4.826 


6.435 


7.239 


8.044 


8.848 


9. 652 1 10. 456 


12.065 


27 


1.733 


4.159 


5.199 


6.932 


7.799 


8.666 


9.532 


10.399ill.265 


12.998 


28 


1.865 


4.477 


5.596 


7.462 


8.395 


9.328 


10.261 


11.193 12.125 


13.991 


■ 29 


2.002 


4.805 


6.006 


8.008 


9.009 


10.01 


11.011 


12.012 13.013 


15.015 


30 


2.142 


5.141 


6.426 


8.568 


9.639 


10.71 


11.781 


12. 852 i 13. 923 


16.065 


31 


2.288 


5.486 


6.865 


9.144 


10.287 


11.43 


12.573 


13.716jl4.866 


17.145 


32 


2.436 


5.846 


7.30S 


9.744 


10.962 


12.18 


13.398 


14.61615.834 


18.270 


33 


2.590 


6.216 


7.770 


10.360 


11.655 


12.959 


14.245 


15.54 16.835 


19.425 


34 


2.746 


6.59 


8.238 


10.984 


12.357 


13.73 


15.103 


16.476ll7.849 


20.595 


35 


2.914 


6.993 


8.742 


11.656 


13.113 


14.57 


16.027 


17. 484 j 18. 941 


21.855 


36 


3.084 


7.401 


9.252 


12.336 


13.878 


15.42 


16.962 


18.504l20.046 


23.130 


37 


3.253 


7.819 


9.774 


13.032 


14.861 


16.29 


17.919 


19.548;21.177 


24.435 


38 


3.436 


8.246 


10.308 


13.744 


15.462 


17.18 


18.898 


20.616 22.334 


25.770 


39 


3.620 


8.648 


10.86 


14 48 


16.29 


18.1 


19.91 


21.62 123.53 


27.150 


40 


3.808 


9.139 


11.424 


15.232 


17.136 


19.04 


20.944 


22.848 24.752 


28.560 


41 


4.00-' 


9.604 


12.006 


16.008 


18.009 


20.00 


22.011 


24.012:26. 013 


30.015 


42 


4.198 


10.065 


12.594 


16.792 


18.901 


20.99 


23.089 


25.188:27.287 


31.485 


43 


4.40 


10.56 


13.20 


17.6 


19.8 


22.00 


24.2 


26.4 J28.6 


33.00 


44 


4.606 


11 046 


13.818 


18.424 


20.727 


23.03 


25.333 


27.636 29.939 


34.545 


45 


4.818 


11.563 


14.454 


19.272 


21.681 


24.09 


26.399 


28.908 31.317 


36.135 


46 


5.043 


12.086 


15.128 


20.1 -14 


22.662 


25.18 


27.698 


30.216 32.754 


37.770 


47 


5.256 


12.614 


15.768 


21.024 


23.652 


26.28 


28.908 


31.536 34.164 


39.420 


48 


5.482 


12.846 


16.446 


21.U2S 


24.669 


27.41 


30.151 


33.152 35.633 


41.115 


49 


5.714 


12.913 


17.142 


22.856 


25.713 


28.57 


31.427 


34.284 37.141 


42.855 


50 


5.950 


14.28 


17.85 


23.8 


26.775 


29.75 


32.725 


35.7 ,38.675 


44.625 


51 


6.180 


14.832 


18.54 


24.76 


27.855 


30.95 


34.045 


37.08 |40.205 


46.425 


52 


6.432 


15.437 


19.296 


25.728 


28.944 


32.16 


35.376 


38.592 41.808 


48.240 


53 


6.684 


16.041 


20.052 


26.736 


30.078 


33.42 


36.762 


40.104 43.446 


50.130 


54 


6.940 


16.656 


20.82 


27.76 


31.23 


34.7 


38.17 


41.64 45.11 


52.05 


55 


7.198 


17.275 


21.594 


28. 792132.391 


35 99 


39.589 


43.188 


46.787 


53.985 


56 


7.462 


17.909 


22.386 29. 84 S 33.579 


37.31 


41.041 


44.772 


48.503 


55.965 


57 


7.732 


18.557 


23.196 30.928 34.794 


38.66 


42.526 


46.392 




57.99 


58 


8.006 


19.214 


24.018 32. 024|36. 027 


40.03 


44.033 


48.036 


52.039 


60.045 


59 


8.284 


19.902 


24.852 33.136 37.278 


41.42 


45.562 


48.704 


53.846 


62.13 


60 


8.566 


20.558 


25.698l34.264 38.547 42.83 


47.113 


51.396 55.679 64.245 






INDICATED HORSE-POWER OF ENGINES. 



759 



To draw the Clearance-line on the Indicator-diagram* 

the actual clearance not being known.— The clearance-line may be obtained 
approximately by drawing a straight line, cbad, across the compression 
curve, first having drawn OX parallel to the atmospheric line and 14.7 lbs. 
below. Measure from a the distance ad, equal to cb, and draw YO perpen- 
dicular to OX through d; then will TB divided by AT be the percentage of 



- 






*C 




Y 


! 

i 
i 
i 


M^^ 


/ 1 1 


fa 


' 




\n 




T 


> 


! 


l! / 




^f\ 


A R 


N 


c 









clearance. The clearance may also be found from the expansion-line by 
constructing a rectangle efhg, and drawing a diagonal gf to intersect the 
line XO. This will give the point O, and by erecting a perpendicular to XO 
we obtain a clearance-line OY. 

Both these methods for finding the clearance require that the expansion 
and compression curves be hyperbolas. Prof. Carpenter (Power, Sept. 
1893) says that with good diagrams the methods are usually very accurate, 
and give results which check substantially. 

The Buckeye Engine Co., however, say that, as the results obtained are 
seldom correct, being sometimes too little, but more frequently too much, 
and as the indications from the two curves seldom agree, the operation lias 
little practical value, though when a clearly defined and apparently undis- 
torted compression curve exists of sufficient extent to admi,t of the applica- 
tion of the process, it may be relied on to give much more correct results 
than the expansion curve. 

To draw the Hyperholic Curve on the Indicator-dia- 
gram.— Select any point /in the actual curve, and from this point draw a 



M 



B 




line perpendicular to the line JB, meet- 
ing the latter in the point J. The line 
JB may be the line of boiler-pressure, 
but this is not material ; it may be drawn 
at any convenient height near the top of 
diagram and parallel to the atmospheric 
line. From J draw a diagonal to K, the 
latter^ point being the intersection of ( 
the vacuum and clearance lines; from / _ 
draw IL parallel with the atmospheric 
line. From L, the point of intersection - 
of the diagonal JK and the horizontal 
line IL, draw the vertical line LM. The 
point M is the theoretical point of cut-off, and LM the cut-off line. Fix 
upon any number of points 1, 2, 3, etc., on the line JB, and from these points 
draw diagonals to K. From the intersection of these diagonals with LM 
draw horizontal lines, and from 1, 2, 3, etc., vertical lines. Where these lines 
meet will be points in the hyperbolic curve. 

Pendulum Indicator Rig.— Power (Feb. 1893) gives a graphical 
representation of the errors in indicator-diagrams, caused by the use of in- 



Ftg. 140. 



760 THE STEAM-ENGIKE. 

correct form of the pendulum rigging. It is shown that the " brumbo " 
pulley on the pendulum, to which the cord is attached, does not genet- 
ally give as good a reduction as a simple piu 
attachment. When the end of the pendulum is 
slotted, working in a pin on the crosshead, the 
error is apt to be considerable at both ends of 
the card. With a vertical slot in a plate fixed 
to the crosshead, and a pin on the pendulum 
working in this slot, the reduction is perfect, 
when the cord is attached to a pin on the pen- 
dulum, a slight error being introduced if the 
brumbo pulley is used. With the connection 
between the pendulum and the crosshead made 
by means of a horizontal link, the reduction is 
nearly perfect, if the construction is such that 
the connecting link vibrates equally above and 
Fig. 141. below the horizontal, and the cord is attached 

by a pin. If the link is horizontal at mid-stroke 
a serious error is introduced, which is magnified if a brumbo pulley also is 
used. The adjoining figures show the two forms recommended. 

Theoretical Water-consumption calculated from the 
Indicator-card.— The following method is given by Prof. Carpenter 
(Poiver, Sept. 18y3) : p = mean effective pressure, I = length of stroke in 
feet, a = area of piston in square inches, a -s- 144 = area in square feet, c = 
percentage of clearance to the stroke, b = percentage of stroke at point 
where water rate is to be computed, n — number of strokes per minute, 
60w = number per hour, w = weight of a cubic foot of steam having a pres- 
sure as shown by the diagram corresponding to that at the point where 
water rate is required, io' — that corresponding to pressure at end of com- 
pression. 



Number of cubic feet per stroke = l(^ ~T )-rrj> 

Corresponding weight of steam per stroke in lbs. — l{^ "t" Jrrrr 



lea 

„ . . L ' , ^ . , Icaio' 

Weight of steam m clearance = - . 

Total weight of | _ rf b+c ^wa _ Icaw' _ la r -, 

steam per stroke) ~ l V ioo / 144 14,400 ~ 14,400|_ ^ ' J' 

Total weight of steam { _ Wnla r , ~\ 

from diagram per hour, ~ i^iOO L "*" ' " ,CW J' 

The indicated horse-power is p I a n -i- 33,000. Hence the steara-consump* 
tion per indicated horse-power is 

lM00L (b + c)w - CW/ J 137.50,- L , n 

-p-Ta~^ = -^~\} b + ° )W ~ CW [ 

33,000 

Changing the formula to a rule, we have: To find the water rate from the 
indicator diagram at any point in the stroke. 

Rule.— To the percentage of the entire stroke which has been completed 
by the piston at the point under consideration add the percentage of clear- 
ance. Multiply this result by the weight of a cubic foot of steam, having a 
pressure of that at the required point. Subtract from this the product of 
percentage of clearance multiplied by weight of a cubic foot of steam hav- 
ing a pressure equal to that at the end of the compression. Multiply this 
result by 137.50 divided by the mean effective pressure.* 

Note.— This method only applies to points in the expansion curve or be- 
tween cut-off and release. 

* For compound or triple-expansion engines read: divided by the equiva- 
lent mean effective pressure, on the supposition that all work is done in one 
cylinder. 



COMPOUND ENGINES. 



761 



The beneficial effect of compression in reducing the water-consumption of 
an engine is clearly shown by the formula. If the compression is carried to 
such a point that it produces a pressure equal to that at the point under 
consideration, the weight of steam per cubic foot is equal, and w = w'. In 
this case the effect of clearance entirely disappears, and the formula 

becomes — —(bw). 
P 
In case of no compression, 10' becomes zero, and the water-rate = 

Prof. Denton (Trans. A. S. M. E., xiv. 1363) gives the following table of 
theoretical water-consumption for a perfect Mariotte expansion with steam 
at 150 lbs. above atmosphere, and 2 lbs. absolute back pressure : 



Ratio of Expansion, r. 


M.E.P., lbs. per sq. in. 


Lbs. of Water per hour 
per horse-power, W. 


10 
15 
20 
25 
30 
35 


52.4 
38.7 
30.9 
25.9 
22.2 
19.5 


9.68 
8.74 
8.20 
7.84 
7.63 
7.45 



The difference between the theoretical water -consumption found by the 
formula and the actual consumption as found by test represents " water not 
accounted for by the indicator, 1 ' due to cylinder condensation, leakage 
through ports, radiation, etc. 

Leakage of Steam.— Leakage of steam, except in rare instances, has 
so little effect upon the lines of the diagram that it can scarcely be detected. 
The only satisfactory way to determine the tightness of an engine is to take 
it when not in motion, apply a full boiler-pressure to the valve, placed in a 
closed position, and to the piston as well, which is blocked for the purpose at 
some point away from the end of the stroke, and see by the eye whether 
leakage occurs. The indicator-cocks provide means for bringing into view 
steam which leaks through the steam-valves, and in most cases that which 
leaks by the piston, and an opening made in the exhaust-pipe or observa- 
tions at the atmospheric escape-pipe, are generally sufficient to determine 
the fact with regard to the exhaust-valves. 

The steam accounted for by the indicator should be computed for both 
the cut-off and the release points of the diagram. If the expansion -line de- 
parts much from the hyperbolic curve a very different result is shown at 
one point from that shown at the other. In such cases the extent of the 
loss occasioned by cylinder condensation and leakage is indicated in a much 
more truthful manner at the cut-off than at the release. (Tabor Indicator 
Circular.) 

COMPOUND ENGINES. 

Compound, Triple- and Quadruple-expansion Engines. 

— A compound engine is one having two or more cylinders, and in which 
the steam after doing work in the first or high-pressure cylinder completes 
its expansion in the other cylinder or cylinders. 

The term "compound" is commonly restricted, however, to engines in 
which the expansion takes place in two stages only— high and low pressure, 
the terms triple-expansion and quadruple-expansion engines being used when 
the expansion takes place respectively in three and four stages. The number 
of cylinders may be greater than the number of stages of expansion, for 
constructive reasons; thus in the compound or two-stage expansion engine 
the low-pressure stage may be effected in two cylinders so as to obtain the 
advantages of nearly equal sizes of cylinders and of three cranks at angles of 
120°. In triple- expansion engines there are frequently two low-pressure 
cylinders, one of them being placed tandem with the high-pressure, and the 
other with the intermediate cylinder, as in mill engines with two cranks at 
90°. In the triple-expansion engines of the steamers Campania and Lucania, 



762 



THE STEAM-ENGINE* 



with three cranks at 120°, there are five cylinders, two high, one intermedi- 
ate, and two low, the high-pressure cylinders being tandem with the low. 

Advantages of Compounding.— The advantages secured by divid- 
ing the expansion into two or more stages are twofold: 1. Reduction of wastes 
of steam by cylinder-condensation, clearance, and leakage; 2. Dividing the 
pressures on the cranks, shafts, etc., in large engines so as to avoid excessive 
pressures and consequent friction. The diminished loss by cylinder-conden- 
sation is effected by decreasing the range of temperature of the metal sur- 
faces of the cylinders, or the difference of temperature of the steam at; 
admission and exhaust. When high-pressure steam is admitted into a single- 
cylinder engine a large portion is condensed by the comparatively cold, 
metal surfaces; at the end of the stroke and during the exhaust the w:ater 
is re-evaporated, but the steam so formed escapes into the atmosphere or 
into the condenser, doing no work; while if it is taken into a second 
cylinder, as in a compound engine, it does work. The steam lost in the first 
cylinder by leakage and clearance also does work in the second cylinder. 
Also, if there is a second cylinder, the temperature of the steam exhausted 
from the first cylinder is higher than if there is only one cylinder, and the 
metal surfaces therefore are not cooled to the same degree. The difference 
In temperatures and in pressures corresponding to the work of steam of 
150 lbs. gauge-pressure expanded 20 times, in one, two, and three cylinders, 
is shown in the following table, by W. H. Weightman, Am. Mach., July 28, 



Single 
Cyl- 
inder. 



Compound 
Cylinders. 



Triple-expansion 
Cylinders. 



Diameter of cylinders, in. . 

Area ratios 

Expansions 

Initial steam - pressures- 
absolute— pounds 

Mean pressures, pounds. . 

Mean effective pressures, 
pounds 

Steam temperatures into 
cylinders 

Steam temperatures out of 
the cylinders 

Difference in temperatures 

Horse-power developed. . . 

Speed of piston ... 

Total initial pressures on 
pistons, pounds 



165 
32.1 



184°. 2 
181.8 



455.218 



33 
1 
5 

165 
86.11 

53.11 

366° 

259°. 9 
106.1 
399 
290 

112,900 



19.68 
15.68 



184°. 2 
175.7 
403 
290 



28 
1 
2.714 



165 
121.44 



293°. 5 
72.5 









46 
2.70 
2.714 


61 

4.746 
2.714 


60.8 
44.75 


22.4 
16.49 


22.35 


12.49 


293°. 5 


234°. 1 


234°. 1 
59.4 

268 


184°. 2 
49.9 
264 



817 53,773 



64 Woolf " and Receiver Types of Compound Engines.- 

The compound steam-engine, consisting of two cylinders, is reducible to two 
forms, 1, in which the steam from the h.p. cylinder is exhausted direct into 
the 1. p. cylinder, as in the Woolf engine; and 2, in which the steam from the 
h. p. cylinder is exhausted into an intermediate reservoir, whence the steam 
is supplied to, and expanded in, the 1. p. cylinder, as in the " receiver- 
engine.' 1 

If the steam be cut off in the first cylinder before the end of the stroke, 
the total ratio of expansion is the product of the ratio of expansion in the 
first cylinder, into the ratio of the volume of the second to that of the first 
cylinder: that is, the product of the two ratios of expansion. 

Thus, let the areas of the first and second cylinders be as 1 to 3J4 the 
strokes being equal, and let the steam be cut off in the first at'J^ stroke ; then 

Expansion in the 1st cylinder „ 1 to 2 

" "2d " ■. lto3^| 

Total or combined expansion, the product of the two ratios. . . 1 to 7 

Woolf Engine, without Clearance— Ideal Diagrams. - 

The diagrams of pressure of an ideal Woolf engine are shown in Fig. 142, as I 
they would be described by the indicator, according to the arrows. In these j 
diagrams pq is the atmospheric line, mn the vacuum line, pc| the admission 



COMPOUKD ENGINES. 



763 



line, dg the hyperbolic curve of expansion in the first cylinder, and gh the con- 



secutive expansion-line of back pressure 
for the return -stroke of the first piston, 
and of positive pressure for the steam- 
stroke of the second piston. At the point 
h, at the end of the stroke of the second 
piston, the steam is exhausted into the 
condenser, and the pressure falls to the 
level of perfect vacuum, mn. 

The diagram of the second cylinder, 
below gh, is characterized by the absence 
of any specific period of admission ; the 
whole of the steam-line gh being expan- 
sional, generated by the expansion of 
the initial body of steam contained in 
the first cylinder into the second. When 
the return-stroke is completed, the 
whole of the steam transferred from 



the first is shut into the second cylin- w 142 WnnTW w wrTwl . t^at 

der Thft final nvossnm pnrl vnlnn p nf * 1G - l?^.— WOOLF ENGINE— IDEAL 



the steam in the second cylinder are the 
same as if the whole of the initial steam had been admitted at once into the 
second cylinder, and then expanded to the end of the stroke in the manner 
of a single-cylinder engine. 

The net work of the steam is also the same, according to both distributions. 

Receiver-engine, without Clearance— Ideal Diagrams.— 
In the ideal receiver-engine the pistons of the two cylinders are con- 
nected to cranks at right angles to each other on the same shaft. The 
receiver takes the steam exhausted from the first cylinder and supplies it to 
the second, in which the steam is cut off and then expanded to the end of 
the stroke. On the assumption that the initial pressure in the second cylin- 
der is equal to the final pressure in the first, and of course equal to the pres- 
sure in the receiver, the volume cut off in the second cylinder must be 
equal to the volume of the first cylinder, for the second cylinder must admit 
as much steam at each stroke as is discharged from the first cylinder. 

In Fig. 143 cd is the line of admission and hg the exhaust-line for the first 








< 


i 


3 




7 




f 


r 




__ 












i ■* 




V 


*^ 






2 

! 



-60 lbs 




d 




- 




\ 




-40 


—-n--> 


< x A-h-> 




-2C 


i 

y 


/ 


h 




_ P 

k 

L 0. 7 






<1 





Fig. 143.— Receiver-engine, Ideal 
Indicator-diagrams. 



Fig. 144.— Receiver Engine, Ideal 
Diagrams reduced and combined. 



cylinder; and dg is the expansion-curve and pq the atmospheric line. In 
the region below the exhaust-line of the first cylinder, between it and the 
line of perfect vacuum, ol, the diagram of the second cylinder is formed; hi, 
the second line of admission, coincides with the exhaust-line hg of the first 
cylinder, showing in the ideal diagram no intermediate fall of pressure, and 
ik is the expansion-curve. The arrows indicate the order in which the dia- 
grams are formed. 

In the action of the receiver-engine, the expansive working of the steam, 
though clearly divided into two consecutive stages, is, as in the Woolf 
engine, essentially continuous from the point of cut-off in the first cylinder 
to the end of the stroke of the second cylinder, where it is delivered to the 
condenser; and the first and second diagrams may be placed together and 



764 



THE STEAM-EXGIHE. 



combined to form a continuous diagram. For this purpose take the second 
diagram as the basis of the combined diagram, namely, hiklo, Fig. 144. The 
period of admission, hi, is one third of the stroke, and as the ratios of the 
cylinders are as 1 to 3, hi is also the proportional length of the first diagram 
as applied to the second. Produce oh upwards, and set off oc equal to the 
total height of the first diagram above the vacuum-line; and, upon the 
shortened base hi, and the height he, complete the first diagram with the 
steam-line cd, and the expansion-line di. 

It is shown by Clark (S. E., p. 432, et seq.) in a series of arithmetical cal- 
culations, that the receiver-engine is an elastic system of compound engine, 
in which considerable latitude is afforded for adapting the pressure in the 
receiver to the demands of the second cylinder, without considerably dimin- 
ishing the effective work of the engine. In the Woolf engine, on the 
contrary, it is of much importance that the intermediate volume of space 
between the first and second cylinders, which is the cause of an interme- 
diate fall of pressure, should be reduced to the lowest practicable amount. 

Supposing that there is no loss of steam in passing through the engine, 
by cooling and condensation, it is obvious that whatever steam passes 
through the first cylinder must also find its way through the second cylin- 
der. By varying, therefore, in the receiver-engine, the period of admission 
in the second cylinder, and thus also the volume of steam admitted for each 
stroke, the steam will be measured into it at a higher pressure and of a less 
bulk, or at a lower pressure and of a greater bulk; the pressure and density 
naturally adjusting themselves to the volume that the steam from the re- 
ceiver is permitted to occupy in the second cylinder. With a sufficiently 
restricted admission, the pressure in the receiver may be maintained at the 
pressure of the steam as exhausted from the first cylinder. On the con- 
trary, with a wider admission, the pressure in the receiver may fall or 
"drop" to three fourths or even one half of the pressure of the exhaust- 
steam from the first cylinder. 

(For a more complete discussion of the action of steam in the Woolf and 
receiver engines, see Clark on the Steam-engine.) 

Combined Diagrams of Compound Engines,— The only way 
of making a correct combined diagram from the indicator-diagrams of the 
several cylinders in a compound engine is to set off all the diagrams on the 
same horizontal scale of volumes, adding the clearances to the cylinder ca- 




Fig. 145. 



pacifies proper. When this is attended to, the successive diagrams fall ex- 
actly into their right places relatively to one another, and would compare 
properly with any theoretical expansion-curve. (Prof. A. B. W. Kennedy, 
Proc. Inst. M. E., Oct. 1886.) 



COMPOUND ENGINES. 



?65 



This method of combining diagrams is commonly adopted, but there are 
objections to its accuracy, since the whole quantity of steam consumed in 
the first cylinder at the end of the stroke is not carried forward to the 
second, but a part of it is retained in the first cylinder for compression. For 
a method of combining diagrams in which compression is taken account of, 
see discussions by Thomas Mudd and others, in Proc. Inst. M. E., Feb., 
1887, p. 48. The usual method of combining diagrams is also criticised by 
Frank H. Ball as inaccurate and misleading (Am. Mach., April 12, 1891; 
Trans. A. S. M. E., xiv. 1405, and xv. 403). 

Figure 145 shows a combined diagram of a quadruple-expansion engine, 
drawn according to the usual method, that is, the diagrams are first reduced 
in length to relative scales that correspond wiih the relative piston-displace- 
ment of the three cylinders. Then the diagrams are placed at such distances 
from the clearance-line of the proposed combined diagram as to correctly 
represent the clearance in each cylinder. 

Calculated Expansions and Pressures in Two-cylinder 
Compound Engines. (James Tribe, Am. Mach., Sept. 8i Oct. 1891.) 

Two-cylinder Compound Non-condensing. 
Back pressure ^3 lb. above atmosphere. 



Initial gauge 

pressure 

Initial absolute 

pressure 

Total expansion . 
Exp a n s i o n s in 

each cylinder.. 
Hyp. log:, plus 1. 
Forward \ High, 
pressures ) Low. . 
Back j High, 
pressures I Low 

Mean 
effective 
pressures , 
Ratio-c y 1 i n d e 1 

areas 



a J High. 
^"JLow. 



100 


110 


120 


130 


140 


150 


115 
7.39 


125 

7.84 


135 

8.41 


145 
9 


155 
9.61 


165 
10.24 


2.7 

1.993 
84.8 
31.3 
42.5 
15.5 


2.8 

2.029 
90.5 
32.3 
44.6 
15.5 


2.9 

2.064 
96 
33.1 
46.5 
15.5 


3 
2.008 
101.4 
33.7 
48.3 
15.5 


3.10 
2.131 

106.5 

34.3 

50 
15.5 


3.2 

2.163 
111.5 
31.8 
51.5 
15.5 


42.3 

15.8 


45.9 
16.8 


49.5 
17.6 


53.1 

18.2 


56.5 

18.8 


60 
19.3 


2.67 


2.73 


2.81 


2.91 


3 


3.11 



lv5 
10.89 

3.3 

2.193 
116.3 
35.2 
53 
15.5 
63.3 
19.7 



185 
11.56 

3.4 

2.223 
120.9 
3.6 
51.4 
15.5 

66.5 
20 1 



2.238 
123.2 
35.7 
55 
15.5 
68.2 
20.2 



Two-cylinder Compound Condensing. 
Back pressure, 6.5 lbs. above vacuum . 



Initial gauge-pressures 

Initial absolute pressures. . 
Probable per cent of loss. . 

Total expansions 

Exps. in each cylinder. . .. 

Hyp. log. plus 1 

Mean forward j High 

pressures 1 Low 

Mean back j High 

pressures ( Low 



Mean 
effective 
pressures 
Terminal 
pressures 



(High., 
| Low . . 



High. 
Low. 



[j 1 cnauyc: 

Initial pressure in 1. p. cyl. 
Ratio of cylinder areas. . 



105 

2.6 
15.7 

3.96 

2.376 
62.9 
15.2; 
5 

4.3 



10.95 
26.5 



25.3 
3.32 



100 

115 

2.9 
17 

4.13 

2.418 
67.3 
15.55 
27.8 

4.3 

19.5 
11.25 

27.8 
6.45 

!6.6 
3.51 



110 

125 

3.3 
18.5 

4.3 

2.458 
71.4 
15.9 
29 

4.3 
42.4 
11.6 

19.0 

6.45 
27.8 

3.66 



120 

135 

3.6 
20 

4.47 

2.49; 
75.4 
16.2 
30.2 

4.3 

45.2 
11.9 
30.2 
6.5 



21.5 
4.64 
2.534 

79.3 

16.5 

31.4 
4.3 

47.9 

12.2 

31.4 
6.55 

;o.2 



140 

155 

4.0 
22.7 

4.77 

2.562 
83.2 
16.75 
32.4 

4.3 
50.8 
12.45 



31.4 

4.08 



150 

165 

4.3 
24.2 

4.92 

2.593 
87 

17.05 
33.5 

4.3 
53.5 
12.75 

33 5 
6.6 
32.4 

4.19 



The probable percentage of loss, line 3, is thus explained : There is always 
a loss of heat due to condensation, and which increases with the pressure of 
steam. The exact percentage cannot be predetermined, as it depends 
largely upon the quality of the non-conducting covering used on the cylin- 
der, receiver, and pipes, etc . but will probably be abonr as shown. 

Proportions of Cylinders in Compound Engines.— Authori- 
ties differ as to the proportions by volume of the high and low pressure 
cylinders v and V. Thus Grashof gives V-+- v = 85 frl Krabak, 0.90 4/rJ 



766 THE STEAM-EHGIHE. 

Werner, |r; and Rankine,|/r2, r being the ratio of expansion. Busley 
makes the ratio dependent on the boiler-pressure thus: 

Lbs. per sq . in 60 90 105 120 

V-i-v = 3 4 4.5 5 

(See Season's Manual, p. 95, etc.. for analytical method; Sennett, p. 496, 
etc. ; Clark's Steam-engine, p. 445, etc; Clark, Rules, Tables, Data, p. 849, etc.) 

Mr. J. McFarlane Gray states that he finds the mean effective pressure in 
the compound engine reduced to the low-pressure cylinder to be approxi- 
mately the square root of 6 times the hoiler-pressure. 

Approximate Horse-power of a Modern Compound 
Marine-engine. (Seaton.)— The following rule will give approximately 
the horse-power developed by a compound engine made in accordance with 

„ . . , ,„„ D* X VpX RX S 
modern marine practice. Estimated H.P. = ~-r . 

D = diameter of l.p. cylinder; p — boiler-pressure by gauge; 
B = revs, per min.; S = stroke of piston in feet. 

Ratio of Cylinder Capacity in Compound Marine En- 
gines. (Seaton.)— The low-pressure cylinder is the measure of the power 
of a compound engine, for so long as the initial steam-pressure and rate of 
expansion are the same, it signifies very little, so far as total power only is 
concerned, whether the ratio between the low and high-pressure cylinders 
is 3 or 4; but as the power developed should be nearly equally divided be- 
tween the two cylinders, in order to get a good and steady working engine, 
there is a necessity for exercising a considerable amount of discretion in 
fixing on the ratio. 

In choosing a particular ratio the objects are to divide the power evenly 
and to avoid as much as possible "drop " and high initial strain. 

If increased economy is to be obtained by increased boiler-pressures, the 
rate of expansion should vary with the initial pressure, so that the pressure 
at which the steam enters the condenser should remain constant. In this 
case, with the ratio of cylinders constant, the cut-off in the high-pressure 
cylinder will vary inversely as the initial pressure. 

Let R be the ratio of the cylinders; r, the rate of expansion; p t the initial 
pressure: then cut-off in high-pressure cylinder = R -r- r; r varies withpj, 
so that the terminal pressure p n is constant, and consequently r — Pt -*- J>n; 
therefore, cut-off in high-pressure cylinder — R X pn -s-Pi- 

Ratios of Cylinders as Found in Marine Practice.— The 
rate of expansion may be taken at one tenth of the boiler-pressure (or about 
one twelfth the absolute pressure), to work economically at full speed. 
Therefore, when the diameter of the low-pressure cylinder does not exceed 
100 inches, and the boiler-pressure 70 lbs., the ratio of the low-pressure to 
the high-pressure cylinder should be 3.5; for a boiler-pressure of 80 lbs., 3.75; 
for 90 lbs., 4.0; for 100 lbs., 4.5. If these proportions are adhered to, there 
will be no need of an expansion-valve to either cylinder. If, however, to 
avoid " drop,' 1 the ratio be reduced, an expansion-valve should be fitted to 
the high-pressure cylinder. 

Where economy of steam is not of first importance, but rather a, large 
power, the ratio of cylinder capacities may with advantage be decreased, 
so that with a boiler-pressure of 100 lbs. it may be 3.75 to 4. 

In tandem engines there is no necessity to divide the work equally. The 
ratio is generally 4, but when the steam-pressure exceeds 90 lbs. absolute 4.5 
is better, and for 100 lbs. 5.0. 

When the power requires that the 1. p. cylinder shall be more than 100 in. 
diameter, it should be divided in two cylinders. In this case the ratio of the 
combined capacity of the two 1. p. cylinders to that of the h. p. may be 3.0 
for 85 lbs. absolute, 3.4 for 95 lbs., 3.7 for 105 lbs., and 4.0 for 115 lbs. 

Receiver Space in Compound Engines should be from 1 to 
1.5 times the capacity of the high-pressure cy Under, when the cranks are at 
an angle of from 90° to 120°. When the cranks are at 180° or nearly this, 
the space may be very much reduced. In the case of triple-compound en- 
gines, with cranks at 120°, and the intermediate cylinder leading the high- 
pressure, a very small receiver will do. The pressure in the receiver should 
never exceed half the boiler-pressure. (Seaton.) 



COMPOUND ENGINES. 767 

Formula for Calculating the Expansion and the Work 
of Steam in Compound Engines. 

(Condensed from Clark on the " Steam-engine.") 

a = area of the first cylinder in square inches; 
a' — area of the second cylinder in square inches; 
r = ratio of the capacity of the second cylinder to that of the first; 
L = length of stroke in feet, supposed to be the same for both cylinders; 
I = period of admission to the first cylinder in feet, excluding clearance; 
c = clearance at each end of the cylinders, in parts of the stroke, in feet; 
L' = length of the stroke plus the clearance, in feet; 
I' = period of admission plus the clearance, in feet; 
s = length of a given part of the stroke of the second cylinder, in feet; 
P = total initial pressure in the first cylinder, in lbs. per square inch, sup- 
posed to be uniform during admission; 
P' = total pressure at the end of the given part of the stroke s; 
p — average total pressure for the whole stroke; 
B = nominal ratio of expansion in the first cylinder, or L -j- Z; 
B' = actual ratio of expansion in the first cylinder, or L' -*- V; 
B" = actual combined ratio of expansion, in the first and second cylinders 
together; 
n = ratio of the final pressure in the first cylinder to any intermediate 

fall of pressure between the first and second cylinders; 
N = ratio of the volume of the intermediate space in the Woolf engine, 
reckoned up to, and including the clearance of, the second piston, 
to the capacity of the first cylinder plus its clearance. The value 
of N is correctly expressed by the actual ratio of the volumes as 
stated, on the assumption that the intermediate space is a vacuum 
when it receives the exhaust-steam from the first cylinder. In point 
of fact, there is a residuum of unexhausted steam in the interme- 
diate space, at low r pressure, and the value of N is thereby prac- 
tically reduced below the ratio here stated. N = — 1. 

n — 1 

to = whole net work in one stroke, in foot-pounds. 
Ratio of expansion in the second cylinder: 

In the Woolf engine, -^ y . 



Total actual ratio of expansion = product of the ratios of the thrse con- 
secutive expansions, in the first cylinder, in the intermediate space, and 
in the second cylinder, 



In the Woolf engine, B' (r — -f- N j . 



' I' 

Combined ratio of expansion behind the pistons = — — — rR' = B". 

n 

Work done in the two cylinders for one stroke, with a given cut-off and a 
given combined actual ratio of expansion: 

Woolf engine, iv = <xP[Z'(l -\- hyp log B") — c]; 

Receiver engine, to — aP\l'(l + hyp log B") — c( 1 -| sr )J' 

when there is no intermediate fall of pressure. 

When there is an intermediate fall, when the pressure falls to %, %, ^ of 
the final pressure in the 1st cylinder, the reduction of work is 0,2$, \.0%, 4.6% 
of that when there is no fall. 



768 THE STEAM-ENGINE. 

Total work in the two cylinders of a receiver-engine, for one stroke for 
any intermediate fall of pressure, 

w m aF [„(«_ti + hyp log B „) I c(l + fit=M^S)J. 

Example— Let a = 1 sq. in., P = 63 lbs., I' = 2.42 ft., n = 4, B" = 5.969, 
c = .42 ft., r = 3, R' = 2.653; 

w=lx 63[2.42(5/4 hyp log 5.969) - .42(1+ 3 * jL )] = 421.55 ft.-lbs. 

Calculation of Diameters of Cylinders of a compound con- 
densing engine of 2000 H.P. at a speed of 700 feet per minute, with 100 lbs. 
boiler-pressure. 

100 lbs. gauge-pressure = 115 absolute, less drop of 5 lbs. between boiler 
and cylinder = 110 lbs. initial absolute pressure. Assuming terminal pres- 
sure in 1. p. cylinder = 6 lbs., and taking the expansion in each cylinder to 
vary as the square root of the total expansion, we have: 

Total expansion of steam in bot h cyl inders = 110 -*- 6 = 18.33. 

Expansion in each cylinder = | / 18.33 = 4.28. 

Point of cut-off in each cylinder, per cent of stroke, -j-^- = 23.36. 

1 -\- hyp log of expansion in each cylinder = 1 + hyp log 4.28 = 2.454. 

Terminal and back pressure of h. p. cyl. and initial of 1. p. cyl., j-53 = 
25.70 lbs. 

Average absolute pressure in h. p. cylinder, 25.7 x 2.454 = 63.07 lbs. 

effective " in " " 63.07 - 25.70 = 37.37 " 

absolute " in 1. p. 6 X 2.454 = 14.72 " 

" effective " in " cyl. assum'g31bs. back pres. = l 1.72 " 

Assuming half the work, or 1000 H.P., to be done in the low-pressure cylin- 
der, 

Area of 1 c 1 = 33000 X H.P. 

' v ' piston-speed x av. effective pressure 

33000 X 1000 . AOO . „,■ _. 

= -^k — tt-^t = 4°23 sq. in. = 71.6 in. diam. 
700 X 11.72 

Area of h. p. cyl. = 4023 X ~~ = ^Ty^lfi = 1262 Sq ' in ' = 4(U in * diam> 

11 72 
Ratios of cylinder areas = ^=^ = 1 to 3.189. 

In this calculation no account is taken of clearance, nor of drop between 
cylinders, nor of area of piston-rod. It also assumes that the diagrams in 
both cylinders are the full theoretical diagrams, with hyperbolic expansion 
curves, with no allowance for rounding of the corners. 

Calculation of Diameters of Cylinders of a 500 H.P. Compound Non-con- 
densing Engine. — Assuming initial pressure 170 lbs. above atmosphere, back 
pressure 15.5 lbs., absolute piston-speed 600 feet per minute. 

Total Expansions -185 h- 15.5 = 11.9. 

Expansions in each cylinder = |/H-9 = 3.45; hyp log = 1.238. 

Terminal pressure h. p. cyl. = 185 -h 3.45 = 53.6 lbs. 

Mean total pressure, " " = 53.6 X (1 + 1.238) = 120.0. 

Back pressure h. p. cyl. = terminal pressure 53.6 lbs. 

Mean effective pressure = 120 - 53.6 = 66.4 lbs. 

Terminal pressure 1. p. cyl. = 53.6 -s- 3.45 = 15.5 lbs. 

Mean total pressure " " = 15.5 X 2.238 = 34.7 lbs. 

Mean effective pressure 1. p. cyl. = 34.7 — 15.5 ^ 19.2 lbs. 
19 2 

Ratio of areas of cylinders = x—, = 1 to 3.46. 

bo. A 



Area of 1. p. cyl. = 

33000 X H.P. 



30.2" 



piston-speed X M.E.P." 600 X 19.2 

Area of h. p. cyl., 716 -=- 3.46 = 207 sq. in. = 16.2 in. diameter, 



TRIPLE-EXPANSION ENGINES. 769 

TRIPLE-EXPANSION ENGINES. 

Proportions of Cylinders.— H. H. Suplee, Mechanics, Nov. 1887, 
ives the following method of proportioning cylinders of triple-expansion 
engines: 

As in the case of compound engines the diameter of the low-pressure 
cylinder is first determined, being made large enough to furnish the entire 
power required at the mean pressure due to the initial pressure and expan- 
; sion ratio given; and then this cylinder is only given pressure enough to per- 
form one third of the work, and the other cylinders are proportioned so as to 
i divide the other two thirds between them. 

Let us suppose that an initial pressure of 150 lbs. is used and that 900 H.P. 
is to be developed at a piston-speed of 800 ft. per min., and that an expan- 
sion ratio of 16 is to be reached with an absolute back pressure of 2 lbs. 

The theoretical M.E.P. with an absolute initial pressure of 150 X 14.7 = 
164.7 lbs. initial at 16 expansions is 

fq+hypio g i6) = 1M7 sjrm 

lb lb 

less 2 lbs. back pressure, = 38.83 - 2 = 36.83. 
In practice only about 0.7 of this pressure is actually attained, so that 
S.83 x 0.7 = 25.781 lbs. is the M.E.P. upon which the engine is to be pro- 
portioned. 

To obtain 900 H.P. we must have 33,000 X 900 = 29,700,000 foot-pounds, and 
this divided by the mean pressure (25.78) and by the speed in feet (800) will 
give 

33000 X 900 
800-X-25T8 =1440Sq - ln ' 

for the area of the 1. p. cylinder, which is about equivalent to 43 in. diam. 

Now as one third of the work is to be done in the 1. p. cylinder, the M.E.P. 
in it will be 25.78 -h 3 = 8.59 lbs. 

The cut-off in the high-pressure cylinder is generally arranged to cut off 
at 0.6 of the stroke, and so the ratio of the h. p. to the 1. p. cylinder is equal 
to 16 X 0.6 = 9.6, and the h. p. cylinder will be 1440 -=- 6 = 150 sq. in. area, or 
about 14 in. diameter, and the M.E.P. in the h. p. cylinder is equal to 
9.6 X 8.59 = 82.46 lbs. 

If the intermediate cylinder is made a mean size between the other two, 
its size would be determined by dividing the area of the 1. p. cylinder by the 
square root of the ratio between the low and the high; but in practice this is 
found to give a result too large to equalize the stresses, so that instead the 
area of the 1. p. cylinder is found by dividing the area of the 1. p. piston by 
1.1 times the square root of the ratio of 1. p. to h. p. cylinder, which in this 
case is 1440 -=- (1.1 V9.6) = 422.5 sq. in., or a little more than 23 in. diam. 

To put the above into the form of rules, we have 

Area of low-pressure piston 
Area h. p. cyl. = — 



Area intermediate cjl 



Cutoff in h. p. cyl. X rate of expansion. 
Area of low-pressure p ; ston 



1.1 X Vratio of 1. p. to h. p. cyl. 



The choice of expansion ratio is governed by the initial pressure, and is 
generally chosen so that the terminal pressure in the 1. p. cylinder shall be 
about 10 lbs. absolute. 

Annular Ring Method.— Jay M. Whitham, Trans. A. S. M. E., x. 

577, give,s the following method of ascertaining the diameter of pistons of 
triple expansion engines: 

Lay down a theoretical indicator-diagram of a simple engine for the par- 
ticular expansion desired. By trial find (with the polar planimeter or other- 
wise) the position of horizontal lines, parallel to the back-pressure line, such 
that the three areas into which they divide the diagram, representing low, 
intermediate, and high pressure diagrams, marked respectively A, B, and C, 
are equal. 

Find the mean ordinate of each area: that of " C ,1 will be the mean un- 
balanced pressure on the small piston; that of " B " will be the mean unbal- 
anced pressure on the area remaining after subtracting the area Of the small 
piston from that of the intermediate; and that of the area "A " will denote 



770 



THE STEAM-ENGINE. 



the mean unbalanced pressure on a square inch of the annular ring of the 
large piston obtained by subtracting tne intermediate from the large piston 
We thus see that the mean ordinates of the two lower cards act on annular 
rings. 
Let H = area of small piston in square inches; 

1 = " " intermediate piston in square inches; 
L = " " large piston in square inches; 
Ph — mean unbalanced pressure per square inch from card "C"; 

Pi = " " " " B"; 

Pi = " " " A"; 

S = piston-speed in feet per minute; 
(I.H.P.) = indicated horse-power of engine. 



Then for equal work in each cylinder we have: 
Area of small piston 



H = 33,000 X 5S-- _=_ {ph x s)\ . 



• (1 



Area of annular ring of j _ , 
intermediate cylinder j — ' 



1,000 x t^ -*■ ( Pi X S); 



Area of interme- { 
diate piston j 



I = H 4- 33,000 X - 



Area of annular ring of large piston = 33,000 x 
Area of large piston = L = I + 33,000 x — ~ 



>- (Pi X S) 
I.H.P. ^ 
3 : 
-*-(PlX8); 



(p X 8); 



(3) 



This method is illustrated by the following example: Given I.H.P. = 3000, 
piston-speed S = 900 ft. permin., ratio of expansion 10, initial steam=pres- 
sure at cylinder 127 lbs. absolute, and back-pressure in large cylinder 4 lbs. 
absolute. Find cylinder diameters for equal work in each. 

The mean ordinate of " C ,1 is found to be ph = 37.414 lbs. per sq. in. 
44 "B" " " " pi = 15.782 " " 
" " " " "A" " " " pi = 11.730 " " " 

Then by (1), (2), and (3) we have: 

H = 33,000 x —^ ■+■ 37.414 x 900 = 980 sq. in., diam. 35%"; 



I = £ 



3 + 33,000 X 



3000 



15.782 X 900 = 



I sq. in., diam. 65" 



- 11.730 X y00 = 6432 sq. in., diam. 90^ 

Mr. Whitham recommends the following cylinder ratios when the piston- 
speed is from 750 to 1000 ft. per min., the terminal pressure in the large 
cylinder being about 10 lbs. absolute. 

Cylinder Ratios recommended for Triple-expansion Engines. 



Boiler-pressure 
(Gauge). 


Small. 


Intermediate. 


Large. 


130 
140 
150 
160 


1 
1 
1 
1 


2.25 
2.40 
2.55 
2.70 


5.00 
5.85 
6.90 
7.25 



170 and upwards— quadruple-expansion engine to be used. 
He gives the following ratios from examination of a number of actual 



engines : 
No. of Engines Steam-boiler 
Averaged. 



3 
11 



Pressure. 
130 
135 
140 
145 
150 
160 



h.p. 



Cylinder Ratios, 
int. 
2.10 
2.07 
2.40 
2.35 
2.54 



4.88 
5.00 
5.84 



TRIPLE-EXPANSION ENGINES. 



m 



A Common Rule for Proportioning the Cylinders of mul- 
tiple-expansion engines is: for two-cylinder compound engines, the cylinder 
ratio is the square root of the number of expansions, and for triple-expansion 
engines the ratios of the high to the intermediate and of the intermediate 
to the low are each equal to the cube root of the number of expansions, the 
ratio of the high to the low being the product of the two ratios, that is, the 
square of the cube root of the number of expansions. Applying this rule to 
the pressures above given, assuming a terminal pressure (absolute) of 10 lbs. 
and 8 lbs. respectively, we have, for triple-expansion engines: 



Boiler- 


Terminal Pressure, 10 lbs. 


(Absolute). 


No. of Ex- 
pansions. 


Cylinder Ratios, 
areas. 


130 
140 
150 
160 


13 
14 
15 
16 


1 to 2.35 to 5.53 
1 to 2.41 to 5.81 
1 to 2.47 to 6.08 
1 to 2.52 to 6.35 



Terminal Pressure, 8 lbs. 



No. of Ex- 
pansions. 


Cylinder Ratios, 
areas. 


16J4 
20 


1 to 2.53 to 6.42 
1 to 2.60 to 6.74 
1 to 2.66 to 7.06 
1 to 2.71 to 7.37 



The ratio of the diameters is the square root of the ratios of the areas, and 
the ratio of the diameters of the first and third cylinders is the same as the 
ratio of the areas of first and second. 

Seaton, in his Marine Engineering, says: When the pressure of steam em- 
ployed exceeds 115 lbs. absolute, it is advisable to employ three cylinders, 
through each of which the steam expands in turn. The ratio of the low- 
pressure to high- pressure cylinder in this system should be 5, when the 
steam-pressure is 125 lbs. absolute; when 135 lbs. absolute, 5.4; when 145 
lbs. absolute, 5.8; when 155 lbs. absolute, 6.2; when 165 lbs. absolute, 6.6. 
The ratio of low-pressure to intermediate cylinder should be about one half 
that between low-pressure and high- pressure, as given above. That is, if 
the ratio of 1. p. to h. p. is 6, that of 1. p. to int. should be about 3, and conse- 
quently that of int. to h. p. about 2. In practice the ratio of int. to h. p. is 
nearly 2.25, so that the diameter of the int. cylinder is 1.5 that of the h. p. 
The introduction of the triple-compound engine has admitted of ships being 
propelled at higher rates of speed than formerly obtained without exceeding 
the consumption of fuel of similar ships fitted with ordinary compound 
engiues; in such cases the higher power to obtain the speed has been devel- 
oped by decreasing the rate of expansion, the low-pressure cylinder being 
only 6 times the capacity of the high-pressure, with a working pressure of 
170 lbs. absolute. It is now a very general practice to make the diameter of 
the low pressure cylinder equal to the sum of the diameters of the h. p. and 
int. cylinders; hence, 

Diameter of int. cylinder = 1.5 diameter of h. p. cylinder; 
Diameter of 1. p. cylinder = 2.5 diameter of h. p. cylinder. 

In this case the ratio of 1. p. to h. p. is 6.25; the ratio of int. to h. p. is 2.25; 
and ratio of 1. p. to int. is 2.78. 
Ratios of Cylinders for Different Classes of Engines. 

(Proc. Inst. M. E., Feb. 1887, p. 36.)— As to the best ratios for the cylinders 
iu a triple engine there seems to be great difference of opinion. Considera- 
ble latitude, however, is due to the requirements of the case, inasmuch as 
it would not be expecied that the same ratio would be suitable for an eco- 
nomical land engaie. where the space occupied and the weight were of 
minor importance, as in a war-ship, where the conditions were reversed. In 
the land engine, for example, a theoretical terminal pressure of about 7 
lbs. above absolute vacuum would probably be aimed at, which would give 
a ratio of capacity of high pressure to low pressure of 1 to 8J^ or 1 to 
9; whilst in a war-ship a terminal pressure would be required of 12 to 13 lbs. 
which would need a ratio of capacity of 1 to 5; yet in both these instances 
the cylinders were correctly proportioned and suitable to the requirements 
of the case. It is obviously unwise, therefore, to introduce any hard-and- 
fast rule. 

Types of Three-stage Expansion Engines.— 1. Three cranks 
at 120 deg. 2. Two cranks with 1st and 2d cylinders tandem. 3. Two 
cranks with 1st and 3d cylinders tandem. The most common type is the 
first, with cylinders arranged in the sequence high, intermediate, low. 



Tel 



THE STEAM-ENGIKE. 



Sequence of Cranks.- Mr. Wyllie (Proc. lust. M. E., 1887) favors the 
sequence high, low, intermediate, while Mr. Mudd favors high, intermediate, 
low. The former sequence, high, low, intermediate, gave an approximately 
horizontal exhaust-line, and thus minimizes the range of temperature and 
the initial load; the latter sequence, high, intermediate, low, increased the 
range and also the load. 

Mr. Morrison, in discussing the question of sequence of cranks, presented 
a diagram showing that with the cranks arranged in the sequence high, 
low, intermediate, the mean compression into the receiver was 19)^ per cent 
of the stroke; with the sequence high, intermediate, low r , it was 5? percent. 

In the former case the compression was just what was required to keep 
the receiver-pressure practically uniform; in the latter case the compression 
caused a variation in the receiver-pressure to the extent sometimes of 
22^ lbs. 

Velocity of Steam through Passages in Compound 
Engines. (l J roc. Inst. M. E., Feb. 1887.)— In the SS. Para, taking the area 
of the cylinder multiplied by the piston-speed in feet per second and 
dividing by the area of the port the velocity of the initial steam through 
the high-pressure cylinder port would be about 100 feet per second; the ex- 
haust w r ould be about 90. In the intermediate cylinder the initial steam 
had a velocity of about 180, and the exhaust of 120. In the low-pressure 
cylinder the initial steam entered through the port with a velocity of 250, 
and in the exhaust-port the velocity was about 140 feet per second. 

QUADRUPLE-EXPANSION ENGINES. 

H. H. Suplee (Trans. A. S. M. E., x. 583) states that a study of 14 different 
quadruple-expansion engines, nearly all intended to be operated at a pres- 
sure of 180 lbs. per sq. in., gave average cylinder ratios of 1 to 2, to 3.78, to 
7.70, or nearly in the proportions 1, 2, 4, 8. 

If we take the ratio of areas of any two adjoining cylinders as the fourth 
root of the number of expansions, the ratio of the 1st to the 4th will be the 
cube of the fourth root. On this basis the ratios of areas for different pres- 
sures and rates of expansion will be as follows : 



Gauge- 


Absolute 


Terminal 


Ratio of 


Ratios of Areas 


pressures. 


Pressures. 


Pressures. 


Expansion. 


of Cylinders. 






\\l 


14.6 


1 : 1.95 :3/81 : 7.43 


160 


175 


17.5 


1 : 2.05: 4.18: 8.55 






i 8 


21.9 


1 : 2.16:4.68: 10.12 






18 


16.2 


1 :2.01 : 4.02: 8.07 


180 


195 


ho 


19.5 


1 : 2.10: 4.42: 9.28 






1 8 


24.4 


1 : 2.22:4.94: 10.98 






112 


17.9 


1 : 2.06: 4.23: 8.70 


200 


215 


ho 


21.5 


1:2.15:4.64: 9.98 






I 8 


26.9 


1 : 2.28: 5.19: 11.81 






112 


19.6 


1:2.10:4.43: 9.31 


220 


235 


ho 


23.5 


1 : 2.20: 4.85: 10.67 






I 8 


29.4 


1 : 2.33: 5.42: 12.62 



Seaton says: When the pressure of steam employed exceeds 190 lbs. abso- 
lute, four cylinders should be employed, with the steam expanding through 
each successively; and the ratio of 1. p. to h. p. should be at least 7.5, and 
if economy of fuel is of prime consideration it should be 8; then the ratio 
of first intermediate to h. p. should be 1.8, that of second intermediate to 
first int. 2, and that of L p. to second int. 2.2. 

In a paper read before the North East Coast Institution of Engineers and 
Shipbuilders, 1890, William Russell Cummins advocates the use of a four- 
cylinder engine with four ctanks as being more suitable for high speeds 
than the three-cylinder three-crank engine. The cylinder ratios, he claims, 
should be designed so as to obtain equal initial loads in each cylinder. The 
ratios determined for the triple engine are 1, 2.01. 6.54, and for the quadru- 
ple 1, 2.08, 4.46, 10.47. He advocates long stroke, high piston- speed, 100 rev- 
olutions per minute, and 250 lbs. boiler-pressure, unjacketed cylinders, and 
separate steam and exhaust valves. 



QUADRUPLE-EXPANSION ENGINES. 



773 



Diameters of Cylinders of Recent Triple-expansion 
Engines, Chiefly Marine. 

Compiled from several sources, 1890-1893. 
Diam. in inches: H — high pressure, J = intermediate, L = low pressure 



H 


I 


L 


H 


I 


L 


H 


I 

36 


L 


H 


I 


L 


3 


5 


8 


16 


25.6 


41 


22 


(40 
J 40 


36 


58 


94 


43/f 


7.5 


13 


16J4 


237^ 


38.5 


38 


61.5 


100 


5 
6.5 


8 
10.5 


12 

16.5 


16.5 


24.5 


j31 
131 


23 


38 
38 


61 

60 


28 1 
28 if 


56 


86 


7 


9 


12.5 


17 


27 


44 


24 


37 


56 


39 


61 


97 


7.1 


1-1.8 


18.9 


17 


26.5 


42 


25 


40 


64 


40 


59 


88 


7.5 


12 


19 


17 


28 


45 


26 


42 


69 


40 


67 


106 


8 


11.5 


16 


18 


27 


40 


26 


42.5 


70 


40 


66 


100 


9 


14.5 


22.5 


18 


29 


48 


28 


44 


72 


41 


66 


101 


9.8 


15.7 


25.6 


18 


305. 


51 


293/ s 


44 


70 


41 3^ 


67 


10634 


10 


16 


25 


18.7 


29.5 


43.3 


29.5 


48 


78 


42 


59 


92 


11 


lij 


24 


18^ 


23.6 


35.4 


30 


48 


77 


43 


66 


9-2 


11 


18 


25 


19.7 


29.6 


47.3 


32 


46 


70 


43 


68 


110 


11 


18 


30 


20 


30 


45 


32 


51 


82 


43% 


67 


106^4 


11.5 


18 


28.5 


20 


32.5 


136 


32 


54 


82 


45 


71 


113 


11.5 


17.5 


30.5 


33 


58 


88 


32.5J 

32.5 j 


68 


J 85.7 


12 


19.2 


30.7 


20 


33 


52 


33.9 


55.1 


84.6 


185.7 


13 


22 


33.5 


21 


32 


48 


34 


54 


85 


47 


r-~ 


J 81.5 


14 


22.4 


36 


21 


36 


51 


34 


50 


90 


to 


] 81.5 


14.5 


24 


39 


21.7 


33.5 


49.2 


34.5 


51 


85 


37| 


79 


j 98 


15 


24 


39 


21.9 


34 


57 


34.5 


57 


92 


37 ( 


)98 


15 


24.5 


38 


22 


34 


51 















"Where the figures are bracketed there are two cylinders of a kind. Two 
28" = one 39.6", two 31" = one 43.8", two 32.5" = one 46.0 ', two 36" = one 
50.9", two 37" = one 52.3", two 40" = one 56.6", two 81.5" = one 115", two 
85.7" = one 121", two 98" = one 140". The average ratio of diameters of 
cylinders of all the engines in the above table is nearly 1 to 1.60 to 2.56 and 
the ratio of areas nearly 1 to 2.56 to 6.55. 

The Progress in Steam-engines between 1876 and 1893 is shown 
in the following comparison of the Corliss engine at the Centennial Exhibi- 
tion in 1876 and the Allis-Corliss quadruple-expansion engine at the Chicago 
Exhibition. 



j Quadruple- ) 



1876. 
. Simple 

2 

40 in. 

72 in. 120 in. 

30 ft. 30 ft. 

76 in. 24 in. 

136,000 lbs. 125,440 lbs. 

60 36 

2000 H.P. 1400 H.P. 

3000 H.P. 2500 H.P. 

650,000 lbs. 1,360,588 lbs. 

The Crank-shaft body or wheel-seat of the All is engine has a diameter of 

21 inches, journals 19 inches, and crank bearings 18 inches, with a total 
length of 18 feet. The crank-disks are of cast iron and are 8 feet in diam- 
eter. The crank-pins are 9 inches in diameter by 9 inches long. 

A Double-tandem Triple-expansion Engine, built by Watts, 
Campbell & Co., Newark, N. J., is described in Am. Alack., April 26, 1894. 
It is two three-cylinder tandem engines coupled to one shaft, cranks at 90°, 
cylinders 21, 32 and 48 by 60 in. stroke, 65 revolutions per minute, rated H.P. 
2000; fly-wheel 28 feet diameter, 12 ft. face, weight 174,000 lbs; main shaft 

22 in. diameter at the swell; main journals 19 X 38 in.; crank-pins 9^ X 10 
in.; distance between centre lines of two engines 24 ft. \% in.; Corliss 
valves, with separate eccentrics for the exhaust-valves of the l.p. cylinder. 



j expansion. 
Cylinders, number 

" diameter 24, 40, 

" stroke 

Fly-wheel, diameter 

" width of face. . 

" weight 

Revolutions per minute. . . 
Capacity, economical 

" maximum 

Total weight 



774 



THE STEAM-ENGINE. 



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SSS'SusS. 



ph" ci|||l §| 1 1, ts gsg^'a 1 1 : ^'s i Its o ill § g|f *3 



ECONOMIC PERFORMANCE OF STEAM-ENGINES. ?70 



ECONOMIC PERFORMANCE OF STEAM-ENGINES. 
Economy of Expansive Working under Various Condi- 
tions, Single Cylinder. 

(Abridged from Clark on the Steam Engine.) 

1. Single Cylinders with Superheated Steam, Noncondensing.— In- 
side cylinder locomotive, cylinders and steam-pipes enveloped by the hot 
gases in the smoke-box. Net boiler pressure 100 lbs. ; net maximum press- 
ure in cylinders 80 lbs. per sq. in. 

Cut-off, per cent 20 25 30 35 40 50 60 70 80 

Actual ratio of expansion 3.91 3.31 2.87 2.53 2.26 1.86 1.59 1.39 1.23 
Water per I.H.P. per hour, 

lbs 18.5 19.4 20 21.2 22.2 24.5 27 30 33 

2. Single Cylinders with Superheated Steam, Condensing.— The best 
results obtained by Hirn, with a cylinder 23% X 67 in. and steam super- 
heated 150° F., expansion ratio 3% to 4}^, total maximum pressure in cylin- 
der 63 to 69 lbs. were 15.63 and 15.69 lbs. of water per I.H.P. per hour. 

3. Single Cylinders of Small Size, 8 or 9 in. Diam., Jacketed, Non- 
condensing.— The best results are obtained at a cut-off of 20 per cent, with 
75 lbs. maximum pressure in the cylinder; about 25 lbs. of water per I.H.P. 
per hour. 

4. Single Cylinders, not Steam-jacketed, Condensing.— Best results. 



Engine. 



Corliss and Wheelock . , 

Hirn, No. 6 

Mair, M 

Bache 

Dexter 

Dallas 

Gallatin 



Cylinder, 

Diam. 

and 

Stroke. 



ins. 

18X48 
23% X 6^ 
32 X 66 

25 X 24 

26 X 36 







Total 




Actual 


Maxi- 


Cut-off. 


Expan- 


Pressure 




Ratio. 


in Cylin- 
der per 
sq. in. 


per cent. 


ratio. 


lbs. 


12.5 


6.95 


104.4 


16.3 


5.84 " 


61.5 


24.6 


3.84 


54.5 


15.5 


5.32 


87.7 


18.3 


4.46 


80.4 


13.3 


5.07 


46.9 


15.0 


4.94 


81.7 



Water as 
Steam 

per 
I.H.P. 

per hour. 



lbs. 

19.58 
19.93 
26.46 
26.25 
23.86 



21.1 



Same Engines, Average Results. 



Long Stroke. 



Corliss and Wheelock. 
Hirn 



Short Stroke. 
Bache 

Dexter, Nos. 20, 21, 22, 23 

Dallas, Nos. 27, 28,29.... 

Gallatin, Nos, 24, 25, 22, { 

26 .y ( 



18 X 48 
23M X 6? 

25 X 24 

26 X36 

36 X30 
30.1 X 3C 



Cut-off, Per cent. 



12.5 
16.3 



15.5 

j 18.3 to 33.3 | 
I average 25 | 
j 13.3 to 26.4 I 
I average 19.8 J 
j 12.3 to 18.5 J 
1 average 15.8 f 



104.4 
61.5 



78.2 



19.58 
19.93 



26.25 
24.05 



:3.50 



Feed-water Consumption of Different Types of Engines. 

—The following tables are taken from the circular of the Tabor Indicator 
(Ashcroft Mfg. Co., 1889). In the first of the two columns under Feed- water 
required, in the tables for simple engines, the figures are obtained by 
computation from nearly perfect indicator diagrams, with allowance for cyl- 
inder condensation according to the table on page 752, but without allow- 
ance for leakage, with back-pressure in the non-condensing table taken at 16 
lbs. above zero, and in the condensing table at 3 lbs. above zero. The com- 
pression curve is supposed to be hyperbolic, and commences at 0.91 of the 
return-stroke, with a clearance of 3% of the piston-displacement. 
Table No, 2 gives the feed-water consumption for jacketed compound-con- 



776 



THE STEAM-ENGINE. 



densing engines of the best class. The water condensed in the jackets is 
included in the quantities given. The ratio of areas of the two cylinders are 
as 1 to 4 for 120 lbs. pressure; the clearance of each cylinder is 3$; and the 
cut off in the two cylinders occurs at the same point of stroke. The initial 
pressure in the 1. p. cylinder is 1 lb. per sq. in. below the back-pressure of the 
h. p. cylinder. The average back pressure of the whole stroke in the 1. p. 
cylinder is 4.5 lbs. for 10$ cut-off; 4.75 lbs. for 20$ cut-off; and 5 lbs. for 30$ 
cut-off. The steam accounted for by the indicator at cut-off in the h. p. 
cylinder (allowing a small amount for leakage) is .74 at 10$ cut-off, .78 at 
20$, and .82 at 30% cut-off. The loss by condensation between the cylinders 
is such that the steam accounted for at cut-off in the 1. p. cylinder, ex- 
pressed in proportion of that shown at release in the h. p. cylinder, is .85 at 
10$ cut-off, .87 at 20$ cut-off, and .89 at 30$ cut-off. 

The data upon which table No. 3 is calculated are not given, but the feed- 
water consumption is somewhat lower than has yet been reached (1894), the 
lowest steam consumption of a triple-exp. engine yet recorded being 11.7 lbs. 
TABLE No. 1. 
Feed-water Consumption, Simple Engines, 
non-condensing engines. condensing engines. 





CO 


®* 


Feed-water Re- 




to 


© 


Feed-water Re- 




o 


3 


quired per I.H.P. 




o 

1 




quired per I.H.P. 




.5 




per Hour. 




«j 


per Hour. 




< 
© 
> 










< 

> 










fi 


c3 ji 


© © d © 




fi 


ciJi 


o © £ © 


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© 


<j.5 3M 




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■8 


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<J.S 3 6C 


© 




•+-' o 


*^t3 OS'S 


ifc5 






*' o 


"^ *j w o3 


o 




bcs 


bc^ ,©, 


o 




o 


fcCS 


6c«1 © 




3 . 


© 


.5 -a 


a a ©m 




3 




.5^3 

IP 


a to ©m 


3 

e> 

a 
© 


fig- 




a £ to 

ail 


orrespondi 
ual Result 
in Practic 
ing Slight 


3 

O 

a 


CD-° 
P-l uj- 


8§ 

3 


spondi 
Result 
Practic 

Slight 


a 


's 53 
1* 


11 3? 


© Ct 01 


6 


3 s 

So, 


03 03 

©£ 


gig" 


S-3 bo 


fi 


§~ 


O 


O 


fi 




§ 


O 


o 


f 


60 


8.70 


37.26 


40.95 


r 


60 


14.42 


18.22 


20.00 


1 


70 


12.39 


30.99 


33.68 


1 


70 


16.96 


17.96 


19.69 


1<M 

{ 


80 


16.07 


27.61 


29.88 


5^ 


80 


19.50 


17.76 


19.47 


90 


19.76 


25.43 


27.43 


1 


90 


22.04 


17.57 


19.27 


100 


23.45 


23.90 


25.73 


100 


24.58 


17.41 


19.07 


r 


60 


21.12 


27.55 


29.43 


f 


60 


22.34 


17.68 


19.34 


i 


70 


26.57 


25.44 


27.04 


70 


26.03 


17 47 


19.09 


20 -j 


80 


32.02 


21.04 


25.68 


10 ; 


80 


29.72 


17 30 


18.89 


i 


90 


37.47 


23.00 


24.57 


1 


90 


33.41 


17.15 


18.70 


100 


42.92 


22.25 


23.77 


I 


100 


37.10 


17.02 


18.56 




60 


30.47 


27.24 


29.10 


1 


60 


29.00 


17.93 


19.51 




70 


37.21 


25.76 


27.43 


1 


70 


33.65 


17.75 


19.27 


30 - 


30 


43.97 


24.71 


26.29 


15H 


80 


38.28 


17.60 


19.09 




90 


50.73 


23.91 


25.38 




90 


42.92 


17.45 


18.91 




100 


57.49 


23.27 


24.68 


I 


100 


47.56 


17.32 


18.74 


f 


60 


37.75 


27.92 


29.63 


r 


60 


34.73 


18.58 


20.09 




70 


45.50 


26.66 


28.18 




70 


40.18 


18.40 


19.85 


40M 


80 


53.25 


25.76 


27.17 


20 -j 


80 


45.63 


18.27 


19.69 




90 


61.01 


25.03 


26.35 


90 


51.08 


18.14 


19.51 


I 


100 


68.76 


24.47 


25.73 


I 


100 


56.53 


18.02 


19.36 


r 


60 


43.42 


28.94 


30.66 


f 


60 


44.06 


20.19 


21.64 


i 


70 


51.94 


27.79 


29.31 


1 


70 


50.81 


20.04 


21.41 


50 ■{ 


80 


60.44 


26.99 


28.38 


30 -; 


80 


57.57 


19.91 


21.25 


1 


90 


68.96 


26.32 


27.62 


1 


90 


64.32 


19.78 


21.06 


I 


100 


77.48 


25.78 


26.99 


1 

[ 

i 


100 

60 
70 


71.08 

51.35 
59.10 


19.67 
21.63 
21.49 


20.93 
22.96 
22.74 












40 : 

| 


80 
90 


66.85 
74.60 


21.36 

21.24 


22.56 
22.41 












1 


100 


82.36 


21.13 


22.24 



CALCULATED PERFORMANCES OF STEAM-ENGIKES. *}"/ 



TABLE No. 2. 
Feed-water Consumption for Compound Condensing Engines. 



Cut-off, 


Initial Pressure above 
Atmosphere. 


Mean Effective Press- 
Atmosphere. 


Feed-water 
Required 


per cent. 


H.P. Cyl., 

lbs. 


L.P. Cyl., 

lbs. 


H.P. Cyl., 

lbs. 


L.P. Cyl., 
lbs. 

2.65 

3.87 
5.23 

5.48 
7.56 
9.74 

7.48 
10.10 
12.26 


per T.H.P. per 
Hour, Lbs. 


,0 { 

20 \ 
30 < 


80 
100 
120 

80 
100 
120 

80 
100 
120 


4.0 

7 3 
11.0 

4.3 

8.1 • 
12.1 

4.6 
8.5 
11.7 


11.67 
15.33 
18.54 
26.73 
33.13 
39.29 
37.61 
46.41 
56.00 


16.92 
15.00 
13.86 
14.60 
13.67 
13.09 ' 

14.99 
14.21 
13.87 



TABLE No. 3. 
Feed-water Consumption for Triple -expansion Condensing Engines. 



Cut-off, 


Initial Pressure above 
Atmosphere. 


Mean Effective Pressure. 


Feed-water 
Required 


cent. 


H.P. Cyl., 
lbs. 


I. Cyl., 

lbs. 


L.P. Cyl., 
lbs. 


H.P. Cyl., 
lbs. 


I. Cyl., 
lbs. 


L.P. Cyl., 
lbs. 


perl.H.P. 

per Hour, 

lbs. 


30 -1 


120 
140 
160 


37.8 
43.8 
49.3 


1.3 

2.8 
3.8 


38.5 
46.5 
55.0 


17.1 

18.6 
20.0 


6.5 
7.1 
8.0 


12.05 
11.4 
10.75 


40 -! 


120 
140 
160 


38.8 
45.8 
51.3 


2.8 
3.9 
5.3 


51.5 
59.5 

70.0 


22.8 
23.7 
25.5 


8.6 
9.1 
10.0 


11.65 
11.4 
10.85 


50 ■) 


120 
140 
160 


39.8 
46.8 

52.8 


3.7 
4.8 
6.3 


60.5 
70.5 
82.5 


26.7 
28.0 
30.0 


10.1 
10.8 
11.8 


12.2 
11.6 
11.15 



Most Economical Point of Cut-off in Steam-engines*, 

(See paper by Wolff and Denton, Trans. A. S. M. E., vol. ii. p. 147-281; also, 
Ratio of Expansion at Maximum Efficiency, R. H. Thurston, vol. ii. p. 128.) 
—The problem of the best ratio of expansion is not one of economy of con- 
sumption of fuel and economy of cost of boiler alone. The question of 
interest on cost of engine, depreciation of value of engine, repairs of engine, 
etc., enters as well; for as we increase the rate of expansion, aud thus, 
within certain limits fixed by the back-pressure and condensation of steam, 
decrease the amount of fuel required aud cost of boiler per unit of work, 
we have to increase the dimensions of the cylinder and the size of the en- 
gine, to attain the required power. We thus increase the cost of the engine, 
etc., as we increase the rate of expansion, while at the same time we de- 
crease the fuel consumption, the cost of boiler, etc. So that there is in 
every engine some point of cut-off, determinable by calculation and graphi- 
cal construction, which will secure the greatest efficiency for a given expen- 
diture or money, taking into consideration the cost of fuel, wages of engineer 
and firemen, interest on cost, depreciation of value, repairs to and insurance 
of boiler and engine, and oil, waste, etc., used for engine. In case of freight- 
carrying vessels, the value of the room occupied by fuel should be consid- 
ered in estimating the cost of fuel. 

Sizes and Calculated Performances of Vertical High- 
speed Engines.— The following tables are taken from a circular of the 
Field Engineering Co., New York, describing the engines made by the Lake 
Erie Engineering Works. Buffalo, N. Y. The engines are fair representatives 
of the type now coining largely into use for driving dynamos directly with- 
out belts. The tables were calculated by E. F. Williams, designer of the 
engines. They are here somewhat abridged t<v save space: 



W& 



THE STEAM-ENGltfE. 









Simple Engi 


nes 


—Non-condensing. 








>-.<v 


<u 


a 


H.P. whgn 


H.P. when 


H.P. when 


Dimen- 


.2 


6 


o& 


o 


£ 


Cutting off 


Cutting off 


Cutting off 


W heels. 

diain. face 


ft 
"p. 


•£• 


°'~ 




Pi 


at 1/5 stroke. 


at J4 stroke. 


at y§ stroke. 


to J 


e CD 


o 


a 


70 


80 


90 


70 


80 


90 


70 


80 


90 


Ft 


Tn 


fca 

c3 


03 ' 

A 1 


Q'~ 


w 


& 


lbs. 
20 


lbs. 

25 


lbs. 
30 


lbs. 
26 


lbs. 
31 


lbs. 
30 


lbs. 
32 


lbs. 

37 


lbs. 
43 


4 


4 


214 


3 




10 


370 


12 


318 


27 


32 


39 


34 


41 


47 


41 


48 


50 


4V. 


5 




8V4 


10 V, 


14 


277 


41 


4!i 


60 


52 


62 


71 


63 


74 


85 


5' 9' 






4 


ia 


16 


246 


53 


6-1 


77 


67 


81 


93 


82 


90 


111 


6'8' 


9 


4 


4U. 


13V6 


18 


222 


66 


81 


96 


84 


100 


• 116 


102 


120 


138 


7Yo 


11 


4 


5 


16 


20 


181 


95 


115 


138 


120 


144 


166 


140 


172 


m 


8'4' 


15 


4U 


6 


18 


24 


158 


119 


144 


173 


151 


181 


208 


183 


215 


248 


10 


19 


5 


7 


22 


28 


138 


179 


21 C 


261 


227 


272 


313 


270 


324 




11 '8' 


28 





8 


24U 


32 


120 


221 


2(17 




2S1 






340 


400 




13'4' 


34 


7 


9 


27 


34 


112 


269 


325 


392 


342 


409 


470 


414 


487 


500 


14'2' 


41 


8 


10 


Mean eff. press, lb 


24 


29 


35 


30.5 


36.5 


42 


37 


43.5 


50 


m. Q 


Ratio of expans'n 


5 


4 


3 


nom 


nal- power 


Terminal pressure 




















rating 0* tne en- 


(about) lbs 


17.9 


20 


22 3 


22 4 


25 


27.0 


29.8 


33.3 


30 8 


gines is at 80 lbs. 


Cyl.condensafn, c , 
Steam per I.H.P 


i 26 


26 


26 


24 


24 


24 


21 


21 


21 


gauge pressure, 
steam cut-off at 


per hour lbs 


32.9 


30 


27.4 


31.2 


29.0 


27.9 


32 


31.4 


30 


J4 stroke. 


Compound Engines — Non-condensing — High - pressure 
Cylinder and Receiver Jacketed. 








H.P. when cutting H.P. when cutting 


H.P.whencutting 




. 


•-< 


off at J4 Stroke | off at J^ Stroke 


off at }4 Stroke 


Diam. 
Cylinder, 
inches. 


0) 


P. 


in h.p. Cylinder. 


in h.p. Cylinder . 


in h.p. Cylinder. 


Cyl. 


Cyl. 


Cyl. 


Cyl. 


Cyl. 


Cyl. 




„ 




Ratio, 


Ratio, 


Ratio, 


Ratio, 


Ratio, 


Ratio, 




p 
02 


> 


3^:1. 


4^:1. 


3^:1. 


4*6 : 1. 


VA • 1. 


4^:1. 


Pk 


Ph 




SO 


90 


130 


150 


80 


90 


130 


150 


80 


90 


130 


150 


W 


w 


J 






lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


bs. 


lbs. 


lbs. 


5% 
63^ 




12 


10 


370 


7 


15 


19 


32 


23 


31 


35 


40 


44 


55 


04 


79 


7W 




12 


318 


9 


19 


24 


40 


29 


39 


45 


59 


56 


70 


81 


101 


r^ 


9 


; 


14 


277 


14 


28 


30 


60 


43 


58 


07 


87 


83 


104 


121 


159 


9 


0U 


19 


16 


246 


1S 


37 


47 


78 


57 


70 


87 


114 


109 




158 


196 


10W 




22 U 


18 


222 


26 


53 


68 


112 


81 


109 


125 


104 


156 


195 


220 


281 


12 


3U 


25 


20 


185 


32 


65 


84 


139 


100 


135 


154 


202 


192 


241 


279 


346 


lWo 






24 


158 


43 


88 


112 


186 


135 


LSI 


200 


271 


258 


323 


374 


464 


16 






28 


138 


57 


118 


151 




180 


242 


277' 


303 


346 


433 


502 


023 


18 i 




38 


32 


120 


74 


1 52 


194 


321 


232 


312 


357 


408 


446 


558 


047 


803 


20 £ 




43 


31 


112 


94 


194 


249 


412 


297 


400 


457 


001 


572 


715 




1030 


-Jf., • 




52 


42 


93 


13S 


285 


3!i5 


003 


430 


587 


070 


WHO 


838 


048 


1215 


1508 


28J^ t 


3 


60 


48 


80 


ISO 


374 


477 


789 


570 


707 


877 


1151 


1096 


370 


1589 


1973 


Mean eff ec. press.. .lbs 


3.3 


6.8 


8.7 


14.4 10.4 


14.0 


16 


21 


20 


25 


29 


36 


Ratio of expansion — 


13*£ 


ism I 10J4 


13M 


6% 


m 


Cyl. condensation, %... 


14 


14 


16 | 16 12 


12 


13 


13 


10 


10 


11 


11 


Ter. press, (about). lbs. 


7.3 


7.7 


7.9, 9 


9 2 


10.4 


10.5 


12 


14 


5.5 


14.0 


17.8 


Loss from expanding 1 
























below atmosphere, % 


34 


15 


17 3 


5 























St per I.H.P. p. hr.lbs 


55 


42 


47 I 29 




27 . 7 


2<s 7 


25.4 30 i 


26.2 


21 


20 


The original table contains figures of horse-power, etc., for 110 and 120 lbs. fi 


cylin 


ler 


rath 


)Of 4 


tol; 


and 


140 1 


bs., 




43^ t 


ol. 















CALCULATED PERFORMANCES OF STEAM-ENGINES. 779 





fomponn 


d-engines- 


Condensing— 


S team-jacket ed. 










H.P. when cutting 


H.P. when cutting 


H.P. when cutting 




03 


53 


off at J4 Stroke 


off at % Stroke 


off at y 2 Stroke 


Diam. 
Cylinder, 
Inches. 


2 


ft 


in h.p. Cylinder. 


in h.p. Cylinder. 


in h.p. Cylinder. 


s 


° s 


Cyl. 


Cyl. 


Cyl. 


Cyl. 


Cyl. 


Cyl. 




- 


^ ° 


Ratio, 


Ratio, 


Ratio, 


Ratio, 


Ratio, 


Ratio, 




o 


> 
a? 


Wz • l. 


4 : 1. 


%:1. 


4 : 1. 


3*6:1. 


4 : 1. 


P-' 


P-i 




80 


110 


115 


125 


80 


110 


115 


125 


80 


110 


115 


125 


X 


a 






lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


6 


H* 


12 


10 


370 


44 


59 


53 


62 


55 


70 


68 


75 


70 


97 


95 


106 


(U-o 


?i.> 


13* 


12 


318 


56 


76 


67 


78 


70 


90 


87 


95 


90 


123 


120 


134 


^1 


9 




14 


277 


83 


112 


100 


116 


104 


133 


129 


141 


133 


183 


1 79 


200 


9i « 


10* 


19 


lfi 


246 


109 


117 


131 


152 


136 


174 


169 


185 


174 


239 


234 


261 


11 


1-2 




IS 


222 


156 


210 


187 


218 


195 


25d 


242 


265 


250 


343 


335 


374 


1SU 




2a 


20 


185 


192 


260 


231 


269 


241 


308 


298 


327 


308 


423 


414 


462 


14 






24 


158 


25S 


348 


310 


361 


323 


413 


400 


439 


413 


568 


555 


619 


17 






28 


138 


840 


467 


415 


484 


433 


554 


536 


588 


554 


761 


744 


830 


19 




38 


35 


120 


446 


602 


535 


624 


558 


714 


691 


758 


714 


981 


059 


1070 


21 




43 


34 


112 


572 


772 


686 


801 


715 


915 


887 


972 


915 


1258 




1373 


26 


55 


42 


93 


83S 


1131 




1174 


1048 


1341 


1299 




1341 


1844 


1801 


2012 


30 


33 


60 


48 


80 


1096 


1480 


1316 


1534 


1370 


1757 


1699 




1757 


2411 


2356 


2632 


Mean effec. press 
Ratio of Expansi 
C3'l. condensatioi 


..lbs. 
on... 

, %. . . 


20 


27 


24 


28 


25 


32 


31 


34 


32 


44 


43 


48 


13* 


mi 


10 


12J4 


6% 


m 


18 I 18 


20 | 20 


15 1 15 


18 1 18 


12 1 12 


14 I 14 


St. per I. H.P. p. h 


r.lbs. 


17.3116.6 


16.6|15.2 


17.0|16.4 


16.3|15.8 


17.5|17.0 


16.8|l6.0 



The original table contains figures for 95 lbs., cylinder ratio 3 
120 lbs., ratio 4 to 1. 



Triple-expansion Engines, Non-condensing.— Receiver 
only Jacketed. 









Horse-power 


Horse-power 


Horse-power 




i. 


CD 


when Cutting 


when Cutting 


when Cutting 


Cylinders, 


i 


off at 42 per 


off at 50 per 


off at 67 per 


■j 




cent of Stroke 


cent of Stroke 


cent of Stroke 




■- 


.2 =,: 


in First Cylin- 


in First Cylin- 


in First Cylin- 




1 




der. 


der. 


der. 


H.P. 


LP. 


L. P. 

12 


180 lbs. 


200 lbs. 


180 lbs. 


200 lbs. 


180 lbs. 


200 lbs 


43/f 


7* 


370 


55 


64 


70 


84 


95 


108 


5* 


8* 


13V, 


12 


318 


70 


81 


90 


106 


120 


137 


6* 


10* 


16* 


14 


27, 


104 


121 


133 


158 


179 


204 


7* 


12 


19 




246 


136 


158 


174 


207 


234 


267 


9 


14* 


22* 


18 


222 


195 


226 


250 


296 


335 


382 


10 


16 


25 


20 


185 


241 


279 


308 


366 


414 


471 


11* 


18 


28* 


24 


158 


323 


374 


413 


490 


555 


632 


13 


22 


33* 


28 


138 


433 


502 


554 


657 


744 


848 


15 


24*, 


38 


32 


120 


558 


647 


714 


847 


959 


1093 


17 


27 


43 


34 


112 


715 


829 


915 


1089 


1230 


1401 


20 


33 


52 


42 


93 


1048 


1215 


1341 


1592 


1801 


2053 


23* 


38 


60 


48 


80 


1370 


1589 


1754 


2082 


2356 


2685 


Mean effective press., lbs. 


25 


29 


32 


38 


43 


49 




16 


13 


10 


Per cent cyl. condens — 


14 


12 


10 


Steam p. I. H.P. p.hr., lbs. 


20.76 1 19.36 


19.25 | 17.00 


17.89 i 17.20 


Lbs. c 


aal at S 


lb. evt 


ip. 


lbs. 


2.59 


2.39 


2.40 


2.12 


2.23 


2.15 



780 



THE STEAM-ENGINE. 



Triple-expansion Engines— Condensing— Steam- 
Jacketed. 



Diameter 

Cylinders, 
inches. 



4M 
JO 

ny 2 

13 

15 

17 
20 

2sy 2 






Horse-power 

when Cut- 
ting off at J4 

Stroke in 
First Cylin- 
der. 



Mean effec. press., lbs. 

No. of expansions 

Percent cyl. condens. 
St.p.I.H.P.p. hr.,lbs. 
Coal at 8 lb. evap., lbs. 



Horse-power 

when Cut- 
ting off at H 

Stroke in 

First Cylin 

der. 



120 140 160 
lbs. lbs. ;lbs 



125 
154 1 

206 245 

27? 329 

35? 424 

458 543 

670 796 

877 1041 



19 19 
14.7 13.9 13.3 
1.8 1.73 1 



Horse-power 

when Cut- 
ting off at Yz 

Stroke in 
First Cylin- 
der. 



„ 140 160 
lbs. lbs. lbs. 



120 140 160 
lbs. lbs. lbs. 



16 | 16 16 | 12 12 
14.3 13.98 13.2 14.3 13.6 13.0 
1.78 1.74 1.65 1.78 1.70 1.62 



Horse-power 
when Cut- 
ting off at y^ 

Stroke in 
First Cylin- 
der. 



120 140 160 
lbs. lb?, lbs. 



110 

140 
208 
272 
390 
481 
645 
865 
1115 
1430 



15.7 14. 
1.96 1.8 



1.7' 



Type of Engine to be used where Exhaust-steam is 
needed for Heating.— In many factories more or less of the steam 
exhausted from the engines is utilized for boiling, drying, heating, etc. 
Where all the exhaust-steam is so used the question of economical use of 
steam in the engine itself is eliminated, and the high-pressure simple engine 
is entirely suitable. Where only part of the exhaust-steam is used, and the 
quantity so used varies at different times, the question of adopting a simple, 
a condensing, or a compound engine becomes more complex. This problem 
is treated by C. T. Main in Trans. A. S. M. E., vol. x. p. 48. He shows that 
the ratios of the volumes of the cylinders in compound engines should vary 
according to the amount of exhaust-steam that can be used for heating. A 
case is given in which three different pressures of steam are required or 
could be used, as in a worsted dye-house: the high or boiler pressure for 
the engine, an intermediate pressure for crabbing, and low-pressure for 
boiling, drying, etc. If it did not make too much complication of parts in 
the engine, the boiler-pressure might be used in the high-pressure cylinder, 
exhausting into a receiver from which steam could be taken for running 
small engines and crabbing, the steam remaining in the receiver passing 
into the intermediate cylinder and expanded there to from 5 to 10 lbs. above 
the atmosphere and exhausted into a second receiver. From this receiver 
is drawn the low-pressure steam needed for drying, boiling, warming mills,- 
etc., the steam remaining in receiver passing into the condensing cylinder. 
Comparison of the Economy of Compound and Single- 
cylinder Corliss Condensing Engines, each expanding 
about Sixteen Times. (D. S. Jacobus, Trans. A. S. M. E., xii. 943.) 

The engines used in obtaining comparative results are located at Stations 
I. and II. of the Paw tucket Water Co. 

The tests show that the compound engine is about 30# more economical 
than the single-cylinder engine. The dimensions of the two engines are as 
follows: Single 20" X 48"; compound 15" and 30^" x 30". The steam 
used per horse-power per hour was: single 20.35 lbs., compound 13.73 lbs. 

Both of the engines are steam-jacketed, practically on the barrels only, 
with steam at full boiler-pressure, viz. single 106.3 lbs., compound 127.5 lbs. 



PERFORMANCES OF STEAM-ENGINES. 781 

The steam-pressure in the case of the compound engine is 127 lbs., or 21 
lbs. higher than for the single engine. If the steam-pressure be raised this 
amount in the case of the single engine, and the indicator-cards be increased 
accordingly, the consumption for the single-C3 T linder engine would be 19.97 
lbs. per hour per horse-power. 

Two-cylinder vs. Three-cylinder Compound Engine.— 
A Wheelock triple-expansion engine, built for the Merrick Thread Co., 
Holyoke, Mass., is constructed so that the intermediate cylinder maybe cut 
out of the circuit and the high-pressure and low-pressure cylinders run as a 
two-cylinder compound, using the same conditions of initial steam-pressure 
and load. The diameters of the cylinders are 12, 16, and 24^§ inches, the 
stroke of the first two being 36 in. and that of the low r -piessure cylinder 48 

i. The results of a test reported by S. M. Green and G. I. Rockwood, Trans. 

. S. M. E., vol. xiii. 647, are as follows: In lbs. of dry steam used per I.H.P. 
per hour, 12 and 24|§ in. cylinders only used, two tests 13.06 and 12.76 lbs., 
average 12.91. All three cylinders used, two tests 12.67 and 12.90 lbs., average 
12.79. The difference is only 1%, and would indicate that more than two cylin- 
ders are unnecessary in a compound engine, but it is pointed out by Prof. 
Jacobus, that the conditions of the test were especially favorable for the 
two-cylinder engine, and not relatively so favorable for the three cylinders. 
The steam-pressure was 142 lbs. and the number of expansions about 25. 
(See also discussion on the Rockwood type of engine, Trans. A. S. M. E., vol. 
cvi.) 

Effect of Water contained in Steam on the Efficiency of 
the Steam-engine. (From a lecture by Walter C. Kerr, before the 
Franklin Institute, 1891.) — Standard writers make little mention of the effect 
of entrained moisture on the expansive properties of steam, but by common 
consent rather than any demonstration they seem to agree that moisture 
produces an ill effect simply to the percentage amount of its presence. 
That is, 5% moisture will increase the water rate of an engine 5%. 

Experiments reported in 1893 by R. C. Carpenter and L. S. Marks, Trans. 
A. S. M. E., xv., in which water in varying quantity w r as introduced into the 
steam-pipe, causing the quality of the steam to range from 99$ to 58$ dry, 
showed that throughout the range of qualities used the consumption of dry 
steam per indicated horse-power per hour remains practically constant, and 
indicated that the water was an inert quantity, doing neither good nor harm. 

It appears thac the extra work done by the heat of the entrained water 
during expansion is sensibly equal to the extra, negative work which it does 
during exhaust and compression, that the heat carried in by the entrained 
water performs no useful function, and that a fair measure of the economy 
of an engine is the consumption of dry and saturated steam. 

Relative Commercial Economy of Best Modern Types of 
Compound and Triple-expansion Engines. (J. E. Denton, 
American Machinist, Dec. 17, 1891.) — The following table and deductions 
show the relative commercial economy of the compound and triple type for 
the best stationary practice in steam plants of 500 indicated horse-power. 
The table is based on the tests of Prof. Schroter, of Munich, of engines built 
at Augsburg, and those of Geo. H. Barrus on the best plants of America, and 
of detailed estimates of cost obtained from several first-class builders. 

Trip motion, or Corliss engines of [^SS l 136 14 "° 
the twin-compound-receiver con- J T £; ^'^ in ^ ^'f^ ie " t ; 

Trip motion, or Corliss engines of [ Lbs. water per hour per | .„ 5g - g0 
the triple-expansion four-cylin- j H. P., by measurement, f 
der-receiver condeusing type, ex- \ Lbs. coal per hour per 1 
panding 2^ times. Boiler pressure, ! H. P., assuming 8.5 lbs. > 1.48 1.50 
150 lbs. i. actual evaporation. ) 

The figures in the first column represent the best recorded performance 
(1891), and tnose in the second column the probable reliable performance. 

Increased cost of triple -expansion plant per horse-power, including 
boilers, chimney, heaters, foundations, piping and erection $4.50 

The following table shows the total annual cost of operation, with coal at 
$4.00 per ton, the plant running 300 days in the year, for 10 hours and for 
24 hours per day : 



782 



THE STEAM-ENGINE. 





10 


24 








Per H.P. 

$9.90 
9.00 
0.90 


Per H.P. 

$28.50 


Expense for coal. Triple plant 

Annual saving of triple plant in fuel 


25.92 

2.60 


Annual interest at 5% on $4.50 

Annual depreciation at 5% on $4.50 

Annual extra cost of oil, 1 gallon per 24-hour 

day, at $0.50, or 15$ of extra fuel cost 

Annual extra cost of repairs at 3% on $4.50 per 


$0.23 
0.23 

0.15 

0.06 


$0.23 
0.23 

0.36 

0.14 








$0.67 


$0.96 


Annual saving per H.P 


$0.23 


$1.64 



The saving between the compound and triple types is much less than that 
involved in the step from the single-expansion condensing to the compound 
engine. The increased cost per horse-power of the triple plant over the 
compound is due almost entirely to the extra cost of the triple engine and 
its foundations, the boilers costing the same or slightly more, owing to their 
extra strength. In the case of the single versus the compound, however, 
about one third of the increased cost of the compound engine is offset by the 
less cost of the latter's boilers. 

Taking the total cost of the plants at $33.50, $36.50 and $41 per horse- 
power respectively, the figures in the table imply that the total annual sav- 
ing is as follows for coal at $4 per ton: 

1. A compound 500 horse-power plant costs $18,250, and saves about $1630 
for 10 hours 1 service, and $4885 for 24 hours 1 service, per year over a single 
plant costing $16,750. That is, the compound saves its extra cost in 10-hour 
service in about one year, or in 24-hour service in four months. 

2. A triple 500 horse-power plant costs $20,500, and saves about $114 per 
year in 10-hour service, or $826 in 24-hour service, over a compound plant, 
thereby saving its extra cost in 10-hour service in about 19% years, or in 24- 
hour service in about 2% years. 

Triple - expansion Pumping-engine at Milwaukee- 
Highest Economy on Record, 1893. (See paper on "Maximum 
Contemporary Economy of the Steam-engine," by R. H. Thurston, Trans. 
A. S. M. E., xv. 313.)— Cylinders 28, 48 and 74 in. by 60 in. stroke; ratios of 
volumes 1 to 3 to 7; total number of expansions 19.55; clearances, h.p. 
1.4$; int. 1.5%; 1. p. 0.77$; volume of receivers: 1st, 101.3 cu. ft.; 2d, 181 cu. 
ft.; steam-pressure gauge during test, average 121.5 lbs.; vacuum 13.84 lbs. 
absolute; revolutions 20.3 per minute; indicated horse-power, h.p. 175.4, int. 
109.6, 1. p. 228.9; total, 573.9; total friction, horse-power 52.91 = 9.22$; dry 
steam per I. H.P. per hour 11.678; B.T.U. per I. H.P. per min. 217.6; duty in 
foot-pounds per 100 lbs. of coal, 143,306,000; per million B.T.U., 137,656,000. 

Steam per I. H.P. per hour, from diagram, at cut-off 9.35 9.12 8.37 

" release.. . 10.1 10.0 8.92 

Steam accounted for by indicator at cut-off, per cent. . . 87.1 85.0 78.2 

" " " " " release, " ... 94.0 93.2 83.2 

Per cent of total steam used by jackets 9.25 

Highest Economy of the Two - cylinder Compound 
Pumping-engines,- Repeated tests of the Pawtucket-Corliss engine, 
15 and '60% by 30 in. stroke, gave a water consumption of 13.69 to 14.16 lbs. 
per I. H.P. per hour. Steam -pressure 123 lbs.; revolutions per min. 48 lbs.; 
expansions about 16. Cylinders jacketed. The lowest water rate was with 
jackets in use; both jackets supplied with steam of boiler pressure. The 
average saving due to jackets was only about 2^ per cent. (Trans. A. S. 
M. E., xi. 328 and 1038; xiii. 176.) 

This record was beaten in 1894 by a Leavitt pumping-engine at Louisville, 
Ky. (Trans. A. S. M. E. xvi.) Cylinders 27.21 and 54.13 in. diam. by 10 ft. 
stroke; revolutions per min. 18.57; piston speed 371.5 ft. ; expansions 20.4; 
steam -pressure, gauge, 140 lbs. Cylinders and receiver jacketed. Steam 



PERFORMANCES OF STEAM-ENGINES. 



783 



used per I.H.P. per hour, 12.223 lbs. Duty per million B.T.U. = 138,126,000 
ft.-lbs. 

Test of a Triple-expansion Funiping-engine with and 
without Jackets, at Laketon, Ind., by Prof. J. E. Denton (Trans. A. 
S. M. E., xiv. 1340).— Cylinders 24, 34 and 54 in. by 36 in. stroke; 28 revs, per 
min. ; H.P. developed about 320; boiler-pressure 150 lbs. Tests made on eight 
different days with different sets of conditions in jackets. At 150 lbs. boiler- 
pressure, and about 20 expansions, with any pressure above 43 lbs. in all of 
the jackets and reheaters, or with no pressure in the high jacket, the per- 
formance was as follows: With 2.5$ of moisture in the steam entering the 
engine, the jackets used 16$ of the total feed-water. About 20$ of the latter 
was condensed during admission to the high cylinder, and about 13.85 lbs. 
of feed-water was consumed per hour per indicated horse-power. With no 
jackets or reheaters in action the feed-water consumption was 14.99 lbs., or 
8.3% more than with jackets and reheaters. The consumption of lubricating 
oil was two thirds of a gallon of machine oil and one and three quarter gal- 
lons of cylinder oil per 24 hours. The friction of the engine in eight tests on 
different days varied from 5.1% to 8.7%. 

If we regard the measurements of indicated horse-power and water as 
liable to an error of one per cent, which is probably a minimum allowance 
for the most careful determinations, the steam economy is the same for the 
following conditions: 

(a) Any pressure from 43 to 131 in the intermediate and low jackets and 
receivers. 

(b) Any pressure from to 151 in the jacket of high cylinder. 

(c) Any cut-off from 21$ to 23$ in high cylinder, from 39$ to 43$ in inter- 
mediate cylinder, from 40$ to 53$ in low cylinder. 

Water Consumption of Three Types of Sulzer Engines. 

(B. Donkin, Jr., Eng'g, Jan. 15, 1892, p. 77.) 

Summary and Averages of Twenty-one Published Experiments op the 

Sulzer Type of Steam-engine. All Horizontal Condensing 

and Steam- jacketed. From 1872 to 1891. 



Single j 

Cyl. 1 

Com. j 

pound. ( 

Triple. . -j 



lbs. 
72 to 

95 
84 to 

104 
104 to 

156 



ft. 
per min 
272 to 



444 to 
607 



157 to 
400 

133 to 
524 

198 to 
615 



Steam Consump 
tion, pounds per 
I.H.P. per hour, 
includingSteam- 
pipe water and 
Jacket Water. 



lbs. 
j 18.7 to 19 8 
( Mean ]9.4 
j 13.35 to 16.0 
1 Mean 14.44 
j 11.85 to 12.86 
j Mean 12.36 



Steam Consump- 
tion, pounds per 
I.H.P. per hour, 
exclud'g Steam- 
pipe water, but 
includingjacket 
Water. 



lbs. 
17.9 to 19.2 
Mean 18.95 
13.4 to 15.5 
Mean 14.3 
11.7 to 12.7 
Mean 12.18 



I 5 exp. 
J 1872-78 
I 10 exp. 
f 1888-91 
I 6 exp. 
) 1888-89 



Triple-expansion Corliss engine at Narragansett E. L. Co., Providence, R. 
I., built by E. P. Allis Co. Cylinder 14, 25 and 33 in. by 48 in. stroke tested at 
99 revs, per min.; 125 lbs. steam-pressure; steam per I.H.P. per hour 12.94 
lbs. ; H.P. 516. A full account of this engine, with records of tests is given by 
J. T. Henthorn, in Trans. A. S. M. E., xii. 643. 

Buckeye-cross compound engine, tested at Chicago Exposition, by Geo. 
H. Barrus (Evg'g Record. Feb. 17, 1894). Cylinder 14 and 28 by 24 in. stroke; 
tested at 165 r. p. m. ; 120 lbs. steam-pressure. I.H.P. in four tests condens- 
ing and one non-condensing 295 224 123 277 267 

Steam per horse-power per hour 16.07 15.71 17.22 16.07 23.24 

Relative Economy of Compound Non-condensing En- 
gines under Variable IiOads.— F. M. Rites, in a paper on the Steam 
Distribution in a Form of Single-acting Engine (Trans. A. S. M. E. xiii. 537), 
discusses an engine designed to meet the following problem : Given an 



784 THE STEAM-EHGIKE. 

extreme range of conditions as to load or steam -pressure, either or both, to 
fluctuate together or apart, violently or with easy gradations, to construct 
an engine whose economical performance should be as good as though the 
engine were specially designed for a momentary condition— the adjustment 
to be complete and automatic. In the ordinary non-condensing compound 
engine with light loads the high -pressure cylinder is frequently forced to 
supply all the power and in addition drag along with it the low-pressure 
piston, whose cylinder indicates negative work. Mr. Rites shows the 
peculiar value of a receiver of predetermined volume which acts as a clear- 
ance chamber for compression in the high-pressure cylinder. The Westiug- 
house compound single-acting engine is designed upon this principle. The 
following results of tests of one of these engines rated at 175 H.P. for most 
economical load are given : 

Watek Rates under Varying Loads, lbs. per H.P. per hour. 

Horse-power 210 170 140 115 100 80 50 

Non-condensing 22.6 21.9 22.2 22.2 22.4 24.6 28.8 

Condensing 18.4 18.1 18.2 18.2 18.3 18.3 20.4 

Efficiency of Non-condensing Compound Engines. (W. 

Lee Church, Am. Mach., Nov. 19, 1891.)— The compound engine, non-con- 
densing, at its best performance will exhaust from the low-pressure ejdin- 
der at a pressure 2 to 6 pounds above atmosphere. Such an engine will be 
limited in its economy to a very short range of power, for the reason that 
its valve-motion will not permit of any great increase beyond its rated 
power, and any material decrease below its rated power at once brings the 
expansion curve in the low-pressure cylinder below atmosphere. In other 
words, decrease of load tells upon the compound engine somewhat sooner, 
and much more severely, than upon the non-compound engine. The loss 
commences the moment the expansion line crosses a line parallel to the 
atmospheric line, and at a distance above it representing the mean effective 
pressure necessary to carry the f fictional load of the engine. When expan- 
sion falls to this point the low-pressure cylinder becomes an air-pump over 
more or less of its stroke, the power to drive which must come from the 
high pressure cylinder alone. Under the light loads common in many 
industries the low-pressure cylinder is thus a positive resistance for the 
greater portion of its stroke. A careful study of this problem revealed the 
functions of a fixed intermediate clearance, always in communication with 
the high-pressure cylinder, and having a volume bearing the same ratio to 
that of the high-pressure cylinder that the high-pressure cylinder bears to 
the low-pressure. Diagrams were laid out on this principle and justified 
until the best theoretical results were obtained. The designs were then laid 
down on these lines, and the subsequent performance of the engines, of 
which some 600 have been built, have fully confirmed the judgment of the 
designers. 

The effect of this constant clearance is to supply sufficient steam to the 
low-pressure cylinder under light loads to hold its expansion curve up to 
atmosphere, and at the same time leave a sufficient clearance volume in the 
high -pressure cylinder to permit of governing the engine on its compression 
under light loads. 

Economy of Engines under Varying Loads. (From Prof. 
W. C. Unwin's lecture before the Society of Arts, London, 1892.)— The gen- 
eral result of numerous trials with large engines was that with a constant 
load an indicated horse-power should be obtained with a consumption of 
1^2 pounds of coal per indicated horse-power for a condensing engine, and 
\% pounds for a non-condensing engine, figures which correspond to about 
1% pounds to ,'.% pounds of coal per effective horse-power. It was much more 
difficult to ascertain the consumption of coal in ordinary everyday work, 
but such facts as were known showed it was more than on trial. 

In electric-lighting stations the engines work under a very fluctuating 
load, and the results are far more unfavorable. An excellent Willans non- 
condensing engine, which on full-load trials worked with under 2 pounds 
per effective horse-power hour, in the ordinary daily working of the station 
used 7^a pounds per effective horse-power hour in 1886, which was reduced 
to 4.3 pounds in 1890 and 3.8 pounds in 1891. Probably in very few cases 
were the engines at electric-light stations working under a consumption of 
4i^> pounds per effective horse-power hour. In the case of small isolated 
motors working with a fluctuating load, still more extravagant results were 
obtained. 



PERFORMANCES OF STEAM-ENGINES. 785 

Engines in Electric Central Stations. 

Year 1886. 1890. 1892. 

Coal used per hour per effective H.P 8.4 5.6 4.9 

" " " " indicated " 6.5 4.35 3.8 

At electric-lighting stations the load factor, viz., the ratio of the average 
load to the maximum, is extremely small, and the engines worked under 
very unfavorable conditions, which largely accounted for the excessive fuel 
consumption at these stations. 

In steam-engines the fuel consumption has generally been reckoned on 
the indicated horse-power. At full-power trials this was satisfactory 
enough, as the internal friction is then usually a small fraction of the total. 
Experiment has, however, shown that the internal friction is nearly con- 
stant, and hence, when the engine is lightly loaded, its mechanical efficiency 
is greatly reduced. At full load small engines have a mechanical efficiency 
of 0.8 to 0.85, and large engines might reach at least 0.9, but if the internal 
friction remained constant this efficiency would be much reduced at low 
powers. Thus, if an engine working at 100 indicated horse power had an effi- 
ciency ot 0.85, then when the indicated horse-power fell to 50 the effective 
horse-power would be 35 horse-power and the efficiency only 0.7. Similarly, 
at 25 horse-power the effective horse-power would be 10 and the efficiency 
0.4 
Experiments on a Corliss engine at Creusot gave the following results : 

Effective power at full load 1.0 0.75 0.50 0.25 0.125 

Condensing, mechanical efficiency 0.82 0.79 0.74 0.63 0.48 

Non condensing, " " 0.86 0.83 0.78 0.67 0.52 

At light loads the economy of gas and liquid fuel engines fell off even 
more rapidly than in steam-engines. The engine friction was large and 
nearly constant, and in some cases the combustion was also less perfect at 
light loads. At the Dresden Central Station the gas-engines were kept 
working at nearly their full power by the use of storage-batteries. The 
results of some experiments are given below : 

Brake load, per Gas-engine, cu. ft. Petroleum Eng., Petroleum Eng., 
cent of full of Gas per Brake Lbs. of Oil per Lbs. of Oil per 

Power. H.P. per hour. B.H.P. per hr. B.H.P. per hr. 

100 22.2 0.96 0.88 

75 23.8 1.11 0.99 

59 28.0 1.44 1.20 

20 40.8 2.38 1.82 

12}^ 66.3 4.25 3.07 

Steam Consumption of Engines of Various Sizes.— W. C. 
Unwin (Cassier's Magazine, 1894) gives a table showing results of 49 tests of 
engines of different types. In non-condensing simple engines, the steam 
consumption ranged from 65 lbs. per hour in a 5-horse-power engine to 22 
lbs. in a 134-H.P. Harris-Corliss engine. In non-condensing compound en- 
gines, the only type tested was the Willans, which ranged from 27 lbs. in a 
10 H.P. slow-speed engine, 122 ft. per minute, with steam-pressure of 84 lbs. 
to 19.2 lbs. in a 40-H.P. engine, 401 ft. per minute, with steam-pressure 165 
lbs. A Willans triple-expansion non-condensing engine, 39 H.P., 172 lbs. 
pressure, and 400 ft. piston speed per minute, gave a consumption of 18.5 lbs. 
In condensing engines, nine tests of simple engines gave results ranging only 
from 18.4 to 22 lbs., and, leaving out a beam pumping-engine running at slow 
speed (240 ft. per minute) and low steam -pressure (45 lbs.), the range is only 
from 18.4 to 19.8 lbs. In compound-condensing engines over 100 H.P., in 13 
tests the range is from 13.9 to 20 lbs. In three triple- expansion engines the 
figures are 11.7, 12 2, and 12.45 lbs., the lowest being a Sulzer engine of 360 
H.P. /In marine compound engines, the Fusiyama and Colchester, tested 
by Prof. Kennedy, gave steam consumption of 21.2 and 21.7 lbs.; and the 
Meteor and Tartar triple-expansion engines gave 15.0 and 19.8 lbs. 

Taking the most favorable results which can be regarded as not excep- , 
tional, it appears that in test trials, with constant and full load, the expen- 
diture of steam and coal is about as follows: 

Per Indicated Horse- Per. Effective Horse- 
power Hour. power Hour. 

Kind of Engine. > * > , ' * 

Coal, Steam, . Coal, Steam, 

lbs. lbs. lbs. lbs. 

Non-condensing 1.80 16.5 2.00 18.0 

Condensing 1.50 13.5 1.75 15.8 



186 



THE STEAM-EKGIKE. 



These may be regarded as minimum values, rarely surpassed by the most 
efficient machinery, and only reached with very good machinery in the 
favorable conditions of a test trial. 

Small Engines and Engines with Fluctuating Loads are 

usually very wasteful of fuel. The following figures, illustrating their low 
economy, are given by Prof. Unwin, Cassier's Magazine, 1894. 

Coal Consumption per Indicated Horse-power in Small Engines. 
In Workshops in Birmingham, Eng. 
Probable I.H.P. at full load... 12 45 60 45 75 60 60 
Average I.H.P. during obser- 
vation 2.96 7.37 8.2 8.6 23.64 19.08 20.08 

Coal per I.H.P. per hour dur- 
ing observation, lbs 36.0 21.25 22.61 18.13 11.68 9.53 8.50 

It is largely to replace such engines as the above that power will be dis- 
tributed from central stations. 

Steam Consumption in Small Engines. 

Tests at Royal Agricultural Society's show at Plymouth, Eng. Engineer- 
ing, June 27, 1890. 



Rated H.P. 


Com- 
pound or 
Simple. 


Diam. of 
Cylinders. 


Stroke, 
ins. 


Max. 
Steam- 
pressure. 


Per Brake H. P., 
per hour. 


%4-i 




h.p. 


l.p. 


Coal. 


Water. 


£&o 


5 
3 
2 


simple 

compound 

simple 


3 


"6 


10 
6 

1Yz 


75 
110 
75 


12.12 

4.82 
11.77 


78.1 lbs. 
42.03 '« 
89.9 " 


6.1 lb. 

8.72" 
7.61" 



Steam-consumption of Engines at Various Speeds. 

(Profs. Denton and Jacobus, Trans. A. S. M. E., x. 722)— 17 X 30 in. engine, 
non-condensing, fixed cut-off, Meyer valve. 

Steam-consumption, lbs. per I.H.P. per Hour. 
Figures taken from plotted diagram of results. 

Revs, per min 8 12 16 20 24 32 40 48 56 72 88 

Ys cut-off, lbs 39 35 32 30 29.3 29 28.7 28.5 28.3 28 27.7 

M " " 39 34 31 29.5 29 28.4 28 27.5 27.1 26.3 25.6 

Y% " " 39 36 34 33 32 30.8 29.8 29.2 28.8 28.7 .... 

Steam-consumption of Same Engine; Fixed Speed, 60 Revs, per Min. 
Varying cut-off compared with throttling-engine for same horse-power 
and boiler-pressures: 
Cut-off, fraction of stroke 0.1 0.15 0.2 0.25 0.3 0.4 0.5 0.6 0.7 0.8 

Boiler-pressure, 90 lbs... 29 27.5 27 27 27.2 27.8 28.5 

60 lbs.. 39 34.2 32.2 31.5 31.4 31.6 32.234.136.5 39 

Throttling-engine, % Cut-off, for Corresponding Horse-powers. 

Boiler-pressure, 90 lbs... 42 37 33.8 31.5 29.8 - 

601bs 50.1 49 46.8 44.6 41 

Some of the principal conclusions from this series of tests are as follows : 

1. There is a distinct gain in economy of steam as the speed increases for 
Yz, %, and Ya cut-off at 90 lbs. pressure. The loss in economy for about £4 
cut-off is at the rate of 1/12 lb. of water per H.P. for each decrease of a 
revolution per minute from 86 to 26 revolutions, and at the rate of % lb. of 
water below 26 revolutions. Also, at all speeds the J4 cut-off is more eco- 
nomical than either the ]^or^ cut-off. 

2. At 90 lbs. boiler-pressure and above Y cut-off, to produce a given H.P. 
requires about 20$ less steam than to cut off at % stroke and regulate by the 
throttle. 

3. For the same conditions with 60 lbs. boiler-pressure, to obtain, by 
throttling, the same mean effective pressure at % cut-off that is obtained by 



PERFORMANCES OF STEAM-ENGINES. 787 

cutting off about ^, requires about 30£ more steam than for the latter 
condition. 

High l»iston-speed in Engines. (Proc. Inst. M. E., July, 1883, p. 
3-21).— The torpedo boat is an excellent example of the advance towards 
high speeds, and shows what can be accomplished by studying lightness 
and strength in combination. In running at 22% knots an hour, an engine 
with cylinders of 16 in. stroke will make 480 revolutions per minute, which 
gives 1280 ft. per minute for piston-speed; and it is remarked that engines 
running at that high rate work much more smoothly than at lower speeds, 
and that the difficulty of lubrication diminishes as the speed increases. 

A High-speed Corliss Engine.— A Corliss engine, 20x42 in., has 
been running a wire-rod mill at the Trenton Iron Co.'s works since 1877, at 
160 revolutions or 1120 ft. piston -speed per minute (Trans. A. S. M. E., ii. 
72). A piston-speed of 1200 ft. per min. has been realized In locomotive 
pmctice. 

The Limitation of Engine-speed. (Chas. T. Porter, in a paper 
on the Limitation of Engine-speed, Trans. A. S. M. E., xiv. 806.)— The 
practical limitation to high rotative speed in stationary reciprocating steam- 
engines is not found in the danger of heating or of excessive wear, nor, as 
is generally believed, in the centrifugal force of the fly-wheel, nor in the 
tendency to knock in the centres, nor in vibration. He gives two objections 
to very high speeds: First, that " engines ought not to be run as fast as 
they can be ;" second, the large amount of waste room in the port, which 
is required for proper steam distribution. In the important respect of 
economy of steam, the high-speed engine has thus far proved a failure. 
Large gain was looked for from high speed, because the loss by condensa- 
tion on a given surface would be divided into a greater weight of steam, but 
this expectaiion has not been realized. For this unsatisfactory result we 
have to lay the blame chiefly on the excessive amount of waste room. The 
ordinary method of expressing the amount of waste room in the percentage 
added by it to the total piston displacement, is a misleading one. It should 
be expressed as the percentage which it adds to the length of steam admis- 
sion. For example, if the steam is cut off at 1/5 of the stroke, 8% added by 
the waste room to the total piston displacement means 40$ added to the 
volume of steam admitted. Engines of four, five and six feet stroke may 
properly be run at from 700 to 800 ft. of piston travel per minute, but for 
ordinary sizes, says Mr. Porter, COO ft. per minute should be the limit. 

Influence of the Steam-jacket.— Tests of numerous engines with 
and without steam-jackets show an exceeding diversity of results, ranging 
all the way from 30$ saving down to zero, or even in some cases showing an 
actual loss. The opinions of engineers at this date (1894) is also as diverse as 
the results, but there is a tendency towards a general belief that the jacket is 
not as valuable an appendage to an engine as was formerly supposed. An ex- 
tensive resume of facts and opinions on the steam-jacket is given by Prof. 
Thurston, in Trans. A. S. M. E., xiv. 462. See also Trans. A. S. M. E., xiv. 
873 and 1340; xiii. 176: xii. 426 and 1340; and Jour. F. I., April, 1891, p. 276. 
The following are a few statements selected from these papers. 

The results of tests reported by the research committee on steam-jackets 
appointed by the British Institution of Mechanical Engineers in 1886, indi- 
cate an increased efficiency due to the use of the steam-jacket of from \% to 
over 30$, according to varying circumstances. 

Sennett asserts that ''it has been abundantly proved that steam- 
jackets are not only advisable but absolutely necessary, in order that high 
rates of expansion may be efficiently carried out and the greatest possible 
economy of heat attained." 

Isherwood finds the gain by its use. under the conditions of ordinary 
practice, as a general average, to be about 2% on small and 8% or 9% on 
large engines, varying through intermediate values with intermediate sizes, 
it being understood that the jacket has an effective circulation, and that 
both heads and sides are jacketed. 

Professor Unwin considers that " in all cases and on all cylinders the 
jacket is useful: provided, of course, ordinary, not superheated, steam is 
used; but the advantages may diminish to an amount not worth the interest 
on extra cost." 

Professor Cotterill says: Experience shows that a steam-jacket is advan- 
tageous, but the amount to be gained will vary according to circumstances. 
In many cases it may be that the advantage is small. Great caution is 
necessary in drawing conclusions from any special set of experiments on 
the influence of jacketing. 



"7SS THE STEAM-ENGINE. 

Mr. E. D. Leavitt has expressed the opinion that, in his practice, steam- 
jackets produce an increase of efficiency of from 15$ to 20$. 

In the Pawtucket pumping engine, 15 and 2,0% x 30 in., 50 revs, per min., 
steam-pressure 125 lbs. gauge, cut-off J4 in h.p. and % in l.p. cylinder, the 
barrels only jacketed, the saving by the jackets was from 1$ to 4$. 

The superintendent of the Holly Mfg. Co. (compound pumping-engines) 
says: "In regard to the benefits derived from steam-jackets on our steam- 
cylinders, I am somewhat of a skeptic. From data taken on our own en- 
gines and tests made I am yet to be convinced that there is any practical 
value in the steam-jacket." . . . " You might practically say that there 
is no difference." 

Professor Schroter from his work on the triple-expansion engines at Augs- 
burg, and from the results of his tests of the jacket efficiency on a small 
engine of the Sulzer type in his own laboratory, concludes: (1) The value 
of the jacket may vary within very wide limits, or even become nega- 
tive. (2) The shorter the cut-off the greater the gain by the use of a 
jacket. (3) The use of higher pressure in the jacket than in the cylinder 
produces an advantage. The greater this difference the better. (4) The 
high-pressure cylinder may be left unjacketed without great loss, but the 
others should always be jacketed. 

The test of the Laketon triple-expansion pumping-engine showed a gain 
of 8.3$ by the use of the jackets, but Prof. Denton points out (Trans. A. .S 
M. E., xiv. 1412) that all but 1.9$ of the gain was ascribable to the greater 
range of expansion used with the jackets. 

Test of a Compound Condensing Engine with and with- 
out Jackets at different Loads. (R. C. Carpenter, Trans. A. S. 
M. E,xiv. 428.)— Cylinders 9 and 1(5 in.Xl4 in. stroke; 112 lbs. boiler-pressure; 
rated capacity 100 H.P. ; 265 revs, per min. Vacuum, 23 in. From the results 
of several tests curves are plotted, from which the following principal figures 
are taken. 

Indicated H.P 30 40 50 60 70 80 90 100 110 120 125 

Steam per I. H.P. per hour: 

With jackets, lbs 22.6 21.4 20.3 19.6 19 18.7 18.6 18.9 19.5 20.4 21.0 

Without jackets, lbs 22. 20.5 19.6 19.2 19.1 19.3 20.1 .... 

Saving by jacket, p. c 10.9 7.3 4.6 3.1 1.0-1.0-1.5 .... 

This table gives a clue to the great variation in the apparent saving due to 
the steam-jacket as reported by different experimenters. With this par- 
ticular engine it appears that when running at its most economical rate of 
100 H.P., without jackets, very little saving is made by use of the jackets. 
When running light the jacket makes a considerable saving, but when over- 
loaded it is a detriment. 

At the load which corresponds to the most economical rate, with no steam 
in jackets, or 100 H.P., the use of the jacket makes a saving of only 1$; but 
at a load of 60 H.P. the saving by use of the jacket is about 11$, and the 
shape of the curve indicates that the relative advantage of the jacket would 
be still greater at lighter loads than 60 H.P. 

Counterbalancing Engines.— Prof. Unwin gives the formula for 
counterbalancing vertical engines: 

W x = W 2 -; (1) 

in which W x denotes the weight of the balance weight and p the radius to 
its centre of gravity, W 2 the weight of the crank-pin and half the weight of 
the connecting-rod, and r the length of the crank. For horizontal engines: 



W 1 =%(W 2 + W 3 )~ to %(W* + W 3 )-, 



in which W 3 denotes the weight of the piston, piston-rod, cross-head, and 
the other half of the weight of the connecting-rod. 

The American Machinist, commenting on these formulae, says: For hori- 
zontal engines formula (2) is often used; formula (1) will give a counter- 
balance too light for vertical engines. We should use formula (2) for 
computing the counterbalance for both horizontal and vertical engines, 
excepting locomotives, in which the counterbalance should be heavier. 



PERFORMANCES OF STEAM-ENGINES. 



789 



Preventing Vibrations of Engines.— Many suggestions have 
been made for remedying the vibration and noise attendant on the working 
of the big engines which are employed to run dynamos. A plan which has 
given great satisfaction is to build hair-felt into the foundations of the 
engine. An electric company has had a DO-horse-power engine removed 
from its foundations, which were then taken up to the depth of 4 feet. A 
layer of felt 5 inches thick was then placed on the foundations and run up 2 feet 
on all sides, and on the top of this the brickwork was built up. — Safety Valve. 

Steam-engine Foundations Embedded in Air.— In the sugar- 
refinery of Claus bpreckels, at Philadelphia, Fa., the engines are distributed 
practically all over the buildings, a large proportion of them being on upper 
floors. Some are bolted to iron beams or girders, and are consequently 
innocent of all foundation. Some of these engines ran noiselessly and sat is- 
factorily, while others produced more or less vibration and rattle. To cor- 
rect the latter the engineers suspended foundations from the bottoms of the 
engines, so that, in looking at them from the lower floors, they were literally 
hanging in the air.— Iron Age. Mar. 13. 1890. 

Cost of Coal for Steam-power.— The following table shows the 
amount and the cost of coal per day and per year for various horse-powers, 
from 1 to 1000, based on the assumption of 4 lbs. of coal being used per hour 
pei 1 horse-power. It is useful, among other things, in estimating the saving 
that may be made in fuel by substituting more economical boilers and 
engines for those already in use. Thus with coal at $3.00 per ton, a saving 
of $9000 per year in fuel may be made by replacing a steam plant of 1000 
H.P., requiring 4 lbs. of coal per hour per horse-power, with one requiring 
only 2 lbs. 





Coal Consumption 


, at 4 lbs. 




















per H.P. per hour ; 


10 hours a 


$1.50. 


$2.00. 


$3.00. 


$4.00. 




day ; 300 days in 


x Yea 
















1 


Lbs. 


Long Tons. 


Short 


Per 


Per 


Per 


Per 


a 
1 




Tons. 


Short Ton. 


Short Ton. 


Short Ton. 


Short Ton. 












Cost in 


Cost in 


Cost in 


Cost in 












Dollars. 


Dollars. 


Dollars. 


Dollars. 




Per 
Day. 


Per 
Day. 


Per 
Year. 


Per 
Day. 


Per 
Year 












































Per 


Per 


Per 


Per 


Per 


Per 


Per 


Per 






.0179 




.02 


6 


Day. 
.03 


Year 
9 


Day. 
.04 


Year. 


Day. 
.06 


Year. 


Day. 
.08 


Year 


l 


40 


5.357 


12 


18 


24 


10 


400 


.1786 


53.57 


.20 


60 


.30 


90 


.40 


120 


.60 


180 


.80 


240 


25 


1,000 


.4464 


133.92 


.50 


150 


.75 


225 


1.00 


300 


1.50 


450 


2.00 


600 


50 


2,000 


.8928 


267.85 


1.00 


300 


1.50 


450 


2.00 


600 


3.00 


900 


4.00 


1,200 


75 


3,000 


1.3393 


401.78 


1.50 


450 


2.'25 


675 


3.00 


900 


4.50 


1,350 


6.00 


1,800 


100 


4,000 


1.7857 


535.71 


2.00 


600 


3.00 


900 


4.00 


1,200 


6.00 


1,800 


8.00 


2,400 


150 


6,000 


2.6785 


803.56 


3.00 


900 


4.50 


1,350 


6.0C 


1,800 


9.00 


2,700 


12.00 


3.000 


200 


8,000 


3.5714 


1,071.42 


4.00 


1,200 


6.00 


1,800 


8.00 


2,400 


12.00 


3,600 


16.00 


1.800 


250 


10,000 


4.4642 


1,339.27 


5.00 


1,500 


7.50 


2.250 


10.00 


3,000 


15.00 


4,500 


20.00 


6,000 


300 


1-2.000 


5 3571 


1,607.13 


6.00 


1,800 


9.00 


2,700 


12.00 


3,600 


18.00 


5,400 


24.00 


7,200 


350 


14,000 


6.2500 




7.00 


2,100 


10.50 


8.150 


14.00 


4,200 


21.00 


6,200 


28.00 


8,400 


400 


16,000 


7.1428 




8.00 


2,400 


12.00 


3,000 


16.00 


4,800 


24.00 


7,200 




9,600 


450 


18,000 


8.0356 




9.00 


2,700 


13.50 


4,050 


18 00 


5,400 




8,100 


36.00 


io.xoo 


500 


20,000 


8.9285 




10.00 


3,000 


15.00 


4,500 


20.00 


6,000 


30.00 


9,000 


40.00 


12,000 


600 


24,000 


10.7142 


3,214.26 


12.00 


3,600 


18.00 


5,400 


24.00 


7,200 


36.00 


10,800 


48.00 


14,400 


700 


28,000 


12.4999 


3,749.97 


14.00 


4,200 


21.00 


6.300 


28.(11) 


8,400 


42.00 


11,600 


56.00 


16,800 


800 


32.000 


14.2856 


4,285.68 


16.00 


4,800 


24.00 


7,200 


32.00 


9,601 


48.00 


12,400 


64.00 




900 


36,000 


16.0713 


4,821.39 


18.00 


5,400 


27.00 


8,100 


36.00 


|o,80( 


54.00 


14,200 


72.00 




1.000 


40,000 


17.8570 


5,357.10 


20.00 


6,000 


30.00 




40.00 


12,001 




18,000 


80.00 24,000 



Storing Steam Heat.— There is no satisfactory method for equalizing 
the load on the engines and boilers in electric-light stations. Storage-batteries 
have been used, but they are expensive in first cost, repairs, and attention. 
Mr. Halpin, of London, proposes to store heat during the day in specially 
constructed reservoirs. As the water in the boilers is raised to 250 lbs. pres- 
sure, it is conducted to cylindrical reservoirs resembling English horizontal 
boilers, and stored there for use when wanted. In this way a comparatively 
small boiler-plant can be used for healing the water to 250 lbs. pressure all 
through the twenty-four hours of the day, and the stored water may be 
drawn on at any time, according to the magnitude of the demand. The 



790 



THE STEAM-ENGINE. 



steam-engines are to be worked by the steam generated by the release of 
pressure from this water, and the valves are to be arranged in such a way 
that the steam shall work at 130 lbs. pressure. A reservoir 8 ft. in diameter 
and 30 ft. long, containing 84,000 lbs. of heated water at 250 lbs. pressure, 
would supply 5250 lbs. of steam at 130 lbs. pressure. As the steam consump- 
tion of a condensing electric -light engine is about 18 lbs. per horse-power 
hour, such a reservoir would supply 286 effective horse-power hours. In 
1878, in France, this method of storing steam was used on a tramway. 
M. Francq, the engineer, designed a smokeless locomotive to work by steam- 
power supplied by a reservoir containing 400 gallons of water at 220 lbs. 
pressure. The reservoir was charged with steam from a stationary boiler 
at one end of the tramway. 

Cost of Steam-power. (Chas. T. Main, A. S. M. E., x. 48.)— Estimated 
costs in .New England in 1S88, per horse-power, tyised on engines of 1000 H. P. 



Compound 
Engine. 

1. Cost engine and piping, complete $2.5.00 

2. Engine-house 8.00 

3. Engine foundations 7.00 

4. Total engine plant 40.00 

5. Depreciation, 4% on total cost 1 .60 

6. Repairs, 2% " " " 0.80 

7. Interest, 5$ " " " 2.00 

8. Taxation, 1.5$ on % cost 0.45 

9. Insurance on engine and house 0.165 

10. Total of lines 5, 6, 7, 8, 9 5.015 

11. Cost boilers, feed-pumps, etc.. 9.33 

12. Boiler-house , 2.92 

13. Chimney and flues 6.11 

14. Total boiler-plant 18.; 

15. Depreciation, 5$ on total cost 

16. Repairs, 2% " " '• 

17. Interest, 5$ " " " .... 

18. Taxation, 1.5$ on % cost 

19. Insurance, 0.5$ on total cost 

20. Total of lines 15 to 19 

21. Coal used per I.H.P. per hour, lbs 

22. Cost of coal per I.H.P. per day of 10J4 cts. 

hours at $5.00 per ton of 2240 lbs 4.00 

23. Attendance of engine per day 0.60 

24. " " boilers " " 0.53 

25. Oil, waste, and supplies, per clay 0.25 

26. Total daily expense 5.38 

27. Yearly running expense, 308 days, per 

I.H.P $16,570 

28. Total yearly expense, lines 10, 20. and 27. . 24.087 

29. Total yearly expense per I.H.P. for power 

if 50$ of exhaust-steam is used for heat- 
ing 12.597 

30. Total if all ex. -steam is used for heating. . . 8.624 



Non-con- 

20.00 $17.50 

7.50 7.50 

5.50 4.50 

53.00 29.50 

7l8 
0.59 
1 .475 
0.332 
0.125 

3.702 

16.00 
5.00 
8.00 



Condens- 



1.32 
0.66 
1.65 
0.371 
0.138 

4.139 



18.36 


24.80 


29.00 


918 


1.240 


1.450 


.367 


.496 


.580 


.918 


1.240 


1.450 


.207 


.279 


.326 


.092 


.124 


.145 


2.502 


3.379 


3.951 


1.75 


2.50 


3.00 



cts. 
5.72 
0.40 
0.75 

0.22 

7.09 



$21,837 
29.355 



14.907 
7.916 



0.35 
0.90 
0.20 



$25,595 
33.248 



16.663 

7.700 



When exhaust- steam or a part of the receiver-steam is used for heating, or 
if part of the steam in a condensing engine is diverted from the condenser, 
and used for other purposes than power, the value of such steam should 



ROTARY STEAM-EKGIKES. ?91 

be deducted from the cost of the total amount of steam generated, in order 
to arrive at the cost properly chargeable to power. The figures in lines 29 
and 30 are based on an assumption made by Mr. Main of losses of heat 
amounting to 25$ between the boiler and the exhaust-pipe, an allowance 
which is probably too large. 

ROTARY STEAM-ENGINES. 

Steam Turbines.— The steam turbine is a small turbine wheel which 
runs with steam as the ordinary turbine does with water. (For description 
of the Parsons and the Dow steam turbines see Modern Mechanism, p. 298, 
etc.) The Parsons turbine is a series of parallel-flow turbines mounted side 
by side on a shaft; the Dow turbine is a series of radial outward-flow tur- 
bines, placed like a series of concentric rings in a single plane, a stationary 
guide-ring being between each pair of movable rings. The speeds of the 
steam turbines enormously exceed those of any form of engine with recip- 
rocating piston, or even of the so-called rotary engines. The three- and four- 
cyliuder engines of the Brotherhood type, in which the several cylinders 
are usually grouped radially about a common crank and shaft, often exceed 
1000 revolutions per minute, and have been driven, experimentally, above 
2000; but the steam turbine of Parsons makes 10.000 and even 20,000 revolu- 
tions, and the Dow turbine is reputed to have attained 25,000. (See Trans. 
A. S. M. E., vol. x. p. 680, and xii. p. 888; Trans. Assoc, of Eng'g Societies, 
vol. viii. p. 583; Eng'g, Jan. 13, 1888, and Jan. 8, 1892; Eng'g Neios, Feb. 27, 
1892.) A Dow turbine, exhibited in 1889, weighed 68 lbs., and developed 10 
HP., with a consumption of 47 lbs. of steam per H.P. per hour, the steam 
pressure being 70 lbs. The Dow turbine is used to spin the fly-wheel of the 
Howell torpedo. The dimensions of the wheel are 13.8 in. diam., 6.5 in. 
width, radius of gyration 5.57 in. The energy stored in it at 10,000 revs, 
per min. is 500,000 ft.-lbs. 

The De Laval Steam Turbine, shown at the Chicago exhibition, 
1893, is a reaction wheel somewhat similar to the Pelton water-wheel. The 
steam jet is directed by a nozzle against the plane of the turbine at quite a 
small angle and tangentially against the circumference of the medium 
periphery of the blades. The angle of the blades is the same at the side of 
admission and discharge. The width of the blade is constant along the 
entire thickness of the turbine. 

The steam is expanded to the pressure of the surroundings before arriv- 
ing at the blades. This expansion takes place in the nozzle, and is caused 
simply by making its sides diverging. As the steam passes through this 
channel its specific volume is increased in a greater proportion than the 
cross section of the channel, and for this reason its velocity is increased, 
and also its momentum, till the end of the expansion at the last sectional 
area of the nozzle. The greater the expansion in the nozzle the greater its 
velocity at this point. A pressure of 75 lbs. and expansion to an absolute 
pressure of one atmosphere give a final velocity of about 2625 ft. per second. 

Expansion is carried further in this steam turbine than in ordinary steam- 
engines. This is on account of the steam expanding completely during its 
work to the pressure of the surroundings. 

For obtaiuing the greatest possible effect the admission to the blades must 
be free from blows and the velocity of discharge as low as possible. These 
conditions would require iu the steam turbine an enormous velocity of 
periphery — as high as 1300 to 1650 ft. per second. The centrifugal force, 
nevertheless, puts a limit to the use of very high velocities. In the 5 horse- 
power turbine the velocity of periphery is 574 ft. per second, and the num- 
ber of revolutions 30,000 per minute. 

However carefully the turbine may be manufactured it is impossible, on 
account of unevenness of the material, to get its centre of gravity to corre- 
spond exactly to its geometrical axle of revolution; and however small this 
difference may be, it becomes very noticeable at such high velocities. De 
Laval has succeeded in solving the problem by providing the turbine with a 
flexible shaft. This yielding shaft allows the turbine at the high rate of 
speed to adjust itself and revolve around its true centre of gravity, the 
centre line of the shaft meanwhile describing "a surface of revolution. 

In the gearing-box the speed is reduced from 30,000 revolutions to 3000 
by means of a driver on the turbine shafts, which sets in motion a cog- 
wheel of 10 times its own diameter. These gearings are provided with spiral 
cogs placed at an angle of about 45°. The shaft of the larger cog-wheel, 
running at a speed of 3000 revolutions, is provided at its outer end with a 
pulley for the further transmission of the power. 



792 THE STEAM-ENGINE. 

Rotary Steam-engines, other thaD steam turbines, have been 
invented by the thousands, but not oue has attained a commercial success. 
The possible advantages, such as saving of space, to be gained by a rotary 
engine are overbalanced by its waste of steam. 
The Tower Spherical Engine, one of the most recent forms of 
rotary-engine, is described in Proc. lust. M. E., 1885, also in Modern 
Mechanism, p. 296. 

DIMENSIONS OF PARTS OF ENGINES. 

The treatment of this subject by the leading authorities on the steam-en- 
gine is very unsatisfactory, being a confused mass of rules and formulae 
based partly upon theory and partly upon practice. The practice of builders 
shows an exceeding diversity of opinion as to correct dimensions. The 
treatment given below is chiefly the result of a study of the works of Rankine, 
Seaton, Unwin, Thurston, Marks, and Whitham, and is largely a condensa- 
tion of a series of articles by the author published in the American Ma- 
chinist, in 1894, with many alterations and much additional matter. In or- 
der to make a comparison of many of the formula? they have been applied 
to the assumed cases of six engines of different sizes, and in some cases 
this comparison has led to the construction of new formulae. 

Cylinder. (Whitham.)— Length of bore = stroke -f- breadth of piston- 
ring — % to 14 ' n ? length between heads = stroke + thickness of piston -f 
sum of clearances at both ends; thickness of piston = breadth of ring -j- 
thickness of flange on one side to carry the ring — + thickness of follower- 
plate. 

Thickness of flange or follower — % to \4> in. % in. 1 in. 

For cylinder of diameter 8 to 10 in. 36 in. 60 to 100 in. 

Clearance of Piston. (Seaton.)— The clearance allowed varies with 
the size of tbe engine from J^ to % in. for roughness of castings and \/\§ to 
"4 bn. for each working joint. Naval and other very fast-running engines 
lave a larger allowance. In a vertical direct-acting engine the parts which 
wear so as to bring the piston nearer the bottom are three, viz., the shaft 
journals, the crank-pin brasses, and oiston-rod gndgeon-brasses. 

Thickness of Cylinder. (Thurston.) — For engines of the older 
types and under moderate steam-pressures, some builders have for many 
years restricted the stress to about 2550 lbs. per sq. in. 

t = apiD + fc (1) 

is a common proportion; t, D, and b being thickness, diam., and a constant 
added quantity varying from to % in., all in inches; p l is the initial unbal- 
anced steam-pressure per sq. in. In this expression b is made larger for 
horizontal than for vertical cylinders, as, for example, in large engines 0.5 
in the one case and 0.2 in the other, the one requiring re-boring more than 
the other. The constant a is from 0.0004 to 0.0005; the first value for verti- 
cal cylinders, or short strokes; the secon-d for horizontal engines, or for 
long strokes. 
Thickness of Cylinder and its Connections for Marine 

Engines. (Seaton).— D = the diam. ot the cylinder in inches: p = load on 

the safety-valves in lbs. per sq. in.; /, a constant multiplier = thickness of 

barrel 4- .25 in. 
Thickness of metal of cylinder barrel or liner, not to be less than p x D ■+■ 

3000 when of cast iron.* (2) 

Thickness of cylinder-barrel = - ■ -f- 0.6 in (3) 

" liner = 1.1 X/. (4) 

Thickness of liner when of steel p x D -i- 6000 + 0.5 
" metal of steam-ports =0.6 x /. 

" " valve-box sides = 0.65 X/. 

* "When made of exceedingly good material, at least twice melted, the 
thickness may be 0.8 of that given by the above rules. 



ha 



DIMENSIONS OF PARTS OF ENGINES. 



793 



Thickness of metal of valve-box covers = 0.7 X f- 

" " cylinder bottom = 1.1 x /, if single thickness. 

" = 0.65 X /, if double 

" " " covers =1.0 x /, if single " 

" " " " =0.6 x/, if double " 

*' cylinder flange =1.4 X /. 

" " cover-flange =1.3 X /. 

" " valve-box" =1.0 x /. 

" " door-flange =0.9 X /. 

" " face over ports = 1.2 X /. 

" " " " =1.0 X /. when there is a false-face. 

" " false-face =0.8 X /, when cast iron. 

" " " =0.6 X /, when steel or bronze. 

Whitham gives the following from different authorities: 

VanBuren:-H = °- 0001 ^+ - 15 ^; & 

I t = 0.03 VDp (6) 

™ d: 1 = <-TO m 

Weisbach: t = 0.8 + 0.00033pD (8) 

Seaton : t = 0.5 -f 0,00Q4pD (9) 

Harwell • }t = 0.0004pD+M (vertical); (10) 

nasweii . ( £ = .0005pD + y 8 (horizontal) (11) 

Whitham recommends (6) where provision is made for the reboring, and 
where ample strength and rigidity are secured, for horizontal or vertical 
cylinders of large or small diameter; (9) for large cylinders using steam 
under 100 lbs. gauge -pressure, and 

t = 0.003D Vp for small cylinders (12) 

Marks gives t = 0.00028pD (13) 

This is a smaller value than is given by the other formulae quoted; but 
Marks says that it is not advisable to make a steam-cylinder less than 0.75 
in. thick under any circumstances. 

The following table gives the calculated thickness of cylinders of engines 
of 10, 30. and 50 in. diam., assuming p the maximum unbalanced pressure on 
the piston = 100 lbs. per sq. in. As the same engines will be used for calcu- 
lation of other dimensions, other particulars concerning them are here 
given for reference. 



Dimensions, etc., of Engines. 



Engine No 

Indicated horse -power I.H.P. 

Diam. of cyl., in D 

Stroke, feet. L 

Revs, per min 

Piston speed, ft. per min S 

Area of piston, sq. in a 

Mean effective pressure . . M.E.P. 

Max. total unbalanced press P 

Max. total per sq. in p 



50 

10 



7854 
100 



.... 5 


4 .... 


... 65 


90 .... 


650 


700 


706.86 


1963.5 


32.3 


30 


70,686 


196,350 


100 


100 



794 



THE STEAM-ENGINE. 



Thickness of Cylinder 

by Formula. 

(1) .OOOipD 4- 0.5, short stroke., 

(1) .OOOopD + 0.5, long stroke . . . 

(2) ,00033pZ> 

(3) .0002pZ> + 0.6 . _ 

(5) .0001pZH-.15 \/D , 

(6) .03 4/Dp 

.. (0+2.5) 

(T) 1 — Z '. p 



1900 

(8) .00033pZ> + 8 

(9) .0004pD-f 05 

( 10) .0004pD + Vh (vertical) . 

(11) .OOOopD + V 8 (horizontal) 

(12) .003D Vp (small engines). 

(13) .00038pD. 



90 
00 
33 
80 


1.70 

2.00 

.99 

1.40 


.57 


1.18 


.95 


1.64 


.66 


1.71 


.13 
.90 
.53 
.63 


1.79 
1.70 
1.33 
1.63 


.30(?) 

.28(?) 


".84 



2.50 
3.00 
1 67 
l.fiO 
1.56 
2.12 



2.45 
2.50 
2.13 



Average of first eleven , 



1.48 



The average corresponds nearly to the formula t = .00037Z)p -4- 0.4 in. A 
convenient approximation is t = .0004Z)p ~f- 0.3 in., which gives for 

Diameters 10 20 30 40 50 60 in. 

Thicknesses 70 1.10 1.50 .1.90 2.30 2.70 in. 

The last formula corresponds to a tensile strength of cast iron of 12,500 
lbs., with a factor of safety of 10 and an allowance of 0.3 in. for reboring. 

Cylinder-heads.— Thurston says: Cylinder-heads may be given a 
thickness, ac l he edges and in the flanges, exceeding somewhat that of the 
cylinder. An excess of not less than 25$ is usual It may be thinner in the 
middle. Where made, as is usual in large engines, of two disks with inter- 
mediate radiating, connecting ribs or webs, that section which is safe 
against shearing is probably ample. An examination of the designs of 
experienced builders, by Professor Thurston, gave 

D being the diameter of that circle in which the thickness is taken. 

Thurston also gives t = .005D Vp _+ °- 25 ( 2 ) 

Marks gives t = 0.003Z) Vp $) 

He also says a good practical rule for pressures under 100 lbs. per sq. in. is 
to make the thickness of the cylinder-heads 1J4 times that of the walls; and 
applying this factor to his formula for thickness of walls, or .00028pZ>, we 
have ... 

t = .00035pD (4) 

Whitham quotes from Seaton, 
. _pD + 500 



(1) 



2000 



-, which is equal to .0005pZ> + .25 inch. 



(5) 



Seaton's formula for cylinder bottoms, quoted above, is 

t = 1.1/, in which / = .0002pZ) + -85 inch, or t = .00022pD + .93. . (6) 
Applying the above formulas to the engines of 10, 30, and 50 inches diame- 
ter, with maximum unbalanced steam-pressure of 100 lbs. per sq. in., we 

Cylinder diameter, inches =10 30 50 

(1) t = .00033Dp_-H .25 

(2) t .= .005D Vp + .25 

(3) t = .003Z) Vp 

(4) t - .00035 Dp 

(5) t = .0005 Dp + .25 

(6) t = .0002 J 'Dp + .93 

Average of 6 . . , 



.53 


1.25 


1.82 


.75 


1.75 


2.75 


.30 

.35 

.75 

1.15 


.90 
1.05 
1.75 
1.59 


1.50 
1.75 
2.75 
2.03 


,65 


1.38 


2.10 



DIMENSIONS OF PARTS OF ENGINES. 795 

The average is expressed by the formula t = .00036Z)p -4- .31 inch. 
Meyer's " Modern Locomotive Construction, 1 ' p. 24, gives for locomotive 
cylinder-heads for pressures up to 120 lbs. : 

For diameters, in 19 to 22 16 to 18 14 to 15 11 to 13 9 to 10 

Thickness, in 134 • * 1 V& % 

Taking the pressure at 120 lbs. per sq. in., the thicknesses 1*4 in- and % in. 
for cylinders 22 and 10 in. diam., respectively, correspond to the formula 
t = ,00035Dp-f .33 inch. 

Web-stiffened Cylinder-covers.— Seaton objects to webs for 
stiffening cast-iron cylinder-covers as a source of danger. The strain on 
the web is one of tension, and if there should be a nick or defect in the 
outer edge of the web the sudden application of strain is apt to start a 
crack. He recommends that high-pressure cylinders over 24 in. and low- 
pressure cylinders over 40 in. diam. should have their covers cast hollow, 
with two thicknesses of metal. The depth of the cover at the middle should 
be about J4 the diam. of the piston for pressures of 80 lbs. and upwards, 
and that of the low-pressure cylinder-cover of a compound engiue equal to 
that of the high-pressure cylinder. Another rule is to make the depth at 
the middle not less than 1.3 times the diameter of the piston-rod. In the 
British Navy the cylinder-covers are made of steel castings, % to 1J4 in. 
thick, generally cast without webs, stiffness being obtained by their form, 
which is often a series of corrugations. 

Cylinder-head Bolts.— Diameter of bolt-circle for cylinder-head = 
diameter of cylinder -f 2 x thickness of cylinder + 2 X diameter of bolts. 
The bolts should not be more than 6 inches' aoart (Whitham).. 

Marks gives for number of bolts b = '' ' , = 

area of a single bolt, p = boiler-pressure in lbs. per sq. in.; 5000 lbs. is taken 
as the safe strain per sq. in. on the nominal area of the bolt. 

Seaton says: Cylinder-cover studs and bolts, when made of steel, should 
be of such a size that the strain in them does not exceed 5000 lbs. per sq. in. 
When of less than % inch diameter it should not exceed 4500 lbs. per sq. in. 
When of iron the strain should be 20$ less. 

Thurston says : Cylinder flanges are made a little thicker than the cylin- 
dei\ and usually of equal thickness with the flanges of the heads. Cylinder- 
bolts should be so closely spaced as not to allow springing of the flanges 
and leakage, say, 4 to 5 times the thickness of the flanges. Their diameter 
should be proportioned for a maximum stress of not over 4000 to 5000 lbs. 
per square inch. 

If D = diameter of cylinder, p = maximum steam -pressure, 6 = number 
of bolts, s — size or diameter of each bolt, and 5000 lbs. be allowed per sq. 
in. of nominal area of the bolt, .7854D 2 p = 39276«2; whence 6s 2 = .00021^; 

, 2 ^£: „ = .01414i/|. 



Diameter of cylinder, inches 

Diameter of bolt-circle, approx . . 
Circumference of circle, approx. 
Minimum No. of bolts, circ. -*- 6. . 

Diam. of bolts, s = .01414D 



10 


30 


13 


35 


40.8 


110 


7 


18 



Vi- 



The diameter of bolt for the 10 inch cylinder is 0.54- in. by the formula, 
but % inch is as small as should be taken, on account of possible overstrain 
by the wrench in screwing: up the nut. 

The Piston. Details of Construction of Ordinary Pis- 
tons. (Seaton.)— Let D be the diameter of the piston in inches, p the effec- 
tive pressure per square inch on it, x a constant multiplier, foundas follows: 

X = £xVp4-l. 



796 THE STEAM-ENGINE. 

The thickness of front of piston near the boss =0.2 X % 

" " " " rim = 0.17 X x. 

" back " =0.18 X x. 

" boss around the rod =0.3 X x. 

" flange inside packing-ring = 0.23 X x. 

at edge = 0.25 x x. 

" packing-ring = 0.15 x x. 

" junk-ring at edge — 0.23 X x. 

" " inside packing-ring =0.21 X x. 

" "at bolt-holes = 0.35 X x. 

"' metal around piston edge = 0.25 X x 

The breadth of packing-ring = 0.63 X x. 

" depth of piston at centre =1.4 x x. 

" lap of junk-ring on the piston = 0.45 X x. 
" space between piston body and packing-ring =0.3 x x. 

" diameter of junk-ringbolts =0.1 X x -f 0.25 in. 

" pitch " " " = 10 diameters. 

" number of webs in the piston = (D + 20) h- 12. 

" thickness " " " = 0.18 X x. 

A. 

Marks gives the approximate rule: Thickness of piston-head= \/ld, in 
which I = length of stroke, and d — diameter of cylinder in inches. Whit- 
ham says in a horizontal engine the rings support the piston, or at least a 
part of it, under ordinary conditions. The pressure due to the weight of 
the piston upon an area equal to 0.7 the diameter of the cylinder X 
breadth of ring-face should never exceed 200 lbs. per sq. in. He also gives 
a formula much used in this country: Breadth of ring-face = 0.15 X diam- 
eter of cylinder. 

For our engines we have diameter = 10 30 50 

Thickness of piston -head. 

Marks, VlO; long stroke 3.31 5.48 7.00 

Marks, % ' ; short stroke 3.94 6.51 8.32 

Seatou, depth at centre = 1.4a? . 4.30 9.80 15.40 

Seaton, breadth of ring = .63.T 1.89 4.41 6.93 

Whitham, breadth of ring = .151) 1.50 4.50 7.50 

Diameter of Piston Packing- rings. — These are generally 
turned, before they are cut, about *4 inch diameter larger than the cylinder, 
for cylinders up to 20 inches diameter, and then enough is cut out of the ring 
to spring them to the diameter of the cylinder. For larger cylinders the 
rings are turned proportionately larger. Seaton recommends an excess 
of \% of the diameter of the cylinder. 

Cross-section of the Rings.— The thickness is commonly made 
l/30th of the diam. of cyl. -j- J^inch, and the width = thickness + % inch. 
For an eccentric ring the mean thickness may be the same as for a ring of 
uniform thickness, and the minimum thickness = % the maximum. 

A circular issued by J. H. Dunbar, manufacturer of packing -rings, 
Youngstown, O., says: Unless otherwise ordered, the thickness of rings will 
be made equal to .03 x their diameter. This thickness has been found 
to be satisfactory in practice. It admits of the ring being made about 3/16" 
to the foot larger than the cylinder, and has. when new, a tension of about 
two pounds per inch of circumference, which is ample to prevent leakage 
if the surface of the ring and cylinder are smooth. 

As regards the width of rings, authorities " scatter " from very narrow to 
verv wide, the latter being fully ten times the former. For instance, Unwin 
gives W= d .014 -|- .08. Whitham's formula is W = d .15. In both for- 
mula? IF is the width of the ring in inches, and d the diameter of the cylinder 
in inches. Unwin's formula makes the width of a 20" ring W = 20 X .014 
-J- .08 = .36", while Whitham's is 20 X .15 = 3" for the same diameter of 
ring. There is much less difference in the practice of engine-builders in thi< 
respect, but there is still room for a standard width ot ring. It is believed 
that for cylinders over 16" diameter %" is a popular and practical width, 
and Vz" for evlinders of that size and under. 

Fit of Piston-rod into Piston. (Seaton.)— The most convenient 
and reliable practice is to turn the piston-rod end with a shoulder of 1/16 
inch for small engines, and % inch for large ones, make the taper 3 in. to 



DIMENSIONS OF PARTS OF ENGINES. 797 

the foot until the section of the rod is three fourths of that of the body, then 
i urn the remaining part parallel; the rod should then fit into the piston so 
as to leave y% incli between it and the shoulder for large pistons, and 1/16 in. 
for small. The shoulder prevents the rod from splitting the piston, and 
allows of the rod being turned true after long wear without encroaching on 
the taper. 

The piston is secured to the rod by a nut, and the size of the rod should 
be such that the strain on the section at the bottom of the thread does not 
exceed 5500 lbs. per sq. in. for iron, 7000 lbs. for steel. The depth of this nut 
need not exceed the diameter which would be found by allowing these 
strains. The nut shouid be locked to prevent its working loose. 

Diameter of Piston-rods.— Unwin gives 

d" = bD Vp, (1 

in which D is the cylinder diameter in inches, p is the maximum unbalanced 
pressure in lbs. per sq. in., and the constant b — 0.0167 for iron, and b = 
0.0144 for steel. Thurston, from an examination of a considerable number 
of rods in use, gives 



(L in feet, D and d in inches), in which a — 10,000 and upward in the various 
types of engines, the marine screw engines or ordinary fast engines on 
shore giving the lowest values, while "low-speed engines" being less 
liable to accident from shock give a = 15,000, often. 

Connections of the piston-rod to the piston and to thecrosshead should 
have a factor of safety of at least 8 or 10. Marks gives 

d" = 0.0179D j/p, for iron; for steel d" = 0.01 05Z> Yp; . . (3) 

and d" ^ 0.03901 \ 'DH*p, for iron; for steel d" = 0.03525 \'DH^p, ^4) 

in which I is the length of stroke, all dimensions in inches. Deduce the 
diameter of piston-rod by (3), and if this diameter is less than 1/12/, then use 
(4). 

~ , . -^. . „ . , , Diameter of cylinder ,_ 
Seaton gives: Diameter of piston-rod = ^ — j/p. 

The following are the values of F: 

Naval engines, direct-acting F — 60 

" " return connuecting-rod, 2 rods F = 80 

Mercantile ordinary stroke, direct-acting F = 50 

long " " " F= 48 

" very long " " F — 45 

" medium stroke, oscillating F — 45 

Note.— Long and very long, as compared with the stroke usual for the 
power of engine or size of cylinder. 

In considering an expansive engine p. the effective pressure should be 
taken as the absolute working pressure, or 15 lbs. above that to which the 
boiler safety-valve is loaded; for a compound engine the value of p for the 
high-pressure piston should be taken as the absolute pressure, less 15 lbs., 
or the same as the load on the safety-valve; for the medium-pressure the 
load may be taken as that due to half the absolute boiler-pressure; and for 
the low-pressure cylinder the pressure to which the escape-valve is loaded 
-J- 15 lbs., or the maximum absolute pressure, which can be got in the re- 
ceiver, or about 25 lbs. It is an advantage to make all the rods of a com- 
pound engine alike, and this is now the rule. 

Applying the above formulae to the engines of 10, 30, and 50 in. diameter, 
both short and long stroke, we have: 



THE STEAM-EKGINE. 



Diameter of Piston-rods. 



Diameter of Cylinder, inches. . 



Thurston 



'A 



Stroke, inches 

Unwin, iron, .0167Z) Yp.. 
Unwin, steel, .0144 D Yp 
/D*pL? D 
10,000 + 80 
Thurston, same with a = 15,000 

Marks, iron, .0179Z> Vp 

Marks, iron, .03901 fW 2 p ... 

Marks, steel, .01052) Yp 

Marks, steel, .035-25 V D*l*p 

Seaton, naval engines, — y'p . . . 



(L in feet), 



1.79 
1.35 
(1.05) 
1.22 
1.67 



D 



Seaton, land engine, — y p 



Average of four for iron. 



1.G7 
1.44 



5.37 

(3.15) 
3.34 
5.01 



2.22 

1.82 



5.01 
4.32 



48 
8.35 
7.20 

5.10 



3.88 
5.37 
5.13 



6.04 

(5.25) 



8.95 
8.54 



The figures in brackets opposite Marks' third formula would be rejected 
since they are less than y$ of the stroke, and the figures derived by his 
fourth formula would be taken instead. The figure 1.79 opposite his first 
formula would be rejected for the engine of 24-inch stroke. 

An empirical formula which gives results approximating the above aver- 
ages is d" = .013 VDlp- 

The calculated results from this formula, for the six engines, are, respec- 
tively, 1.42, 1.88, 3.90, 5.61, 6.37, 9.01. 

Piston-rod. Guides.— The thrust on the guide, when the connecting- 
rod is at its maximum angle with the line of the piston-rod, is found from 
the formula: Thrust = total load on piston X tangent of maximum angle 
of connecting-rod — p tan 0. This angle is the angle whose tangent = half 
stroke of piston -*- length of connecting-rod. 

Ratio of length of connecting-rod to stroke 2 2)4 3 

Maximum angle of connecting-rod with line of 

piston-rod 14° 29' 11° 19' 9° 36' 

Tangent of the angle 25 .20 .1667 

Secant of the angle 1.0308 1.0198 1.0138 

Seaton says: The area of the guide-block or slipper surface on which the 
thrust is taken should in no case be less than will admit of a pressure of 400 
lbs. on the square inch; and for good working those surfaces which take the 
thrust when going ahead should be sufficiently large to prevent the maxi- 
mum pressure exceeding 100 lbs. per sq. in. When the surfaces are kept 
well lubricated this allowance may be exceeded. 

Thurston says: The rubbing surfaces of guides are so proportioned that 
if Vbe their relative velocity in feet per minute, and p be ihe intensity of 
pressure on the guide in lbs. per sq. in., pV < 60,000 and pV > 40,000. 

The lower is the safer limit; but for marine and stationary engines it is 
allowable to take p = 60,000 -s- V. According to Rankine, for locomotives, 

p = where p is the pressure in lbs. per sq. in. and F"the velocity of 

rubbing in feet per minute. This includes the sum of all pressures forcing 
the two rubbing surfaces together. 

Some British builders of portable engines restrict the pressure between 
the guides and cross-heads to less than 40, sometimes 35 lbs. per square inch. 

For a mean velocity of 600 feet per minute, Prof. Thurston's formulas 
give, p < 100, p > 66.7; Rankine's gives p = 72.2 lbs. per sq. in. 



DIMENSIONS OF PARTS OF ENGINES. 799 

Whitham gives, 

A = area of slides in square inches = — = — — ===z, 

p Vn* - 1 p Vn* - 1 

in which P = total unbalanced pressure, p^ = pressure per square inch 
on piston, d = diameter of cylinder, p = pressure allowable per square inch 
on slides, and n = length of connecting-rod -f- length of crank. This is 
equivalent to the formula, A = P tan -f- 2> . For n = 5, p l = 100 and p 
= 80, A = .2004d 2 . For the three engines 10, 30 and 50 in. diam., this would 
give for area of slides, A = 20, 180 and 500 sq. in., respectively. Whitham 
says: The normal pressure on the slide may be as high as 500 lbs. per sq. in., 
but this is when there is good lubrication and freedom from dust. Station- 
ary and marine engines are usually designed to carry 100 lbs. per sq. in., 
and the area in this case is reduced from 50$ to 60$ by grooves. In locomo- 
tive engines the pressure ranges from 40 to 50 lbs. per sq. in. of slide, on ac- 
count of the inaccessibility of the slide, dirt, cinder, etc. 

There is perfect agreement among the authorities as to the formula for 
area of the slides, A = P tan -*- p ; but the value given to p , the allow- 
able pressure per square inch, ranges all the way from 35 lbs. to 500 lbs. 

The Connecting-rod. Ratio of length of connecting-rod to length 
of stroke.— Experience has led generally to the ratio of 2 or 2% to 1, the 
latter giving a long and easy-working rod, the former a rather short, but 
yet a manageable one (Thurston). Whitham gives the ratio of from 2 to 4J^, 
and Marks from 2 to 4. 

Dimensions of the Connecting-rod. — The calculation of the diameter of 
a connecting-rod on a theoretical basis, considering it as a strut subject to 
both compressive and bending stresses, and also to stress due to its inertia, 
in high-speed engines, is quite complicated. See Whitham, Steam-engine 
Design, p. 217; Thurston, Manual of S. E., p. 100. Empirical formulas are as 
follows: For circular rods, largest at the middle, D — diam. of cylinder, I ~ 
length of connecting-rod in inches, p = maximum steam-pressure per sq. in. 



(1) Whitham, diam. at middle, d" - 0.0272 V Dl Vp. 

(2) Whitham, diam. at necks, d" = 1.0 to 1.1 x diam. of piston-rod. 

(3) Sennett, diam. at middle, d" = — Vv- 

(4) Sennett, diam. at necks, d" — — Vp- 

60 

(5) Marks, diam., d" — 0.0179Z) Vp. if d iam. is greater than 1/24 length. 

(6) Marks, diam., d" = 0.02758 \/ Dl Vp if diam. found by (5) is less than 
1/24 length. 

(7) Thurston, diam. at middle, d" = a yD L Vp + C, D in inches, L in 
feet, a = 0.15 and C = % inch for fast engines, a — 0.08 and C — % inch for 
moderate speed. 

(8) Seaton says: The rod may be considered as a strut free at both ends, 
and, calculating its diameter accordingly, 

diameter at middle = — — ^-^-r > 

4o.5 

where R = the total load on piston P multiplied by the secant of the maxi- 
mum angle of obliquity of the connecting-rod. 

For wrought iron and mild steel a is taken at 1/3000. The following are 
the values of r in practice: 
Naval engines— Direct-acting r = 9 toll; 

" " Return connecting-rod r = 10 to 13, old; 

" " " " r = 8 to 9, modern; 

" " Trunk r= 11.5 to 13. 

Mercantile " Direct-acting, ordinary r — 12. 
" " " long stroke r = 13 to 16. 

(9) The following empirical formula is given by Seaton as agreeing closely 
with good modern practice: 

Diameter of conne cting-rod at middle — X'lK-±- 4, I = length of rod in 
inches, and K= 0.03 V effective loacTon piston in pounds, 



800 



THE STEAM-ENGINE. 



The diam. at the ends may be 0.875 of the diam. at the middle. 

Seaton's empirical formula when translated into terms of D andp is the 

same as the second one by Marks, viz., d" = 0.02758 V Dl Vp~- Whitham's 
(1) is also practically the same. 

(10) Taking Seaton's more complex formula, with length of connecting- 
rod = 2.5 x length of stroke^and r = 12 and 16, respectively, it reduces to: 
Diam. at middle = .02294 VP and .02411 VP for short and long stroke en- 
gines, respectively. 

Applying the above formulas to the engines of our list, we have 

Diameter of Connecting-rods. 



Diameter of Cylinder, inches. . 



Stroke, inches 

Length of connecting-i-od I 

(3) d" = ;| Vp = .0182Z) |/p.... 

(5) d" = .0179D Vp 

(6) d" = .02758^ Dl Vp 

(7) d" = OAbVDL Vp + %, 

(7) d" = 0.08 V DL Vp + U 

(9) d" = .03 VF. 

(10) d" = .02294 VP; .02411 VP. . 



1.82 
1.79 



2.67 
2.03 



Average . 



2.24 2. 



5.46 
5.37 



2.54 
2.67 
2.14 



13.29 
10.16 



6.38 6.27 10.52 10 



240 
9.09 



13.29 
10.68 



Formulae 5 and 6 (Marks), and also formula 10 (Seaton), give the larger 
diameters for the long-stroke engine; formulas? give the larger diameters 
for the short-stroke engines. The average figures show but little difference 
in diameter between long- and short-stroke engines; this is what might be 
expected, for while the connecting-rod, considered simply as a column, 
would require an increase of diameter for an increase of length, the load 
remaining the same, yet in an engine generally the shorter the connecting- 
rod the greater the number of revolutions, and consequently the greater the 
strains due to inertia. The influences tending to increase the diameter 
therefore tend to balance each other, and to render the diameter to some 
extent independent of the length. The average figures correspond nearly 
to the simple formula d" = .021 D Vp. The diameters of rod for the three 
diameters of engine by this formula are, respectively, 2.10, 6.30, and 10.50 in. 
Since the total pressure on the piston P — .7854D 2 p, the formula is equiva 
lent to d' = .0287 VP. 

Connecting-rod. Ends.— For a connecting-rod end of the marine 
type, where the end is secured with two bolts, each bolt should be propor- 
tioned for a safe tensile strength equal to two thirds the maximum pull or 
thrust in the connecting-rod. 

The cap is to be proportioned as a beam loaded with the maximum pull 
of the connecting-rod, and supported at both ends. The calculation should 
be made for rigidity as well as strength, allowing a maximum deflection of 
1/100 inch. For a strap-and-key connecting-rod end the strap is designed for 
tensile strength, considering that two thirds of the pull on the conuecting- 
rod may come on one arm. At the point where the metal is slotted for the 
key and gib, the straps must be thickened to make the cross-section equal 
to that of the remainder of the strap. Between the end of the strap and the 
slot the strap is liable to fail in double shear, and sufficient metal must be 
provided at the end to prevent such failure. 

The breadth of the key is generally one fourth of the width of the strap, 
and the length, parallel to the strapj should be such that the cross-section 
will have a shearing strength equal to the tensile strength of the section of 
the strap. The taper of the key is generally about % inch to the foot. 



DIMENSIONS OF PARTS OF ENGINES. 801 

Tapered Connecting-rods.— In modem high-speed engines it is 
customary to make the connecting-rods of rectangular instead of circular 
section, the sides being parallel, and the depth increasing regularly from 
the crosshead end to the crank-pin end. According to Grashof, the bending 
action on the rod due to its inertia is greatest at 6/10 the length from the 
crosshead end, and, according to this theory, that is the point at which the 
section should be greatest, although in practice the section is made greatest 
at the crank-pin end. 

Professor Thurston furnishes the author with the following rule for tapered 
connecting-rod of rectangular section : Take the section as computed by the 

formula d" = O.lV DL Vp + 3/4 for a circular section, and for a rod 4/3 the 
actual length, placing the computed section at 2/3 the length from the small 
end, and carrying the taper straight through this fixed section to the large 
end. This brings the computed section at the surge point and makes it 
heavier than the rod for which a tapered form is not required. 
Taking the above formula, multiplying L by 4/3, and changing it to I in 

inches, it becomes d = 1/30 V Dl Vp -|- 3/4". Taking a rectangular section 
of the same area as the round section whose diameter is d, and making the 
depth of the section h = twice the thickness t, we have .7854d 2 = lit = 2i 2 , 

whence t = .627d = .0209 V Dl Vp + .47", which is the formula for the thick- 
ness or distance between the parallel sides of the rod. Making the depth at 
the crosshead end = IM, and at 2/3 the length = 2t, the equivalent depth at 
the crank end is 2.25t. Applying the formula to the short-stroke engines of 
our examples, we have 



Diameter of cylinder, inches 

Stroke, inches 

Length of connecting-r od . 

Thickness, t = .0209 VdI Vp -f .47 =.. 

Depth at crosshead end, \M — 

Depth at crank end, 2%t 



10 
12 
30 


30 

30 
75 


1 61 


3.60 


2.42 
3.62 


5.41 
8.11 



48 
120 

5.59 
8.39 

12.58 



The thicknesses t, found by the formula t = .0209 V Dl Vp + .47, agree 
closely with the more simple formula t = MD Vp -f- .60", the thicknesses 
calculated by this formula being respectively 1.6, 3.6, and 5.6 inches. 

The Crank-pin.— A crank-pin should be designed (1) to avoid heating, 
(2) for strength, (3) for rigidity. The heating of a crank-pin depends on the 
pressure on its rubbing-surface, and on the coefficient of friction, which 
latter varies greatly according to the effectiveness of the lubrication. It also 
depends upon the facility with which the heat produced may be carried 
away: thus it appears that locomotive crank-pins may be prevented to some 
degree from overheating by the cooling action of the air through which they 
pass at a high speed. 

Marks gives 1 = .0000247 fpNW = 1.038/^^^ (1) 

Whitham gives I = 0.9075/ - J ^ P, - ) , (2) 

in which I = length of crank-pin journal in inches, f — coefficient of friction, 
which may be taken at .03 to .05 for perfect lubrication, and .08 to .10 for im- 
perfect; p = mean pressure in the cylinder in pounds per square inch; D 
= diameter of cylinder in inches; N = number of single strokes per minute; 
I.H.P. = indicated horse-power; L = length of stroke in feet. These 
formulEe are independent of the diameter of the pin, and Marks states as a 
general law, within reasonable limits as to pressure and speed of rubbing, 
the longer a bearing is made, for a given pressure and number of revolutions, 
the cooler it will work; and its diameter Las no effect upon its heating. 
Both of the above formulae are deduced empirically from dimensions of 
crank-pins of existing marine engines. Marks says that about one-fourth 
the length required for crank-pins of propeller engines will serve for the pins 
Of side-wheel engines, and one tenth for locomotive engines, making the 



802 THE STEAM-ENGINE. 

formula for locomotive crank-pins I = .00000247/piVD 2 , or if p = 150, / 
= .08, and N= 600, / = .013D 2 . 

Whitham recommends for pressure per square inch of projected area, for 
naval engines 500 pounds, for merchant engines 400 pounds, for paddle-wheel 
engines 800 to 900 pounds. 

Thurston says the pressure should, in the steam engine, never exceed 500 
or 600 pounds per square inch for wrought-iron pins, or about twice that 
figure for steel. He gives the formula for length of a steel pin, in inches, 

1 = PR+- 600.000, (3) 

in which P and R are the mean total load on the pin in pounds, and the 
number of revolutions per minute. For locomotives, the divisor may be 
taken as 500,000. Where iron is used this figure should be reduced to 300,000 
and 250,000 for the two cases taken. Pins so proportioned, if well made and 
well lubricated, may always be depended upon to run cool; if not well 
formed, perfectly cylindrical, well finished, and kept well oiled, no crank-pin 
can be relied upon. It is assumed above that good bronze or white-metal 
bearings are used. 

Thurston also says : The size of crank-pins required to prevent heating of 
the journals may be determined with a fair degree of precision hy either of 
the formulae given below : 

l = l t^T ^nkine,1865); (4) 

^-60^J (ThUrStOI1 ' 1862); (5) 

< = 3l|o¥o^ anB ^ en < 1866) (6) 

The first two formulas give what are considered by their authors fair work- 
ing proportions, and the last gives minimum length for iron pins. (V — 
velocity of rubbing-surface in feet per minute.) 

Formula (1) was obtained by observing locomotive practice in which great 
liability exists of annoyance by dust, and great risk occurs from inaccessi- 
bility while running, and (2) by observation of crank-pins of naval screw- 
engines. The first formula is therefore not well suited for marine practice. 

Steel can usually be worked at nearly double the pressure admissible with 
iron running at similar speed. 

Since the length of the crank- pin will be directly as the power expended 
upon it and inversely as the pressure, we may take it as 

l=a^,. . . (7) 

in which a is a constant, and L the stroke of piston, in feet. The values of 
the constant, as obtained by Mr. Skeel, are about as follows: a — 0.04 where 
water can be constantly used; a = 0.045 where water is not generally used; 
a = 0.05 where water is seldom used; a - 0.06 where water is never needed. 
Unwin gives 

; IH.P. rR 

I = a (8 

r ' 

in which r = crank radius in inches, a = 0.3 to a = 0.4 for iron and for marine 
engines, and a = 0.066 to a — 0.1 for the case of the best steel and for loco- 
motive work, where it is often necessary to shorten up outside pins as much 
as possible. 

J. B. Stanwood (Eng^g, June 12, 1891), in a table of dimensions of parts of 
American Corliss engines from 10 to 30 inches diameter of cylinder, gives 
sizes of crank-pins which approximate closely to the formula 

1= .275D" + .5 in.; d = .25D" (9) 

By calculating lengths of iron crank-pins for the engines 10. 30, and 50 inches 
diameter, long and short stroke, by the several formulae above given, it is 
found that there is a great difference in the results, so that one formula in 
certain cases gives a length three times as great as another. Nos. (4). (5), and 
(6) give lengths much greater than the others. Marks (1), Whitham (2), 
Thurston (7), I = .06 I.H.P. -h L, and Unwin (8), I = 0.4 I.H.P. -*- r, give re- 
sults which agree more closely. 



DIMENSIONS OF PARTS OF ENGINES. 



803 



The calculated lengths of iron crank pins for the several cases by formulae 
(1), (2), (7), and (8) are as follows: 

Length of Cranlc-pins. 

Diameter of cylinder D 

Stroke I. (ft.) 

Revolutions per minute R 250 125 130 65 90 45 

Horse-power I.H.P. 50 50 450 450 1,250 1,250 

Maximum pressure lbs. 7,854 7,854 70,686 70,686 196,350 196,350 

Mean pressure per cent of max 42 42 32.3 32.3 30 30 

Mean pressure P. 3,299 3,299 22,832 22,832 58,905 58, 905 

Length of crank-pin 

(1) Whitham, I = .9075 X .05 I.H.P. -s- L. 

(2) Marks, I = 1.038 X .05 I.H.P. -s- L. 

(7) Thurston, I = .06 I.H.P. +L 

(8)Unwin, I = .4 I.H.P. -s- r 3.33 1.67 12.0 6.0 20.83 10.42 

(8) " I = .3 I.H.P. -r- r 2.50 1.25 9.0 4.5 15.62 7.81 

Average 

(8) Unwin, best steel, I = .ilJ^J 83 .42 3.0 1.5 5.21 2.61 

r 

(3) Thurston, steel, I = -**L 1.37 .69 4.95 2.47 8.84 4.42 

o00,000 

The calculated lengths for the long-stroke engines are too low to prevent 
excessive pressures. See " Pressures on the Crank-pins," below. 

The Strength of the Crank-pin is determined substantially as is' 
that of the crank. In overhung cranks the load is usually assumed as 
carried at its extremity, and, equating its moment with that of the resist- 
ance of the pin, 



10 


10 


30 


30 


50 


1 


2 


W» 


5 


4 


250 


125 


130 


65 


90 


50 


50 


450 


450 


1,250 


7,854 


7.854 


70,686 


70,686 


196,350 


42 


42 


32.3 


32.3 


30 


3,299 


3,299 


22,832 


22,832 


58,905 


2.18 


1.09 


8.17 


4.08 


14.18 


2.59 


1.30 


9.34 


4.67 


16.22 


3.00 


1.50 


10.80 


5.40 


18.75 


3.33 


1.67 


12.0 


6.0 


20.83 


2.50 


1.25 


9.0 


4.5 


15.62 


2.72 


1.36 


9.86 


4.93 


17.12 


.83 


.42 


3.0 


1.5 


5.21 


1.37 


.69 


4.95 


2.47 


8.84 



J4PI = l/32trrd*, and d 



-{/^ 



in which d = diameter of pin in inches, P = maximum load on the piston, 
t = the maximum allowable stress on a square inch of the metal. For iron 
it may be taken at 9000 lbs. For steel the diameters found by this formula 
may be reduced 10%. (Thurston.) 
Unwin gives the same formula in another form, viz. : 



4/x^W* 



the last form to be used when the ratio of length to diameter is assumed. 
For wrought iron, t = 6000 to 9000 lbs. per sq. in., 



0- 



For steel, t - 9000 to 13,000 lbs. per sq. in,, 



Whitham gives d = 0.0827 \/Pl = 2.1058, 



. 1/?= 

sq. in,, 



0291 to .0238. 



for strength, and 



d = 0.405 yPl 3 for rigidity, and recommends that the diameter be calculated 
by both formulae, and the largest result taken. The first is the same as 
Unwinds formula, with t taken at 9000 lbs. per sq. in. The second is based 
upon an erroneous assumption. 



804 



THE STEAM-EtfGItfE. 



Marks, calculating the diameter for rigidity, gives 
d = O.OffifypPD 2 = 0.' 



. 4 /(H.P.)/ 3 



p = maximum steam-pressure in pounds per square inch, D — diameter of 
cylinder in inches, L — length of stroke in feet, N= number of single strokes 
per minute. He says there is no need of an investigation of the strength of 
a crank-pin, as the condition of rigidity gives a great excess of strength. 

Marks's formula is based upon the assumption that the whole load may be 
concentrated at the outer end, and cause a deflection of .01 inch at that 
point. 

It is serviceable, he says, for steel and for wrought iron alike. 

Using the average lengths of the crank-pins already found, we have the 
following for our six engines : 

Diameter of Crank-pins. 



Diameter of cylinder. . . 

Stroke, ft 

Length of crank-pin. . . 

TT . , ' 3/5.IPZ 
Unwin, d = a/ — j- ... 

Marks, d = .066 typPD*. 



10 

1 

2.72 


10 

2 

1.36 


30 
9.86 


30 

5 

4.93 


50 

4 

17.12 


2.29 


1.82 


7.34 


5.82 


12.40 


1.39 


.85 


6.44 


3.78 


12.41 



50 

8 

8.56 

9.84 
7.39 



Pressures on the Crank-pins.— If we take the mean pressure upon 
the crank-pin = mean pressure on piston, neglecting the effect of the vary- 
ing angle of the connecting-rod, we have the following, using the average 
lengths already found, and the diameters according to Unwin and Marks: 



Engine No 


1 


2 


3 


4 


5 


6 




10 
1 
3,299 
6.23 
3.78 
530 
873 


10 

2 

3,299 

236 
1.16 
1,398 
2,845 


30 
2% 

72 .4 
63.5 
315 
360 


30 

5 

22,832 
28.7 
18.6 
796 
1,228 


50 
4 

58,905 
212.3 
212.5 
277 

277 


50 




8 


Mean pressure on pin, pounds 


58,905 

84.2 




63.3 


Pressure per square inch, Unwin 

" " " " Marks 


700 
930 



The results show that the application of the formulae for length and diam- 
eter of crank-pins give quite low pressures per square inch of projected 
area for the short-stroke high-speed engines of the larger sizes, but too high 
pressures for all the other engines. It is therefore evident that after calcu- 
lating the dimensions of a crank-pin according to the formulae given that the 
results should be modified, if necessary, to bring the pressure per square 
inch down to a reasonable figure. 

In order to bring the pressures down to 500 pounds per square inch, we 
divide the mean pressures by 500 to obtain the projected area, or product 
of length by diameter. Making I = 1.5d for engines Nos. 1, 2, 4 and 6, the 
revised table for the six engines is as follows : 



Engine, No". 1 2 

Length of crank- pin, inches 3.15 3.15 

Diameter of crank-pin 2.10 2.10 



.34 



4 5 6 

3.37 17.12 13.30 
5.58 12.40 8.87 



Crosshead-pin or Wrist-pin.— Whitham says the bearing surface 
for the wrist-pin is found by the formula for crank-pin design. Seaton says 
the diameter at the middle must, of course, be sufficient to withstand the 
bending action, and general^ from this cause ample surface is provided for 
good working; but in any case the area, calculated by multiplying the diam- 
eter of the journal by its length, should be such that the pressure does not 
exceed 1200 lbs. per'sq. in., taking the maximum load on the piston as the 
total pressure on it. 

For small engines with the gudgeon shrunk into the jaws of the connect- 



DIMENSIONS OF PARTS OF ENGINES. 805 

ing-rod, and working in brasses fitted into a recess in the piston-rod end and 
secured by a wrought- iron cap and two bolts, Seaton gives: 

Diameter of. gudgeon = 1.25 X diam. of piston-rod. 
Length of gudgeon = 1.4 x diam. of piston-rod. 

If the pressure on the section, as calculated by multiplying length by 
diameter, exceeds 1200 lbs. per sq. in., this length should be increased. 

J. B. Stanwood, in his "Ready Reference 1 ' book, gives for length of 
crosshead-pin 0.25 to 0.3 diam. of piston, and diam. = 0.18 to 0-.2 diam. of 
piston. Since he gives for diam. of piston-rod 0.14 to 0.17 diam. of piston, 
his dimensions for diameter and length of crosshead-pin are about 1.25 and 
1.8 diam. of piston-rod respectively. Taking the maximum allowable press- 
ure at 1200 lbs. per sq. in. and making the length of the crosshead-pin = 
4/3 of its diameter, we have d = l/P-^40, I = \/~P -s- 30, in which P = max- 
imum total load on piston in lbs., d = diam. and Z = length of pin in inches. 
For the engines of our example we have: 

Diameter of piston, inches 10 30 50 

Maximum load on piston, lbs. 7854 70,686 196,350 

Diameter of crosshead-pin, inches 2.22 6.65 11.08 

Length of crosshead-pin, inches 2.96 *8.86 14.77 

Stanwood's rule gives diameter, inches 1.8to2 5.4 to 6 9.0 to 10 

Stan wood's rule gives length, inches 2.5 to 3 7.5 to 9 12.5 to 15 

Stan wood's largest dimensions give pressure 

per sq. in., lbs 1309 1329 1309 

Which pressures arc greater than the maximum allowed by Seaton. 

The Crank-arm.— The crank-arm is to be treated as a lever, so that 
if a is the thickness in direction paral.el to the shaft-axis and b its breadth 
at a section x inches from the crank-pin centre, then, bending moment M 
at that section = Px, P being the thrust of the connecting-rod, and / the 
safe strain per square inch, 



„ fab* . aXb* T 6T , 

P*=- W - and -g- = j, or a = ^^ f ; b 



w%- 



If a crank-arm were constructed so that b varied as Vx (as given by the 
above rule) it would be of such a curved form as to be inconvenient to man- 
ufacture, and consequently it is customary in practice to find the maxi- 
mum value of b and draw tangent lines to the curve at the points ; these 
lines are generally, for the same reason, tangential to the boss of the crank- 
arm at the shaft. 

The shearing strain is the same throughout the crank-arm; and, conse- 
quently, is large compared with the bending strain close to the crank-pin ; 
and so it is not sufficient to provide there only for bending strains. The 
section at this point should be such that, in addition to what is given by the 
calculation from the bending moment, there is an extra square inch for 
every 8000 lbs. of thrust on the connecting-rod (Seaton). 

The length of the boss h into which the shaft is fitted is faoin 0.75 to 1.0 
of the diameter of the shaft D, and its thickness e must be calculated from 
the twisting strain PL. (L = length of crank.) 

For different values of length of boss h, the following values of thickness 
of boss e are given by Seaton: 

When h = D, then e = 0.35 D; if steel, 0.3. 
h = 0.9 D, then e = 0.38 D, if steel, 0.32. 
h = 0.8 D, then e = 0.40 D, if steel, 0.33. 
h = 0.7 D. then e = 0.41 D, if steel, 0.34. 

The crank-eye or boss into which the pin is fitted should bear the same 
relation to the pin that the boss does to the shaft. 

The diameter of the shaft-end onto which the crank is fitted should be 
1.1 X diameter of shaft. 

Thurston says: The empirical proportions adopted by builders will com- 
monly be found to fall well within the calculated safe margin. These pro- 
portions are, from the practice of successful designers, about as follows : 

For the wrought-iron crank, the hub is 1.75 to 1.8 times the least diameter 
of that part of the shaft carrying full load; the eye is 2.0 to 2.25 the diame- 
ter of the inserted portion of the pin, and their depths are, for the hub, 1.0 
to 1.2 the diameter of shaft, and for the eye, 1.25 to 1.5 the diameter of pin. 



806 



THE STEAM-EKGIXE. 



The web is made 0.7 to 0.75 the width of adjacent hub or eye, and is given a 
depth of 0.5 to 0.6 that of adjacent hub or eye. 

For the cast-iron crank the hub and eye are a little larger, ranging in 
diameter respectively from 1.8 to 2 and from 2 to 2.2 times the diameters of 
shaft and pin. The flanges are made at either end of nearly the full depth 
of hub or eye. Cast-iron has, however, fallen very generally into disuse. 

The crank-shaft is usually enlarged at the seat of the crank to about 1.1 
its diameter at the journal. The size should be nicely adjusted to allow for 
the shrinkage or forcing on of the crank. A difference of diameter of one 
fifth of l$,"wiil usually suffice ; and a common rule of practice gives an 
allowance of but one half of this, or .001. 

The formulas given by different writers for crank-arms practically agree, 
since they all consider the crank as a beam loaded at one end and fixed at 
the other. The relation of breadth to thickness may vary according to the 
taste of the designer. Calculated dimensions for our six'engines are as fol 
lows : 

©line u si oiis of Crank-arms, 



Diam. of cylinder, ins.. . 

Stroke S, ins * 

Max. pressure on pin P, 

(approx.) lbs 

Diam. crank-pin d 



7T.H.P. 



Diam. shaft, 



(a = 4.69, 5.09 and 5.22). 

Length of boss, .8D 

Thickness of boss, AD. 

Diam. of boss, 1.8D 

Length crank-pin eye,. 8d 
Thickness of crank-pin 

eye, Ad 

Max. mom. Tat distance 

}4S - y%D from centre 

of pin, inch-lbs 

Thickness of crank-arm 

a = .75D 

Greatest breadth, 



: y 9001 



6r 

9000a 

Min.mom. T at distance 
d from centre of pin^l'd 
Least breadth, 



10 
12 


10 
24 


30 
30 


30 
60 


50 

48 


7854 
2.10 


7854 
2.10 


70,686 
7.34 


70,686 
5.58 


196,350 
12.40 


1-2.74 

2.19 
1.10 
4.93 
1.76 


3.46 


7.70 


9.70 


12.55 


2.77 
1.39 
6.23 
1.76 


6.16 
3.08 
13.86 

5.87 


7.76 
3.88 
17.46 
4.48 


10.04 
5.02 

22.59 
9.92 


,88 


.88 


2.94 


2.23 


4.46 


37, 149 


80,661 


788,149 


1,848,439 


3,479,322 


2.05 


2.60 


5.78 


7.28 


9.41 


3.48 


4.55 


9.54 


13.0 


15.7 


16,493 


16,493 


528,835 


394,428 


2,434,740 


2.32 


2.06 


7.81 


6.01 


13.13 



196,350 

8.87 



28.47 
7.10 



7,871,671 
11.87 



1,741,6! 



The Shaft.— Twisting Resistance. 

for torsion, we have: T= — d 3 S - 



-From the g enera l formula 

.19635d 3 S, whence d = 1/ ^—, in which 

T = torsional moment in inch-pounds, d = diameter in inches, and S = the 
shearing resistance of the material in pounds per square inch. 

If a constant force P were applied to the crank-pin tangentially to its path, 
the work done per minute would be 

13,000 XLH.P., 

in which L = length of cank in inches, and R = revs, per min., and the 
mean twisting moment T = ' x 63,025. Therefore 



3 /s.ijr y 



?I.H.P. 



DIMENSIONS OF PARTS OF ENGINES. 



807 



This may take the form 

3 /l.H.P. 



X F, or d - c 



in which .Fand a are factors that depend on the strength of the material 
and on the factor of safety. Taking S at 45,000 pounds per square inch for 
wrought iron, and at 60,000 for steel, we have, for simple twisting by a uni- 
form tangential force, 



Factor of safety = 5 

Iron F= 35.7 

Steel F= 26.8 



6 8 10 
42.8 57.1 71.4 
32.1 42.8 53.5 



6 8 
3.5 3.85 
3.18 3.5 



10 
4.15 



Unwin, taking for safe working strength of wrought iron 9000 lbs., steel 
13.500 lbs., and cast iron 4500 lbs., gives a = 3.294 for wrought iron, 2.877 for 
steel, and 4.15 for cast iron. Thurston, for crank-axles of wrought iron, 
gives a = 4.15 or more. 

Seaton says: For wrought iron, /, the safe strain per square inch, should 
not exceed 9000 lbs., and when the shafts are more than 10 inches diameter, 
8000 lbs. Steel, when made from the ingot and of good materials, will ad- 
mit of a stress of 12,000 lbs. for small shafts, and 10,000 lbs. for those above 
10 inches diameter. 

The difference in the allowance between large and small shafts is to com- 
pensate for the defective material observable in the heart of large shafting, 
owing to the hammering failing to affect it. 
a / i h p 

The formula d = ai/ ' assumes the tangential force to be uniform 

and that it is the only acting force. For engines, in which the tangential 
force varies with the angle between the crank and the connecting-rod, and 
with the variation in steam-pressure in the cylinder, and also is influenced 
by the inertia of the reciprocating parts, and in which also the shaft may be 
subjected to bending as well as torsion, the factor a must be increased, to 
provide for the maximum tangential force aud for bending. 

Seaton gives the following table showing the relation between the maxi- 
mum and mean twisting moments of engines working under various condi- 
tions, the momentum of the moving parts being neglected, which is allow- 
able: 



Description of Engine. 



Steam Cut-off 



Max. 

Twist 

Divided 

by 

Mean 

Twist. 
Mome't 



Cube 
Root 
of the 
Ratio. 



Single-crank expansive 

Two-cylindev expansive, cranks at 90° — 



Three-cylinder compound, cranks 120°. . . 

" '■' 1. p. cranks ] 

opposite one another, andh.p. midway j 



0.7 

0.8 

h.p.0.5, l.p. 0.6 



2.625 
2.125 
1.835 
1.698 
1.616 
1.415 
1.298 
1.256 
1.270 
1.329 
1.357 
1.40 

1.26 



1.38 
1.29 
1.22 
1.20 
1.17 
1.12 



1.08 
1.10 
1.11 
1.12 



Seaton also gives the following rules for ordinary practice for ordinary 
two-cylinder marine engines: 



Diameter of the tunnel-shafts = a/ ' • 



XF, or a. 



;/ I.H P. 



808 THE STEAM-ENGINE. 

Compound engines, cranks at right angles: 
Boiler pressure 70 lbs., rate of expansion 6 to 7, F = 70, a = 4.12. 
Boiler pressure 80 lbs., rate of expansion 7 to 8, F = 72, a — 4.16. 
Boiler pressure 90 lbs., rate of expansion 8 to 9, F = 75, a — 4.22. 

Triple compound, three cranks at 120 degrees: 
Boiler pressure 150 lbs., rate of expansion 10 to 12, F = 62, a = 3.96. 
Boiler pressure 160 lbs., rate of expansion 11 to 13, F = 64, a = 4. 
Boiler pressure 170 lbs., rate of expansion 12 to 15, F — 67, a — 4.06. 

Expansive engines, cranks at right angles, and the rate of expansion 5, 
boiler-pressure 60 lbs., F — 90, a — 4.48. 

Single-crank compound engines, pressure 80 lbs., F — 96, a — 4.58. 

For the engines we are considering it will be a very liberal allowance for 
ratio of maximum to mean twisting moment if we take it as equal to the 
ratio of the maximum to the mean pressure on the piston. The factor a, 
then, in the formula for diameter of the shaft will be multiplied by the cube 

root of this ratio, or a/— =1.34, i/^- = 1.45, and a/~ = 1.49 for the 
y 42 ' |/ 32.3 y .30 

10, 30, and 50-in. engines, respectively. Taking a = 3.5, which corresponds 
to a shearing strength of 60,000 and a factor of safety of 8 for steel, or to 
45,000 and a factor of 6 for iron, we have for the new coefficient cij in the 



3 /l H P 
formula d x = a x A/ ■' ' * , the values 4.69, 5.08, and 5.22, from which we 

obtain the diameters of shafts of the six engines as follows: 

Engine No 1 2 3 4 5 6 

Diana.- of cyl 10 10 30 30 50 50 

Horse-power, I.H.P 50 50 450 450 1250 1250 

Revs, per min., R 250 125 130 65 90 45 

Diam. of shaft d = a 1 i/ I J5l!:.... 2.74 3.46 7.67 9.70 12.55 15.82 

y r 

These diameters are calculated for twisting only. When the shaft is also 
subjected to bending strain the calculation must be modified as below : 

Resistance to Bending.— The strength of a circular-section shaft 
to resist bending is one half of that to resist twisting. If B is the bending 
moment in inch-lbs., and d the diameter of the shaft in inches, 



B= -^ x/; and d = i/y X 10.2; 






f is the safe strain per square inch of the material of which the shaft is 
composed, and its value may be taken as given above for twisting (Seaton). 

Equivalent Twisting Moment.— When a shaft is subject to 
both twisting and bending simultaneously, the combined strain on any sec- 
tion of it may be measured by calculating what is called the equivalent 
twisting moment; that is, the two strains are so combined as to be treated 
as a twisting strain only of the same magnitude and the size of shaft cal- 
culated accordingly. Rankine gave the following solution of the combined 
action of the two strains. 

If T = the twisting moment, and B — the bending mome nt on a se ction of 
a shaft, then the equivalent twisting moment T± = B-\- YB 2 -f- T 2 . 

Seaton says: Crank-shafts are subject always to twisting, bending, and 
shearing strains; the latter are so small compared with the former that 
they are usually neglected directly, but allowed for indirectly by means of 
the factor /. 

The two principal strains vary throughout the revolution, and the maxi- 
mum equivalent twisting moment can only be obtained accurately by a 
series of calculations of bending and twisting moments taken at fixed inter- 
vals, and from them constructing a curve of strains. 

Considering the engines of our examples to have overhung cranks, the 
maximum bending moment resulting from the thrust of the connecting-rod 
on the crank-pin will take place when the engine is passing its centres 
(neglecting the effect of the inertia of the reciprocating parts), and it will 
be the product of the total pressure on the piston by the distance between 



DIMENSIONS OP PARTS OP ENGINES. 



809 



two parallel lines passing through the centres of the crank-pin and of the 
shaft bearing, at right angles to their axes; which distance is equal to 
y& length of crank-pin bearing -f length of hub -f-i^ length of shaft bearing -f 
any clearance that may be allowed between the crank and the two bearings. 
For our six engines we may take this distance as equal to Yo, length of 
crank-pin -f- thickness of crank-arm-)- 1.5 X the diameter of the shaft as 
already found by the calculation for twisting. The calculation of diameter 
is then as below: 



Engine No. 


1 


2 


3 


4 


5 


Diain. of cyl., in. . 


10 


10 


30 


30 


50 


Horse-power 


50 


50 


450 


450 


1250 


Revs, per min.. .. 


250 


125 


130 


65 


90 


Max. press, on pis,P 


7,854 


7,854 


70,686 


70,686 


196,350 


Leverage,* L in 


6.32 


7.94 


22.20 


26.00 


36.80 


Bd.mo.PL^Pin.-lb 


49,637 


62,361 


1,569,222 


1,837,836 


7,225,680 


Twist, morn. T. 


47,124 


94,248 


1,060,290 


2,120,580 


4,712,400 


Equiv. Twist, mom. 












T!=B+ VB^+T 2 












(approx.) 


118.000 


175.000 


3,463.000 


4.647.000 


15,840,000 



1250 

45 

196,350 

42.25 

8,295,788 
9,424,800 



* Leverage = distance between centres of crank-pin and shaft bearing = 

Having already found the diameters, on the assumption that the shafts 
were subjected to a twisting moment T only, we may find the diameter for 
resisting combined bending and twisting by multiplying the diameters 
already found by the cube roots of the ratio Ti -5- T, or 



Giving corrected diameters d 1 =. 



1.40 



1.27 1.46 1.34 1.64 1.36 
4.39 11.35 12.99 20.58 21.52 



By plotting these results, using the diameters of the cylinders for abscissas 
and diameters of the shafts for ordinates, we find that for the long-stroke 
engines the results lie almost in a straight line expressed by the formula, 
diameter of shaft = .43 X diameter of cylinder; for the short -stroke engines 
the line is slightly curved, but does not diverge far from a straight line 
whose equation is, diameter of shaft = .4 diameter of cylinder. Using these 
two formulas, the diameters of the shafts will be 4.0, 4.3, 12.0, 12.9, 20.0, 21.5. 

J. B. Stanwood, in Engineering, June 12, 1891, gives dimensions of shafts 
of Corliss engines in American practice for cylinders 10 to 30 in. diameter. 
The diameters range from 4 15/16 to 14 15/ 16, following precisely the equation, 
diameter of shaft = % diameter of cylinder - 1/16 inch. 

Fly-wtieel Shafts.— Thus far we have considered the shaft as resist- 
ing the force of torsion and the bending moment produced by the pressure 
on the crank-pin. In the case of fly-wheel engines the shaft on the opposite 
side of the bearing from the crank pin has to be designed with reference to 
the bending moment caused by the weight of the fly wheel, the weight of 
the shaft itself, and the strain of the belt. For engines in which there is an 
outboard bearing, the weight of fly-wheel and shaft being supported by 
two bearings, the point of the shaft at which the bending moment is a 
maximum may be taken as the point midway between the two bearings or 
at the middle of the fly-wheel hub, and the amount of the moment is the 
product of the weight supported by one of the bearings into the distance 
from the centre of that bearing to the middle point of the shaft. The shaft 
is thus to be treated as a beam supported at the ends and loaded in the 
middle. In the case of an overhung fly-wheel, the shaft having only one 
bearing, the point of maximum moment should be taken as the middle of 
the bearing, and its amount is very nearly the product of half the weight 
of the fly-wheel and the shaft into the distance from the middle of its hub 
from the middle of the bearing. The bending moment should be calculated 
and combined with the twisting moment as above shown, to obtain the 
equivalent twisting moment, and the diameter necessary at the point of 
maximum moment calculated therefrom. 

In the case of our six engines we assume that the weights of the fly- 
wheels, together with the shaft, are double the weight of fly-wheel rim 

obtained from the formula] W— 785,400 -^r^r' (given under Fly-wheels); 



810 



THE STEAM-ENGINE. 



that the shaft is supported by an outboard bearing, the distance between 
the two bearings being 2%, 5, and 10 feet for the 10-in., 30-in., and 50-in. 
engines, respectively. The diameters of the fly-wheels are taken such 
that their rim velocity will be a little less than 6000 feet per minute. 

Engine No 1 2 3 4 5 6 

Diam. of cyl., inches 10 10 30 30 50 50 

Diam. of fly-wheel, ft 7.5 15 14.5 29 21 42 

Revs, per min 250 125 130 65 90 45 

Half wt.fly-wh'l and shaft,lb. ■ 268 536 5,963 11,936 26,470 52,940 

Lever arm for max.mom.,in. 15 15 30 30 60 60 

Max. bending moment, in.-lb. 4020 8040 179,040 358,080 1,588,200 3,176,409 

As these are very much less than the bending moments calculated from 
the pressures on the crank-pin, the diameters already found are sufficient 
for the diameter of the shaft at the fly-wheel hub. 

In the case of engines with heavy band fly-wheels and with long fly-wheel 
shafts it is of the utmost importance to calculate the diameter of the shaft 
with reference to the bending moment due to the weight of the fly-wheel 
and the shaft. 

B. H. Coffey (Power, October, 1892) gives the formula for combined bend- 
ing and twisting resistance, T t = .196d3£, in which T x = B + j/^ + r 2 ; T 
being the maximum, not the mean twisting moment; and finds empirical 
working values for .196S as below. He says: Four points should be consid- 
ered in determining this value: First, the nature of the material; second, 
the manner of applying the loads, with shock or otherwise; third, the ratio 
of the bending moment to the torsional moment— the bending moment in a 
revolving shaft produces reversed strains in the material, which tend to rup- 
ture it; fourth, the size of the section. Inch for inch, large sections are 
weaker than small ones. He puts the dividing line between large and small 
sections at 10 in. diameter, and gives the following safe values of S X .196 for 
steel, wrought iron, and cast iron, for these conditions. 

Value of S X .196. 



Ratio. 


Heavy Shafts 
with Shock. 


Light shafts with 
Shock. Heavy 
Shafts No Shock. 


Light Shafts 
No Shock. 


Bto T. 


Steel . 


Wro't 
Iron. 


Cast 
Iron. 


Steel. 


Wro't 
Iron. 


Cast 
Iron. 


Steel. 


Wro't 
Iron. 


Cast 
Iron. 


3 to 10 or less 

3 to 5 or less 

1 to 1 or less 

B greater than T. . 


1045 
941 

855 

784 


880 
785 
715 
655 


440 
393 
358 

328 


1566 
1410 
1281 
1176 


1320 
1179 
1074 
984 


660 
589 
537 
492 


2090 
1882 
1710 
1568 


1760 
1570 
1430 
1310 


880 
785 
715 
655 



Mr. Coffey gives as an example of improper dimensions the fly-wheel 
shaft of a 1500 H.P. engine at Willimantic, Conn., which broke while the en- 
gine was running at 425 H.P. The shaft was 17 ft. 5 in. long between centres 
of bearings, 18 in. diam. for 8 ft. in the middle, and 15 in. diam. for the re- 
mainder, including the bearings. It broke at the base of the fillet connect- 
ing the two large diameters, or 56% in. from the centre of the bearing. He 
calculates the mean torsional moment to be 446,654 inch-pounds, and the 
maximum at twice the mean; and the total weight on one bearing at 87,530 
lbs., which, multiplied by 56^ in., gives 4,945,445 in. -lbs. bending moment at 
the fillet. Applying the formula 7\ = B + VB* + T 2 , gives for equivalent 
twisting moment 9,971,045 in.-lbs. Substituting this value in the formula 
T x = .196, Sd 3 gives for S the shearing strain 15,070 lbs. per sq. in., or if the 
metal had a shearing strength of 45,000 lbs., a factor of safety of only 3. 
Mr. Coffey considers that 6000 lbs. is all that should be allowed for S under 
these circumstances. This would give d = 20.35 in. If we take from Mr. 
Coffey's table a value of .1965 = 1100, we obtain d 3 = 9000 nearly, or d — 20.8 
in., instead of 15 in., the actual diameter. 

Length of Shaft-bearings.— There is as great a difference of 
opinion among writers, and as great a variation in practice concerning length 
of journal-bearings, as there is concerning crank-pins. The length of a 



DIMENSIONS OF PARTS OF ENGINES. 811 

journal being determined from considerations of its heating, the observa- 
tions concerning heating of crank-pins apply also to shaft-bearings, and the 
formulae for >ength of crank-pins to avoid heating may also be used, using 
for the total load upon the bearing the resultant of all the pressures brought 
upon it, by the pressure on the crank, by the weight of the fly-wheel, and by 
the pull of the belt. After determining this pressure, however, we must 
resort to empirical values for the so-called constants of the formulae, really 
variables, which depend on the power of the bearing to carry away heat, 
and upon the quantity of heat generated, which latter depends on the pres- 
sure, on the number of square feet of rubbing surface passed over in a 
minute, and upon the coefficient of friction. This coefficient is an exceed- 
ingly variable quantity, ranging from .01 or less with perfectly polished 
journals, having end-play, and lubricated by a pad or oil-bath, to .10 or more 
with ordinary oil-cup lubrication. 

For shafts resisting torsion only, Marks gives for length of bearing I = 
.0000247 fpND 2 , in which /is the coefficient of friction, p the mean pressure 
in pounds per square inch on the piston, JVthe number of single strokes per 
minute, and D the diameter of the piston. For shafts under the combined 
stress due to pressure on the crank-pin, weight of fly-wheel, etc., he gives 
the following: Let Q = reaction at bearing due to weight, S = stress due 
steam pressure on piston , and R x — the resultant force ; for horizontal engines, 
Ri = YQ 2 -f- S 2 , for vertical engines R x = Q + S, when the pressure on the 
crank is in the same direction as the pressure of the shaft on its bearings, 
and R t — Q - S when the steam pressure tends to lift the shaft from its 
bearings. Using empirical values for the work of friction per square inch 
of projected area, taken from dimensions of crank-pins in marine vessels, 
he finds the formula for length of shaft-journals I — .0000325/RiiV, and 
recommends that to cover the. defects of workmanship, neglect of oiling, 
and the introduction of dust, / be taken at .16 or even greater. He says 
that 500 lbs. per sq. in. of projected area may be allowed for steel or wrought- 
iron shafts in brass bearings with good results if a less pressure is not attain- 
able without inconvenience. Marks says that the use of empirical rules that 
do not take account of the number of turns per minute has resulted in bear- 
ings much too long for slow- speed engines and too short for high-speed 
engines. 

Whitham gives the same formula, with the coefficient .00002575. 

Thurston says that the maximum allowable mean intensity of pressure 

PV 
may be, for all cases, computed by his formula for journals, I = — , or 

b(J,u(J0ct 
P(V + 20) 
by Rankine's, I — -, in which Pis the mean total pressure in pounds, 

44,o00u 
Fthe velocity of rubbing surface in feet per minute, and d the diameter of 
the shaft in inches. It must be borne in mind, he says, that the friction work 
on I he main bearing next the crank is the sum of that due the action of the 
piston on the pin, and that due that portion of the weight of wheel and 
shaft and of pull of the belt which is carried there. The outboard bearing 
carries practically only the latter two parts of the total. The crank-shaft 
journals will be made longer on one side, and perhaps shorter on the other, 
than that of the crank-pin, in proportion to the work falling upon each, i.e., 
to their respective products of mean total pressure, speed of rubbing sur- 
faces, and coefficients of friction. 

Unwin says: Journals running at 150 revolutions per minute are often 
only one diameter long. Fan shafts running 150 revolutions per minute have 
journals six or eight diameters long. The ordinary empirical mode of pro- 
portioning the length of journals is to make the length proportional to the 
diameter, and to make the ratio of length to diameter increase with the 
speed. For wrought-iron journals: 

Revs, per min. = 50 100 150 200 250 500 1000 -i = .004R + 1. 

Length -f- diam. = 1.2 1.4 1.6 1.8 2.0 3.0 5.0. 

Cast-iron journals may have I -*- d — 9/10, and steel journals I -h d = 3J4, 
of the above values. 

0.4 H. P. 
Unwin gives the following, calculated from the formula I = ; , in 

which r is the crank radius in inches, and H.P. the horse-power transmitted 
to the crank- pin. 



812 



THE STEAM-ENGINE. 

Theoretical Journal Length in Inches. 



Load on 




Revolutions of Journal per minute. 












in 
pounds. 


50 


100 


200 


300 


500 


1000 


1,000 


.2 


.4 


.8 


1.2 


2. 


4. 


2,000 


.4 




8 


1.6 


2.4 


4. 


8. 


4,000 


.8 


1 


6 


3.2 


4.8 


8. 


16. 


5,000 


1.0 


2 




4. 


6. 


10. 


20. 


10,000 


2. 


4 




8. 


12. 


20. 


40. 


15,000 


3. 


6 




12. 


18. 


30. 




20,000 


4. 


8 




16. 


24. 


40. 




30,000 


6. 


12 




24. 


36. 






40,000 


8. 


16 




32. 








50,000 


10. 


20 




40. 









Applying these different formluae to our six engines, we have: 



Engine No.. 



1 


2 


3 


4 


5 


10 

50 
250 
3,299 
268 


10 

50 
125 
3,299 
536 


30 
450 
130 

23,185 
5,968 


30 
450 
65 

23,185 
11,936 


50 
1,250 

90 
58,905 
26,470 


3,310 

3.84 


3,335 
4.39 


23,924 
11.35 


26,194 
12.99 


64,580 
20.58 


5.38 


2.71 


20.87 


11.07 


37.78 


4.27 


2.15 


16.53 


8.77 


29.95 


3.61 


1.82 


14.00 


7.43 


25.36 


5.22 


2.78 


21.70 


10.85 


35.16 


7.68 


6.59 


17.25 


16.36 


27.99 


3.33 


1.60 


12.00 


6.00 


20.83 


4.92 


2.99 


17.05 


10.00 


29.54 



Diam. cyl 

Horse -power 

Revs, per min 

Mean pressure on crank-pin = S... 
Half wc. of fly-wheel and shaft = ^ 
Resultant press, on bearing 



Diam. of shaft journal 

Length of shaft journal: 

Marks, I = .0000325/S 1 2V(/=.10) 

Whitham, I = .0000515/i?,B(/= 10). 

PV 
Thurston, Z 



Rankine, I 
Unwin, 



" 60,000d" ' " ' 

_P(F+20) 



44,800d "■* 

I = (.004B + l)d 

TT . , 0.4 H.P. 
Unwin, 1= 



Average 4. 92 



50 

1,250 
45 

58,905 
52,940 



23.17 

18.35 

15.55 

22.47 
25.39 
10.42 

19.22 



If we divide the mean resultant pressure on the bearing by the projected 
area, that is, by the product of the diameter and length of the journal, using 
the greatest and smallest length out of the seven lengths for each journal 
given above, we obtain the pressure per square inch upon the bearing, as 
follows: 



Engine No . 



1 



3 


4 


5 


6 


176 


336 


151 


353 


97 


123 


83 


145 


124 


202 


106 


191 




155 




175 



Pressure per sq. in., shortest journal 

Longest journal .... 

Average journal 

Journal of length = diam 



Many of the formulae give for the long-stroke engines a length of journal 
less than the diameter, but such short journals are rarely used in practice. 
The last line in the above table has been calculated on the supposition that 



DIMENSIONS OF PARTS OF ENGINES. 813 

the journals of the long-stroke engines are made of a length equal to the 
diameter. 

In the dimensions of Corliss engines given hy J. B. Stan wood (Eng., June 
12, 1891), the lengths of the journals for engines of diam. of cyl. 10 to 20 in. 
are the same as the diam. of the cylinder, and a little more than twice the 
diam. of the journal. For engines above 20 in. diam. of cyl. the ratio of 
length to diam. is decreased so that an engine of 30 in. diam. has a journal 
26 in. long, its diameter being 14£f in. These lengths of journal are greater 
than those given by anj^ of the formulae above quoted. 

There thus appears to be a hopeless confusion in the various formulae for 
length of shaft journals, but this is no more than is to be expected from the 
variation in the coefficient of friction, and in the heat-conducting power of 
journals in actual use, the coefficient varying from .10 (or even .16 as given 
by Marks) down to .01, according to the condition of the bearing surfaces 

PV 
and the efficiency of lubrication. Thurston's formula, I =rrrrr,-, reduces to 

oU,UUUa 
the form I - .000004363PP, in which P = mean total load on journal, and 
R = revolutions per minute. This is of the same form as Marks' and 
Whitham's formulae, in which, if /the coefficient of friction be taken at .10, 
the coefficients of PR are, respectively, .0000065 and .00000515. Taking the 
mean of these three formulae, we have I - .0000053PP, if / = .10 or I = 
.000053/PP for any other value of/. The author believes this to be as safe 
a formula as any for length of journals, with the limitation that if it brings 
a result of length of journal less than the diameter, then the length should 
be made equal to the diameter. Whenever with / = .10 it gives a length 
which is inconvenient or impossible of construction on account of limited 
space, then provision should be made to reduce the value of the coefficient 
of friction below .10 by means of forced lubrication, end play, etc.. and to 
carry away the heat, as by water-cooled journal-boxes. The value of P 
should be taken as the resultant of the mean pressure on the crank, and the 
load brought on the bearing by the weight of the s haft, fl y-wheel, etc., as 
calculated by the formula already given, viz., R x = YQ' 2 + S 2 for horizontal 
engines, and P x = Q 4- S for vertical engines. 

For our six engines the formula I — .0000053PP gives, with the limitation 
for the long-stroke engines that the length shall not be less than the diam- 
eter, the following: 

Engine No 1 

Length of journal 4.39 

Pressure per square inoh on journal. . 196 

Crank - shafts with Centre-crank and Double-crank 
Arms,— In centre-crank engines, one of the crank-arms, and its adjoining 
journal, called the after journal, usually transmit the power of the engine 
to the work to be done, and the journal resists both twisting and bending 
moments, while the other journal is subjected to bending moment only. 
For the after crank-journal the diameter should be calculated the same as 
for an overhung crank, using: the formula for combined bending and twist- 
ing moment, Tj = B + VB 2 -f T 2 , in which T x is the equivalent twisting 
moment, B the bending moment, and T the twisting moment. This value 

3/5 17 
of T, is to be used in the formula diameter —a/ — The bending mo- 

y 8 

ment is taken as the maximum load on piston multiplied by one fourth of 
the length of the crank-shaft between middle points of the two journal 
bearings, if the centre crank is midway between the bearings, or by one 
half the distance measured parallel to the shaft from the middle of the 
crank-pin to the middle of the after bearing. This supposes the crank- 
shaft to be a beam loaded at its middle and supported at the ends, but 
Whitham would make the bending moment only one half of this, consider- 
ing the shaft to be a beam secured or fixed at the ends, with a point of con- 
traflexure one fourth of the length from the end. The first supposition is 
the safer, but since the bending moment will in any case be much less than 
the twisting moment, the resulting diameter will be but little greater than 
if Whitham's supposition is used. For the forward journal, which is sub- 

3/jo 2B 
jected to bending moment only, diameter of shaft = a/ — ! , in which B 



3 


4 


5 6 


16.48 


12.99 


30.80 21.52 


128 


155 


102 171 



814 



THE STEAM-ENGINE. 



is the maximum bending moment and S the safe shearing strength of the 
metal per square inch. 

For our six engines, assuming them to be centre-crank engines, and con- 
sidering the crank-shaft to be a beam supported at the ends and loaded in 
the middle, and assuming lengths between centres of shaft bearings as 
given below, we have: 



Engine No 

Length of shaft, assumed, 
inches, L 

Max. press, on crank-pin, P 

Max. bending moment. 
B = 14PL, inch-lbs 

Twisting moment, T 

Equiv. twisting moment, 

B + VB* + T* 

Diameter of after journal. 

d = l/^ 

V 8000 

Diam. of forward journal, 

d^// 1 ^ 

V 8000 



20 

7,854 

39,270 
47,124 

101,000 
3. 



49,637 

94, "" 



156,000 
4.60 



848,232 
1,060,290 



2,208,000 
11.15 



60 

70,61 



1.060,290 
2,120,580 



3,430,000 
13.00 



76 
196,350 



729,750 
712,400 



,740,000 
18.25 



196,350 



4,712.400 
9,424,800 



15,240,000 
21.20 



The lengths of the journals would be calculated in the same manner as in 
the case of overhung cranks, by the formula I = .000053/Pi?, in which P is 
the resultant of the mean pressure due to pressure of steam on the piston, 
and the load of the fly-wheel, shaft, etc., on each of the two bearings. 
Unless the pressures are equally divided between the two bearings, the 
calculated lengths of the two will be different; but it is usually customary 
to make them both of the same length, and in no case to make the length 
less than the diameter. The diameters also are usually made alike for the 
two journals, using the largest diameter found by calculation. 

The crank-pin for a centre crank should be of the same length as for an 
overhung crank, since the length is determined from considerations of 
heating, and not of strength. The diameter also will usually be the same, 
since it is made great enough to make the pressure per square inch on the 
projected area (product of length by diameter) small enough to allow of 
free lubrication, and the diameter so calculated will be greater than is re- 
quired for strength. 

C rant- shaft with Two Cranks coupled at 90°. — If the 
whole power of the engine is transmitted through the after journal of the 
after crank-shaft, the greatest twisting moment is equal to 1.414 times the 
maximum twisting moment due to the pressure on one of the crank-pins. 
If T — the maximum twisting moment produced by the steam-pressure on 
one of the pistons, then Tj the maximum twisting moment on the after part 
of the crank-shaft, and on the line-shaft, produced when each crank makes 
an angle of 45° with the centre line of the engine, is 1.41421 Substituting 
this value in the formula for diameter to resist simple torsion, viz., d = 



, we have d 



= f 



5.1 Xl.4142 7 



or d = 1 . 932 



n- 



, in which T is 



the maximum twisting moment produced by one of the pistons, d = diam- 
eter in inches, and S = safe working shearing strength of the material. 
For the forward journal of the after crank, and the after journal of the 
forward crank, the torsional moment is that due to the pressure of steam 
on the forward piston only, and for the forward journal of the forward 
crank, if none of the power of the engine is transmitted through it, the 
torsional moment is zero, and its diameter is to be calculated for bending 
moment only. 

For Combined Torsion and Flexure.— Let B x = bending mo- 
ment on either j nirual of the forward crank due to maximum pressure on 



DIMENSIONS OF I'ARTS OF ENGINES. 815 

forward piston, i? 2 = bending moment on either journal of the after crank 
due to maximum pressure on after piston, 1\ = maximum twisting momeno 
on after journal of forward crank, and T 2 — maximum twisting moment on 
after journal of after crank due to pressure on the after piston. 

Then equivalent twisting moment on after journal of forward crank = B } 
+ VBJ -f 2\ 2 . _____ 

On forward journal of after crank = _ a + Vb 3 * + TJ. 

On after journal of after crank = £ 2 + VBf + {T x + T,) 2 . 

These values of equivalent twisting moment are to be used in the formula 

for diameter of journals d = A/ - — . For the forward journal of the 

3 /l0 2B 
forward crank-shaft d = 4/ — : — - 1 . 

V s 

It is customary to make the two journals of the forward crank of one 
diameter, viz., that calculated for the after journal. 

For a Three-cylinder Engine with cranks at 120°, the greatest 
twisting moment on the after part of the shaft, if the maximum pressures 
on the three pistons are equal, is equal to twice the maximum pressure on 
any one piston, and it takes place when two of the cranks make angles of 
30° with the centre line, the third crank being at right angles to it. (For de- 
monstration, see WhithanVs " Steam-engine Design, 11 p. 252.) For combined 
torsion and flexure the same method as above given for two crank engines 
is adopted for the first two cranks; and for the third, or after crank, if all 
the power of the three cylinders is transmitted through it, we have the 
equivalent twisting momenton the forward journal = B 3 -\- | / 5 3 2 -|-(T' 1 +T 2 ) 2 , 
and on the after journal = B 3 + \ B^ + (T x -f 2 T 2 + r 3 ) 2 , B 3 and T s being 
respectively the bending and twisting moments due to the pressure on the 
third piston. 

Crank - shafts for Triple-expansion marine Engines, 
according to an article in The Engineer, April 25, 1890, should be made 
larger than the formulae would call for, in order to provide for the stresses 
due to the racing of the propeller in a sea-way, which can scarcely be cal- 
culated. A kind of unwritten law has sprung up for fixing the size of a 
crank-shaft, according to which the diameter of the shaft is made about 
0.45Z) , where D is the diameter of the high-pressure cylinder. This is for 
solid shafts. When the speeds are high, as in war-ships, and the stroke 
short, the formula becomes 4D, even for hollow shafts. 

The Valve-stem or Valve-rod.— The valve-rod should be designed 
to move the valve under the most unfavorable conditions, which are when 
the stem acts by thrusting, as a long column, when the valve is unbalanced 
(a balanced valve may become unbalanced by the joint leaking) and when it 
is imperfectly lubricated. The load on the valve is the product of the ar Q a 
into the greatest unbalanced pressure upon it per square inch, and the co- 
efficient of friction may be as high as 20%. The product of this coefficient 
and the load is the force necessary to move the valve, w T hich equals the 
maximum thrust on the valve-rod. From this force the diameter of the 
valve-rod may be calculated by Hodgkinson's formula for columns. An 



•V: 



Ibp 



empirical formula given by Sea ton is: Diam. of rod = d 

I = length and 6 = breadth of valve, in inches; p = maximum absolute 
pressure on the valve in lbs. per sq in., and Fa coefficient whose values are, 
for iron: long rod 10,000, short 12,000; for steel: long rod 12,000, short 14,500. 

Whitham gives the short empirical rule: Diam. of valve-rod — 1/30 diam. 
of cyl. = % diam of piston-rod. 

Size of Slot-link. (Seaton.)— Let D be the diam. of the valve-rod 



»Vi5< 



then Diameter of block-pin when overhung = D. 

" " " secured at both ends = 0.75 x D. 

" eccentric-rod pins = 0.7 x Z). 

suspension-rod pins = 0.55 x D. 

" " '• pin when overhung = 0.75 x D. 



816 THE STEAM-ENGINE. 

Breadth of link = 0.8 to 9 x D. 

length of block = 1.8 to 1.6 x D. 

Thickness of bars of link at middle = 0.7 x D. 

If a single suspension rod of round section, its diameter = 0.7 X D. 
If two suspension rods of round section, their diameter = 0.55 x D. 
Size of I>ouble-bar liinlts.— When the distance between centres of 
eccentric pins = 6 to 8 times throw of eccentrics (throw = eccentricity = 
half- travel of valve at full gear) D as before : 

Depth of bars = 1.25 X D -f- % in. 

Thickness of bars =0.5 x D + Vi ^ 

Length of sliding-block = 2.5 to 3 X D. 

Diameter of eccentric-rod pins = 0.8 x D + \i in. 
" centre of sliding-block = 1.3 x D. 

When the distance between eccentric -rod pins = 5 to 5J/£ times throw of 
eccentrics: 

Depth of bars = 1.25 x D + Ja in. 

Thickness of bars =0.5 X D -j- J4 in. 

Length of sliding-block = 2.5 to 3 x D. 

Diameter of eccentric-rod pins = 0.75 X D. 
Diameter of eccentric bolts (top end) at bottom of thread = 0.42 X D when 
of iron, and 0.38 x D when of steel. 

The Eccentric.— Diam. of eccentric-sheave = 2.4 X throw of eccentric 
+ 1.2 X diam. of shaft. D as before 

Breadth of the sheave at the shaft = 1.15 X D + 0.65 inch 

Breadth of the sheave at the strap = D 4- 0.6 inch. 

Thickness of metal around the shaft = 0.7 x D + 0.5 inch. 

Thickness of metal at circumference = 0.6 x D 4- 0.4 inch. 

Breadth of key # = 0.7 X D+ 0.5 inch. 

Thickness of key = 0.25 x D -f 0.5 inch. 

Diameter of bolts connecting parts of strap = 0.6x^+0.1 inch. 

Thickness of Eccentric-strap. 

When of bronze or malleable cast iron: 

Thickness of eccentric-strap at the middle = 0.4 x D + 0.6 inch. 

" '•' " " " sides = 0.3 X D + 0.5 inch. 

When of wrought iron or cast steel: 

Thickness of eccentric-strap at the middle = 0.4 X D + 0.5 inch. 

" " " sides = 0.27 X D + 0.4 inch. 

The Eccentric-rod.— The diameter of the eccentric-rod in the body 
and at the eccentric end may be calculated in the same way as that of the 
connecting-rod, the length being taken from centre of strap to centre of 
pin. Diameter at the link end = 0.8D -f 0.2 inch. 

This is for wrought-iron; no reduction in size should be made for steel. 

Eccentric-rods are often made of rectangular section. 

Reversing-gear should be so designed as to have more than sufficient 
strength to withstand the strain .of both the valves and their gear at the 
same time under the most unfavorable circumstances; it will then have the 
stiffness requisite for good working. 

Assuming the work done in reversing the link-motion, W, to be only that 
due to overcoming the friction of the valves themselves through their whole 
travel, then, if T be the travel of valves in inches; for a compound engine 

T/ l x bxp \ T ( V X& 1 Xp' \ 
12V 5 / ■+" 12\ 5 /' 

l l , b 1 andp 1 being length, breadth and maximum steam-pressure on valve 
of the second cylinder; and for an expansive engine 



|(Lx|x,) ; or | (IX6XP) . 



12 

To provide for the friction of link-motion, eccentrics and other gear, and 
for abnormal conditions of the same, take the work at one and a half times 
the above amount. 



FLY-WHEELS. 817 

To find the strain at any part of the gear having motion when reversing, 
divide the work so found by the space moved through by that part in feet; 
the quotient is the strain in pounds; and the size may be found from the 
ordinary rules of construction for any of the parts of the gear. (Seaton.) 

Engine-frames or Red-plates.— No definite rules for the design 
of engine-frames have been given by authors of works on the steam-engine. 
The proportions are left to the designer who uses " rule of thumb," or 
copies from existing engines. F. A. Halsey {Am. Mach., Feb. 14, 1895) has 
made a comparison of proportions of the frames of horizontal Corliss 
engines of several builders. The method of comparison is to compute from 
the measurements the number of square inches in the smallest cross-sec- 
tion of the frame, that is, immediately behind the pillow-block, also to 
compute the total maximum pressure upon the piston, and to divide the 
latter quantity by the former. The result gives the number of pounds 
pressure upon the piston allowed for each square inch of metal in the 
frame. He finds that the number of pounds per square inch of smallest 
section of frame ranges from 217 for a 10 X 30-in. engine up to 575 for a 
28 X 48-inch. A 30 X 60-inch engine shows 350 lbs., and a 32-iuch engine 
which has been running for many years shows 667 lbs. Generally the 
strains increase with the size of the engine^ and more cross-section of metal 
is allowed with relatively long strokes than with short ones. 

From the above Mr. Halsey formulates the general rule that in engines 
of moderate speed, and having strokes up to one and one-half times the 
diameter of the cylinder, the load per square inch of smallest section 
should be for a 10-inch engine 300 pounds, which figure should be increased 
for larger bores up to 500 pounds for a 30- inch cylinder of same relative 
stroke. For high speeds or for longer strokes the load per square inch 
should be reduced. 

FL.Y-WHEEL.S. 

The function of a fly-wheel is to store up and to restore the periodical fluc- 
tuations of energy given to or taken from an engine or machine, and thus 
to keep approximately constant the velocity of rotation. Rankine calls the 

AE 
quantity — =- the coefficient of fluctuation of speed or of unsteadiness, in 

which E is the mean actual energy, and AS the excess of energy received or 
of work performed, above the mean, during a given interval. The ratio of 
the periodical excess or deficiency of energy AS to the whole energy exerted 
in one period or revolution General Morin found to be from 1/6 to J4 for 
single-cylinder engines using expansion; the shorter the cut-off the higher 
the value. For a pair of engines with cranks coupled at 90° the value of the 
ratio is about J4, and for three engines with cranks at 120°, 1/12 of its value 
for single cylinder engines. For tools working at intervals, such as punch- 
ing, slotting and plate-cutting machines, coining-presses, etc., AS is nearly 
equal to the whole work performed at each operation. 
AE 
A fly-wheel reduces the coefficient jr— to a certain fixed amount, being 

4Eq 

about 1/32 for ordinary machinery, and 1/50 or 1/60 for machinery for fine 
purposes. 

If m be the reciprocal of the intended value of the coefficient of fluctua- 
tion of speed, AE the fluctuation or energy, /the moment of inertia of the 

fly-wheel alone, and a its mean angular velocity, I = — — *. As the rim of 

a fly-wheel is usually heavy in comparison with the arms, /may be taken 
to equal Wr*, in which W — weight of rim in pounds, and r the radius of the 

wheel; then W = — —-r- = — ''— — , if v be the velocity of the rim in feet per 

second. The usual mean radius of the fly-wheel in steam-engines is from 
three to five times the length of the crank. The ordinary values of the prod- 
uct m g, the unit of time being the second, lie between 1000 and 2000 feet. 
(Abridged from Rankine, S E., p. 62.) 
Thurston gives for engines with automatic valve-gear W = 250,000 

'ig , in which A = area of piston in square inches, S = stroke in feet, p = 

mean steam pressure in lbs. per sq. in., R = revolutions per minute, D — out- 
side diameter of wheel in feet. Thurston also gives for ordinary forms of 



818 THE STEAM-EtfGitfE. 

non- condensing engine with a ratio of expansion between 3. and 5, W =. 

j^r%, in which a ranges from 10,000,000 to 15,000,000, averaging 12,000,000. 

For gas-engines, in which the charge is fired with every revolution, the Amer- 
ican Machinist gives this latter formula, with a doubled, or 24,000,000. 
Presumably, if the charge is fired every other revolution, a should be again 
doubled. 

Rankine C" Useful Rules and Tables," p. 247) gives W = 475,000 ^SL^ , in 

which Fis the variation of speed per cent, of the mean speed. Thurston's 
first rule above given corresponds with this if we take Fat 1.9 per cent. 

Hartnell (Proc. Inst., M. E. 1882. 427) says : The value of V, or the 
variation permissible in portable engines, should not exceed 3 per cent, with 
an ordinary load, and 4 per cent when heavily loaded. In fixed engines, for 
ordinary purposes, V = 2^ to 3 per cent. For good governing or special 
purposes, such as cotton -spinning, the variation should not exceed \% to 2 
per cent. 

F. M. Rites (Trans. A. S. M. E., xiv. 100) develops a new formula for weight 
C X I H P C 

of rim, viz., W — pi"?™" - " - "' an ^ wei g Qt of rim per horse-power = pj™, in 

which C varies from 10,000,000,000 to 20,000,000,000; also using the latter value 

Mv 2 W 3 14 2 Z) 2 i? 2 
of C, he obtains for the energy of the fly-wheel — — = ^j— nnnn — = 

£ t>4 .4 ooOO 
CxH.P.(3 .14) 2 P 2 i Z 2 _ 850,000 H.P. 
B 3 D 2 X 64.4 X 3600 ~ R 

The limit of variation of speed with such a weight of wheel from excess of 
power per fraction of revolution is less than .0023. 

The value of the constant C given by Mr. Rites was derived from practice 
of the Westinghouse single-acting engines used for electric-lighting. For 
double-acting engines in ordinary service a value of C = 5,000,000,000 would 
probably be ample. 

From these formulae it appears that the weight of the fly-wheel for a given 
horse-power should vary inversely with the cube of the revolutions and the 
square of the diameter. 

J. B. Stanwood {Eng'g, June 12, 1891) says: Whenever 480 feet is the 
lowest piston-speed probable for an engine of a certain size, the fly-wheel 
weight for that speed approximates closely to the formula 

r/ 2 <} 

W = weight in pounds, d = diameter of cylinder in inches, s = stroke in 
inches, D = diameter of wheel in feet, R = revolutions per minute, corre- 
sponding to 480 feet piston -speed. 

In a Ready Reference Book published by Mr. Stanwood, Cincinnati, 1892, 
he gives the same formula, with coefficients as follows: For slide-valve en- 
gines, ordinary duty, 350,000; same, electric-lighting, 700,000; for automatic 
high-speed engines, 1,000,000; for Corliss engines, ordinary duty 700,000, 
electric-fighting 1,000,000. 

Thurston's formula above given, W = ^pjr 2 , with a = 12,000,000, when re- 

d 2 s 
duced to terms of d and s in inches, becomes W = 785,400 ^,, ' .. . 



33,000 
in which P = mean effective pressure. Taking this at 40 lbs., we obtain 

W = 5,000,000,000^^. If mean effective pressure = 30 lbs., then W = 
6,666,000,000^^. 

Emil Theiss {Am. Much., Sept. 7 and 14, 1893) gives the following values 
or d, the coefficient of steadiness, which is the reciprocal of what Rankine 
calls the coefficient of fluctuation : 



FLY-WHEELS. 



819 



For engines operating — 

Hammering and crushing machinery d = 5 

Pumping and shearing machinery d = 20 to 30 

Weaving and paper-making machinery d = 40 

Milling machinery d = 50 

Spinning machinery d = 50 to 100 

Ordinary driving-engines (mounted on bed-plate), 

belt transmission d = 35 

Gear-wheel transmission d = 50 

Mr. Theiss's formula for weight of fly-wheel in pounds is W= i X 2 ' — —'■> 

vhere d is the coefficient of steadiness, V the mean velocity of the fly- 
wheel rim in feet per second, n the number of revolutions per minute, i = 
a coefficient obtained by graphical solution, the values of which for dif- 
ferent conditions are given in the following table. In the lines under "cut- 
off," p means " compression to initial pressure," and " no compression ": 

Values of i. Single-cylinder Non-condensing Engines. 



5 73 


3 

3 


Cut-off, 1/6. 


Cut-off, y A . 


Cut-off, %. 


Cut-off, X. 


* So. 


Comp. 
P 





Comp. 
P 





Comp. 
P 


O 


Comp. 

P 


200 
400 
600 


272.690 
240,810 
194,670 
158,200 


218.580 
187,430 
145,400 
108,690 


242.010 

208,200 
168.590 
162,070 


209,170 
179,460 
136,460 
135,260 


220,760 
188,510 
165,210 


201,920 
170.040 
146,610 


193,340 
174,630 


182,840 
167,860 


800 






Single-cylinder Condensing Engines. 


a £.5 


Cut-off, ^. 


Cut-off, 1/6. 


Cut-off, y A . 


Cut-off, %. 


Cut-off, 14 




Comp- 
P 


O 


Comp 
P 


O 


Oomp. 
P 


O 

167,140 
133,080 


Comp. 
P 


O 


Comp 
P 


O 


200 
400 


265.560 
194,550 

148.780 


176,560 
117,870 


234, 16C 

174,38( 


173.660 
118,350 


204,210 
164,720 


189,600 
174,630 


161.830 
151,680 


172,69 


D 156,990 


600 


140,090 






TWO- CYLINDER ENGINES, CRANKS AT 90°. 




Cut-off, 1/6. 


Cut-off, y A . 


Cut-off, %. 


Cut-off, fc. 


03 0> . 


Comp. 
P 





Comp. 
V 


O 


Comp. 
P 





Comp. 
P 


O 


200 
400 
600 

800 


71,080 
70,160 
70,040 
70,040 


! Mean 
(60,140 
J 


59,420 
57,000 
57,480 
60,140 


\ Mean 
( 54,340 
J 


49,272 
49.150 
49,220 


\ Mean 
f 50,000 


37,920 
35,500 


[ Mean 
j 36,950 


Three-cylinder Engines, Cranks at 120°. 




Cut-off, 1/6. 


Cut-off, J4. 


Cut-off, y B . 


Cut-off, jjg; 


s|| 


Comp. 
P 


O 


Comp. 
P 


O 


Comp. 
V 


O 


Comp. 
P 


O 


200 
800 


33.810 
30,190 


32,240 
31,570 


33,810 
35,140 


35,500 
33,810 


34,540 
36,470 


33.450 
32,850 


35,260 
33,810 


32,370 
32,370 



As a mean value of'/ for these engines we may use 33,810. 



820 THE STEAM-ENGINE. 

Centrifugal Force in Fly-wheels. — Let W = weight of rim in 
pounds; R — mean radius of rim in feet; r — revolutions per minute, g — 
32.16: v — velocity of rim in feet per second = 2wRr -=- 60. 

Centrifugal force of whole rim = F= -^r- = ," - = .000341 WiJr*. 

b gR 3600g 

The resultant, acting at right angles to a diameter of half of this force, 
tends to disrupt one half of the wheel from the other half, and is resisted by 
the section of the rim at each end of the diameter. The resultant of half the 

2 
radial forces taken at right angles to the diameter is 1 h- y$n = - of the sum 

of these forces; hence the total force F is to be divided by 2 x 2 X 1.5108 
= 6.2832 to obtain the tensile strain on the cross-section of the rim,. or, total 
strain on the cross-section = S = .00005427 WRr*. The weight W, of a 
rim of cast iron 1 inch square in section is 2nR X 3.125 = 19.635.R pounds, 
whence strain per square inch of sectional area of rim = Si = .0010656.K 2 j 2 
= .0002664D 2 r 2 _ .0000270 V, in which D = diameter of wheel in feet, and V 
is velocity of rim in feet per minute. Si = .0972v 2 , if v is taken in feet per 
second. 

For wrought iron S x = .0011366i?2,-2 _ .0002842Z> 2 v 2 = .0000288F 2 . 

For steel S x = .001l593i? 2 r 2 = .0002901 Z) 2 ? 2 = .0000294 F 2 . 

For wood S x = .0000888E 2 r 2 = .0000222.D 2 ?- 2 = .00000225F 2 . 

The specific gravity of the wood being taken at 0.6 = 37.5 lbs. per cu. ft., 
or 1/12 the weight of cast iron. 

Example.— Required the strain per square inch in the rim of a cast-iron 
wheel 30 ft. diameter, 60 revolutions per minute. 

Answer. 15 2 X 60 2 X .0010656 = 863.1 lbs. 

Required the strain per square inch in a cast-iron wheel-rim running a 
mile a minute. Answer. .000027 X 5280 2 = 752.7 lbs. 

In cast-iron fly-wheel rims, on account of their thickness, there is difficulty 
in securing soundness, and a tensile strength of 10,000 lbs. per sq. in. is as 
much as can be assumed with safety. Using a factor of safety of 10 gives a 
maximum allowable strain in the rim of 1000 lbs. per sq. in., which corre- 
sponds to a rim velocity of 6085 ft. per minute. 

For any given material, as cast iron, the strength to resist centrifugal force 
depends only on the velocity of the rim, and not upon its bulk or weight. 

Chas. E. Emery (Cass. Mag., 1892) says: By calculation half the strength 
of the arms is available to strengthen the rim, or a trifle more if the fly- 
wheel centres are relatively large. The arms, however, are subject to trans- 
verse strains, from belts and from changes of speed, and there is, moreover, 
no certainty that the arms and rim will be adjusted so as to pull exactly 
together in resisting disruption, so the plan of considering the rim by itself 
and making it strong enough to resist disruption by centrifugal force within 
safe limits, as is assumed in the calculations above, is the safer way. 

It does not appear that fly-wheels of customary construction should be 
unsafe at the comparatively low speeds now in common use if proper 
materials are used in construction. The cause of rupture of fly-wheels that 
have failed is usually either the " running away " of the engine, such as may 
be caused by the breaking or slackness of a governor-belt, or incorrect 
design or defective materials of the fly-wheel. 

Chas. T. Porter (Trans. A. S. M. E., xiv. 808) states that no case of the 
bursting of a fly-wheel with a solid rim in a high-speed engine is known. He 
attributes the bursting of wheels built in segments to insufficient strength 
of the flanges and bolts by which the segments are held together. (See also 
Thurston, " Manual of the Steam-engine.' 1 Part II, page 413, etc.) 

Arms of Fly-wheels and Pulleys. — Professor Torrey (Am. 
Mack , July 30, 1891) gives the following formula for arms of elliptical cross- 
section of cast-iron wheels : 

W = load in pounds acting on one arm; .9 = strain on belt in pounds per 
inch of width, taken at 56 for single and 112 for double belts; v = width of 
belt in inches; n — number of arms; L = length of arm in feet; b = breadth 

of arm at hub; d = depth of arm at hub, both in inches : W = 

n ' 

WL 
b = T^Tr. ■ The breadth of the arm is its least dimension = minor axis of 

dOa 2 
the ellipse, and the depth the major axis. This formula is based on a factor 
of safety of 10- 



FLY-WHEELS. 821 

In using the formula, first assume some depth for the arm, and calculate 
fthe required breadth to go with it. If it gives too round an arm, assume 
the breadth a little greater, and repeat the calculation. A second trial will 
ialmost always give a good section. 

The size of the arms at the hub having been calculated, they may be 
somewhat reduced at the rim end. The actual amouut cannot be calculated, 
as there are too many unknown quantities. However, the depth and 
breadth can be reduced about one third at the rim without danger, and this 

ill give a well-shaped arm. 

Pulleys are often cast in halves, and bolted together. When this is done 
the greatest care should be taken to provide sufficient metal in the bolts. 
This is apt to be the very weakest point in such pulleys. The combined area 
of the bolts at each joint should be about 28/100 the cross-section of the pul 
ley at that point. (Torrey.) 



Unwin gives d = O.GSST'i/ ~^r~ for single belts ; 



3/BD 



/BD 
IT 



for double belts; 



D being the diameter of the pulley, and B the breadth of the rim, both in 
inches. These formulae are based on an elliptical section of arm in which 
b = OAd or d = 2.5b on a width of belt = 4/5 the width of the pulley rim, 
a maximum driving force transmitted by the belt of 56 lbs. per inch of width 
for a single belt and 112 lbs. for a double belt, and a safe working stress of 
cast iron of 2250 lbs. per square inch. 
If in Torrey 's formula we make b = OAd, it reduces to 



<-tf 



WL s/WL 

187. 5 ; d= V 12 • 



Example.— Given a pulley 10 feet diameter; 8 arms, each 4 feet long; face, 
6 inches wide; belt, 30 inches: required the breath and depth of the arm at 
the hub. According to Unwin, 

3 /BD s /36 X 120 
d = 0.6337 4/ — = 0.633 j/ g = 5 - 16 for sin S le belt » b = 2 - 06 ! 



3/36 
«j/ ~ 



X 120 

-g — ■ = 6.50 for double belt, 6 = 2.60. 

WL 

According to Torrey, if we take the formula b ■= -— and assume d = 5 

and 6.5 inches, respectively, for single and double belts, we obtain b = 1.08 
and 1.33, respectively, or practically oidy one half of the breadth according 
to Unwin. and, since transverse strength is proportional to breadth, an arm 
only one half as strong. 

Torrey 's formula is said to be based on a factor of safety of 10, but this 
factor can be only apparent and not real, since the assumption that the 
strain on each area is equal to the strain on the belt divided by the number 
of arms, is, to say the least, inaccurate. It would be more nearly correct to 
say that the strain of the belt is divided among half the number of arms. 
Unwin makes the same assumption in developing his formula, but says it is 
only in a rough sense true, and that a large factor of safety must be allowed. 
He therefore takes the low figure of 2250 lbs. per square inch for the safe 
working strength of cast iron. Unwin says that his equations agree well 
with practice. 

Diameters off Fly-wheels for Various Speeds.— If 6000 feet 
per minute be the maximum velocity of rim allowable, then 6000 = nRD, in 
which B = revolutions per minute, and D — diameter of wheel in feet, 

^ 6000 1910 
whence D = — — = -— -. 
nR K 



822 



THE STEAM-ENGINE. 



Maximum Diameter of Fly-wheel Allowable for Different Numbers 
of Revolutions. 





Assuming Maxi 


mum Speed of 


Assuming Maximum Speed 


Revolutions 


5000 feet per minute. . 


of 6000 feet per minute. 


per minute. 












Circum. ft. 


Diam. ft. 


Circum. ft. 


Diam. ft. 


40 


125 


39.8 


150. 


47.7 


50 


100 


31.8 


120. 


38.2 


60 


83.3 


26.5 


100. 


31.8 


70 


71.4 


22.7 


85.72 


27.3 


80 


62.5 


19.9 


75.00 


23.9 


90 


55.5 


17.7 


66.66 


21.2 


100 


50. 


15.9 


60.00 


19.1 


120 


41.67 


13.3 


50.00 


15.9 


140 


35.71 


11.4 


42.86 


13.6 


160 


31.25 


9.9 


37.5 


11.9 


180 


27.77 


8.8 


33.33 


10.6 


200 


25.00 


8.0 


30.00 


9.6 


220 


22.73 


7.2 


27.27 


8.7 


240 


20.83 


6.6 


25.00 


8.0 


260 


19.23 


6.1 


23.08 


7.3 


280 


17.86 


5.7 


21.43 


6.8 


300 


16.66 


5.3 


20.00 


6.4 


350 


14.29 


4.5 


17.14 


5.5 


400 


12.5 


4.0 


15.00 


4.8 


450 


11.11 


3.5 


13.33 


4.2 


500 


10.00 


3.2 


12.00 3.8 



Strains in the Rims of Fly-band Wheels Produced by 
Centrifugal Force. (James B. Stanwood, Trans. A. S. M. E., xiv. 251.) 
—Mr. Stanwood mentions one case of a fly-band wheel where the periphery 
velocity on a 17' 9" wheel is over 7500 ft. per minute. 

In band saw-mills the blade of the saw is operated successfully over 
wheels 8 and 9 ft. in diameter, at a periphery velocity of 9000 to 10,000 ft. per 
minute. These wheels are of cast iron throughout, of heavy thickness, with 
a large number of arms. 

In shingle-machines and chipping-machines where cast-iron disks from 2 to 
5 ft. in diameter are employed, with knives inserted radially, the speed is 
frequently 10,000 to 11,000 ft. per minute at the periphery. 

If the rim of a fly-wheel alone be considered, the tensile strain in pounds 

F 2 
per square inch of the rim section is T — —— — nearly, in which V = velocity 

in feet per second; but this strain is modified by the resistance of the arms, 
which prevent the uniform circumferential expansion of the rim, and induce 
a bending as well as a tensile strain. Mr. Stanwood discusses the strains in 
band-wheels due to transverse bending of a section of the rim between a 
pair of arms. 

When the arms are few in number, and of large cross-section, the ring 
will be strained transversely to a greater degree than with a greater number 
of lighter arms. To illustrate the necessary rim thicknesses for various 
rim velocities, pulley diameters, number of arms, etc., the following table 
is given, based upon the formula 



* = - 



.475d 



V p-a 10/ 



in which t = thickness of rim in inches, d = diameter of pulley in inches, 
JV= number of arms, V= velocity of rim in feet per second, and F — the 
greatest strain in pounds per square inch to which any fibre is subjected. 
The value of F is taken at 6000 lbs. per sq. in. 



ELY-WHEELS. 



823 



Thickness of Rims in Solid Wheel 


s. 


Diameter of 
Pulley in 
inches. 


Velocity of 

Rim in feet per 

second. 


Velocity of 

Rim iii feet per 

minute. 


No. of 
Arms. 


Thickness in 
inches. 


24 
24 

48 
108 
108 


50 

88 
88 
184 

184 


3,000 

5,280 
5,280 
11,040 
11,040 


6 
6 
6 

16 
36 


2/10 
15/32 

15/16 



If the limit of rim velocity for all wheels be assumed to be 88 ft. per sec- 
ond, equal to 1 mile per minute, F — 6000 lbs., the formula becomes 

t-'^L -0 7-*-. 

.67iV 2 N 2 

When wheels are made in halves or in sections, the bending strain may 
be such as to make t greater than that given above. Thus, when the joint 
comes half way between the arms, the bending action is similar to a beam 
supported simply at the ends, uniformly loaded, and t is 50$ greater. Then 
the formula becomes 

.713d 

Vp 10/ 

or for a fixed maximum rim velocity of 88 ft. per second and F = 6000 lbs., 

t = ' ' . In segmental wheels it is preferable to have the joints opposite 

the arms. Wheels in halves, if very thin rims are to be employed, should 
have double arms along the line of separation, 

Attention should be given to the proportions of large receiving and tight- 
ening pulleys. The thickness of rim for a 48-in. wheel (shown in table) with 
a rim velocity of 88 ft. per second, is 15/16 in. Many wrecks have been 
caused by the failure of receiving or tightening pulleys whose rims have been 
too thin. Fly-wheels calculated for a given coefficient of steadiness are fre- 
quently lighter than the minimum safe weight. This is true especially of 
large wheels. A rough guide to the minimum weight of wheels can be de- 
duced from our formulae. The arms, hub, lugs, etc., usually form from one 
quarter to one third the entire weight of the wheel. If b represents the fact 
of a wheel in inches, the weight of the rim (considered as a simple annular 
ring) will be w = .82dtb lbs. If the limit of speed is 88 ft. per second, then 
for solid wheels t = 7d -e- N 2 . For sectional wheels (joint between arms) 
t = 1.05d h-F. Weight of rim for solid wheels, w =. .57d 2 b -*- N 2 in pounds. 
Weight of rim in sectional wheels with joints between arms, w = .8Qd 2 b -*- 
JV 2 in pounds. Total weight of wheel: for solid wheel, W = .76d 2 6 -s- N 2 to 
.86<2 2 6 -T- N 2 , in pounds. For segmental wheels with joint between arms, 
W = 1.05d 2 b -h N 2 to 1.3d 2 b -f- N 2 , in pounds. 

(This subject is further discussed by Mr. Stanwood, in vol. xv., and by 
Prof. Gaetano Lanza, in vol. xvi.. Trans. A. S. M. E.) 

A Wooden Rim Flywheel,, built in 1891 for a pair of Corliss en- 
gines at the Amoskeag Mfg. Co.'s mill, Manchester, N. H., is described by 
C H. Manning in Trans. A. S. M. E., xiii. 618. It is 30 ft. diam. and 108 in. face. 
The rim is 12 inches thick, and is built up of 44 courses of ash plank, 2, 3, 
and 4 inches thick, reduced about % inch in dressing, set edgewise, so as to 
break joints, and glued and bolted together. There are two hubs and two 
sets of arms, 12 in each, all of cast iron. The weights are as follows: 

Weight (calculated) of ash rim 31 ,855 lbs. 

" of 24 arms (foundry 45,0'20) 40,349 " 

" 2hubs( " 35,030) 31,394^ " 

Counter- weights in 6 arms 664 " 

Total, excluding bolts and screws 104,262± " 

The wheel was tested at 76 revs, per min., being a surface speed of nearly 
7200 feet per minute. 



824 THE STEAM -ENGlM. 

]\Ir. Manning discusses the relative safety of cast iron and of wooden 
wheels as follows: As for safety, the speeds being the same in both 
cases, i he hoop tension in the rim per unit of cross-section would be directly 
as the weight per cubic unit; and its capacity to stand the strain directly as 
the tensile strength per square unit; therefore the tensile strengths divided 
by the weights will give relative values of different materials. Cast iron 
weighing 450 lbs. per cubic foot and with a tensile strength of 1,440,000 lbs. 
per square foot would give a value of 1,440,000 -=- 450= 3200, whilst ash, of 
which the rim was made, weghing 34 lbs. per cubic foot, and with 1,152,000 
lbs. tensile strength per square foot, gives a result 1,152,000 -=- 34 = 33,882, 
and 33,882-^-3200 = 10.58, or the wood-rimmed pulley is ten times safer 
than the cast-iron when the castings are good. This would allow the wood- 
rimmed pulley to increase its speed to 4/10.58 =3.25 times that of a sound 
cast-iron one with equal safety. 

"Wooden Fly-wneel of tlie Willimantic Linen Co. (Illus- 
trated in Power, March, 1893.) — Rim 28 ft. diam., 110 in. face. The rim is 
carried upon three sets of arms, one under the centre of each belt, with 12 
arms in each set. 

The material of the rim is ordinary whitewood, % in. in thickness, cut into 
segments not exceeding 4 feet in length, and either 5 or 8 inches in width. 
These were assembled by building a complete circle 13 inches in width, first 
with the 8 inch inside and the 5-inch outside, and then beside it another cir- 
cle with the widths reversed, so as to break joints. Each piece as it was 
added was brushed over with glue and nailed with three-inch wire nails to 
the pieces already in position. The nails pass through three and into the 
fourth thickness. At the end of each arm four 14-inch bolts secure the 
rim, the ends being covered by wooden plugs glued and driven into the face 
of the wheel. 

Wire-wound Fly-wlieels for Extreme Speeds. (Eng'gNews, 
August 2, 1890.)— The power required to produce the Mannesmann tubes is 
very large, varying from 2000 to 10,000 H.P., according to the dimensions of 
the tube. Since this power is only needed for a short time (it takes only 30 
to 45 seconds to convert a bar 10 to 12 ft. long and 4 in. in diameter into a 
tube), and then some time elapses before the next bar is ready, an engine of 
1200 HP. provided with a large fly-wheel for storing the energy will supply 
power enough for one set of rolls. These fly-wheels are so large and run at 
such great speeds that the ordinary method of constructing them cannot be 
followed. A wheel at the Mannesmann Works, made in Komotau, Hungary, 
in the usual manner, broke at a tangential velocity of 125 ft. per second. 
The fly-wheels designed to hold at more than double this speed consist of a 
cast-iron hub to which two steel disks, 20 ft. in diameter, are bolted; around 
the circumference of the wheel thus formed 70 tons of No. 5 wire are wound 
under a tension of 50 lbs. In the Mannesmann Works at Landore, Wales, 
such a wheel makes 240 revolutions a minute, corresponding to a tangential 
velocity of 15,080 ft. or 2.85 miles per minute. 

THE SLIDE-VALVE. 

Definitions.*— Travel = total distance moved by the valve. 

Throw of the Eccentric = eccentricity of the eccentric = distance from the 
centre of the shaft to the centre of the eccentric disk = J^ the travel of the 
valve. (Some writers use the term " throw " to mean the whole travel of 
the valve.) 

Lap of the valve, also called outside lap or steam-lap = distance the outer 
or steam edge of the valve extends beyond or laps over the steam edge of 
the port when the valve is in its central position. 

Inside lap, or exhaust-lap ■— distance the inner or exhaust edge of the 
valve extends beyond or laps over the exhaust edge of the port when the 
valve is in its central position. The inside lap is sometimes made zero, or 
even negative, in which latter case the distance between the edge of the 
valve and the edge of the port is sometimes called exhaust clearance, or 
inside clearance. 

Lead of the valve = the distance the steam-port is opened when the engine 
is on its centre and the piston is at the beginning of the stroke. 

Lead-angle = the angle between the position of the crank when the valve 
begins to be opened and its position when the piston is at the beginning of 
the stroke. 

The valve is said to have lead when the steam-port opens before the piston 



THE SLIDE-VALYE. 



825 



begins its stroke. If the piston begins its stroke before the admission of 
steam begins the valve is said to have negative lead, and its amount is the 
lap of the edge of the valve over the edge of the port at the instant when 
"ie piston stroke begins. 

Lap-angle = the angle through which the eccentric must be rotated to 
cause the steam edge to travel from its central position the distance of the 
lap. 

Angular advance of the eccentric = lap-angle -\- lead angle. 

Linear advance = lap -f- lea<l. 

Effect of I*ap, JLeacL, etc., upon the Steam Distribution.— 
Given valve-travel 2% in., hip % in., lead 1/16 in., exhaust-lap % in., re- 
quired crank position for admission, cut-off, release and compression, and 
greatest port-opening. (Halsey on Slide-valve Gears.) Draw a circle of 
liameter fh = travel of valve. From O the centre set off 0« = lap and ab 
- lead, erect perpendiculars Oe, ac, bd; then ec is the lap-angle and cd the 
lead-angle, measured as arcs. Set off fg ■ = cd, the lead-angle, then Og is 
the position of the crank for steam admission. Set off 2ec + cd from h to i; 
then Oi is the crank-angle for cut-off, and/fc-f-//i is the fraction of stroke 
completed at cut-off. Set off 01 = exhaust-lap and draw lm; em is the 
exhaust-lap angle. Set off hn = ec — cd — em, and On is the position of 
crank at release. Set off fp = ec -j- cd — em, and Op is the position of crank 
• compression, fo -*-fh is the fraction of stroke completed at release, and 
hq -s- hf is the fraction of the return stroke completed when compression 
begins; Oh, the throw of the eccentric, minus Oa the lap, equals ah the 
maximum port-opening. 

If a valve has neither lap nor lead, the line joining the centre of the eccen- 




Fig. 146. 
trie disk and the centre of the shaft being at light angles to the line of the 
crank, the engine would follow full stroke, admission of steam beginning at 
the beginning of the stroke and ending at the end of the stroke. 

Adding lap to the valve enables us to cut off steam before the end of the 
stroke; the eccentric being advanced on the shaft an amount equal to the 
lap-angle enables steam to be admitted at the beginning of the stroke, as 



826 THE STEAM-ENGINE. 

before lap was added, and advancing it a further amount equal to the lead 
angle causes steam to be admitted before the beginning of the stroke. 

Having given lap to the valve, and having advanced the eccentric on the 
shaft from its central position at right angles to the crank, through the 
angular advance = lap-angle and lead-angle, the four events, admission, 
cut-off, release or exhaust-opening, and compression or exhaust-closure, 
take place as follows: Admission, when the crank lacks the lead-angle of 
having reached the centre; cut-off, when the crank lacks two lap-angles and 
one lead-angle of having reached the centre. During the admission of 
steam the crank turns through a semicircle less twice the lap-angle. The 
greatest port opening is equal to half the travel of the valve less the lap. 
Therefore for a given port-opening the travel of the valve must be in- 
creased if the lap is increased. When exhaust-lap is added to the valve it 
delays the opening of the exhaust and hastens its closing by an angle of 
rotation equal to the exhaust- lap angle, which is the angle through which 
the eccentric rotates from its middle position while the exhaust edge of the 
valve uucovers its lap. Release then takes place when the crank lacks one 
lap-angle and one lead-angle minus one exhaust-lap angle of having reached 
the centre, and compression when the crank lacks lap-angle -4- lead-angle + 
exhaust-lap angle of having reached the centre. 

The above discussion of the relative position of the crank, piston, and 
valve for the different points of the stroke is accurate only with a connect- 
ing-rod of infinite length. 

For actual connecting-rods the angular position of the rod causes a 
distortion of the position of the valve, causing the events to take place too 
late in the forward stroke and too early in the return. The correction of 
this distortion may be accomplished to some extent by setting the valve so 
as to give equal lead on both forward and return stroke, and by altering 
the exhaust-lap on one end so as to equalize the release and compression. 
F. A. Halsey, in his Slide-valve Gears, describes a method of equalizing the 
cut-off without at the same time affecting the equality of the lead. In 
designing slide-valves the effect of angularity of the connecting-rod should 
be studied on the drawing-board, and preferably by the use of a model. 

Sweet's Valve-diagram.— To find outside and inside lap of valve 
for different cut-offs and compressions (see Fig. 147): Draw a circle whose 



A 1 M 




P G 

Fig. 147.-— Sweet's Valve-diagram. 

diameter equals travel of valve. Draw diameter BA and continue to A 1 , 
so that the length AA 1 bears the same ratio to XA as the length of connect- 
ing-rod does to length of engine-crank. Draw small circle E with a diam- 
eter equal to lead. Lay off AC so that ratio of AC to AB = cut-off in 
parts of the stroke. Erect perpendicular CD. Draw DL tangent to E\ 
draw XS perpendicular to DL; XS is then outside lap of valve. 

To find release and compression: If there is no inside lap, draw FE 
through X parallel to DL. F and E will be position of crank for release 
and compression. If there is an inside lap, draw a circle about X, in which 
radius XY equals inside lap. Draw i?G tangent to this circle and parallel 
to DL; then H and G are crank position for release and compression. 
Dva.w HN and MG, then AN is piston position at release and AM piston 
position at compression, AB being considered stroke of engine. 

To make compression alike on each stroke it is necessary to increase the 
inside lap on crank end of valve, and to decrease by the same amount the 



THE SLIDE-VALVE. 



82? 



inside lap on back end of valve. To determine this amount, through AT with 
a radius MM 1 = A A 1 , draw arc MP, from Pdraw FT perpendicular to AB, 
then TM is the amount to be added to inside lap ou crank end, and to be 
deducted from inside lap on back end of valve, inside lap being XY. 

For the Rilgram Valve Diagram, see Halsey on Slide-valve Gears. 

Tlie Ze uner Valve -diagram is given in most of the works on the 
steam-engine, and in treatises on valve-gears, as Zeuner's, Peabody's, and 



Fig. 148.— Zeuner's Valve-diagram. 

Spangler's. The following is condensed from Holmes on the Steam-engine: 
Describe a circle, with radius OA equal to the half travel of the valve. 
From O measure off OB equal to the outside lap, and BC equal to the lead. 
When the crank-pin occupies the dead centre A, the valve has already 
moved to the right of its central position by the space OB -(- BC. From C 
erect the perpendicular CE and join OE. Then will OE be the position 
occupied by the line joining the centre of the eccentric with the centre of 
the crank-shaft at the commencement of the stroke. On the line OE as 
diameter describe the circle OCE ; then any chords, as Oe, OE, Oe', will 
represent the spaces travelled by the valve from its central position when 
the crank-pin occupies respectively the positions opposite to D, E, and F. 
Before the port is opened at all the valve must have moved from its central 
position by an amount equal to the lap OB. Hence, to obtain the space by 
which the port is opened, subtract from each of the arcs Oe. OE, etc., a 
length equal to OB This is represented graphically by describing from 
centre a circle with radius equal to the lap OB ; then the spaces fe, gE, 
etc., intercepted between the circumferences of the lap-circle Bfe' and the 
valve-circle OCE, will give the extent to which the steam-port is opened. 

At the point k, at which the choi 1 Ok is common to both valve and lap 
circles, it is evident that the valve las moved to the right by the amount of 
the lap, and is consequently just, on the point of opening the steam-port. 
Hence the steam is admitted before the commencement of the stroke, when 
the crank occupies the position OH, and while the portion HA of the revo- 



328 THE STEAM-EHGINE. 

lution still remains to be accomplished. When the crank-pin reaches the 
position A, that is to say, at the commencement of the stroke, the port is 
already opened by the space 00 - OB — BO, called the lead. From this 
point forward till the crank occupies the position 01? the port continues to 
open, but when the crank is at OE the valve has reached the furthest limit 
of its travel to the right, and then commences to return, till when in the 
position OF the edge of the valve just covers the steam-port, as is shown 
by the chord Oe', being again common to both lap and valve circles. Hence 
when the crank occupies the position OF the cut-off takes place and the - 
steam commences to expand, and continues to do so till the exhaust opens. 
For the return stroke the steam-port opens again at H.' and closes at F' . 

There remains the exhaust to be considered. When the line joining the 
centres of the eccentric and crank-shaft occupies the position opposite to 
OG at right angles to the line of dead centres, the crank is in the line OP at 
right angles to OE ; and as OP does not intersect either valve-circle the 
valve occupies its central position, and consequently closes the port by the 
amount of the inside lap. The crank must therefore move through such ' 
an angular distance that its line of direction OQ must intercept a chord on 
the valve-circle OK equal in length to the inside lap before the port can be 
opened to the exhaust. This point is ascertained precisely in the same 
manner as for the outside lap, namely, by drawing a circle from centre O, 
with a radius equal to the inside lap; this is the small inner circle in the 
figure. Where this circle intersects the two valve-circles we get four points 
which show the positions of the crank when the exhaust opens and closes 
during each revolution. Thus at Q the valve opens the exhaust on the side 
of the piston which we have been considering, while at R the exhaust closes 
and compression commences and continues till the fresh steam is read- 
mitted at H. 

Thus the diagram enables us to ascertain the exact position of the crank 
when each critical operation of the valve takes place. Making a resume of 
these operations of one side of the piston, we have: Steam admitted before 
the commencement of the stroke at H. At the dead centre A the valve is 
already opened by the amount BO. At E the port is fully opened, and 
valve has reached one end of its travel. At F steam is cut off, consequently 
admission lasted from if to F. At P valve occupies central position, and 
ports are closed both to steam and exhaust. At Q exhaust opened, conse- 
quently expansion lasted from F to Q. At K exhaust opened to maximum 
extent", and valve reached the end of its travel to the left. At R exhaust 
closed, and compression begins and continues till the fresh steam is admitted 
a,tH. 

Problem.— The simplest problem which occurs is the following : Given 
the length of throw, the angle of advance of the eccentric, and the laps of 
the valve, find the angles of the crank at which the steam is admitted and 
cut off and the exhaust opened and closed. Draw the line OE, representing 
the half -travel of the valve or the throw of the eccentric at the given angle 
of advance with the perpendicular OG. Produce OEJto K. On Oi? and OK 
as diameters describe the two valve-circles. - With centre and radii equal to 
the given laps describe the outside and inside lap-circles. Then the inter- 
section of these circles with the two valve-circles give points sth rough which 
the lines OH, OF, OQ, and OR can be drawn. These lines give the required 
positions of the crank. 

Numerous other problems will be found in Holmes on the Steam-engine, 
including problems in valve-setting and the application of the Zeuner dia- 
gram to link motion and to the Meyer valve-gear. 

Port Opening. —The area of port opening should be such that the ve- 
locity of the steam in passing through it should not exceed 6000 ft. per min. 
The ratio of port area to piston area will then vary with the piston-speed as 
follows: 
For speed of piston, ) 100 200 300 40Q 50Q 600 70Q 800 900 1000 120Q 

tt. per mm. s 
Port area = piston \ m Q33 Q5 m m j m 133 ig m 2 
ai ea x ' 

For a velocity of 6000 ft. per min., 



Port area 



sq. of diam. of cyl. X piston speed 



The length of the port opening may be equal to or something less than the 
diameter of the cylinder, and the width = area of portopening -=- its length. 

The bridge between steam and exhaust ports should be wide enough to 
prevent a leak of steam into the exhaust due to overtravel of the valve. 



THE SLIDE-VALVE. 



829 



Auchincloss gives: Width of exhaust port = width of steam port + 
J^ travel of valve — width of bridge. 

Lead. (From Pea-body's Valve-gears.)— The lead, or the amount that 
the valve is open when the engine is on a dead point, varies, with the type 
and size of the engine, from a very small amount, or even nothing, up to % 
of an inch or more. Stationary-engines running at slow speed may have 
from 1/64 to 1/16 inch lead. The effect of compression is to fill the waste 
space at the end of the cylinder with steam; consequently, engines having 
much compression need less lead Locomotive-engines having the valves 
controlled by the ordinary form of Stephenson link-motion may have 
a small lead when running slowly and with along cut-off, but when at speed 
with a short cut-off the lead is at least J4 inch; and locomotives that have 
valve-gear which gives constant lead commonly have J4 nich lead. The 
lead angle is the angle the crank makes with the line of dead points at 
admission. It may vary from 0° to 8°. 

Inside Lead.— Weisbach (vol. ii. p. 296) says: Experiment shows that 
the earlier opening of the exhaust ports is especially of advantage, and in 
the best engines the lead of the valve upon the side of the exhaust, or the 
inside lead; is 1/25 to 1/15; i.e., the slide-valve at the lowest or highest posi- 
tion of the piston has made an opening whose height is 1/25 to 1/15 of the 
whole throw of the slide-valve. The outside lead of the slide-valve or the 
lead on the steam side, on the other hand, is much smaller, and is often 
only 1/100 of the whole throw of the valve. 

Effect of Changing Outside Lap, Inside Lap, Travel 
and Angular Advance. (Thurston.) 





Admission 


Expansion 


Exhaust 


Compression 


Incr. 
O.T. 


is later, 
ceases sooner 


occurs earlier, 
continues longer 


is unchanged 


begins at 
same point 


Incr. 
I.L. 


unchanged 


begins as before, 
continues longer 


occurs later, 
ceases earlier 


begins sooner, 
continues longer 


Incr. 
T. 


begins sooner, 
continues longer 


begins later, 
ceases sooner 


begins later, 
ceases later 


begins later, 
ends sooner 


Incr. 
A. A. 


begins earlier, 
period unaltered 


begins sooner, 
per. the. same 


begins earlier, 
per. unchanged 


begins earlier, 
p.n*. the same 



Zeuner gives the following relations (Weisbach-Dubois, vol. ii. p. 307): 
If »<? = travel of valve, p = maximum port opening; 

L = steam-lap, I = exhaust-lap; 

L D 

R = ratio of steam-lap to half travel =- 



: ratio of exhaust lap to half travel 



' .5S' 
I 



1x8; 



S = 2p + 2L-- 



i + 2R + S; S = 



2p 



1 - 



R' 



If a = angle HOF between positions of crank at admission and at cut-off, 
and j8 = angle QOR between positions of crank at release and at 
., _, .,sin(180°-o) , .sin (180° - j3 

compression, then R = y> : — — ; r = X6- : — —- . 

sin \4>a ' '* sin J^|8 

Ratio of Lap and of Port-opening to Valve-travel .—The 

table on page 831, giving the ratio of lap to travel of valve and ratio of travel 
to port opening, is abridged from one given by Buel in Weisbach-Dubois, 
vol. ii. It is calculated from the above formulae. Intermediate values may 
be found by the formulae, or with sufficient accuracy by interpolation from 
the figures in the table. By the table on page 830 the crank-angle may be 
found, that is, the angle between its position when the engine is on the 
centre and its position at cut-off, release, or compression, when these are 
known in fractions of the stroke. To illustrate the use of the tables the 
following example is given by Buel: width of port = 2.2 in. ; width of port 
opening = width of port + 0.3 in.; over overtravel = 2.5 in.; length of 
connecting-rod = 2% times stroke; cut-off, .75 of stroke; release, .95 of 
stroke; lead-angle, 10°. From the first table we find crank-angle = 114.6; 



830 



THE STEAM-ENGIJSTE. 



)etweeti 
; = 3.72, 



add lead-angle, making 121.6.° From the second table, for angle beti 
admission and cut-off, 125°, we have ratio of travel to port-opening = 
or for 124.6° — 3.74, which, multiplied by port-opening 2.5, gives 9.45 in 
travel. The ratio of lap to travel, by the table, is .2324, or 9.45 X .2324 = 2.2 
in. lap. For exhaust-lap we have, for release at .95, crank-angle = 151.3; 
add lead-angle 10° = 161.3°. From the second table, by interpolation, ratio 
of lap to travel = .0811, and .0811 X 9.45 — 0.77 in., the exhaust-lap. 

Lap-angle = y> (180° — lead- angle — crank- angle at cut-off); 

= y z (180° - 10 - 114.6; = 27.7°. 
Angular advance = lap-angle X lead-angle = 27.7 -f- 10 = 37.7°. 
Exhaust lap-angle = crank-angle at release -j- lap-angle -f lead-angle — 180°; 

=-. 151.3 -f 27.7 -f 10 - 180° = 9°. 
Crank-angle at com- 1 
pression measured >- = 180° — lap -angle — lead-angle — exhaust lap-angle; 
on return stroke ) 

= 180 -27.7-10-9= 133.3° ; corresponding, by 
table, to a piston position of .81 of the return stroke; or 
Crank-angle at compression = 180° — (angle at release - angle at cut-off) 

■+ lead-angle; 
= 180 - (151.3 - 114.6)+10 = 133.3°. 
The positions determined above for cut-off and release are for the forward 
stroke of the piston. On the return stroke the cut-off will take place at 
the same angle, 114.6°, corresponding by table to 66.6$ of the return 
stroke, instead of 75$. By a slight adjustment of the angular advance 
and the length of the eccentric rod the cut-off can be equalized. The 
width of the bridge should be at least 2.5 -j- 0.25 — 2.2 = 0.55 in. 

Crank Angles for Connecting-rods of Different Length, 

Forward and Return Strokes. 



t? s § 


Ratio of Length of Connecting-rod to Length of Stroke. 


c2S 




2 


®/% 


3 


3^ 


4 


5 


Infi- 
nite. 


























For. 


^02 s 
o 
O 


For. 


Ret. 


For. 


Ret. 


For. 


Ret. 


For. 


Ret. 


For. 


Ret. 


For. 


Ret. 


or 


























Ret. 


.01 


10.3 


13.2 


10.5 


12.8 


10.6 


12 6 


10.7 


12.4 


10.8 


12.3 


10.9 


12.1 


11.5 


.02 


14.6 


18.7 


14.9 


18.1 


15.1 


17.8 


15.2 


17.5 


15.3 


17.4 


15.5 


17.1 


16.3 


.03 


17.9 


22.9 


18.2 


22.2 


18.5 


21.8 


18.7 


21.5 


18.8 


21.3 


19.0 


21.0 


19.9 


.04 


20.7 


26.5 


21.1 


25.7 


21.4 


25.2 


21.6 


24.9 


21.8 


24.6 


22.0 


24.3 


23.1 


.05 


23.2 


29.6 


23.6 


28.7 


24.0 


28.2 


24.2 


27.8 


24.4 


27.5 


24.7 


27.2 


25.8 


.10 


33.1 


41.9 


33.8 


40.8 


34.3 


40.1 


34 6 


39.6 


34.9 


39.2 


35.2 


38.7 


36.9 


.15 


41 


51.5 


41.9 


50.2 


42.4 


49.3 


42.9 


48.7 


43.2 


48.3 


43.6 


47.7 


45.6 


.20 


48 


59.6 


48.9 


58.2 


49.6 


57.3 


50.1 


56.6 


50.4 


56.2 


50.9 


55.5 


53.1 


.25 


54.3 


66.9 


55.4 


65.4 


56.1 


64.4 


56.6 


63.7 


57.0 


63 3 


57.6 


62.6 


60.0 


.30 


60.3 


73.5 


61.5 


72.0 


62.2 


71.0 


62.8 


70.3 


63.3 


69.8 


63.9 


69.1 


66.4 


.35 


66.1 


79 8 


67.3 


78.3 


68.1 


77.3 


68.8 


76.6 


69 2 


76.1 


69.9 


75.3 


72.5 


.40 


71.7 


85.8 


73.0 


84.3 


73.9 


83.3 


74.5 


82.6 


75.0 


82.0 


75.7 


81.3 


78.5 


.45 


77.2 


91.5 


78.6 


90.1 


79.6 


89.1 


80.2 


88.4 


80.7 


87.9 


81.4 


87.1 


84.3 


.50 


82.8 


97.2 


84.3 


95.7 


85.2 


94.8 


85.9 


94.1 


86.4 


93.6 


87.1 


92.9 


90.0 


.55 


88.5 


102.8 


89.9 


101.4 


90.9 


100.4 


91.6 


99.8 


92.1 


99.3 


92.9 


98. C 


95.7 


.60 


94.2 


108.3 


95.7 


107.0 


96.7 


106.1 


97.4 


105.5 


98.0 


105.0 


98.7 


104.3 


101.5 


.65 


100.2 


113.9 


101.7 


112.7 


102.7 


111.9 


103.4 


111.2 


103.9 


110.8 


104.7 


110.1 


107.5 


.70 


106.5 


119.7 


108.0 


118.5 


109.0 


117.8 


109.7 


117.2 


110.2 


116.7 


110.9 


116.1 


113.6 


.75 


113.1 


125.7 


114.6 


124.6 


115.6 


123.9 


116.3 


123.4 


116.7 


123.0 


117.4 


122.4 


120.0 


.80 


120.4 


132 


121.8 


131.1 


122.7 


130.4 


123.4 


129.9 


123.8 


129.6 


124.5 


129.1 


126.9 


.85 


128.5 


139 


129.8 


138.1 


130.7 


137.6 


131.3 


137.1 


131.7 


136.8 


132.3 


136.4 


134.4 


.90 


138.1 


146.9 


139.2 


146.2 


139.9 


145.7 


140.4 


145.4 


140.8 


145.1 


141.3 


144.8 


143.1 


.95 


150.4 


156.8 


151.3 


156.4 


151.8 


156.0 


152.2 


155.8 


152.5 


155.6 


152.8 


155.3 


154.2 


.96 


153.5 


159.3 


154.3 


158.9 


154.8 


158.6 


155.1 


158.4 


155.4 


158.2 


155.7 


158.0 


156.9 


.97 


157.1 


162.1 157.8 


161.8 


158.2 


161.5 


158.5 


161.3 


158.7 


161.2 


159.0 


161.0 


160.1 


.98 


161.3 


165.41161.9 


165.1 


162.2 


164.9 162.5 


164.8 


162.6 


164.7 


162.9 


164.5 


163.7 


.99 


166.8 


169.7 167.2 


169.5 


167.4 


169.4167.6 


169.3 


167.7 


109.21167.9 


169.1 


168.5 


1.00 


ISO 


180 1180 


180 


180 


180 Il80 


180 


180 


180 Il80 


180 


180 



THE SLIDE-VALVE. 



831 



Relative Motions of Cross-head and Crank.— If L = length 
of connecting-rod, R — length of crank, 6 = angle of crank with centre line 
of engine, D — displacement of cross-head from the beginning of its stroke, 



= R(l - cos 9) = L - VL* - R* sin 2 6. 
Lap and Travel of Valve. 



a Otl £ 







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7.61 


135° 


.1913 


3.24 


35 


.4769 


43.22 


90 




6.83 


140 


.1710 


3.04 


40 


.4699 


33.17 


95 




6.17 


145 


.1504 


2.86 


45 


.4619 


26.27 


100 




5.60 


150 


.1294 


2.70 


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21.34 


105 




5.11 


155 


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55 


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17.70 


110 




4.69 


160 


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14.93 


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4.32 


165 


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12.77 


120 


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170 


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2.19 


70 


.4096 


11.06 


125 




3.72 


175 


.0218 


2.09 


75 


.3967 


9.68 


130 


2118 


3.46 


180 


.0000 


2.00 


80 


.3830 


8.55 















PERIODS OF ADMISSION, OR CUT-OFF, FOR VARIOUS 
LAPS AND TRAVELS OF SLIDE-VALVES. 

The two following tables are from Clark on the Steam-engine. In the first 
table are given the periods of admission corresponding to travels of valve 
of from 12 in. to 2 in., and laps of from 2 in. to % in., with J4 hi. and y& in. of 
lead. With greater leads than those tabulated, the steam would be cut off 
earlier than as shown in the table. 

The influence of a lead of 5/16 in. for travels of from 1% in. to 6 in., and 
laps of from y^ in. to 1% hx, as calculated for in the second table, is exhibited 
by comparison of the periods of admission in the table, for the same lap and 
travel. The greater lead shortens the period of admission, and increases the 
range for expansive working. 

Periods of Admission* or Points of Cnt-off, for Given 
Travels and Laps of Slide-valves. 



^ 




Periods of Admission, or Points of Cut-off, 


for the following 




o3 






Laps of Valves in inches. 




















H > 


t-J 


2 




1^ 


m 


1 


Vs 


% 


% 


^ 


¥s 


in. 


in. 


% 


% 


% 


% 


% 


% 


% 


% 


% 


12 


Va 


88 


90 


93 


95 


96 


97 


98 


98 


99 


99 


10 


A 


82 


87 


89 


92 


95 


96 


97 


98 


98 


99 


8 


Va 


72 


78 


84 


88 


92 


94 


95 


96 


98 


98 


6 


M 


50 


62 


71 


79 


86 


89 


91 


94 


96 


97 


Wz 


A 


43 


56 


68 


77 


85 


88 


91 


94 


96 


97 


5 


** 


32 


47 


61 


72 


82 


86 


89 


92 


95 


97 


4^ 


H 


14 


35 


51 


66 


78 


83 


87 


90 


94 


96 


4 


H 




17 


39 


57 


72 


78 


83 


88 


92 


95 


3 


% 
% 






20 


44 
23 


63 
50 


71 
61 


79 
71 


84 

79 


90 
86 


94 






91 


2^ 
2 


y 8 










27 


43 


57 


70 


80 


88 














33 


52 


70 


81 



832 



THE STEAM-ENGINE. 



Periods of Admission, or Points of Cut-off, for given I 
Travels and Laps of Slide-valves. 

Constant lead, 5/16. 



Travel. 








Lap. 










Inches. 


% 


% 


% 


% 


1 


V/a 


m 


Wa 


VA 


Wa 

m 
m 

2 


19 


















39 


















47 


17 
















55 
61 
65 
68 

74 
76 


34 
42 
50 
55 
59 
63 
67 
















2% 

2\i 

m 


14 
30 
38 
45 
49 
56 


























13 
27 
36 
43 






















2% 


12 
26 










m 

3 










78 
80 
81 


70 
73 

74 


59 
62 
65 


47 
50 

55 


32 

38 
44 


11 
23 
30 














&/a 


10 






m 


83 


76 


68 


59 


48 


34 


22 






3% 


84 


78 


71 


62 


51 


40 


29 


9 




sy 2 


85 


80 


73 


64 


53 


45 


34 


20 




m 


86 


81 


75 


66 


57 


49 


38 


26 


9 


m 

3% 


87 


82 


76 


68 


60 


52 


42 


32 


19 


87 


83 


78 


70 


63 


55 


46 


36 


25 


4 


88 


84 


79 


72 


66 


58 


49 


40 


29 


4J4 


89 


86 


81 


76 


70 


63 








4^ 


90 


87 


83 


79 


73 


67 


61 


54 


45 


92 


89 


85 


81 


76 


70 


65 


58 


51 


5 


93 


90 


87 


83 


78 


73 


67 


62 


56 


5^ 


94 


92 


89 


86 


82 


78 


73 


68 




6 


95 


93 


91 


88 


85 


82 


78 


74 


69 



Diagram for Port-opening, Cut-off, and Lap.— The diagram 
on the opposite page was published in Power, Aug., 1893. It shows at a 
glance the relations existing between the outside lap, steam port-opening, 
and cut-off in slide valve engines. 

In order to use the diagram to find the lap, having given the cut-off and 
maximum port-opening, follow the ordinate representing the latter, taken 
on the horizontal scale, until it meets the oblique line representing the given 
cut-off. Then read off this height on the vertical lap scale. Thus, with a 
port-opening of 1J4 i Q ch and a cut-off of .50, the intersection of the two lines 
occurs on the horizontal 3. The required lap is therefore 3 in. 

If the cut off and lap are given, follow the horizontal representing the 
latter until it meets the oblique line representing the cut-off. Then vertically 
below this read the corresponding port-opening on the horizontal scale. 

If the lap and port-opening are given, the resulting cut-off may be ascer- 
tained by finding the point of intersection of the ordinate representing the 
port-opening with the horizontal representing the lap. The oblique line 
passing through the point of intersection will give the cut-off. 

If it is desired to take lead into account, multiply the lead in inches by the 
numbers in the following table corresponding to the cut-off, and deduct the 
result from the lap as obtained from the diagram: 



Cut-off. 


Multiplier. 


Cut-off. 


Multiplier. 


.20 


4.717 


.60 


1.358 


.25 


3.731 


.625 


1.288 


.30 


3.048 


.65 


1.222 


.33 


2.717 


.70 


1.103 


.375 


2.381 


.75 


1.000 


.40 


2.171 


.80 


0.904 


.45 


1.930 


.85 


0.815 


.50 


1.706 


.875 


0.772 


.55 


1.515 


.90 


0.731 



THE SLIDE-VALVE. 



833 



Cut-off 
.20 .25 .30 .35 .375.40 .45 .50 .55 .60 


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12 


3 4 5 



Maximum Steam Port opening in Inches. 
DIAGRAM FOR SLIDE VALVES. 



834 THE STEAM-ENGINE. 

Piston-valve.— The piston-valve is a modified form of the slide valve. 
The lap, lead, etc., are calculated in the same manner as for the common 
slide-valve. The diameter of valve and amount of port-opening are calcu- 
lated on the basis that the most contracted portion of the steam-passage 
between the valve and the cylinder should have an area such that the 
velocity of steam through it will not exceed 6000 ft. per minute. The area 
of the opening around the circumference of the valve should be about double 
the area of the steam-passage, since that portion of the opening that is 
opposite from the steam-passage is of little effect. 

Setting the Valves of an Engine. — The principles discussed 
above are applicable not only to the designing of valves, but also to adjust- 
ment of valves that have been improperly set; but the final adjustment of 
the eccentric and of the length of the rod depend upon the amount of lost 
motion, temperature, etc., and can be effected ouly after trial. After the 
valve has been set as accurately as possible when cold, the lead and lap for 
the forward and return strokes being equalized, indicator diagrams should 
be taken and the length of the eccentric-rod adjusted, if necessary, to cor-' 
rect slight irregularities. 

To Put an Engine on its Centre.— Place the engine in a posi- 
tion where the piston will have nearly completed its outward stroke, and 
opposite some point on the cross-head, such as a corner, make a mark upon 
.the guide. Against the rim of the pulley or crank-disk place a pointer and 
mark a line with it on the pulley. Then turn the engine over the centre until 
the cross-head is again in the same position on its inward stroke. This will 
bring theci»ank as much below the centre as it was above it before. With the 
pointer in the same position as before make a second mark on the pulley- 
rim. Divide the distance between the marks in two and mark the middle 
point. Turn the engine until the pointer is opposite this middle point, and 
it will then be on its centre. To avoid the error that may arise from the 
looseness of crank-pin and wrist-pin bearings, the engine should be turned 
a little above the centre and then be brought up to it, so that the crank pin 
will press against the same brass that it does when the first two marks are 
made. 

Iiink-motion.— Link-motions, of which the Stephenson link is the 
most commonly used, are designed for two purposes: first, for reversing the 
motion of the engine, and second, for varying the point of cut-off by varying 
the travel of the valve. The Stephenson link-motion is a combination of 
two eccentrics, called the forward and back eccentric, with a link connect- 
ing the extremeties of the eccentic-rods; so that by varying the position of 
the link the valve-rod may be put in direct connection with either eccentric, 
or may be given a movement controlled in part by one and in part bjvthe 
other eccentric. When the link is moved by the reversing lever into a posi- 
tion such that the block to which the valve-rod is attached is at either end 
of the link, the valve receives its maximum travel, and when the link is in 
mid-gear the travel is the least and cut-off takes place early in the stroke. 

In the ordinary shifting-link with open rods, that is, not crossed, the lead 
of the valve increases as the link is moved from full to mid-gear, that is, as 
the period of steam admission is shortened. The variation of lead is equa- 
lized for the front and back strokes by curving the link to the radius of the 
eccentric-rods concavely to the axles. With crossed eccentric-rods the lead 
decreases as the link is moved from full to mid-gear. In a valve-motion 
with stationary link the lead is constant. (For illustration see Clark's Steam- 
engine, vol. ii. p. 22.) 

The linear advance of each eccentric is equal to that of the valve in full 
gear, that is, to lap -|- lead of the valve, when the eccentric-rods are attached 
to the link in such position as to cause the half- travel of the valve to equal 
the eccentricity of the eccentric. 

The angle between the two eccentric radii, that is, between lines drawn 
from the centre of the eccentric disks to the centre of the shaft equals 180° 
less twice the angular advance. 

Buel, in Appleton's Cyclopedia of Mechanics, vol. ii. p. 316, discusses the 
Stephenson link as follows: " The Stephenson link does not give a perfectly 
correct distribution of steam; the lead varies for different points of cut-off. 
The period of admission and the beginning of exhaust are not alike for both 
ends of the cylinder, and the forward motion varies from the backward. 

" The correctness of the distribution of steam by Stephenson's link-motion 
depends upon conditions which, as much as the circumstances will permit, 
ought to be fulfilled, namely: 1. The link should be curved in the arc of a 
circle whose radius is equal to the length of the eccentric- rod. 2, The 



THE SLIDE-VALVE. 



830 



eccentric-rods ought to be long ; the longer they are in proportion to the 
eccentricity the more symmetrical will the travel of the valve be on both 
sides of the centre of motion. 3. The link ought to be short. Each of its 
points describes a curve in a vertical plane, whose ordinatesgrow larger the 
farther the considered point is from the centre of the link; and as the hori- 
zontal motion only is transmitted to the valve, vertical oscillation will cause 
irregularities. 4. The link-hanger ought to be long. The longer it is the 
nearer will be the arc in which the link swings to a straight line, and thus 
the less its vertical oscillation. If the link is suspended in its centre, the 
curves that are described by points equidistant on both sides from the centre 
are not alike, and hence results the variation between the forward and back- 
ward gear. If the link is suspended at its lower end, its lower half will have 
less vertical oscillation and the upper half more. 5. The centre from which 
the link-hanger swings changes its position as the link is lowered or raised, 
and also causes irregularities. To reduce them to the smallest amount the 
arm of the lifting-shaft should be made as long as the eccentric-rod, and the 
centre of the lifting-shaft should be placed at the height corresponding to 
the central position of the centre on which the link- hanger swings." 

All these conditions can never be fulfilled in practice, and the variations 
in the lead and the period of admission can be somewhat regulated in an 
artificial way, but for one gear only. This is accomplished by giving differ- 
ent lead to the two eccentrics, which difference will be smaller the longer the 
eccentric-rods are and the shorter the link, and by suspending *he link not 
exactly on its centre line but at a certain distance from it, giving what is 
called " the offset." 

For application of the Zeuner diagram to link-motion, see Holmes on the 
Steam-engine, p. 290. See also Clark's Railway Machinery (1855), Clark's 
Steam-engine and Zeuner's and Auchincloss's Treatises on Slide-valve 
Gears. 

The following rules are given by the American Machinist for laying out a 
link for an upright slide-valve engine. By the term radius of link is meant 
the radius of the link-arc ab, Fig. 150, drawn through the centre of the slot; 




(cf ^y ~~ -Q -^ka^ rrg^ 




Fig. 150. 

this radius is generally made equal to the distance from the centre of shaft 
to centre of the link-block pin P when the latter stands midway of its travel. 
The distance between the centres of the eccentric-rod pins e x e 2 should not 
be less than 2J^ times, and, when space will permit, three times the throw of 
the eccentric. "By the throw we mean twice the eccentricity of the eccentric. 
The slot link is generally suspended from the end next to the forward eccen- 
<ric at a point in the link-arc prolonged. This will give comparatively a 
j-mall amount of slip to the link-block when the link is in forward gear: bur, 
tli is slip will be increased when the link is in backward gear. This increase 



836 THE STEAM-ENGINE. 

of slip is, however, considered of little importance, because marine engines, 
as a rule, work but very little in the backward gear. When it is necessary 
that the motion shall be as efficient in backward gear as in forward gear, 
then the liuk should be suspended from a point midway between the two 
eccentric-rod pins; in marine engine practice this point is generally located 
on the link-arc; for equal cut-offs it is better to move the point of suspen- 
sion a small amount towards the eccentrics. 

For obtaining the dimensions of the link in inches : Let L denote the 
length of the valve, B the breadth, p the absolute steam-press ure per sq. 
in., and R a factor of computation used as below; then R = .01 \ ' L x B Xp. 

Breadth of the link = R x 1.6 

Thickness T of the bar = R X .8 

Length of sliding-block = R X 2.5 

Diameter of eccentric-rod pins = (R X •?) + 34 

Diameter of suspension -rod pin = (R X .6) + 34 

Diameter of suspension-rod pin when overhung.. = (R X .8) -j- 34 

Diameter of block-pin when overhung = R -f- 34 

Diameter of block-pin when secured at both ends = {R X .8) + 34 

The length of the link, that is, the distance from a to b, measured on a 
straight line joining the ends of the link-arc in the slot, should be such as to 
allow the centre of the link-block pin Pto be placed in a line with the eccen- 
tric-rod pins, leaving sufficient room for the slip of the block. Another type 
of link frequently used in marine engines is the double bar link, and this 
type is again divided into two classes: one class embraces those links which 
have the eccentric-rod ends as well as the valve-spindle end between ihe 
bars, as shown at B (with these links the travel of the valve is less than 
the throw of the eccentric); the other class embraces those links, shown at 
C, for which the eccentric-rods are made with fork-ends, so as to connect to 
studs on the outside of the bars, allowing the block to slide to the end of the 
link, so that the centres of the eccentric-rod ends and the block-pin are in 
line when in full gear, making the travel of the valve equal to the throw of 
the eccentric. The dimensions of these links when the distance between 
the eccentric-rod pins is 2*^ to 2% times the throw of eccentrics can be 
found as follows: 

Depth of bars = (R X 1.25) +%" 

Thickness of bars = (R X .5) -f 34" 

Diameter of centre of sliding-block — R X 1.3 

When the distance between the eccentric-rod pins is equal to 3 or 4 times 
the throw of the eccentrics, then 

Depth of bars = (R X 1.25) + %" 

Thickness of bars = (R X .5) -f 34" 

All the other dimensions may be found by the first table. These are em- 
pirical rules, and the results may have to be slightly changed to suit given 
conditions. In marine engines the eccentric-rod ends for all classes of links 
have adjustable brasses. In locomotives the slot- link is usually employed, 
and in these the pin-holes have case-hardened bushes driven into the pin- 
holes, and have no adjustable brasses in the ends of the eccentric- rods. The 
link in B is generally suspended by one of the eccentric-rod pins; and the 
link in C is suspended by one of the pins in the end of the link, or by one of 
the eccentric-rod pins. 

Other Forms of Valve-Gear, as the Joy, Marshall, Hackworth, 
Bremme, Walschaert, Corliss, eic, are described in Clark's Steam-engine, 
vol. ii. The design of the Reynolds- Corliss valve-gear is discussed by A. H. 
Eld ridge in Power, Sep. 1893. See also Henthorn on the Corliss engine. 
Rules for laying down the centre lines of the Joy valve -gear are given in 
American Machinist, Nov. 13, 1890. For Joy's " Fluid- pressure Reversing- 
valve," see Eng'g, May 25, 1894. 

GOVERNORS. 

Pendulum or Fly-ball Governor.— The inclination of the arms 
of a revolving pendulum to a vertical axis is such that the height of the 
point of suspension It above the horizontal plane in which the centre of 
gravity of the balls revolve (assuming the weight of the rods to be small 



GOVERNORS. 837 

compared with the weight of the balls) bears to the radius r of the circle 
described by the centres of the balls the ratio 

h weight to or 



r centrifugal force wv* i; 2 ' 
gr 

which ratio is independent of the weight of the balls, v being the velocity 
of the centres of the balls in feet per second. 

If T = number of revolutions of the balls in 1 second, v — 2nrT = ar, in 
which a — the angular velocity, or 2nT, and 

gr' 2 g . 0.8146 . . 9.775 . , 

h = V = i^f 2' or h = -pr- feet = ~W mches ' 

g being taken at 32.16. If N ■ 

inches 

For revolutions per minute. 40 45 50 60 75 

The height in inches will be 21.99 17.38 14.08 9.775 6.256 

Number of turns per minute required to cause the arms to take a given 
angle with the vertical axis: Let I — length of the arm in inches from the 
centre of suspension to the centre of gyx-ation, and a the required angle; 
then 



/WW = 187 . 6 ,/_L_ = buu/1 

y i cos a y i cos a y h 

The simple governor is not isochronous; that is, it does not revolve at a 
uniform speed in all positions, the speed changing as the angle of the arms 
changes. To remedy this defect loaded governors, such as Porter's, are 
used. From the balls of a common governor whose collective weight is A 
let there be hung by a pair of links of lengths equal to the pendulum arms 
a load B capable of sliding on the spindle, having its centre of gravity in 
the axis of rotation. Then the centrifugal force is that due to A alone, and 
the effect of gravity is that due to A + 25; consequently the altitude for a 
given speed is increased in the ratio {A + 2B) : A, as compared with that of 
a simple revolving pendulum, and a given absolute variation in altitude pro- 
duces a smaller proportionate variation in speed than in the common gover- 
nor. (Rankine, S. E., p. 551.) 

For the weighted governor let I — the length of the arm from the point of 
suspension to the centre of gravity of the ball, and let the length of the sus- 
pending-link, l x = the length of the portion of the arm from the point of 
suspension of the arm to the point of attachment of the link; G — the weight 
of one ball, Q = half the weight of the sliding weight, h = the height of the 
governor from the point of snspension to the plane of revolution of the 
balls, a — the angular velocity = 2itT, T being the number of revolutions per 

second; then a =i/ — — y\ + -j ^J; h = — — yi X -~ f,) i n feet, or 

35190/ 
iV 2 v T i Q J 
minute. 

For various forms of governor see App. Cycl. Mech., vol. ii. 61, and Clark's 
Steam-engine, vol. ii. p. 65. 

To Change the Speed of an Engine Having a Fly-ball 
Governor.-A slight difference in the speed of a governor changes the 
position of its weights from that required for f nil load to that required for 
no load. It is evident therefore that, whatever the speed of the engine, the 
normal speed of the governor must be that for which the governor was de- 
signed ; i.e., the speed of the governor must be kept the same. To change the 
speed of the engine the problem is to so adjust the pulleys which drive the 
governor that the engine at its new speed shall drive it just as fast as it was 
driven at its original speed. In order to increase the engine-speed we must 
decrease the pulley upon the shaft of the engine, i.e., the driver, or increase 
that on the governor, i.e., the driven, in the proportion that the speed of the 
engine is to be increased. 



f 27 O ~\ 2 
h = ^^- ( l + -y j^) in inches, N being the number of revolutions per 



838 THE STEAM-EHGLNE. 

Fly-wheel or Shaft Governors.— At the Centennial Exhibition 
in 187b there were shown a few siearn-engines in which the governors were 
contained in the fry-wheel or band-wheel, the fly-balls or weights revolving 
around the shaft in a vertical plane with the wheel and shifting the eccen- 
tric so as automatically to vary the travel of the valve and the point of cut- 
off. This form of governor has since come into extensive use, especially for 
high-speed engines. In its usual form two weights are carried on arms the 
ends of which are pivoted to two points on the pulley near its circum- 
ference, 180° apart. Links connect these arms to the eccentric. The 
eccentric is not rigidly keyed to the shaft but is free to move trans- 
versely across it for a certain distance, having an oblong hole which allows 
of this movement. Centrifugal force causes the weights to fly towards the 
circumference of the wheel and to pull the eccentric into a position of min- 
imum eccentricity. This force is resisted by a spring attached to each arm 
which tends to pull the weights towards the shaft and shift the eccentric to 
the position of maximum eccentricity. The travel of the valve is thus 
varied, so that it tends to cut off earlier in the stroke as the engine increases 
its speed. Many modifications of this general form are in use. For discus- 
sions of this form of governor see Hartnell, Proc. Inst. M. E., 1882, p. 408; 
Trans. A. S. M. E., ix. 300; xi. 1081 ; xiv. 9^; xv. 929 ; Modern Mechanism. 
p. 399; Whitham's Constructive Steam Engineering; J. Begtrup, Am. Much., 
Oct. 19 and Dec. 14, 1893, Jan. 18 and March 1, 1894. 

Calculation of Springs for Shaft-governors. (Wilson Hart- 
nell, Proc. Inst. M. E., Aug. 1882.)— The springs for shaft-governors may be 
conveniently calculated as follows, dimensions being in inches: 

Let W = weight of the balls or weights, in pounds; 

r x and r 2 = the maximum and minimum radial distances of the centre 
of the balls or of the centre of gravity of the weights; 

lx and Z 2 = the leverages, i.e., the perpendicular distances from the cen- 
tre of the weight-pin to a line in the direction of the centrifugal force, 
drawn through the centre of gravity of the weights or balls at radii 
r x and r 2 ; 

m x and m 2 = the corresponding leverages of the springs; 

Cx and C 2 = the centrifugal forces, for 100 revolutions per minute, at 
radii r x and r 2 ; 

Pi and P 2 = the corresponding pressures on the spring; 

(It is convenient to calculate these and note them down for reference.) 

C 3 and C 4 = maximum and minimum centrifugal forces; 

S = mean speed (revolutions per minute); 

Si and /S 2 = the maximum and minimum number of revolutions per 
minute; 

P 3 and P 4 = the pressures on the spring at the limiting number of revo- 
lutions (Si and <S 2 ); 

P 4 - P 3 = D — the difference of the maximum and minimum pressures 
on the springs; 

V — the percentage of variation from the mean speed, or the sensitive- 
ness; 

t — the travel of the spring; 

^t = the initial pressure on the spring; 

v = the stiffness in pounds per inch; 

w — the maximum pressure = u -f- 1. 

The mean speed and sensitiveness desired are supposed to be given. Then 



Sx 


- 6 ~ 100' 




^-^ + Ioo' 


Cx 


= 0.28 X rx X W; 




C 2 = 0.28 X r„ X W; 


Px 


= ^4 ; 




P* = C 2 X - 2 ; 
m 2 ' 


p 3 


^x(f-)'; 




*=**(*# 




D 

v = -. , u = 


P 3 


Pa 

IV = — \ 



It is usual to give the spring-maker the values of P 4 and of v or w. To 
ensure proper space being provided, the dimensions of the spring should be 



CONDENSERS, AIR-PUMPS, ETC. 



839 



calculated by the formulae for strength and extension of springs, and the 
least length of the spring as compressed be determined. 



The governor-power 



.Pm + Pa 



With a straight centripetal line, the governor-power 



. C 3 + C t 



( r *-. r *\ 

V 12 /' 



For a preliminary determination of the governor-power it may be taken 
as equal to this in all cases, although it is evident that with a curved cen- 
tripetal line it will be slightly less. The difference D must be constant for 
the same spring, however great or little its initial compression. Let the 
spring be screwed up until its minimum pressure is I\. Then to find the 
speed P 6 = P k + D, 



»-.«»^ 



sI Viv 



The speed at which the governor would be isochronous would be 
100* 



Wtt 



Suppose the pressure on the spring with a speed of 100 revolutions, at the 
maximum and minimum radii, was 500 lbs. and 100 lbs., respectively, then 
the pressure of the spring to suit a variation from 95 to 105 revolutions will 

/ 95 \ 2 /105\ 2 

be 100 X {rfc) = 90 - 2 and 200 X \tkq) = 220 - 5 - That is, the increase 

of resistance from the minimum to the maximum radius must be 220 — 90 = 
130 lbs. 

The extreme speeds due to such a spring, screwed up to different press- 
ures, are shown in the following table: 



Revolutions per minute, balls shut 

Pressure on springs, balls shut 

Increase of pressure when balls open fully 

Pressure on springs, balls open fully 

Revolutions per minute, balls open fully. 
Variation, per cent of mean speed 



80 


90 


95 


100 


110 


fi4 


HI 


90 


100 


121 


130 


180 


ISO 


130 


180 


194 


211 


220 


23o 


25] 


9S 


102 


105 


107 


112 


10 


6 


5 


3 


1 



120 
144 
130 
274 
117 
-1 



The speed at which the governor would become isochronous is 114. 

Any spring will give the right variation at some speed; hence in experi- 
menting with a governor the correct spring may be found from any wrong 
one by a very simple calculation. Thus, if a governor with a spring whose 
stiffness is 50 lbs. per inch acts best when the engine runs at 95, 90 being its 

/90V 
proper speed, then 50 X ( qk/ = ^5 * DS - * s tne stiffness of spring required. 

To determine the speed at which the governor acts best, the springs may 
be screwed up until it begins to " hunt " and then slackened until the gov- 
ernor is as sensitive as is compatible with steadiness. 



CONDENSERS, AIR-PUMPS CIRCULATING- 
PUMTPS, ETC. 

The Jet Condenser. (Chiefly abridged from Seaton's Marine Engi- 
neering.)— The jet condenser is now uncommon, being generally supplanted 
by the surface condenser. With the jet condenser a vacuum of 24 in. was 
considered fairly good, and 25 in. as much as was possible with most conden- 
sers; the temperature corresponding to 24 in. vacuum, or 3 lbs. pressure ab- 
solute, is 140°. In practice the temperature in the hot-well varies from 110° 
to 120°, and occasionally as much as 130° is maintained. To find the quantity 
of injection-water per pound of steam to be condensed: Let T x = tempera- 
ture of steam at the _exhaust pressure; T q = temperature of the cooling- 



840 THE STEAM-ENGINE. 

water; T 2 = temperature of the water after condensation, or of the hot-well; 
Q = pounds of the cooling-water per lb. of steam condensed; then 

_ 1114° + O.SiT, - r a ) 
V T 2 -T Q ■ 

WH 
Another formula is: Q - -— , in which W is the weight of steam con- I 

densed, iJ the units of heat given up by 1 lb. of steam in condensing, and I 
R the rise in temperature of the cooling- water. 

This is applicable both to jet and to surface condensers. The allowance made I 
for the injection- water of engines working in the temperate zone is usually 
27 to 30 times the weight of steam, and for the tropics 80 to 35 times; 30 I 
times is sufficient for ships which are occasionally in the tropics, and this is 
what was usual to allow for general traders. 

Area of injection orifice = weight of injection- water in lbs. per min. -s- 650 
to 780. 

A rough rule sometimes used is: Allow one fifteenth of a square inch for 
every cubic foot of water condensed per hour. 

Another rule: Area of injection orifice = area of piston -=- 250. 

The volume of the jet condenser is from one fourth to one half of that of 
the cylinder. It need not be more than one third, except for very quick- 
running engines. 

Ejector Condensers.— For ejector or injector condensers (Bulkley's, 
Scbutte's, etc.) the calculations for quantity of condensing-water is the same 
as for jet condensers. 

The Surface Condenser-Cooling Surface.— Peclet found that 
with cooling water of an initial temperature of 68° to 77°, one sq. ft. of copper 
plate condensed 21.5 lbs. of steam per hour, while Joule states that 100" lbs. 
per hour can be condensed. In practice, with the compound engine, brass 
condenser -tubes, 18 B.W.G thick, 13 lbs. of steam per sq. ft. per hour, with 
the cooling-water at an initial temperature of £0°, is considered very fair 
work when the temperature of the feed- water is to be maintained at 120°. 
It has been found that the surface in the condenser may be half the heating 
surface of the boiler, and under some circumstances considerably less than 
this. In general practice the following holds good when the temperature of 
sea- water is about 60° : 

Terminal pres., lbs., abs.... 30 20 15 12^ 10 8 6 

Sq. ft. per I H.P 3 2.50 2.25 2.00 1.80 1.60 1.50 

For ships whose station is in the tropics the allowance should be increased 
by 20$?, and for ships which occasionally visit the tropics 10% increase will 
give satisfactory results. If a ship is constantly employed in cold climates 
10$ less suffices 

Wbitham (Steam-engine Design, p. 283, also Trans. A. S. M. E., ix. 431) 

gives the following: S — -, in which <S = condensing-surfacein sq. 

ft.; T x = temperature Fahr. of steam of the pressure indicated by the 
vacuum-gauge; t = mean temperature of the circulating water, or the 
arithmetical mean of the initial and final temperatures; L — latent heat of 
saturated steam at temperatare T x \ k — perfect conductivity of 1 sq. ft. of 
the metal used for the condensing-surface for a range cf 1° F. (or 557 B.T.U. 
per hour for brass, according to Isherwood's experiments); c = fraction de- 
noting the efficiency of the condensing surface; W — pounds of steam con- 
densed per hour. From experiments by Loring and Emery, on U.S.S. Dallas, 
c is found to be 0.323, and ck = 180; and the equation becomes 

WL 



180^ - t) * 

Whitham recommends this formula for designing engines having indepen- 
dent circulating pumps. When the pump is worked by the main engine the 
value of S should be increased about 10$. 

Taking T x at 135° F., and L — 1020, corresponding to 25 in. vacuum, and t 

* . *r. to 1020 PF 17W 

for summer temperatures at 75°, we have: £ = — — -^--. 

Condenser Tubes are generally made of solid-drawn brass tubes, and 
tested both by hydraulic pressure and steam. They are usually made of a 
composition of 68$ of best selected copper and %2% of best Silesian spelter. 



CONDENSERS, AIR-PUMPS, ETC. 



841 



The Admiralty, however, always specify the tubes to be made of 70% of best 
selected copper and to have 1% of tin in the composition, and test the tubes 
to a pressure of 300 lbs. per sq. in. (Seaton.) 

The diameter of the condenser tubes varies from 14 inch in small conden- 
sers, when they are very short, to 1 inch in very large condensers and long 
tubes. In the mercantile marine the tubes are, as a rule, % inch diameter 
externally, and 18 B.W.G. thick (0.049 inch); and 16 B.W.G. (0.065), under 
some exceptional circumstances. In the British Navy the tubes ai-e also, 
as a rule, % inch diameter, and 18 to 19 B.W.G. thick, tinned on both sides; 
when the condenser is made of brass the.Admiralty do not require the tubes 
to be tinned. Some of the smaller engines have tubes % inch diameter, and 
19 B.W.G. thick. The smaller the tubes, the larger is the surface which 
can be got in a certain space. 

In the merchant service the almost universal practice is to circulate the 
water through the tubes. 

Whitham says the velocity of flow through the tubes should not be less 
than 400 nor more than 700 ft. per min. 

Tube-plates are usually made of brass. Rolled-brass tube-plates 
should be from 1.1 to 1.5 times the diameter of tubes in thickness, depending 
on the method of packing. When the packings go completely through the 
plates the latter, but when only partly through the former, is sufficient. 
Hence, for %-inch tubes the plates are usually % to 1 inch thick with glands 
and tape-packings, and 1 to 1J4 inch thick with wooden ferrules. 

The tube-plates should be secured to their seatings by brass studs and 
nuts, or brass screw-bolts; in fact there must be no wrought iron of any 
kind inside a condenser. When the tube-plates are of large area it is advis- 
able to stay them by brass-rods, to prevent them from collapsing. 

Spacing of Tubes, etc.— The holes for ferrules, glands, or india- 
rubber are usually J4 inch larger in diameter than the tubes; but when ab- 
solutely necessary the wood ferrules may be only 3/32 inch thick. 

The pitch of tubes when packed with wood ferrules is usually 14 mcu 
more than the diameter of the ferrule-hole. For example, the tubes are 
generally arranged zigzag, and the number which may be fitted into a 
square foot of plate is as follows: 



Pitch of 
Tubes. 


No. in a 
sq. ft. 


Pitch of 
Tubes. 


No. in a 
sq. ft. 


Pitch of 
Tubes. 


No. in a 
sq. ft. 


1" 

1 1/16" 


172 
150 

137 


1 5/32" 
1 3/16" 
1 7/32" 


128 
121 
116 


l l 4" 
1 9/32" 
1 5/16" 


110 
106 

99 



Quantity of Cooling Water.— The quantity depends chiefly upon 
its initial temperature, which in Atlantic practice may vary from 40° in the 
winter of temperate zone to 80° in subtropical s^as. To raise the tempera- 
ture to 100° in the condenser will require three times as many thermal units 
in the former case as in the latter, and therefore only one third as much 
cooling- water will be required in the former case as in the latter. 

T x = temperature of steam entering the condenser; 
T = " " circulating-water entering the condenser; 

T 2 = " " " " leaving the condenser; 

T 3 = " " water condensed from the steam; 



Q = quantity of circulating water in lbs. 



. 1114 + 0.3(2*! - T 9 ) 



It is usual to provide pumping power sufficient to supply 40 times the 
weight of steam for general traders, and as much as 50 times for ships sta- 
tioned in subtropical seas, when the engines are compound. If the circulat- 
ing-pump is double-acting, its capacity may be 1/53 in the former and 1/42 
in the latter case of the capacity of the low-pressure cylinder. 

Air-pump.— The air-pump in all condensers abstracts the water con- 
densed and the air originally contained in the water when it entered the 
boiler. In the case of jet-condensers if, also pumps out the water of con- 
densation and the air which it contained. The size of the pump is calculated 
from these conditions, making allowance for efficiency of the pump. 



842 



THE STEAM-EXGIHE. 



Ordinary sea- water contains, mechanically mixed with it, 1/20 of its vol 
ume of air when under the atmospheric pressure. Suppose the pressure in 
the condenser to be 2 lbs. and the atmospheric pressure 15 lbs., neglecting 
the effect of temperature, the air on entering the condenser will be expanded 
to 15/2 times its original volume; so that a cubic foot of sea-water, when it 
has entered the condenser, is represented by 19/20 of a cubic foot of water 
and 15/40 of a cubic foot of air. 

Let q be the volume of water condensed per minute, and Q the volume of 
sea- water required to condense it; and let T. z be the temperature of the 
condenser, and 2\ that of the sea-water. 

Then 19/20 (q + Q) will be the volume of water to be pumped from the 
condenser per minute, 



and ^q+Q)X 



T. 2 + 461° 
2\ + 461° 



the quantity of air. 



If the temperature of the condenser be taken at 120°, and that of sea- 
water at 60°, the quantity of air will then be .418(g + Q), so that the total 
volume to be abstracted will be 



.95(g + Q) + -418(2 +Q) = U 



-Q). 



If the average quantity of injection-water he taken at 26 times that con- 
densed, q + Q will equal 27q. Therefore, volume to be pumped from the 
condenser per minute = 37g, nearly. 

In surface condensation allowance must be made for the water occasion- 
ally admitted to the boilers to makeup for w-aste, and the air contained in 
it, also for slight leak in the joints and glands, so that the air-pump is made 
about half as large as for jet-condensation. 

The efficiency of a single-acting air-pump is generally taken at 0.5, and 
that of a double-acting pump at 0.35. When the temperatur of the sea is 
60°, and that of the (jet) condenser is V20°. Q being the volume of the cooling 
water and q the volume of the condensed water in cubic feet, and n the 
number of strokes per minute, 

The volume of the single-acting pump = 2.74 ( J • 



The volume of the double-acting pump = 4( ^" l ~- j> 



The following table gives the ratio of capacity of cylinder or cylinders to 
that of the air-pump; in the case of the compound engine, the low-pressure 
cylinder capacity only is taken. 



Description of Pump. 


Description of Engine. 


Ratio. 


Single-acting vertical 

Double-acting horizontal. . 


Jet-condensing, expansion 1^ to 2 — 
Surface " " 1)4 to 2.... 
Jet " " 3 to 5.... 
Surface " " 3 to 5 — 

Surface " compound 

Jet " expansion 1J^ to 2 

Surface " " l^to2.... 
Jet " " 3 to 5 ... 
Surface " " 3 to 5 ... 
Surface " compound 


6 to 8 

8 to 10 
10 to 12 

12 to 15 

15 to 18 
10 to 13 

13 to 16 

16 to 19 
19 to 24 
24 to 28 



The Area tlirougli Valve-seats and past the valves should not be 
less than will admit the full quantity of water for condensation at a velocity 
not exceeding 400 ft. per minute. In practice the area is generally in 
excess of this. 

Area through foot- valves = D 2 X S-*- 1000 square inches. 
Area through head-valves = D 2 X Sj- 800 square inches. 
Diameter of discharge-pipe = D x VS -s- 35 inches. 
D — diam. of air-pump in inches, S = its speed in ft. per min. 

James Tribe (Am. Much., Oct. 8, 1891) gives the following rule for air- 



CONDENSERS, AIR-PUMPS, ETC. 843 

pumps used with jet-condensers: Volume of single-acting- air-pump driven 
by main engine = volume of low-pressure cylinder in cubic feet, multiplied 
by 3.5 and divided by the number of cubic feet contained in one pound of 
exhaust-steam of the given density. For a double-acting air-pump the 
same rule will apply, but the volume of steam for each stroke of the pump 
will be but one half. Should the pump be driven independently of the 
engine, then the relative speed must be considered. Volume of jet-con- 
denser = volume of air-pump X 4. Area of injection valve = vol. of air- 
pump in cubic inches -^ 520. 

Circulating-pump. — Let Q be the quantity of cooling water in cubic 
fe*t, n the number of strokes per minute, and S the length of stroke in feet. 

Capacity of circulating-pump = Q -=- n cubic feet. 



3.55i/— 

y n : 



The following table gives the ratio of capacity of steam-cylinder or cylin- 
ders to that of the circulating-pump : 

Description of Pump. Description of Engine. Ratio. 

Single-acting. Expansive 1^ to 2 times. 13 to 16 

3 to 5 " 20 to 25 

" " Compound. 25 to 30 

Double " Expansive 1]4 to 2 times. 25 to 30 

3 to 5 " 36 to 46 

" " Compound. 46 to 56 

The ctear area through the valve-seats and past the valves should be such 
that the mean velocity of flow does not exceed 450 feet per minute. The 
flow through the pipes should not exceed 500 ft. per min. in small pipes and 
600 in large pipes. 

For Centrifugal Circulating -pumps, the velocity of flow in the inlet and 
outlet pipes should not exceed 400 ft. permiu. The diameter of the fan- wheel 
is from 2^ to 3 times the diam. of the pipe, and the speed at its periphery 
450 to 500 ft. per min. If W = quantity of water per minute, in American 
gallons, d = diameter of pipes in inches, R — revolutions of wheel per min., 

- , Jiam. of fan-wheel = not less than -!=—. Breadth of blade at 
16.44 R 



--■/: 



w 



tip = -r^-r. Diam. of cylinder for driving the fan = about 2.8 V'diain. of pipe, 

and its stroke = 0.28 X diam. of fan. 

Feed-pumps for Marine Engines.— With surface-condensing 
engines the amount of water to be fed by the pump is the amount condensed 
from the main engine plus what may be needed to supply auxiliary engines 
and to supply leakage and waste. Since an accident may happen to the 
surface-condenser, requiring the use of jet-condensation, the pumps of 
engines fitted with surface-condensers must be sufficiently large to do duty 
under such circumstances. With jet-condensers and boilers using salt water 
the dense salt water in the boiler must be blown off at intervals to keep the 
density so low that deposits of salt will not be formed. Sea-water contains 
about 1/32 of its weight of solid matter in solution. The boiler of a surface- 
condensing engine may be worked with safety when the quantity of salt is 
four times that in sea-water. If Q — net quantity of feed-water required in 
a given time to make up for what is used as steam, n = number of times the 
saltness of the water in the boiler is to that of "sea- water, then the gross feed- 
water = — Z~jQ- * n orc * er to De capable of filling the boiler rapidly each 
feed-pump is made of a capacity equal to twice the gross feed-water. Two 
feed-pumps should be supplied, so that one may be kept in reserve to be 
used while the other is out of repair. If Q be the quantity of net feed- water 
in cubic feet, I the length of stroke of feed-pump in feet, and n the num- 
ber of strokes per minute, 



Diameter of each feed-pump plunger in inches 



/550 



xQ 



844 



THE STEAM-ENGINE. 



W 



If Wbe the net feed- water in pounds, 

/g yy j] 
Diameter of each feed-pump plunger in inches = A/ — — — — 

y n X l 

An Evaporative Surface Condenser built at the Virginia Agri 
cultural College is described by James H. Fitts (Trans. A. S. M. E., xiv. 690). 
It consists of two rectangular end chambers connected by a series of hori- 
zontal rows of tubes, each row of tubes immersed in a pan of water. 
Through the spaces between the surface of the water in each pan and the 
bottom of the pan above air is drawn by means of an exhaust-fan. At the 
top of one of the end-chambers is an inlet for steam, and a horizontal dia- 
phragm about midway causes the steam to traverse the upper half of the 
tubes and back through the lower. An outlet at the bottom leads to the air- 
pump. The condenser, exclusive of connection to the exhaust fan, occupies 
a floor space of 5' 4}4" x 1' 9%", and 4' 1%" high. There are 27 rows of 
tubes, 8 in some and 7 in others; 210 tubes" in all. The tubes are of brass, 
No. 20 B.W G., %" external diameter and 4' 9J4" in length. The cooling sur- 
face (internal) is 176.5 sq. ft. There are 27 cooling pans, each 4' 9J4" X 1' 9«%", 
and 1 7/16" deep. These pans have galvanized iron bottoms which slide 
into horizontal grooves 34" wide and J4" deep, planed into the tube-sheets. 
The total evaporating surface is 234.8 sq. ft. Water is fed to every third pan 
through small cocks, and overflow-pipes feed the rest. A wood casing con- 
nects one side with a 30" Buffalo Forge Co.'s disk- wheel. This wheel is 
belted to a 3" x 4" vertical engine The air-pump is 5%" diameter with a 
6" stroke, is vertical and single-acting. 

The action of this condenser is as follows: The passage of air over the 
water surfaces removes the vapor as it rises and thus hastens evaporation. 
The heat necessary to produce evaporation is obtained from the steam in the 
tubes, causing the steam to condense. It was designed to condense 800 lbs. 
steam per hour and give a vacuum of 22 in., with a terminal pressure in the 
cylinder of 20 lbs. absolute. 

Results of tests show that the cooling-water required is practically equal in 
amount to the steam used by the engine. And since consumption of steam 
is reduced by the application of a condenser, its use will actually reduce the 
total quantity of water required. From a curve showing the rate* of evapora- 
tion per square foot of surface in still air, and also one show ng the rate 
when a current of air of about 2300 ft. per min. velocity is passed over its 
surface, the following approximate figures are taken: 



Temp. 
F. 


Evaporation, lbs. per 
sq. ft. per hour. 


Temp. 
F. 


Evaporation, lbs. per 
sq. ft. per hour. 


Still Air. 


Current. 


Still Air. 


Current. 


100° 
110 
120 
130 


0.2 
0.25 
0.4 
0.6 


1.1 
1.6 
2.5 

3.5 


140° 
150 
160 
170 


0.8 
1.1 
1.5 
2.0 


5.0 
6.7 
9.5 



fhe Continuous Use of Condensing-water is described in a 
series of articles in Poiver, Aug.-Dec, 1892. It finds its application in situa- 
tions where water for condensing purposes is expensive or difficult to obtain. 

In San Francisco J. C. H. Stut cools the water after it has left the bot- 
well by means of a system of pans upon the roof. These pans are shallow 
troughs of galvanized iron arranged in tiers, on a slight incline, so that the 
water flows back and forth for 1500 o* ?000 ft., cooling by evaporation and 
radiation as it flows. The pans are about 5 ft. in width, and the water as it 
flows has a depth of about half an inch, the temperature being reduced from 
about 140° to 90°. The water from the hot-well is pumped up to the highest 
point of the cooling system and allowed to flow as above described, discharg- 
ing finally into the main tank or reservoir, whence it again flows to the con- 
denser as required. As the water in the reservoir lowers from evaporation, an 
auxiliary feed from the city mains to the condenser is operated, thereby 
keeping the amount of water in circulation practically constant. An accu- 
mulation of oil from the engines, with dust from the surrounding streets, 
makes a cleaning necessary about once in six weeks or two months. It is 
found by comparative trials, running condensing and non condensing, that 



CONDENSERS, AIR-PUMPS, ETC. 845 

about 50$ less water is taken from the city mains when the whole apparatus 
is in use than when the engine is run non-condensing. 22 to 23 in. of vacuum 
are maintained. A better vacuum is obtained on a warm day with a brisk 
breeze blowing than on a cold day with but a slight movement of the air. 

In another plant the water from the hot- well is sprayed from a number of 
fountains, and also from a pipe extending around its border, into a large 
pond, the exposure cooling it sufficiently for the obtaining of a good vacuum 
by its continuous use . 

In the system patented by Messrs. See, of Lille, France, the water is dis- 
charged from a pipe laid in the form of a rectangle and elevated above a 
pond through a series of special nozzles, by which it is projected into a fine 
spray. On coming into contact with the air in this state of extreme divi- 
sion the water is cooled 40° to 50°, with a loss by evaporation of only one 
tenth of its mass, and produces an excellent vacuum. A 3000-H.P. cooler 
upon this system has been erected at Lannoy, one of 2500 H.P. at Madrid, and 
one of 1200 H.P. at Liege, as well as others at Roubaix and Tourcoing. The 
system could be used upon a roof if ground space were limited. 

In an arrangement adopted by the Worthington Pump Co. for supplying 
water to condensers attached to vacuum pans, the injection-water is taken 
from a tank, and after having passed through the condenser is discharged in 
a heated condition to the top of a cooling tower, where it is scattered by 
means of distributing-pipes. The water falling from top to bottom of the 
tower is lowered in temperature by the cooling effect of the atmosphere and 
the absorption of heat caused by a portion of the water being vaporized, and 
is led to the tank to be again started, on its circuit. 

In the evaporative condenser of T. Ledward & Co. of Brockley, London, 
the water trickles over the pipes of the large condenser or radiator, and by 
evaporation carries away the heat necessary to be abstracted to condense 
the steam inside. The condensing pipes are fitted with corrugations 
mounted with circular ribs, whereby the radiating or cooling j-urface is 
largely increased. The pipes, which are cast in sections about 76 in. long by 
Zy 2 ha. bore, have a cooling surface of 26 sq. ft., which is found sufficient 
under favorable conditions to permit of the condensation of 20 to 30 lbs. 
of steam per hour when producing a vacuum of 13 lbs. per sq. in. In a 
condenser of this type at Rixdorf , near Berlin, a vacuum ranging from 24 
to 26 in. of mercury was constantly maintained during the hottest weather 
of August. The initial temperature of the cooling-water used in the appara- 
tus under notice ranged from 80° to 85° F., and the temperature in the sun, 
to which the condenser was exposed, varied each day from 100° to 115° F. 
During the experiments it was found that it was possible to run one engine 
under a load of 100 horse-power and maintain the full vacuum without the 
use of any cooling water at all on the pipes, radiation afforded by the pipes 
alone sufficing to condense the steam for this power. 

In Klein's condensing water-cooler, the hot water coming from the con- 
denser enters at the top of a w T ooden structure about twenty feet in height, 
and is conveyed into a series of parallel narrow metal tanks. The water 
overflowing from these tanks is spread as a thin film over a series of wooden 
partitions suspended vertically about 3^ inches apart within the tower. 
The upper set of partitions, corresponding to the number of metal tanks, 
reaches half-way down the tower. From there down to the well is sus- 
pended a second set of partitions placed at right angles to the first set. This 
impedes the rapidity of the downflow of the water, and also thoroughly 
mixes the water, thus affording a better cooling. A fan-blower at the base of 
the tower drives a strong current of air with a velocity of about twenty feet 
per second against the thin film of water running down over the partitions. 
It is estimated that for an effectual cooling two thousand times more air 
than water must be forced through the apparatus. With such a velocity 
the air absorbs about two per cent of aqueous vapor. The action of the 
strong air-current is twofold: first, it absorbs heat from the hot water by 
being itself warmed by radiation; and, secondly, it increases the evapora- 
tion, which process absorbs a great amount of heat. These two cooling 
effects are different during the different seasons of the year. During the 
winter months the direct cooling effect of the cold air is greater, while 
during summer the heat absorption by evaporation is the more important 
factor. Taking all the year round, the effect remains very much the same. 
The evaporation is never so great that the deficiency of water would not 
be supplied by the additional amount of water resulting from the condensed 
steam, whiie in very cold winter months it may be necessary to occasionally 
rid the cistern of surplus water. It was found that the vacuum obtained by 



846 THE STEAM-ENGINE. 

this eontinual use of the same condensing-water varied during the year 
between 27.5 and 28.7 inches. The great saving of space is evident from 
the fact that only the five-hundredth part of the floor-space is required as 
if cooling tanks or ponds were used. For a 100-horse-power engine the 
floor-space required is about four square yards by a height of twenty feet. 
For one horse-power 3.6 square yards cooling-surface is necessary. The 
vertical suspension of the partitions is very essential. With a ventilator 50 
inches in diameter and a tower 6 by 7 feet and 20 feet high, 10,500 gallons of 
water per hour were cooled from 104° F. to 68° F. The following record 
was made at Mannheim, Germany: Vacuum in condenser, 28.1 inches; tem- 
perature of condensing-water entering at top of tower, 104° to 108° F.; 
temperature of water leaving the cooler. 6(5.2° to 71.6° F. The engine was 
of the Sulzer compound type, of 120 horse-power. The amount of power 
necessary for the arrangement amounts to about three per cent of the total 
horse-power of the engine for the ventilator, and from one and one half to 
three per cent for the lifting of the water to the top of the' cooler, the total 
being four and one half to six per cent. 

A novel form of condenser has been used with considerable success in 
Germany and other parts of the Continent. The exhanst-steam from the 
engine passes through a series of brass pipes immersed in water, to which 
it gives up its heat. Between each section of tubes a number of galvanized 
disks are caused to rotate. These disks are cooled by a current of air 
supplied by a fan and pass down into the water, cooling it by abstract- 
ing the heat given out by the exhaust-steam and carrying it up where it is 
driven off by the air-current. The disks serve also to agitate the water and 
thus aid it in abstracting the heat from the steam. With 85 per cent 
vacuum the temperature of the cooling water was about 130° F., and a 
consumption of water for condensing is guaranteed to be less than a pound 
for each pound of steam condensed. For an engine 40 in. X 50 in., 70 revo- 
lutions per minute, 90 lbs. pressure, there is about 1150 sq. ft. of condensing- 
surface. Another condenser, 1600 sq. ft. of condensing-surface, is used for 
three engines, 32 in. x 48 in., 27 in. X 40 in., and 30 in. X 40 in., respectively. 
— The Steamship. 

The Increase of Power that may be obtained by adding a condenser 
giving a vacuum of 26 inches of mercury to a non-condensing engine may be 
approximated by considering it to be equivalent to a net gain of 12 pounds 
mean effective pressure per square inch of piston area. If A — area of piston 

in square inches, S = piston-speed in ft. per minute, then = — — - = H.P. 

made available by the vacuum. If the vacuum = 13.2 lbs. per sq. in. = 27.9 
in. of mercury, then H.P. = AS -+- 2500. 

The saving of steam for a given horse-power will be represented approxi- 
mately by the shortening of the cut-off when the engine is run with the 
condenser. Clearance should be included in the calculation. To the mean 
effective pressure non-condensing, with a given actual cut-off, clearance 
considered, add 3 lbs. to obtain the approximate mean total pressure, con- 
densing. From tables of expansion of steam find what actual cut-off will 
give this mean total pressure. The difference between this and the original 
actual cut-off, divided by the latter and by 100, will give the percentage of 
saving. 

The following diagram (from catalogue of H. R. Worthington) shows the 
percentage of power that may be gained by attaching a condenser to a non- 
condensing engine, assuming that the vacuum is 12 lbs. per sq. in. The 
mean effective pressures are those of a non-condensing engine exhausting 
at atmospheric pressure, clearance and compression not considered. 

The left-hand vertical column of figures are the initial steam-pressures 
(above the atmosphere), and the upper horizontal column the several 
points of cut-off that represent the point of the stroke at which the steam 
is shut off and admission ceases; directly under this column is a similar one 
of the mean effective pressures. To determine the mean effective pressure 
produced by 90 pounds steam, cut-off at one quarter, find 90 in the initial- 
pressure column, and follow the line to the rig..t until it intersects the 
oblique line that corresponds to the 34 cut-off. Now read the mean effective 
pressure from the figures directly above, which in this case is 49 pounds. 
By glancing down and reading on the lower scale the figure that corresponds 
with this point of intersection the percentage of gain in power will be seen 
to be between 25 and 30 per cent of the power of the engine when running 
non-condensing. 



GAS, PETROLEUM, AND HOT-AIR ENGINES. 



847 



(' Point of Cut-off 

VJfefoflafe f/9 Vs f/s 'M ' -t/3 'fa 

j j j 7 Mean j Effect j\fe Pressure 




Per Cenl or Power gamed by Vacuum 



Fig. 151. 

Evaporators and Distillers are used with marine engines for the 
purpose of providing fresh water for the boilers or for drinking purposes. 

Weirds Evaporator consists of a small horizontal boiler, contrived so as 
to be easily taken to pieces and cleaned. The water in it is evaporated by 
the steam from the main boilers passing through a set of tubes placed in its 
bottom. The steam generated in this boiler is admitted to the low- 
pressure valve-box, so that there is no loss of energy, and the water con- 
densed in it is returned to the main boilers. 

In Weir's Feed-heater the feed-water before entering the boiler is heated 
up very nearly to boiling-point by means of the waste water and steam 
from the low-pressure valve-box of a compound engine. 

GAS, PETROLEUM, AND HOT-AIR ENGINES. 

Gas-engines. — For theory of the gas-engine, see paper by Dugald 
Clerk, Proc. Inst. C. E. 1882, vol. lxix.; and Van Nostrand's Science Series, 
No. 62. See also Wood's Thermodynamics. For construction of gas-engines, 
see Robinson's Gas and Petroleum Engines; articles by Albert' Spies in 
Cassier's Magazine, 1893; also Appleton's Cyc. of Mechanics, and Modern 
Mechanism. 

In the ordinary type of single-cylinder gas-engine (for example the Otto) 
known as a four-cycle engine one ignition of gas takes place in one end of 
the cylinder every two revolutions of the fly-wheel, or every two double 
strokes. The following sequence of operations takes place during four con- 
secutive strokes: (a) inspiration during an entire stroke; (b) compression 
during the second (return) stroke; (c) ignition at the dead-point, and expan- 
sion during the third stroke; (d) expulsion of the burnt gas during the fourth 
(return) stroke. Beau de Rochas in 1862 laid down the law that there are 



848 GAS, PETROLEUM, AHD HOT-AIR ENGIHES. 

four conditions necessary to realize the best results from the elastic force 
of gas: (1) The cylinders should have the greatest capacity with the smallest 
circumferential surface; (2) the speed should be as high as possible; (3) the 
cut-off should be as early as possible; (4) the initial pressure should be as 
high as possible. In modern engines it is customary for ignition to take 
place, not at the dead point, as proposed by Beau de Rochas, but somewhat 
later, when the piston has already made part of its forward stroke. At first 
sight it might be supposed that this would entail a loss of power, but experi- 
ence shows that though the area of the diagram is diminished, the power 
registered by the friction-brake is greater. Starting is also made easier by 
this method of working. (The Simplex Engine, Proc. Inst. M. E. 1889.) 

In the Otto engine the mixture of gas and air is compressed to about 3 
atmospheres. When explosion takes place the temperature suddenly rises 
to somewhere about 2900° F. (Robinson.) 

The two great sources of waste in gas-engines are: 1. The high tempera- 
ture of the rejected products of combustion ; 2. Loss of heat through the 
cylinder walls to the water-jacket. As the temperature of the water-jacket 
is increased the efficiency of the engine becomes higher. 

With ordinary coal-gas the consumption maybe taken at 20 cu. ft. per 
hour per I.H.P., or 24 cu. ft. per brake H.P. The consumption will vary with 
the quality of the gas. When burning Dowson producer-gas the consump- 
tion of anthracite (Welsh) coal is about 1.3 lbs. per I.H.P. perhourfor 
ordinary working. With large twin engines, 100 H.P., the consumption is 
reduced to about 1.1 lb. The mechanical efficiency or B. H.P. -+- I.H.P. in 
ordinary engines is about 85%; the friction loss is less in larger engines. 

Efficiency of the Gas-engine. (Thurston on Heat as a Form of 
Energy.) 

Heat transferred into useful work 17# 

" to the jacket-water 52 

" lost in the exhaust-gas 16 

" " by conduction and radiation 15 

- 83# 

This represents fairly the distribution of heat in the best forms of gas- 
engine. The consumption of gas in the best engines ranges from a mini- 
mum of 18 to 20 cu. ft. per I.H.P. per hour to a maximum exceeding in the 
smaller engines 25 cu. ft. or 30 cu. ft. In small engines the consumption per 
brake horse-power is one third greater than these figures. 

The report of a test of a 170-H.P. Crossley (Otto) gas-engine in England, 
1892, using producer-gas, shows a consumption of but ,85 lb. of coal per H.P. 
hour, or an absolute combined efficiency of 21.3$ for the engine and pro- 
ducer. The efficiency of the engine alone is in the neighborhood of 25$. 

The Taylor gas-producer is used in connection with the Otto gas-engine at 
the works of Schleicher, Schumm & Co., of Philadelphia. The only loss is due 
to radiation through the walls of the producer and a small amount of heat 
carried off in the water from the scrubber. Experiments on a 100-H.P. 
engine show a consumption of 97/100 lb. of carbon per I.H.P. per hour. This 
result is superior to any ever obtained on a steam-engine. (Iron Age, 1893.) 

Tests of the Simplex Gas-engine. (Proc. Inst. M. E. 1889.)— 
Cylinder 7% X 15% in., speed 160 revs, per min. Trials were made with town 
gas of a heating value of 607 heat-units per cubic foot, and with Dowson 
gas, rich in CO, of about 150 heat-units per cubic foot. 

Town Gas. Dowson Gas. 

T 2. 3. 7. 2. 3? 

Effective H.P 6.70 8.67 9.28 7.12 3.61 5.26 

Gas per H.P. per hour, cu. ft.. 21.55 20.12 20.73 88.03 114.85 97.88 

Water per H.P. per hour, lbs. 54.7 44.4 43.8 58.3 

Temp, water entering, F 51° 51° 51° 48° 

" effluent 135° 144° 172° 144° 

The gas volume is reduced to 32° F. and 30 in barometer. A 50-H.P. engine 
working 35 to 40 effective H.P. with Dowson generator consumed 51 lbs. 
English anthracite per hour, equal to 1.48 to 1.3 lbs. per effective H.P. A 16- 
H.P. engine working 12 H.P. used 19.4 en. ft. of gas per effeciive H.P. 

A 320-H.F". Gas-engine.— The flour-mills of M. Leblanc, at Pantin, 
France, have been provided with a 320-horse-power fuel-gas engine of the 
Simplex type. With coal-gas the machine gives 450 horse-power. There is 
one cylinder, 34.8 in. diaui. ; the piston-stroke is 40 in.; and the speed 100 revs. 



GAS-ENG1XES. 



849 



per min. Special arrangements have been devised in order to keep the 
different parts of the machine at appropriate temperatures. The coal used 
is 0.81 J lb. per indicated or 1 .03 lb. per brake horse-power. The water used 
is 8% gallons per brake horse-power per hour. 

Test ofam Otto Gas-engine. (Jour. F. /., Feb. 1890, p. 115.)— En- 
gine 7 11. P. nominal; working capacity of cylinder .2594 cu. ft.; clearance 
space .179(3 cu. ft. 

Per cent 

Heat-units. of Heat 

received. 

Transferred into work 22.84 

Taken by jacket- water 49 . 94 

" " exhaust 27.22 



Temperature of gas supplied . . 62.2 
'_' " " exhaust... 774.3 

" " enteringwater 50.4 

" " exit water 89.2 

Pressure of gas, in. of water.. 3.06 

Revolution per min., av'ge 161.6 

Explosions missed per min., 

average 6.8 

Mean effective pressure, lbs. 

per sq. in 59. 

Horse -power, indicated 4.94 

Work per explosion, foot- 
pounds 2204. 

Explosions per minute 74. 

Gas used perl.H.P. per hour, 
cu. ft 23.4 



Composition of the gas: 



C0 2 ... 
C 3 H 4 .. 

O 

CO.... 
CH 4 . 
H .... 
N... . 



r olume. 


By Weight. 


0.50# 


1.923# 


4.32 


10.520 


1.00 


2.797 


5.33 


15.419 


27.18 


38.042 


51.57 


9.021 


9.06 


22.273 



Temperatures and Pressures developed in a Gas-engine. 

(Clerk on the Gas-engine.)— Mixtures of air and Oldham coal-gas. Temper- 
ature before explosion, 17° C. 



Mixture. 


Max. Press 






Gas. Air. 


lbs. 


per sq. in 


1 vol. 14 vols. 




40. 


1 " 13 " 




51.5 


1 " 12 " 




60. 


1 •* 11 " 




61. 


1 " 9 " 




78. 


1 " 7 " 




87. 


1 " 6 " 




90. 



91. 



Temp, of Explo- 
sion calculated 
from observed 
Pressure. 
806° C. 

1033 

1202 

1220 

1557 

1733 

1792 

1812 

1595 



Theoretical 
Temp, of Explo- 
sion if all Heat 
were evolved. 
1786° C. 
1912 



Test of the Clerk Gas-engine. (Proc Inst. C. E. 1882, vol. lxix.)— 
Cylinder 6 X 12 in., 150 revs, per min.; mean available pressure 70.1 lbs., 9 
I.H.P.; maximum pressure, 220 lbs. per sq. in. above atmosphere; pressure 
before ignition, 41 lbs. above atm. ; temperature before compression 60° 
F., after compression, 313° F.; temperature after ignition calculated from 
pressure, 2800° F. ; gas required per I.H.P. per hour, 22 cu. ft. 

Combustion of the Gas in the Otto Engine.— John Imray, in 
discussion of Mr. Clerk's paper on Theory of the Gas-engine, says: The 
change which Mr. Otto introduced, and which rendered the engine a success, 
was that, instead of burning in the cylinder an explosive mixture of gas and 
air, he burned it in company with, and arranged in a certain way in respect 
of, a large volume of incombustible gas which was heated by it, and which 
diminished the speed of combustion. W. R. Bousfield, in the same discus- 
sion, says: In the Otto engine the charge varied from a charge which was 
an explosive mixture at the point of ignition to a charge which was merely 
an inert fluid near the piston. When ignition took place there was n explo- 
sion close to the point of ignition that was gradually communicated through- 
out the mass of the cylinder. As the ignition got farther away from the 
primary point of ignition the rate of transmission became slower, and if the 
engine were not worked too fast the ignition should gradually catch up to 
the piston during its travel, all the combustible gas being thus consumed. 
This theory of slow combustion is, however, disputed by Mr. Clerk, who 
holds that the whole quantity of combustible gas is ignited in an instant. 

Use of Carburetted Air in Gas-engines.— Air passed over 



850 GAS, PETROLEUM, AtfD HOT-AIR E^GItfES. 






gasoline or volatile petroleum spirit of low sp. gr., 0.65 to 0.70, liberates 
some of the gasoline, and the air thus saturated with vapor is equal in heat- 
ing or lighting power to ordinary coal-gas. It may therefore be used as a 
fuel for gas-engines. Since the vapor is given off at ordinary temperatures 
gasoline is very explosive and dangerous, and should be kept in an under- 
ground tank out of doors. A defect in the use of carburetted air for gas- 
engines is that the more volatile products are given off first, leaving an oily 
residue which is often useless. Some of the substances in the oil that are 
taken up by the air are apt to form troublesome deposits and incrustations 
when burned in the engine cylinder. 

The Otto Gasoline-engine. (Eng'g Neivs, May 4, 1893.)— It is 
claimed that where but a small gasoline-engine is used and the gasoline 
bought at retail the liquid fuel will be on a par with a steam-engine using 6 
lbs. of coal per horse -power per hour, and coal at $3.50 per ton, and will 
besides save all the handling of the solid fuel and ashes, as well as the at- 
tendance for the boilers. As very few small steam-engines consume less 
than 6 lbs. of coal per hour, this is an exceptional showing for economy. At 
8 cts. per gallon for gasoline and 1/10 gal. required per H.P. per hour, the 
cost per H.P. per hour will be 0.8 cent. 

The Priestman Petroleum-engine. (Jour. Frank. Inst., Feb. 
1893 )— The following is a description of the operation of the engine: Any 
ordinary high -test (usually 150° test) oil is forced under air-pressure to an 
atomizer, where the oil is met by a current of air and broken up into atoms 
and sprayed into a mixer, where it is mixed with the proper proportion of 
supplementary air and sufficiently heated by the exhaust from the cylinder 
passing around this chamber. The mixture is then drawn by suction into 
the cylinder, where it is compressed by the piston and ignited by an electric 
spark, a governor controlling the supply of oil and air proportionately to 
the work performed. The burnt products are discharged through an ex- 
haust-valve which is actuated by a cam. Part of the air supports the com- 
bustion of the oil, and the heat generated by the combustion of the oil 
expands the air that remains and the products resulting from the explosion, 
and thus develops its power from air that it takes in while running. In 
other words, the engine exerts its power by inhaling air, heating that air, 
and expelling the products of combustion when done with. In the largest 
engines only the 1/250 part of a pint of oil is used at any one time, and in 
the smallest sizes the fuel is prepared in correct quantities varying from 
1/7000 of a pint upward, according to whether the engine is running on light 
or full duty. The cycle of operations is the same as that of the Otto gas- 
engine. 

Trials of a 5-H.P. Priestman Petroleum-engine. (Prof. 
W. C. Unvvin, Proc. Inst. C. E. 1892.)— Cylinder, 8% X 12 in., making normally 
200 revs, per min. Two oils were used, Russian and American. The more 
important results were given in the following table: 





Trial V. 

Full 
Power. 


Trial I. 

Full 
Power. 


Trial IV. 

Full 
Power. 


Trial II. 

Half 
Power. 


Trial III. 
Light. 


Oil used | 

Brake H.P 


Day- 
light. 

7.722 
9.3G9 
0.824 

0.842 

0.694 
33.4 

151.4 

35.0 

35.4 


Russo- 
lene. 
6.765 
7.408 
0.91 

0.946 

0.864 
31.7 

134.3 

27.6 

23.7 


Russo- 
lene. 

6.882 
8.332 
0.876 

0.988 

0.816 
43.2 

128.5 

26.0 

25.5 


Russo- 
lene. 
3.62 
4.70 
0.769 

1.381 

1.063 
21.7 

48.5 

14.8 

15.6 


Russo- 
lene. 


I.H.P 

Mechanical efficiency. . . 
Oil used per brake H.P. 


0.889 


Oil used per brake H.P. 

hour, lb 

Lb. of air per lb. of oil. . 
Mean explosion pressure, 


5.734 
10.1 

9.6 


Mean compression pres- 
sure, lbs. per sq. in .. 

Mean terminal pressure, 
lbs. per sq in 


6.0 



To compare the fuel consumption with that of a steam-engine, 1 lb. of 
oil might be taken as equivalent to 1*4 lbs. of coal. Then the consumption 



EFFICIENCY OF LOCOMOTIVES. 851 

in the oil-engine was equivalent, in Trials I., IV., and V., to 1.18 lbs., 1.23 lbs., 
and 1.02 lbs. of coal per brake horse-power per hour. From Trial IV. the 
following values of the expenditure of heat were obtained: 

Per cent. 

Useful work at brake 13.31 

Engine friction 2.81 

Heat shown on indicator-diagram. 16.12 

Rejected in jacket-water 47.54 

" in exhaust -gases 26.72 

Radiation and unaccounted for 9.61 

Total 99.99 

Naphtha-engines are in use to some extent in small yachts and 
launches. The naphtha is vaporized in a boiler, and the vapor is used ex- 
pansively in the engine-cylinder, as steam is used; it is then condensed and 
returned to the boiler. A portion of the naphtha vapor is used for fuel un- 
der the boiler. According to the circular of the builders, the Gas Engine 
and Power Co. of New York, a 2-H.P. engine requires from 3 to 4 quarts of 
naphtha per hour, and a 4-H.P. engine from 4 to 6 quarts. The chief advan- 
tages of the naphtha-engine and boiler for launches are the saving of weight 
and the quickness of operation. A 2-H.P. engine weighs 200 lbs., a 4-H.P. 300 
lbs. It takes only about two minutes to get under headway. (Modern 
Mechanism, p. 270.) 

Mot-air (or Caloric) Engines.— Hot-air engines are used to some 
extent, but their bulk is enormous compared with their effective power. For 
an account of the largest hot-air engine ever built (a total failure) see 
Church's Life of Ericsson. For theoretical investigator, see Rankine's 
Steam-engine and Rontgen's Thermodynamics. For description of con- 
structions, see Appletoifs Cyc. of Mechanics and Modern Mechanism, and 
Babcock on Substitutes for Steam, Trans. A. S. M. E., vii., p. 693. 

Test of a Hot-air Engine (Robinson).— A vertical double-cylinder 
(Caloric Engine (Jo.'s) 12 nominal H.P. engine gave 20.19 I.H.P. in the work- 
ing cylinder and 11.38 I.H.P. in the pump, leaving 8.81 net I.H.P.; while the 
effective brake H.P. was 5.9, giving a mechanical efficiency of 67^. Con- 
sumption of coke, 3.7 lbs. per brake H.P. per hour. Mean pressure on 
pistons 15.37 lbs. per square inch, and in pumps 15.9 lbs., the area of working 
cylinders being twice that of the pumps. The hot air supplied was about 
1160° F. and that rejected at end of stroke about 890° F. 

The b st result of Stirling's he?t-engine was 2.7 lbs. per brake H.P. per 
hour. Bailey's hot-air engine, 2 H.P. nominal, gave 4.2 I.H.P., 2.6 B.H.P.; 
mechanical efficiency 62^; estimated temperature at highest pressure 1500° 
F., and at atmospheric pressure 700° F. Highest pressure, 14 lbs. per square 
inch above atmosphere. Consumption of fuel, 7 lbs. per hour per brake 
H.P., and of cooling water, 30 lbs. 



LOCOMOTIVES. 

Efficiency of Iiocomotives and Resistance of Trains. 

(George R. Henderson, Proc. Engrs. Club of Phila. 1886.)— The efficiency of 
locomotives can be divided into two principal parts : the first depending 
upon the size of the cylinders and wheels, the valve-gear, boiler and steam- 
passages, of which the tractive power is a function; and the second upon 
the speed, grade, curvature, and friction, which combine to produce the 
resistance. 
The tractive pow r er may be determined as follows : 

Let P = tractive power; 

p = average effective pressure in cylinder; 
£? = stroke of piston ; 
d = diameter of cylinders; 
D — diameter of driving-wheels. Then 

47rrf2 p g _ tftpS 
4nD '" D ' 



852 



LOCOMOTIVES. 



The average effective pressure can be obtained from an indicator-dia- 
gram, or by calculation, when the initial pressure and ratio of expansion are 
known, together with the other properties of the valve-motion. The sub- 
joined table from " Auchincloss " gives the proportion of mean effective 
pressure to boiler-pressure above atmosphere for various proportions of 
cut-off. 



Stroke, 
Cut off at— 


M.E.P. 
(Boiler- 
pres. = 1). 


Stroke, 
Cut off at— 


(M.E.P. 

Boiler- 

pres. = 1). 


Stroke, 
Cut off at— 


M.E.P. 

(Boiler- 

pres. = 1). 


A25 = H 

.15 

.175 

'.25 = H 
.3 


.15 
.2 
.24 

.28 
.32 
.4 
.46 


.333 = y s 
.375 = % 
.4 
.45 

.55 


.5 = % 
.55 

.57 
.62 

.67 

.72 


.625 = % 
.666 = %j 

•875 = % 


.79 

.82 
.85 
.89 
.93 
.98 



These values were deduced from experiments with an English locomotive 
by Mr. Gooch. As diagrams vary so much from different causes, this table 
will only fairly represent practical cases. It is evident that the cut-off must 
be such that the boiler will be capable of supplying sufficient steam at the 
given speed. 

, In the following calculations it is assumed that the adhesion of the engine 
is at least equal to the tractive power, which is generally the case — if the 
engine be well designed— except when starting, or running at a very low 
rate of speed, with a small expansive ratio. When running faster, economy, 
and also the size of the boiler, necessitate a higher ratio of expansion, thus 
reducing the tractive power below the adhesion. If the adhesion be less 
than the tractive power, substitute it for the latter in the following for- 
mulae. 

The resistances can be computed in the following manner, first consider- 
ing the train: 

There is a resistance due to friction of the journals, pressure of wind, etc., 
which increases with the speed. Most of the experiments made with a view 
of determining the resistance of trains have been with European rolling-stock 
and on European railways. The few trials that have been made here seem 
to prove that with American systems this resistance is less. 

The following table gives the resistance at different speeds, assumed for 
American practice : 

Speed in miles per hour : 
s = 5 10 15 20 25 30 35 40 45 50 55 60 

Resistance in pounds per ton of 2240 lbs. : 
y = 3.1 3.4 4. 4.8 5.8 7.1 8.6 10.2 12.1 14.3 16.8 19.2 

Coefficient of resistance in terms of load : 

I = .0015 .0017 .0020 .0024 .0029 .0035 .0043 .0051 .0060 .0071 .0084 .0096 



0+£> 



I = .0015 

The resistance due to curvature is about .5 lb. per ton per degree of cur- 
vature, or the coefficient = .00025c, where c = the curvature in degrees. 

The effect of grades may be determined by the theory of the inclined 
plane. 

Consider a load Lona grade of m feet per mile. The component of the 
weight L acting in the line of traction, or parallel to the track, is 

L sin d = ^ = -00019 Lot. 
52b0 

To combine these coefficients in one equation representing the resistance 
of the train : 
Let L = weight of train, exclusive of engine, in pounds; 

R = resistance of train, in pounds. 

s, c } and m, as above. Then 

B = L [.0015(l + J^)+ .00025c ± .00019m], 



INERTIA AND RESISTANCES OF RAILROAD TRAINS. 853 

the ± sign meaning that this coefficient is positive for ascending and nega- 
tive for descending grades. 

To find a grade upon which a train would descend by itself, take the last 
coefficient minus and make R — U, whence 

" i = r - 9 ( 1 + bfo) + 1 - 3c - 

As locomotives usually have a long rigid wheel-base, the coefficient for 
curvature had better be doubled. The resistance due to the friction of the 
working parts will be considered as being proportional to the tractive power, 
so that the effective tractive power will be represented by itP, the resistance 
being (1 - it) P. 

Combining all these values, there results the equation between the trac- 
tive power and the weight of the train and engine: 

uP- W (.0005c ± .00019m) = Ll+ .00025c ± .00019m, 

IF being weight of engiue aud tender, and u being probably about .8. 
Transforming, we have 

_ uF - TF(. 0005c ± ,00019m) 
Z+ .00025c ± .00019m ' 
and 

L(l + .00025c ± .00019m) + IF(.0005c ± .00019m) 

These deductions, says Mr. Henderson, agree well with railroad practice. 
The figures given above for resistances are very much less than those 
given by the old formulge (which were certainly wrong), but even Mr. Hen- 
derson's figures for high speed are too high, according to a diagram given by 
D. L. Barnes in Eng'g M'tg., June, 1894, from which the following figures are 
derived: 

Speed, miles per hour 50 60 70 80 90 100 

Resistance, pounds per gross ton . . 12 12.4 13.5 15 17 20 

Eng'g News, March 8, 1894, gives a formula which for high speeds gives 
figures for resistance between those of Mr. Barnes and Mr. Henderson. See' 
tests reported in Eng'g News of June 9, 1892. The formula is, resistance in 
pounds per ton = J4 velocity in miles per hour -\- 2. This gives for 

Speed 5 10 15 20 25 30 35 40 45 50 60 70 80 90 100 

Resistance.. 3J4 4.5 0% 7 8}/ 4 9.5 10^12 13^14.5 17 19.5 22 24.5 27 

For tables showing that the resistance varies with the area exposed to the 
resistance and friction of the air per ton of load, see Dashiell, Trans. A. S. 
M. E., vol. xiii. p. 371. 

Inertia and Resistances of Railroad Trains at Increasing 
Speeds.— A series of tables and diagrams is given in R. R. Gaz., Oct. 31, 
15*90, to show the resistances due to inertia in starting trains and accelerat- 
" ig their speeds. 

The mechanical principles and formulae from which these data were cal- 
culated are a.s follows: 

iS = speed in miles per hour to be acquired at the end of a mile. 

S -5- 2 = average speed in miles per hour during the first mile run. 

V = velocity in feet per second at the end of a mile; then F-h 2 = aver- 
age velocity in feet per second during the first mile run. 

5280 -i- V/2 = time in seconds required to run first mile = 10560 -h V. 

F-r (10560 ■+- V) - V 2 -4- 10560 = .0000947 F 2 = Constant gain in velocity or 
acceleration in feet per second necessary to the acquirement of a velocity V 
at the end of a mile. 

g — acceleration due to the force of gravity, i.e., 32.2 feet per second. 

The forces required to accelerate a given mass in a given time to different 
velocities are in proportion to those velocities. The weight of a body is the 
measure of the force which accelerates it in the case of gravity, and as we 
are considering 1 lb., or the unit of weight, as the mass to be accelerated, 
we have g: (F 2 -7- 10560) : : 1 is to the force required to accelerate 1 lb. to the 
velocity Fat the end of a mile run, or, what is the same, to accelerate it at 
the rate of F 2 -r- 10560 feet per second. 

From this the pull on the drawbar— it is the same as the force just men- 
tioned, and is properly termed the inertia— in pounds per pound of train 
weight is F 2 -r- (10560 X 32.2), which equals .00000294F 2 . 



854 LOCOMOTIVES. 

This last formula also gives the grade in per cent which will give a resist- 
ance equal to the inertia due to acceleration. 

The grade in feet per mile is .00000294 F 2 X 5280 = . 01553 F 2 . 

The resistance offered in pounds per ton is 2000 times as much as per 
pound, or .00588F 2 . 

When the adhesion of locomotive drivers is 600 lbs. per ton of weight 
thereon— this is about the maximum— then the tons on drivers necessary to 
overcome the inertia of each ton of total train load are .00588F 2 h- 600 = 
. 0000098 F 2 . In this determination of resistances no account has been taken 
of the rotative energy of the wheels. 

Efficiency of the Mechanism of a Locomotive. — Druitt 
Halpin (Proc. Inst. M. E., January, 1889,) writes as follov\s, concerning the 
tractive efficiency of locomotives; With simple two-cylinder engines, hav- 
ing four wheels coupled, experiments have been made by the late locomo- 
tive superintendent of the Eastern Railway of Fiance, M. Regray, with the 
greatest possible care and with the best apparatus, and the result arrived at 
was that out of 100 I.H.P in the cylinders 43 H.P. only was available on the 
draw-bar. The loss of 57% was rather a high price to pay for the efficiency 
of the engine. How much of that loss was due to coupling-rods no one 
could yet say; but a considerable amount of it must be due to the rods, be- j 
cause it was known that large engines with a single pair of driving-wheels 
not coupled were doing their work more economically, while advanced loco- 
motive engineers who had not yet gone in for compounding were at any rate 
going back to the single pair of driving-wheels. Moreover, that astonishing 
loss of b'i% had been confirmed independently on the Pennsylvania Railroad, 
trials made with an engine having 18*4 X 24-in. cylinders and 6 ft. in. 
wheels four-coupled; by taking indicator diagrams up to 65 miles an hour, 
which were professed to be taken correctly, the power on the draw-bar 
was found to be only 42$ of that in the cylinders, or only \% less than in the 
French experiments. 

The Size of Locomotive Cylinders is usually taken to be such 
that the engine will just overcome the adhesion of its wheels to the rails un- 
der favorable circumstances. 

The adhesion of the wheel is about one third the weight when the rail is 
clean and sanded, but is usually assumed at 0.25. (Thurston.) 

A committee of the American Association of Master Mechanics, after 
studying the performance reports of the best engines, proposes the follow- 

SdCcI^P"^ 
iug formula for weight on driving-w r heels: W = — in which the 

mean pressure in the cylinder is taken at 0.85 of the boiler-pressure at 
starting, C is a numerical coefficient of adhesion, d the diameter of cylinder 
in inches, D that of the drivers in inches, P the pressure in the boiler in 
pounds per square inch. S the stroke of piston in inches. C is taken as 0.25 
for passenger engines, 0.24 for freight, and 0.22 for " switching " engines. 

The common builder's rule for determining the size of cylinders for the 
locomotive is the following, in which we accept Mr. Forney's assumption 
that the steam-pressure at the engine may be taken as nine tenths that in 

the boiler: The tractive force is, approximately, F — - — -= where 

C is the circumference of tires of driving-wheels, S — the stroke in inches, 
Pi = the initial unbalanced steam-pressure in the cylinder in pounds per 
square inch, and A = the area of one cylinder in square inches. If D — 

9p v d 2 S 
diameter of driving wheel and d = diameter of cylinder, F = 

Taking the adhesion at one fourth the weight W, 



0.9p 1 X A X iS _ O.Vp^S , 
C D ' 



whence the area of each piston i 



0.25CIF . /0.25DW 



/ 0.25£>H 

y o.9p 1 s 



0.9 X 4 X piiS' y o.^s ' 

The above formulae give the maximum tractive force; for the mean trac- 
tive force substitute for pi in the formulae the mean effective pressure. 



BOILERS, GRATE-SURFACE, SMOKE-STACKS, ETC. 855 

Von Borries's rule for the diameter of the low-pressure cylinder of a com- 
pound locomotive is d 2 = — — , 
ph 

where d — diameter of l.p. cylinder in inches; 
D = diameter of driviiig-wheel in inches; 
p = mean effective pressure per sq. in., after deducting internal 

machine friction; 
h = stroke of piston in inches; 
Z = tractive force required, usually 0.14 to 0.16 of the adhesion. 

The value of p depends on the relative volume of the two cylindei-s, and 
from indicator experiments may be taken as follows: 
nioco ^f -Pnonno Ratio of Cylinder p in percentage p for Boiler-press 
Ulass oi engine. Volumes. of Boiler-pressure. ureofl761bs. 

Large-tender eng's 1 : 2 or 1 : 2.05 42 74 

Tank-engines l:2orl:2.2 40 71 

The Size oi Locomotive Boilers. (Forney's Catechism of the 
Locomotive.)— They should be proportioned to the amount of adhesive 
weight and to the speed at which the locomotive is intended to work. Thus 
a locomotive with a great deal of weight on the driving-wheels could pull a 
heavier load, would have a greater cylinder capacity than one with little ad- 
hesive weight, would consume more steam, and therefore should have a 
larger boiler. 

The weight, and dimensions of locomotive boilers are in nearly all cases 
determined by the limits of weight and space to which they are necessarily 
confined. It may be stated generally that within these limits a locomotive 
boiler cannot be made too large. In other words, boilers for locomotives 
should always be made as large as is possible under the conditions that de - 
termine the weight and dimensions of the locomotives. 

Wootten's Locomotive. (Clark's Steam-engine ; see also Jour. 
Frank. Inst. 1891, and Modern Mechanism, p. 485.) — J. E. Wootten designed 
and constructed a locomotive boiler for the combustion of anthracite and 
lignite, though specially for the utilization as fuel of the waste produced in 
the mining and preparation of anthracite. The special feature of the engine 
is the fire-box, which is made of great length and breadth, extending clear 
over the wheels, giving a grate-area of from 64 to 85 sq. ft. The draught 
diffused over these large areas is so gentle as not to lift the fine panicles of 
the fuel. A number of express-engines having this type of boiler are engaged 
on the fast trains between Philadelphia and Jersey City. The fire-box shell 
is 8 ft. 8 in. wide and 10 ft. 5 in. long ; the fire-box is 8x9^ ft., making 76 sq. 
ft. of grate-area. The grate is composed of bars and water-tubes alternately. 
The regular types of cast-iron shaking grates are also used. The height of 
the fire-box is only 2 ft. 5 in. above the grate. The grate is terminated by 
a bridge of fire-brick, beyond which a combustion-chamber, 27 in. long, 
leads to the flue-tubes, about 184 in number, 1% in. diam. The cylinders are 
21 in. diam., with a stroke of 22 inches. The driving-wheels, four-coupled, 
are 5 ft. 8 in. diam. The engine weighs 44 tons, of which 29 tons are on driv- 
ing wheels. The heating-surface of the fire-box is 135 sq. ft., that of the 
flue-tubes is 982 sq. ft.; together, 1117 sq. ft., or 14.7 times the grate-area. 
Hauling 15 passenger-cars, weighing with passengers 360 tons, at an average 
speed of 42 miles per hour, over ruling gradients of 1 in 89, the engine con- 
sumes 62 lbs. of fuel per mile, or 34*4 lbs. per sq. ft. of grate per hour. 

Qualities Essential for a Free-steaming Locomotive. 
(From a paper by A. E. Mitchell, read before the N. Y. Railroad Club; 
Eng'g Neivs, Jan. 24, 1891.)— Square feet of boiler-heating surface for bitu- 
minous coal should not be less than 4 times the square of the diameter in 
inches of a cylinder 1 inch larger than the cylinder to be used. One tenth 
of this should be in the fire-bOx. On anthracite locomotives more heating- 
surface is required in the fire-box, on account of the larger grate-area 
required, but the heating-surface of the flues should not be materially 
decreased. 

Grate-surface, Smoke-stacks, and Exhaust-nozzles for 
Locomotives. (Am. Mac//,., Jan. 8, 1891.)— For grate-surface for anthra- 
cite coal: Multiply the displacement in cubic feet of one piston during a 
stroke by 8.5; the product will be the area of the grate in square feet. 

For bituminous coal : Multiply the displacement in feet of one piston 
during a stroke by 6^; the product will be the grate-area in square feet for 
engines with cylinders 12 in. in diameter and upwards. For engines with 



856 



LOCOMOTIVES. 



smaller cylinders the ratio of grate-area to piston-displacement should be 7J^ 
to 1, or even more, if the design of the engine will admit this proportion. 

The grate-areas in the following table have been found by the foregoing 
rules, and agree very closely with the average practice : 

Smoke-stacks.— The internal area of the smallest cross-section of the stack 
should be 1/1? of the area of the grate in soft-coal-burning engines. 

A. E. Mitchell, Supt. of Motive Power of the N. Y. L. E. & W. R. R., says 
that recent practice varies from this rule. Some roads use the same size of , 
stack, 13J4 in. diam. at throat, for all engines up to 20 in. diam. of cylinder. 

The area of the orifices in the exhaust-nozzles depends on the quantity and 
quality of the coal burnt, size of cylinder, construction of stack, and the 
condition of the outer atmosphere. It is therefore impossible to give rules 
for computing the exact diameter of the orifices. All that can be done is to 
give a rule by which an approximate diameter can be found. The exact 
diameter can only be found by trial. Our experience leads us to believe that 
the area of each orifice in a double exhaust-nozzle should be equal to 1/400 
part of the grate-surface, and for single nozzles 1/200 of the grate-surface. 
These ratios have been used in finding the diameters of the nozzles given in 
the following table. The same sizes are often used for either hard or soft 
coal-burners. 











Double 


Single 










Nozzles. 


Nozzles. 


Size of 


for Anthra- 
cite Coal, in 
sq. in. 


for Bitumin- 
ous Coal, in 
sq. in. 


Diameter 






Cylinders, 
in inches. 


of Stacks, 
in inches. 


Diam. of 
Orifices, in 


Diam. of 
Orifices, in 










inches. 


inches. 


12 X 20 


1591 


1217 


&A 


2 


2 13/16 


13 X 20 


1873 


1432 


io*2 


2^ 


3 


14 X 20 


2179 


1666 


HJ4 


2 5/16 


314 


15 X 22 


2742 


2097 


12^ 


2 9/16 


3 11/16 


16 X 24 


3415 


2611 


14 


2% 


4 1/16 


17 X 24 


3856 


2948 


15 


3 1/16 


4 5/16 


18 X 24 


4321 


3304 


15% 


3^ 


4% 


19 X 24 


4810 


3678 


16£ 


3 7/16 


4 13/16 


20 X 24 


5337 


4081 


1-T& 


Ws 


5 1/16 



Exhaust-nozzles in Locomotive Boilers.— A committee of 
the Am. Ry. Master Mechanics' Assn. in 1890 reported that they had, after 
two years of experiment and research, come to the conclusion that, owing 
to the great diversity in the relative proportions of cylinders and boilers, 
together with the difference in the quality of fuel, any rule which does not 
recognize each and all of these factors would be worthless. 

The committee was unable to devise any plan to determine the size of the 
exhaust-nozzle in proportion to any other part of the engine or boiler, and 
believes that the best practice is for each user of locomotives to adopt a 
nozzle tliat will make steam freely and fill the other desired conditions, best 
determined by an intelligent use of the indicator and a check on the fuel 
account. The conditions desirable are : That it must create draught enough 
on the fire to make steam, and at the same time impose the least possible 
amount of work on the pistons in the shape of back pressure. It should be 
large enough to produce a nearly uniform blast without lifting or tearing 
the fire, and he economical in its use of fuel. 

Fire-brick Arches in Locomotive Fire-boxes.— A com- 
mittee of the Am. Ry. Master Mechanics' Assn. in 1S90 reported strongly in 
favor of the use of brick arches in locomotive fire-boxes. They say : It is 
the unanimous opinion of all who use bituminous coal and brick arch, that 
it is most efficient in consuming the various gases composing black smoke, 
and by impeding and delaying their passage through the tubes, and ming- 
ling: and subjecting them to the heat of the furnace, greatly lessens the 
volume ejected, and intensifies combustion, and does not in the least check 
but rather augments draught, with the consequent saving of fuel and in- 
creased steaming capacity that might be expected from such results. This 
in particular when used in connection with extension front. 

Size, Weight, Tractive Power, etc., of Different Sizes of 
Locomotives. (J. G. A. Meyer, Modern Locomotive Construction, Am. 



SIZE, WEIGHT, TRACTIVE POWER, ETC. 



857 



Mach.< Aug. 8, 1885.)— The tractive power should not be more or less than 
the adhesion. In column 3 of each table the adhesion is given, and since the 
adhesion and tractive power are expressed by the same number of pounds, 
these figures are obtained by finding the tractive power of each engine, for 
this purpose always using the small diameter of driving-wheels given in 
column 2. The weight on drivers is shown in column 4, which is obtained by 
multiplying the adhesion by 5 for all classes of engines. Column 5 gives the 
weights on the trucks, and these are based upon observations. Thus, the 
weight on the truck for an eight-wheeled engine is about one half of that 
placed on the drivers. 

For Mogul engines we multiply the total weight on drivers by the decimal 
2, and the product will be the weight on the truck. 

For ten-wheeled engines the total weight on the drivers, multiplied by the 
decimal .32, will he equal to the weight on the truck. 

And lastly, for consolidation engines, the total weight on drivers multi- 
plied by the decimal .16, will determine the weight on the truck. 

In column 6 the total weight of each engine is given, which is obtained by 
adding the weight on the drivers to the weight on the truck. Dividing the 
adhesion given in column 1 by 7% will give the number of tons of 2000 lbs. 
that the engine is capable of hauling on a straight and level track, column 7. 

The weight of engines given in these tables will be found to agree gen- 
erally with the actual weights of locomotives recently built, although it 
must not be expected that these weights will agree in every case with the 
actual weights, because the different builders do not build the engines alike. 

The actual weight on trucks for eight-wheeled or ten-wheeled engines will 
not differ much from those given in the tables, because these weights depend 
greatly on the difference between the total and rigid wheel-base, and these 
ai-e not often changed by the different builders. The proportion between 
the rigid and total wheel-base is generally the same. 

The rule for finding the tractive power is : 



( Square of dia. of | 
| piston in inches j 



( Mean effect, steam | 
I press, per sq. in. j 



i stroke | 
i in feet j 



Diameter of wheel in feet. 



tractive power. 



Eight wheeled Locomotives. 




Ten- 


5VHEELED 


Engines. 






so 








el t« 


iT 


& 










a m 

i-sl 


q| 


o 


ri 


ft 
o 


3 


1 


§lil 


i 
I 


> 

ft 

o 


C 


ft 
C 
o 


p 

H 

C 














lilt 












p 


11 


■d 


1 


1 


H 


£ O 


II 

ft ? 


"0 

< 


1 




3| 


1 


a 


S 


4 


'» 


6 


7 


1 


2 


3 


4 


5 


6 


7 


in. 


in. 


lbs. 


lbs. 


lbs. 


lbs. 




in. 


in 


lbs 


lbs. 


lbs. 


lbs. 




10x20 


45-51 


4000 


20000 


10000 


30000 


. 533 


12X18 


39-43 5981 


29907 


9570 


39477 


797 


11x22 


45-51 


5324 


2C620 


13310 


39930 


709 


13X18 


41-45 6677 


33387 


II 


44070 


890 




48-54 


5940 


29700 


14850 


44550 


792 


U -20 


43-47 t 8205 


41023 


13127 


54150 


1093 




49-57 


6828 


34140 


17070 


51210 


910 


15 ,-22 


45-50 1 9900 


49500 




65340 


1320 




55-61 


7697 


384S5 


19242 


57727 


1026 




48-54 11520 


57600 




76032 


1536 


15X24 


55-66 


8836 


44180 


22090 


66270 


1178 


17X24 


51-56 12240 


' 




80784 


1632 


16X24 


58-66 


9533 


47665 




71497 


1271 




51-56 13722 


68611 


219o5 


90566 


1829 


17X24 


60-66 


10404 


52020 


26010 


78030 


1387 


19X24 


54-60 14440 


72200 


2310+ 


95304 


1925 


18X24 


61-66 


11472 


57360 


28680 


86640 


1529 




1 











Mogul Engines. 



Consolidation Engines. 



in. 


in. 


lbs. 


lbs. 


lbs. 


lbs. 




in. 


in. lbs. 


lbs. lbs. lbs. 




11x16 


35-40 


4978 


24891 


4978 




663 


14x16 


36-38 1 7840 


39200 ' 6272 45472 


1045 


12 > 18 


36-41 


6480 


32400 


6480 


38880 


864 


15xlS 


30-38 10125 


50625 8100 58725 


1350 


13X18 


37-42 


7399 


36997 


7399 


44396 


986 


20X24 


48-50 18000 


90000 14 400 104400 


2400 


14X20 


39-43 


9046 


45230 


9046 


54270 


1206 


22x24 


50 52 i 20909 


:::; 121271 


2787 


15. 22 


42-47 


10607 


53035 


looo; 


03,1 12 


1414 






16X24 


45-51 


12288 


61440 


12288 


73738 


1638 






17x24 


49 54 


127 39 


63697 


12739 


76436- 


1698 






18X24 


51-50 


13722 


68611 


13722 


82333 


1829 






19X24 54 00 14440 


72200 


14440 


86640 


1925 







858 LOCOMOTIVES. 

Leading American Types of Locomotive for Freight and 
Passenger Service. 

1. The eight-wheel or "American" passenger type, having four coupled 
driving-wheels and a four-wheeled truck in front. 

2. The •' ten- wheel " type, for mixed traffic, having six coupled drivers and 
a leading four-wheel truck. 

3. The " Mogul " freight type, having six coupled driving-wheels and a 
pony or two-wheel truck in front. 

4. The " Consolidation " type, for heavy freight service, having eight 
coupled driving-wheels and a ponj' truck in front. 

Besides these there is a great variety of types for special conditions of 
service, as four-wheel and six- wheel switching-engines, without trucks; the 
Forney type used on elevated railroads, with four coupled wheels under the 
engine and a four-wheeled rear truck carrying the water-tank and fuel; 
locomotives for local and suburban service with four coupled driving-wheels, 
with a two-wheel truck front and rear, or a two-wheel truck front and a 
four-wheel truck rear, etc. "Decapod 1 ' engines for heavy freight service 
have ten coupled driving-wheels and a two-wheel truck in front. 

Steam-distribution for High-speed Locomotive©. 

(C. H. Quereau, Eng'g News, March 8, 1894.) 

Balanced Valves.— Mr. Philip "Wallis, in 1886, when Engineer of Tests for 
the C, B. & Q. R. R., reported that while 6 H.P. was required to work un- 
balanced valves at 40 miles per hour, for the balanced valves 2.2 H.P. only 
was necessary. 

Effect of Speed on Average Cylinder-pressure.— Assume that a locomotive 
has a train in motion, the reverse lever is placed in the running notch, and 
the track is level ; by what is the maximum speed limited ? The resistance 
of the train and the load increase, and the power of the locomotive de- 
creases with increasing speed till the resistance and power are equal, when 
the speed becomes uniform. The power of the engine depends on the 
average pressure in the cylinders. Even though the cut-off and boiler- 
pressure remain the same, this pressure decreases as the speed increases; 
because of the higher piston-speed and more rapid valve-travel the steam 
has a shorter time in which to enter the cylinders at the higher speed. The 
following table, from indicator-cards taken from a locomotive at varying 
speeds, shows the decrease of average pressure with increasing speed: 

Miles per hour 46 51 51 53 54 57 60 66 

Speed, revolutions 224 248 248 258 263 277 292 321 

Average pressure per sq. in.: 

Actual 51.5 44.0 47.3 43.0 41.3 42.5 37.3 36.3 

Circulated 46.5 46.5 44.7 43.8 41.6 39.5 35.9 

The "average pressure calculated" was figured on the assumption that 
the mean effective pressure would decrease in the same ratio that the speed 
increased. The main difference lies in the higher steam-line at the lower 
speeds, and consequent higher expansion-line, showing that more steam 
entered tne cylinder. The back pressure and compression-lines agree quite 
closely for all the cards, though they are slightly better for the slower 
speeds. That the difference is not greater may safely be attributed to the 
large exhaust-ports, passages, and exhaust tip, which is 5 in. diameter. 
These are matters of great importance for high speeds. 

Boiler-pressure.— The increase of train resistance with increased speed is 
not as the square of the velocity, as is commonly supposed. It is more likely 
that it increases as the speed after about 20 miles an hour is reached. As- 
suming that the latter is true, and that an average of 50 lbs. per square inch 
is the greatest that can be realized in the cylinders of a given engine at 40 
miles an hour, and that this pressure furnishes just sufficient power to keep 
the train at this speed, it follows that, to increase the speed to 50 miles, the 
mean effective pressure must be increased in the same proportion. To in- 
crease the capacity for speed of any locomotive its power must be increased, 
and at least by as much as the speed is to be increased. One way to accom- 
plish this is to increase the boiler-pressure. That this is generally realized, 
is shown by the increase in boiler-pressure in the last ten years. For twenty- 
three single-expansion locomotives described in the railway journals this 
year the steam-pressures are as follows; 3, 160 lbs.; 4, 165 lbs ■ 2, 170 lbs, ; 
13, 180 lbs.; 1, 190 lbs, 



SOME LARGE AMERICAN LOCOMOTIVES, 1893. 859 

Valve-travel. — An increased average cylinder-pressure may also be 
obtained by increasing the valve-travel without raising the boiler-pressure, 
and better results will be obtained by increasing both. The longer travel 
gives a higher steam-pressure in the cylinders, a later exhaust-opening, 
later exhaust-closure, and a larger exhaust-opening— all necessary for high 
speeds and economy. I believe that a 20-in. port and 6^-in. (or even 7-in.) 
travel could be successfully used for high-speed engines, and that frequently 
by so doing the cylinders could be economically reduced and the counter- 
balance lightened. Or, better still, the diameter of the drivers increased, 
securing lighter counterbalance and better steam-distribution. 

Size of Drivers.— Economy will increase with increasing diameter of 
drivers, provided the work at average speed does not necessitate a cut-off 
longer than one fourth the stroke. The piston-speed of a locomotive with 
62-in. drivers at 55 miles per hour is the same as that of one with 68-in. 
drivers at 61 miles per hour. 

Steam-ports.— The length of steam-ports ranges from 15 in. to 23 in., and 
lias considerable influence on the power, speed, and economy of the loco- 
motive. In cards from similar engines the steam-line of the card from the 
engine with 23-in. ports is considerably nearer boiler-pressure than that of 
the card from the engine with 17J4-in. ports. That the higher steam-line is 
due to the greater length of steam-port there is little room for doubt. The 
23-in. port produced 531 H.P. in an 18^-m. cylinder at a cost of 23.5 lbs. of 
indicated water per I. H.P. per hour. The lTJ^-in. port, 424 H.P., at the rate 
of 22 9 lbs. of water, in a 19-in. cylinder. 

Allen Valves. — There is considerable difference of opinion as to the advan- 
tage of the Allen pnrted-valve (See Eng. Netvs, July 6, 1893.) 

Speed of Railway Trains.— In 1834 the average speed of trains on 
the Liverpool and Manchester Railway was twenty miles an hour; in 1838 it 
was twenty-five miles an hour. But by 1840 tnere'were engines on the Great 
Western Railway capable of running fifty miles an hour with a train, and 
eighty miles an hour without. A speed of 86 miles per hour was made in 
England with the T. W. Worsdell compound locomotive. The total weight 
of the engine, tender, and train was 695,000 lbs.; indicator-cards were taken 
showing 1068.6 H.P. on the level. At a speed of 75 miles per hour on a 
level, and the same train, the indicator-cards showed 1040 H.P. developed. 
(Trans. A. S. M. E., vol. xiii., 363.), 

The limitation to the increase of speed of heavy locomotives seems at 
present to be the difficulty of counterbalancing the reciprocating parts. The 
unbalanced vertical component of the reciprocating parts causes the pres- 
sure of the driver on the rail to vary with every revolution. Whenever the 
speed is high, it is of considerable magnitude, and its change in direction is 
so rapid that the resulting effect upon the rail is not inappropriately called 
a "hammer blow. 1 ' Heavy rails have been kinked, and bridges have been 
shaken to their fall under the action of heavily balanced drivers revolving 
at high speeds. The means by which the evil is to be overcome has not yet 
been made clear. See paper by W. F. M. Goss, Trans. A. S. M. E.. vol. xvi. 

Engine No. 999 of the New York Central Railroad ran a mile in 32 seconds, 
equal to 112 miles per hour, May 11, 1893. 

Speed in miles | _ circum. of driving-wheels in in. x no. of rev, per min. x 60 
per hour f - 63,360 

= diam, of driving-wheels in in. X no. of rev. per min. X .003 
(approximate, giving result 8/10 of 1 per cent too great). 

DIMENSIONS OF SOME LARGE AMERICAN 
LOCOMOTIVES, 1893. 

The four locomotives described below were exhibited at the Chicago 
Exposition in 1893. The dimensions are from Engineering News, June, 1893. 
The first, or Decapod engine, has ten-coupled driving-wheels. It is one of 
the heaviest and most powerful engines ever built for freight service. The 
Philadelphia & Reading engine is a new type for passenger service, with four- 
coupled drivers. The Rhode Island engin-e has six drivers, with a 4-wheel 
leading truck and a 2-wheel trailing truck. These three engines have all 
compound cylinders. The fourth is a simple engine, of the standard Ameri- 
can 8-wheel type, 4 driving-wheels, and a 4-wheel truck in front. This 
engine holds the world's record for speed (1893) for short distances, having 
run a mile in 32 seconds. 



860 



LOCOMOTIVES. 



Baldwin. 
N. Y., L. E. 
& 
W. R. R. 
Decapod 
Freight. 



Baldwin. 

Phila. 

& 

Read. R. R 

Express 

Passenger. 



Rhode Id. 

Loeomoti 1 e 

Works. 

Heavy 

Express. 



N. Y. C. & 

H. R. R. 

Empire 

State 

Express, 
No. 999. 



Running-gear : 

Driving-wheels, diam 

Truck " " 

Journals, driving-axles... 
" truck- " ... 
" tender- " , .. 

Wheel-base : 

Driving 

Total engine 

" tender 

" engine and tender. . . 
Wt. in working-order: 

On drivers 

On truck-wheels 

Engine, total. 

Tender " 

Engine and tender, loaded 
Cylinders : 

h.p. (2) 

lp. (2) 

Distance centre to centre. 

Piston-rod, diam — 

Connecting-rod, length... 

Steam-ports 

Exhaust-ports 

Slide-valves, out. lap, h.p. 

" " out. lap, l.p.. 

" " in. lap, h.p... 

" " in. lap, l.p. . . 

" " max. travel . 

" " lead, h.p 

lead, l.p 



9 x 10 in. 

5 xlO " 
4^x 9 " 



8 " 

4 " 



170,000 lbs. 
29,500 " 
192,500 " 
117,500 " 
310,000 '.« 

16x28 in. 
27x28 " 
7 ft. 5" 

4 in. 
9' 8 7/16'" 
28^ x 2 in. 

28^x8 " 
%in. 



Boiler— Type 

Diam. of barrel inside. . 

Thickness of barrel-plates 

Height from rail to centre 
line 

Length of smoke-box 

Working steam-pressure.. 
Firebox— type 

Length inside 

Width " 

Depth at front 

Thickness of side plates . . 
" " back plate. . . 

Thickness of crown-sheet. 
" " tube " 

Grate-area 

Stay-bolts, diam., \% in. 
Tubes— iron 

Pitch 

Diam., outside 

Length betw'n tube-plates 
Heating-surface : 

Tubes, exterior 

Fire-box 

Miscellaneous : 

Exhaust-nozzle, diam 

Sniokestack,smarst diam. 

" height from 

rail to top 



1/16 in. 

5/16 " 

Straight 

5 ft. 2}4 in. 



8 ft. in. 

5 " 7% " 

180 lbs. 

Wootten 

10' 11 9/16' 

8 ft. 2y 8 in 

4 " 6 " 

5/16 in. 

5/16 '• 

Vs " 

M " 

89.6 sq. ft. 

pitch,4*4 in 

354 

234 in. 

2 " 

11 ft. 11 in 

2,208.8 ft. 
234.3 " 

5 in. 
1 ft. 6 " 

15 " 6^ " 



6 ft. 6 in. 
4 " " 

8V£xl2in. 
6'^xl0 " 
4J/ 2 x 8 " 

6 ft. 10 in. 
23 " 4 " 
16 " " 

47 " 3 " 

82,700 lbs. 

47,000 " 
129,700 " 

80,573 " 
210,273 " 

13x24 in. 
22x24 " 

7 ft. 4}^ in. 

3^ in. 

8 ft, 0^ in. 

24x1^ in. 

24x4J^ " 

%in. 

. %" 

(neg.) y H in. 

None 

5 in. 



Straight 

4 ft. 8J4 in. 

% i". 



180 lbs. 
Wootten 

9 ft. 6 in. 

8 " oy 8 " 

3 " 234 " 

5/16 in. 
5/16 " 
5/16 * 

76.8 sq. ft. 

'324*'" 
2 1/16 in. 
11/2 in. 

10 ft. in. 

1,262 sq. ft, 
173 " " 

5^ in. 
1 ft. 6 in. 

14 ft. 0M in. 



x- 834 in. 
5^x10 " 

41/4 x 8 " 



15 " " 
50 ',' 6% " 

88,500 lbs. 

54,500 " 
143,000 " 

75,000 " 
218,000 " 

one 21 x26 
one 31 x 26 
7 ft. 1 iu. 
3'^ in. 
10 ft. 3)4 in. 
11^x20 and 



xl2^in. 
6^x10 " 

4% x 8 " 

8 ft. 6 in. 
23 " 11 " 
15 ft. 2}4 " 

47 " m " 

84,000 lbs. 

40,000 " 
121,000 " 

80,000 " 
204,000 " 

19x24 in. 



8 ft. l^in. 

1^x18 in. 

234x18 " 
lin. 



1/10 in. 
"5^'in." 



Wagon top 
5 ft. 2 in. 



ft. 11 in. 

'.* 1 " 

200 lbs. 

Radial stay 

10 ft. in. 

2 '• 9% » 

6 " 1034 " 

5/16 in. 



28 sq. ft. 
4 in. 
272 



12 ft. 8% in 



Wagon top 
4 ft. 9 in. 
9/16 in. 

7 ft. 11^ in. 

1 " 8 r 

190 lbs. 
Buchanan 
9 ft. 15% in. 
3 " 1% " 
6 " iy 4 " 

5/16 in. 

5/16 il 



1 ft. 3 in. 
15 »* 2 " 



12 ft. in. 
1,697 sq.ft. 



3^ in. 
1 ft. 3^ in. 



DIMENSIONS OF AMERICAN LOCOMOTIVES- 



861 



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862 



LOCOMOTIVES. 



Dimensions of Some American Locomotives.— The table on 
page 861 is condensed from one given by D. L. Barnes, in his paper on 
" Distinctive Features and Advantages of American Locomotive Practice,' 1 
Trans. A.S.C.E., 1893. The formula from which column marked "Ratio of 
cylinder-power to weight available for adhesion" is calculated as follows: 
2 X cylinder area X boiler-pressure x stroke 
Weight on drivers X diameter of driving-wheel' 

(Ratio of cylinder-power of compound engines cannot be compared with 
that of the single-expansion engines.) 

Where the boiler-pressure could not be determined from the description 
of the locomotives, as given by the builders and operators of the locomotives, 
it has been assumed to be 160 lbs. per sq. in. above the atmosphere. 

For compound locomotives the figures in the last column of ratios are 
based on the capacity of the low-pressure cylinders only, the volume of the 
high-pressure being omitted. This has been done for the purpose of com- 
parison, and because there is no accurate simple way of comparing the 
cylinder-power of single-expansion and compound locomotives. 

Dimensions of Standard Locomotives on tlie N. Y. C. & 
II. It. R. and Penna. It. It., 1882 and 1893. 



C. H. Quereau, 


Eng'g News 


March 8, 189 


4. 








N. Y. C. & H. R. R. 


Pennsylvania R. R. 




Through 
Passenger. 


Through 
Freight. 


Through 
Passenger. 


Through 
Freight. 




1882. 


1893. 


1882. 


1893. 


1882. 


1893. 


1882. 


1893. 


Grate surface, sq. ft " 

Heating surface, sq. ft.. 


17.87 

1353 

50 

70 

150 

17X24 

hy A 

1/16 

% 



15« 

Am. 


27.3 
1821 

58 
78,86 

180 
19X24 

1/16 
1 


18 

1U 

Am. 


17.87 
1353 

50 

64 

150 

17X24 

5M 

1/16 

% 
1/162 

15V6 

m 

Am. 


29.8 
1763 
58 
67 
160 
19X26 

1/16 

% 

3/32Z 

18 

Mog. 


17.6 
1057 
50 
62 
125 
17X24 

5 
1/16 

% 


16 

m 

Am. 


33.2 

1583 
57 
78 
175 
I8ix2± 



V&cl 

Am. 


23. 
1260 

54 

50 

125 

-.20X24 

5 

Va 

U 
1/221 

16 

Cons. 


31.5 
1498 
60 


Driver, diam., in 

Steam- pressure, lbs. .. 
Cylin., diam. and stroke. 

Valve-travel, ins . . . 

Lead at full gear, ins 


50 

140 

20X24 

5 

1/16 

1/32Z 

16 

1% 

Cons. 


Inside lap or clearance. . 

Sream-ports, length 

width 

Type of engine 



Two-cylinder Compound. 


Single-expansion. 


Revolu- 
tions. 


Speed, 

miles per 

hour. 


Water 
per I.H.P. 
per hour. 


Revolu- 
tions. 


Miles per 
Hour. 


Water. 


100 to 150 
150 " 200 
200 " 250 
250 " 275 


21 to 31 
31 " 41 
41 " 51 
51 " 56 


18.33 lbs. 
18.9 " 
19.7 " 
21.4 " 


151 
219 
253 
307 
321 


31 
45 
52 
63 
66 


21.70 
20.91 
20.52 
20.23 
20.01 



Indicated Water Consumption of Single and Compound 
Locomotive Engines at Varying Speeds. 

C . H. Quereau, Eng'g News, March 8, 1894. 



It appears that the compound engine is the more economical at low speeds, 
the economy decreasing as the speed increases, and that the single engine 
increases in economy with increase of speed within ordinary limits, becom- 
ing more economical than the compound at speeds of more than 50 miles 
per hour. 

The C, B. & Q. two-cylinder compound, which was about 30$ less eco- 
nomical than simple engines of the same class when tested in passenger 
service, has since been shown to be 15$ more economical in freight service 



ADVANTAGES OF COMPOUNDING, 863 

than the best single-expansion engine, and 29#~more economical than the 
average record of 40 simple engines of the some class on the same division. 
Indicator-tests of a Locomotive at High Speed. (Locomo- 
tive Eng'g, June, 189:1)— Cards were taken by Mr. Angus Sinclair on the 
locomotive drawing the Empire State Express. 

Results of Indicator-diagrams. 



Card No. 



3VS. 


Miles 
per hour. 


I.H.P. 


Card No 


160 


37.1 


648.3 


V 


260 


60.8 


728 


8 


190 


44 


551 


9 


250 


58 


891 


10 


260 


60 


960 


11 



Revs. -'- LH.P. 



Miles, 
per hour. 
304 70.5 977 

296 68.6 972 

300 69.6 1,045 

304 70.5 1,059 

340 78.9 1,120 

12 310 71.9 1,026 

The locomotive was of the eight-wheel type, built by the Schenectady 
Locomotive Works, with 19 X 24 in. cylinders, 78-in. drivers, and a large 
boiler and fire-box. Details of important dimensions are as follows : 
Heating-surface of fire-box, 150.8 sq. ft.: of tubes, 1670.7 sq. ft.; of boiler, 
1821.5 sq ft. Grate area, 27.3 sq. ft. Fire-box: length, 8 ft.; width, 3 ft 4% 
in. Tubes, 268; outside diameter, 2 in. Ports: steam, 18 X % in.; exhaust, 
18 X 2% in. Valve-travel, 5^ in. Outside lap, 1 in.; inside lap, 1/64 in. 
Journals: driving-axle, 8l£ X 10^2 in.; truck-axle, 6 x 10 in. 

The train consisted of four coaches, weighing, with estimated load, 340,000 
lbs. The locomotive and tender weighed in working order 200,000 lbs,, 
making the total weight of the train about 270 tons. During the time that 
the engine was first lifting the train into speed diagram No. 1 was taken. It 
shows a mean cylinder-pressure of 59 lbs. According to this, the power 
exerted on the rails to move the train is 6553 lbs., or 24 lbs. per ton. The 
speed is 37 miles an hour. When a speed of nearly 60 miles an hour was 
reached the average cylinder-pressure is 40.7 lbs., representing a total 
traction force of 4520 lbs., without making deductions for internal friction. 
If we deduct 10^ for friction, it leaves 15 lbs. per ton to keep the train going 
at the speed named. Cards 6, 7, and 8 represent the work of keeping the 
train running 70 miles an hour. They were taken three miles apart, when 
the speed was almost uniform. The average cylinder-pressure for the three 
cards is 47.6 lbs. Deducting \0% again for friction, this leaves 17.6 lbs. per 
ton as the pow r er exerted in keeping the train up to a velocity of 70 miles. 
Throughout the trip 7 lbs. of water were evaporated per lb. of coal. The 
work of pulling the train from New York to Albany was done on a coal con- 
sumption of about 3% lbs. per H.P. per hour. The highest power recorded 
was at the rate of 1120 H.P. 

Locomotive-testing Apparatus at the Iiaboratory of 
Purdue University. (W. F. M. Goss, Trans. A. S. M. E., vol. xiv. 826 )— 
The locomotive is mounted with its drivers upon supporting wheels which 
are carried by shafts turning in fixed bearings, thus allowing the engine to 
be run without changing its position as a whole. Load is supplied by four 
friction-brakes fitted to the supporting shafts and offering resistance to the 
turning of the supporting wheels. Traction is measured by a dynamometer 
attached to the draw-bar. The boiler is fired in the usual way, and an 
exhaust-blower above the engine, but not in pipe connection with it, carries 
off all that may be given out at the stack. 

A Standard Method of Conducting Locomotive-tests is given in a report 
by a Committee of the A. S. M. E. in vol. xiv. of the Transactions, page 1312. 
'"Waste of Fuel in Locomotives.— In American practice economy 
of fuel is necessarily sacrificed to obtain greater economy due to heavy 
train-loads. D. L. Barnes, in Eng. Mag., June, 1894, gives a diagram showing 
the reduction of efficiency of boilers due to high rates of combustion, from 
which the following figures are taken: 

Lbs. of coal per sq. ft. of grate per hour 12 40 80 120 160 200 

Per cent efficiency of boiler 80 75 67 59 51 43 

A rate of 12 lbs. is given as representing stationary-boiler practice, 40 lbs. 
is English locomotive practice, 120 lbs. average American, and 200 lbs. max- 
imum American, locomotive practice. 

Advantages of Compounding.— Report of a Committee of the 
American Railway Master Mechanics' Association on Compound Locomotives 
(Am. Mach., July 3, 1890) gives the following summary of the advantages 
gained by compounding: (a) It has achieved a saving in the fuel burnt 
averaging 18$ at reasonable boiler pressures, with encouraging possibilities 



864 LOCOMOTIVES. 

of further improvement in pressure and in fuel and water economy. (6) It 
has lessened the amount of water (dead weight) to be hauled, so that (c) the 
tender and its load are materially reduced in weight, (d) It has increased 
the possibilities of speed far beyond 60 miles per hour, without unduly 
straining the motion, frames, axles, or axle-boxes of the engine, (e) It has 
increased the haulage-power at full speed, or, in other words, has increased 
the continuous H.P. developed, per given weight of engine and boiler. (/) In 
some classes has increased the starting-power, (g) It has materially lessened 
the slide-valve friction per H.P. developed, (h) It has equalized or distrib- 
uted the turning force on the crank-pin, over a longer portion of its path, 
which, of course, tends to lengthen the repair life of the engine, (i) In the 
two-cylinder type it has decreased the oil consumption, and has even done 
so in the Woolf four-cylinder engine. (,;') Its smoother and steadier draught 
on the fire is favorable to the combustion of all kinds of soft coal; and the 
sparks thrown being smaller and less in number, it lessens the risk to prop- 
erty from destruction by fire, (k) These advantages and economies are 
gained without having to improve the man handling the engine, less being 
left to his discretion (or careless indifference) than in the simple engine. (I) 
Valve-motion, of every locomotive type, can be used in its best working and 
most effective position, (m) A wider elasticity in locomotive design is per- 
mitted; as, if desired, side-rods can be dispensed with, or articulated engines 
of 100 tons weight, with independent trucks, used for sharp curves on moun- 
tain service, as suggested by Mallet and Brunner. 

Of 2? compound locomotives in use on the Phila. and Reading Railroad (in 
1892), 12 are in use on heavy mountain grades, and are designed to be the 
equivalent of 22 X 24 in. simple consolidations; 10 are in somewhat lighter 
service and correspond to 20 x 24 in. consolidations; 5 are in fast passenger 
service. The monthly coal record shows: 

Class of Engine. No. «£"£-' 

Mountain locomotives 12 25fctoS0% 

Heavy freight service 10 12% to 11% 

Fast passenger 5 % to 11$ 

(Report of Com. A. R. M. M. Assn. 1892.) For a description of the various 
types of compound locomotive, with discussion of their relative merits, see 
paper by A. Von Borries, of Germany, The Development of the Compound 
Locomotive, Trans. A. S. M. E. 1893, vol. xiv., p. 1172. 

Counterbalancing Iioconiotives.— The following rules, adopted 
by different locomotive- builders, are quoted in a paper by Prof. Lanza 
(trans. A. S. M. E., x. 302): 

A. " For the main drivers, place opposite the crank-pin a weight equal to 
one half the weight of the back end of the connecting-rod plus one half the 
weight of the front end of the connecting-rod, piston, piston-rod, and cross- 
head. For balancing the coupled wheels, place a weight opposite the crank- 
pin equal to one half the parallel rod plus one half of the weights of the 
front end of the main-rod, piston, piston-rod, and cross-head. The centres 
of gravity of the above weights must be at the same distance from the 
axles as the crank-pin." 

B. The rule given by D. K. Clark : " Find the separate revolving weights 
of crank-piu boss, coupling-rods, and connecting-rods for each wheel, also 
the reciprocating weight of the piston and appendages, and one half the 
connecting-rod, divide the reciprocating weight equally between each wheei 
and add the part so allotted to the revolving weight on each wheel: the 
sums thus obtained are the weights to be placed opposite the crank-pin, and 
at the same distance from the axis. To find the counterweight to be used 
when the distance of its centre of gravity is known, multiply the above 
weight by the length of the crank in inches and divide by the given dis- 
tance. 11 This rule differs from the preceding in that the same weight is 
placed in each wheel. 

C. " W= ^ — — , in which S = one half the stroke, G = distance 

from centre of wheel to centre of gravity in counterbalance, w -■ weight at 
crank-pin to be balanced, W = weight in counterbalance, / = coefficient of 
friction so called, — 5 in. ordinary practice. The reciprocating weight is 
found by adding together the weights of the piston, piston-rod, cross-head, 
and one half of the main rod. The revolving weight for the main wheel is 
found by adding together the weights of the crank-pin hub, crank-pin, one 



PETROLEUM-BURNING LOCOMOTIVES. 865 

half of the main rod, and one half of each parallel-rod connecting to this 
wheel; to this add the reciprocating weight divided by the number of 
wheels. The revolving weight for the remainder of the wheels is found in 
the same manner as for the main wheel, except one half of the main rod is 
not added. The weight of the crank-pin hub and the counterbalance does 
not include the weight of the spokes, but of the metal inclosing them. This 
calculation is based for one cylinder and its corresponding wheels. 1 ' 

D. "Ascertain as nearly as possible the weights of crank-pin, additional 
weight of wheel boss for the same, add side rod, and main connections, 
piston-rod and head, with cross-head on one side: the sum of these multi- 
plied by the distance in inches of the centre of the crank-pin from the centre 
of the wheel, and divided by the distance from the centre of the wheel to 
the common centre of gravity of the counterweights, is taken for the total 
counterweight for that side of the locomotive which is to be divided among 
the wheels on that side." 

E. " Balance the wheels of the locomotive with a weight equal to the 
weights of crank-pin, crank-pin hub, main and parallel rods, brasses, etc., 
plus two thirds of the weight of the reciprocating parts (cross-head, piston 
and rod and packing)'" 

F. '• Balance the weights of the revolving parts which are attached to 
each wheel with exactness, and divide equally two thirds of the weights of 
the reciprocating parts between all the wheels. One half of the main rod is 
computed as reciprocating, and the other as revolving weight.' 1 

See also articles on Counterbalancing Locomotives, in R. R. & Eng. Jour., 
March and April, 1890, and a paper by W. F. M. Goss, in Trans. A. S. M. E., 
vol xvi. 

Maximum Safe Load for Steel Tires on Steel Rails. 
(A. S. M. E., vii., p. 786.)— Mr. Chanute's experiments led to the deduction 
that 12,000 lbs. should be the limit of load for any one driving-wheel. Mr. 
Angus Sinclair objects to Mr. Chanute's figure of 12,000 lbs., and says that 
a locomotive tire which has a light load on it is more injurious to the rail 
than one which has a heavy load. In English practice 8 and 10 tons are 
safely used. Mr. Obeiiin Smith has used steel castings for cam-rollers 4 in. 
diam. and 3 in. face, which stood well under loads of from 10,000 to 20,000 
lbs. Mr. C. Shaler Smith proposed a formula for the rolls of a pivot- bridge 
which may be reduced to the form : Load = 1760 x face X Vdiam., all in 
lbs. and inches. 

See dimensions of some large American locomotives on pages 860 and 861. 
On the " Decapod " the load on each driving-wheel is 17,000 lbs., and on 
"No. 999," 21.000 lbs. 

Narrow-gauge Railways in Manufacturing Works.— 
A tramway of 18 inches gauge, several miles in length, is in the works of 
tb,e Lancashire and Yorkshire Railway. Curves of 13 feet radius are used. 
The locomotives used have the following dimensions (Proc. Inst. M. E.. July, 
1888): The cylinders were 5 in. diameter with 6 in. stroke, and 2 ft. 3*4 in. 
centre to centre. The wheels were 16J4 in- diameter, the wheel-base 

2 ft. 9 in. ; the frame 7 ft. 4J4 in. long, and the extreme width of the engine 

3 feet. The boiler, of steel, 2 ft. 3 in. outside diameter and 2 ft. long between 
tube plates, containing 55 tubes of 1% in. outside diameter; the fire-box, of 
iron and cylindrical, 2 ft. 3 in. long and 17 in. inside diameter. The heating- 
surface 10*42 sq. ft. in the fire-box and 36.12 in the tubes, total 46.54 sq. ft.; 
the grate-area, 1.78 sq. ft.; capacity of tank, 26J/£ gallons; working-pressure, 
170 lbs. per sq. in.; tractive power, say, 1412 lbs., or 9.22 lbs. per lb. of effec- 
tive pressure per sq. in. on the piston. Weight, when empty, 2.80 tons; 
When full and in working order, 3.19 tons. 

For description of a system of narrow-gauge railways for manufactories, 
see circular of the C. W. Hunt Co., New York. 

Light Locomotives.— For dimensions of light ocomotives used for. 
mining, etc., and for much valuable information concerning them, see cata- 
logue of H K. Porter & Co., Pittsburgh. 

Petroleum-burning Locomotives. (From Clark's Steam-en- 
gine.)— The combustion of petroleum refuse in locomotives has been success 
f idly practised by Mr. Thos. Urquhart. on the Grazi and Tsaritsin Railway, 
Southeast Russia. Since November, 1884, the whole stock of 143 locomotives 
under his superintendence has been fired with petroleum refuse. The oil is 
injected from a nozzle through a tubular opening in the back of the fire-box, 
by means of a jet of steam, with an induced current of air. 

A brickwork cavity or "regenerative or accumulative combustion-cham- 
ber" is formed in the fire-box, into which the combined current breaks as 



866 LOCOMOTIVES. 

spray against the rugged brickwork slope. In this arrangement the brick- 
work is maintained at a white heat, and combustion is complete and smoke- 
less. The form, mass, and dimensions of the brickwork are the most im- 
portant elements in such a combination. 

Compressed air was tried instead of steam for injection, but no appreciable 
reduction in consumption of fuel was noticed. 

The heating-power of petroleum refuse is given as 19,832 heat-units, 
equivalent to the evaporation of 20.53 lbs. of water from and at 212° F., or to 
17.1 lbs. at 8^ atmospheres, or 125 lbs. per sq. in., effective pressure. The 
highest evaporative duty was 14 lbs. of water under 8J^ atmospheres per lb. 
of the fuel, or nearly 82$ efficiency. 

There is no probability of any extensive use of petroleum as fuel for loco- 
motives in the United States, on account of the unlimited supply of coal and 
the comparatively limited supply of petroleum. 

Fireless Ijocomotive.— The principle of the Francq locomotive is 
that it depends for the supply of steam on its spontaneous generation from 
a body of heated water in a reservoir. As steam is generated and drawn 
off the pressure falls; but by providing a sufficiently large volume of water 
heated to a high temperature, at a pressure correspondingly high, a margin 
of surplus pressure may be secured, and means may thus be provided for 
supplying the required quantity of steam for the trip. 

The fireless locomotive designed for the service of the Metropolitan Rail- 
way of Paris has a cylindrical reservoir having segmental ends, about 5 ft. 
7 in. in diameter, 26J4 ft. in length, with a capacity of about 620 cubic feet. 
Four fifths of the capacity is occupied by water, which is heated by the aid 
of a powerful jet of steam supplied from stationary boilers. The water is 
heated until equilibrium is established between the boilers and the reser- 
voir. The temperature is raised to about 390° F., corresponding to 225 lbs. 
per sq. in. The steam from the reservoir is passed through a reducing- 
valve, by which the steam is reduced to the required pressure. It is then 
passed through a tubular superheater situated within the receiver at the 
upper part, and thence through the ordinary regulator to the cylinders. 
The exhaust-steam is expanded to a low pressure, in order to obviate noise 
of escape. In certain cases the exhaust-steam is condensed in closed 
vessels, which are only in part filled with water. In the upper free space a 
pipe is placed, into which the steam is exhausted. Within this pipe another 
pipe is fixed, perforated, from which cold water is projected into the sur- 
rounding steam, so as to effect the condensation as completely as may be. 
The heated water falls on an inclined plane, and flows off without mixing 
with the cold water. The condensing water is circulated by means of a 
centrifugal pump driven by a small three -cylinder engine. 

In working off the steam from a pressure of 225 lbs. to 67 lbs., 530 cubic 
feet of water at 390° F.as sufficient for the traction of the trains, for working 
the circulating-pump for the condensers, for the brakes, and for electric- 
lighting of the train. At the stations the locomotive takes from 2200 to 3300 
lbs. of steam — nearly the same as the weight of steam consumed during the 
run between two consecutive charging stations. There is 210 cubic feet of 
condensing water. Taking the initial temperature at 60° F., the tempera- 
ture rises to about 180° F. after the longest runs underground. 

The locomotive has ten wheels, on a base 24 ft. long, of which six are 
coupled, 4]4 ft- in diameter. The extreme wheels are on radial axles. The 
cylinders are 23^j in. in diameter, with a stroke of 23^ in. 

The engine weighs, in working order, 53 tons, of which 36 tons are on the 
coupled wheels. The speed varies from 15 miles to 25 miles per hour. The 
trains weigh about 140 tons. 

Compressed-air Iiocomotives.— For an account of the Mekarski 
system of compressed-air locomotives see page 509, ante. 



SHAFTIKG. 



867 



SHAFTING. 

(See also Torsional Strength; also Shafts op Steam-engines.) 
For diameters of shafts to resist torsional strains only, Molesworth gives 

3/pf 

d = A/ — , in which d = diameter in inches, P = twisting force in pounds 

applied at the end of a lever-arm whose length is I in inches, K = a coeffi- 
cient whose values are, for cast iron 1500, wrought iron 1700, cast steel 3200, 
gun-bronze 460, brass 425, copper 380, tin 220, lead 170. The value given for 
cast steel probably applies only to high-carbon steel. 
Thurston gives: 



For head shafts well 
supported against 
springing: 



For line shafting, m 
hangers 8 ft. apart: 



For transmission sim- 
ply, no pulleys: 



H.P. = 



d*R . 
125' 

d a R 



3/125 H.P. 
-V-R-' 



for iron; 



-*, for cold-rolled iron. 



H. P . = *^ = ;/?^P-, for iron; 

90 y R 



, 3 /55 H.P 



for cold-rolled iron. 



?:R = i | tf= |/^, for)ron; 



-; d 



=f 3 -^. 



for cold-rolled iron. 



H.P. = horse-power transmitted, d = diameter of shaft in inches, R = rev- 
olutions per minute. 

.-,. , 3 /100 H.P. 
J. B. Francis gives for turned-iron shafting d = ju — — — . 

Jones and Laughlins give the same formulae as Prof. Thurston, with the 
following exceptions: For line shafting, hangers 8 ft. apart: 

c oW -.-onedi,-on,H.P. = ^,<i = |/^. 

For simply transmitting power and short counters: 

„,, d*R , V50H.P.. 
turned iron, H.P. = — , d = 4/ — — — ; 

,, „ A . „ „ dm , V30H.P. 

cold-rolled iron, H.P. = -^- , d = A/ — — — . 

They also give the following notes: Receiving and transmitting pulleys 
should always be placed as close to hearings as possible; and it is good prac- 
tice to frame short " headers " between the main tie-beams of a mill so as 
to support the main receivers, carried by the head shafts, with a bearing 
close to each side as is contemplated in the formulae. But if it is preferred, 
or necessary, for the shaft to span the full width of the " bay " without in- 



868 



SHAFTING. 



termediate bearings, or for the pullej r to be placed away from the bearings 
towards or at the middle of the bay, the size of the shaft must be largely 
increased to secure the stiffness necessary to support the load without un- 
due deflection. Shafts may not deflect more than 1/80 of an inch to each 
foot of clear length with safety. 

To find the diameter of shaft necessary to carry safely the main pulley at 
the centre of a bay: Multiply the fourth' power of the diameter obtained by 
above formulae by the length of the " bay," and divide this product by the 
distance from centre to centre of the bearings when the shaft is supported 
as required by the formula. The fourth root of this quotient will be the 
diameter required. 

The following table, computed by this rule, is practically correct and safe. 



rieter of 
ift given 
the For r 
lee for 
d Shafts. 


Diameter of Shaft necessary to carry the Load at the Centre of 
a Bay, which is from Centre to Centre of Bearings 




m ft. 


3 ft. 


m ft. 


4 ft. 


5 ft. 


6 ft. 


8 ft, 


10 ft, 


in. 
2 

m 

4 


in. 

m 

2^ 
3 


in. 

2% 

4 


in. 

m 
m 
m ' 

5 


in. 

2^ 
2V 8 

m 

3M 

m 

4V 8 
6 


in. 

2% 
3 

4 
Ws 

m 


in. 

2% 

m 

4H 

m 

5% 

6 

6% 


in. 

2% 
3% 
4 

5*1 

5^ 
6 

ey 2 


in. 
3 

m 

5% 
5% 
6^ 
6% 

m 






6 























As the strain upon a shaft from a load upon it is proportional to the 
product of the parts of the shaft multiplied into each other, therefore, 
should the load be applied near one end of the span or bay instead of at the 
centre, multiply the fourth power of the diameter of the shaft required to 
carry the load at the centre of the span or bay by the product of the two 
parts of the shaft when the load is near one end, and divide this product by 
the product of the two parts of the shaft when the load is carried at the 
centre. The fourth root of this quotient will be the diameter required. 

The shaft in a line which carries a receiving-pulley, or which carries a 
transmitting-pulley to drive another line, should always be considered a 
head -shaft, and should be of the size given by the rules for shafts carrying 
main pulleys or gears. 

Deflection of Shafting. (Pencoyd Iron Works.)— As the deflection 
of steel and iron is practically alike under similar conditions of dimensions 
and loads, and as shafting is usually determined by its transverse stiffness 
rather than its ultimate strength, nearly the same dimensions should be 
used for steel as for iron. 

For continuous line shafting it is considered good practice to limit the 
deflection to a maximum of 1/100 of an inch per foot of length. The weight 
of bare shafting in pounds = 2.6d 2 L = W, or when as fully loaded with 
pulleys as is customary in practice, and allowing 40 lbs. per inch of width 
for the vertical pull of the belts, experience shows the load in pounds to be 
about 13d 2 £ = W. Taking the modulus of transverse elasticity at 26,000,000 
lbs., we derive from authoritative formulae the following: 



^873^2, d = a / ~, for bare shafting; 



73' 



L = \ I75d 2 , d = if — , for shafting cai-rying pulleys,'etc. ; 

L being the maximum distance in feet between bearings for continuous 
shafHnsr subjected to bending stress alone, d = diam. in inches. 

The torsional stress is inveiselv proportional to the velocity of rotation, 
while the bending stress will not be reduced in the same ratio. It is there- 
fore impossible to write a formula covering the whole problem and sufiri- 



HORSE-POWER AT DIFFERENT SPEEDS. 



869 



ciently simple for practical application, but the following rules are correct 
within the range of velocities usual in practice. 

For continuous shafting so proportioned as to deflect not more than 1/100 
of an inch per foot of length, allowance being made for the weakening 
effect of key-seats, 

d = I/ 50H ' F -, L=V 720d 2 , for bare shafts; 

d = a/ -— — , L = \ 140d 2 , for shafts carrying pulleys, etc. 

d = diam. in inches, L = length in feet, R = revs, per min. 

The following table (by J. B. Francis) gives the greatest admissible dis- 
tances between the hearings of continuous shafts subject to no transverse 
strain except from their own weight, as would be the case were the power 
given off from the shaft equal on all sides, and at an equal distance from 
the hanger-bearings. 



Distance between 
Bearings, in ft. 

Diam. of Shaft, Wrought-iron Steel 

in inches. Shafts. Shafts 

2 15.46 15.89 

3 17.70 18.19 

4 19.48 20.02 

5 2C.99 21.57 



Distance between 

k Bearings, in ft. 

Diam.of Shaft, Wrought-iron Steel 

in inches. Shafts. Shafts. 

6 22.30 22.92 

7 23.48 24.13 

8 24.55 25.23 

9 25.53 26.24 



These conditions, however, do not usually obtain in the transmission of 
power by belts and pulleys, and the varying circumstances of each case 
render it' impracticable to give any rule which would be of value for univer- 
sal application. 

For example, the theoretical requirements would demand that the bear- 
ings be nearer together on those sections of shafting where most power 
is delivered from the shaft, while considerations as to the location and 
desired contiguity of the driven machines may render it impracticable to 
separate the driving-pulleys by the intervention of a hanger at the theo- 
retically required location. (Joshua Rose.) 

Horse-power Transmitted by Turned Iron Shafting at 
Different Speeds. 

As Prime Mover or Head Shaft carrying Main Driving-pulley or Gear, 
well supported by Bearings. Formula : H.P. == d 3 R -s- 125. 



F\,,£ 






Number of Revolutions pei 


Minute. 






.2 °S 


60 


80 


100 


125 


150 


175 


200 


225 


250 


275 


300 


Ins. 


H.P. 


H.P. 


H.P. 


H.P. 


H.P. 


H.P. 


H.P. 


H.P. 


H.P. 


H.P. 


H.P. 


m 


2.6 


3.4 


4.3 


5.4 


6.4 


7.5 


8.6 


9.7 


10.7 


11.8 


12.9 


2 


3.8 


5.1 


6.4 


8 


9.6 


11.2 


12.8 


14.4 


16 


17.6 


19.2 


2H 


5.4 


7.3 


8.1 


10 


12 


14 


16 


18 


20 


22 


24 


2J4 


7.5 


10 


12.5 


15 


18 


22 


25 


28 


31 


34 


37 


2% 


10 


13 


16 


20 


24 


28 


32 


36 


40 


44 


48 


3 


13 


17 


20 


25 


30 


35 


40 


45 


50 


55 


60 


334 


16 


22 


27 


34 


40 


47 


54 


61 


67 


74 


81 


3^2 


20 


27 


34 


42 


51 


59 


68 


76 


85 


93 


102 


3% 


25 


33 


42 


52 


63 


73 


84 


94 


105 


115 


126 


4 


30 


41 


51 


64 


76 


89 


102 


115 


127 


140 


153 


4)4 


43 


58 


72 


90 


108 


126 


144 


162 


180 


198 


216 


5 


60 


80 


100 


125 


150 


175 


200 


225 


250 


275 


300 


534 


80 


106 


133 


166 


199 


233 


266 


299 


333 


366 


400 



870 








SHAFTIKG 












As Second Movers or Line-shafting, Bearings 8 ft. apart. 


Formula : H.P. = d 3 R -s- 90. 


s\,e 


Number of Revolutions per Minute. 


.5 ©J 






















O 02 


100 


125 


150 


175 


200 


225 


250 


275 


300 


325 
H.P. 


350 


Ins. 


H.P. 


H.P. 


H.P. 


H.P. 


H.P. 


H.P. 


H.P. 


H.P. 


H.P. 


H.P. 


1% 

l 7 ^ 


6 


7.4 


8.9 


10.4 


11.9 


13.4 


14.9 


16.4 


17.9 


19.4 


20.9 


7.3 


9.1 


10.9 


12.7 


14.5 


16.3 


18.2 


20 


21.8 


23.6 


25.4 


a 


8.9 


11.1 


13.3 


15.5 


17.7 


20 


22.2 


24.4 


26.6 


28.8 


31 


m 


10.6 


13.2 


15.9 


18.5 


21.2 


23.8 


26.5 


29.1 


31.8 


34.4 


37 


2y 4 


12.6 


15.8 


19 


22 


25 


28 


31 


35 


38 


41 


44 


2% 


15 


18 


22 


26 


29 


33 


37 


41 


44 


48 


52 


2^ 


17 


21 


26 


30 


34 


39 


43 


47 


52 


56 


60 


2%j 


23 


29 


34 


40 


46 


52 


58 


64 


69 


75 


81 


3 


30 


37 


45 


52 


60 


67 


75 


82 


90 


97 


105 




38 


47 


57 


66 


76 


85 


95 


104 


114 


123 


133 


47 


59 


71 


83 


95 


107 


119 


131 


143 


155 


167 


m 


58 


73 


88 


102 


117 


132 


146 


162 


176 


190 


205 


4 


71 


89 


107 


125 


142 


160 


178 


196 


213 


231 


249 


For Simply Transmitting Power. 


Formula : H.P. = d 3 R -v- 50. 


B^4 


Number of Revolutions per Minute. 


cS o 5 


100 


125 


150 


175 


200 


233 


267 


300 


333 


367 


400 


Ins. 


H.P. 


H.P. 


H.P. 


H.P. 


H.P. 


H.P. 


H.P. 


H.P. 


H.P. 


H.P. 


H.P. 


1^ 


6.7 


8.4 


10.1 


11.8 


13.5 


15.7 


17.9 


20.3 


22.5 


24.8 


27.0 


1% 


8.6 


10.7 


12.8 


15 


17.1 


20 


22.8 


25.8 


28.6 


31.5 


34.3 


1M 


10.7 


13.4 


16 


18.7 


21.5 


25 


28 


32 


36 


39 


43 


i% 


13.2 


16.5 


19.7 


23 


26.4 


31 


35 


39 


44 


48 


52 


2 


16 


20 


24 


28 


32 


37 


42 


48 


53 


58 


64 


2V« 


19 


24 


29 


33 


38 


44 


51 


57 


63 


70 


76 


m 


22 


28 


34 


39 


45 


52 


60 


68 


75 


83 


90 


2% 


27 


33 


40 


47 


53 


62 


70 


79 


88 


96 


105 


2J/ 2 


31 


39 


47 ■ 


54 


62 


73 


83 


93 


104 


114 


125 


2% 


41 


52 


62 


73 


83 


97 


111 


125 


139 


153 


167 


3 


54 


67 


81 


94 


108 


126 


144 


162 


180 


198 


216 


m 


68 


86 


103 


120 


137 


160 


182 


205 


228 


250 


273 


Wz 


85 


107 


128 


150 


171 


200 


228 


257 


285 


313 


342 


Horse-power Transmitted by Cold-rolled Iron Shafting 


at Different Speeds. 


As Prime Mover or Head Shaft carrying Main Driving-pulley or 


Gear, well supported by Bearings. Formula : H.P. = d a B -s- 75. 


F : -,,^ 


Number of Revolutions per Minute. 


SO J 

O 02 


60 


80 


100 


125 
H.P. 


150 


175 


200 


225 


250 


275 


300 


Ins. 


H.P. 


H.P. 


H.P. 


H.P. 


H.P. 


H.P. 


H.P. 


H.P. 


H.P. 


H.P. 


W> 


2.7 


3.6 


4.5 


5.6 


6.7 


7.9 


9.0 


10 


11 


12 


13 


m 


4.3 


5.6 


7.1 


8.9 


10.6 


12.4 


14.2 


16 


18 


19 


21 


2 


6.4 


8.5 


10.7 


13 


16 


19 


21 


24 


26 


29 


32 


2^ 


9 


12 


15 


19 


23 


26 


30 


34 


38 


42 


46 


2^2 


12 


17 


21 


26 


31 


36 


41 


47 


52 


57 


62 


2% 


16 


22 


27 


35 


41 


48 


55 


62 


70 


76 


82 


3 


21 


29 


36 


45 


54 


63 


72 


81 


90 


98 


108 


3M 


27 


36 


45 


57 


68 


80 


91 


103 


114 


126 


136 


3J^ 


34 


45 


57 


71 


86 


100 


114 


129 


142 


157 


172 


3% 


42 


56 


70 


87 


105 


123 


140 


158 


174 


193 


210 


4 


51 


69 


85 


106 


128 


149 


170 


192 


212 


244 


256 


4^ 


73 


97 


121 


151 


182 


212 


243 


273 


302 


333 


364 



























HORSE-POWER AT DIFFERENT SPEEDS. 



As Second Movers or Line-shafting, Bearings 8 ft. apart. 
Formula : H.P. = d s R -f- £0. 



s £ 






Number of Revolutions per 


Minute. 




cS O Jj 
























5 </2 


100 


125 


150 


175 


200 


225 


250 


275 


300 


325 


350 


Ins. 


H.P. 


H.P. 


H.P. 


H.P. 


H.P. 


H.P. 


H.P. 


H.P. 


H.P. 


H.P. 


H.P. 


IV 9 . 


6.7 


8.4 


10.1 


11.8 


13.5 


15.2 


16.8 


18.5 


20.2 


21.9 


23.6 


Wh 


8.6 


10.7 


12.8 


15 


17.1 


19.3 


21.5 


23.6 


25.7 


28.9 


31 




10.7 


13.4 


16 


18.7 


21.5 


24.2 


26.8 


29.5 


32.1 


34.8 


39 


13.2 


16.5 


19.7 


23 


26.4 


29.6 


32.9 


36.2 


39.5 


42.8 


46 


2 


16 


20 


24 


28 


32 


36 


40 


44 


48 


52 


56 


2*4 


19 


24 


29 


33 


38 


43 


48 


52 


57 


62 


67 




22 


28 


34 


39 


45 


50 


56 


61 


68 


74 


80 


27 


33 


40, 


47 


53 


60 


67 


73 


80 


86 


94 


m 


31 


39 


47 


54 


62 


69 


78 


86 


93 


101 


109 


41 


52 


62 


73 


83 


93 


104 


114 


125 


135 


145 


3 


54 


67 


81 


94 


108 


121 


134 


148 


162 


175 


189 


334 


68 


86 


103 


120 


137 


154 


172 


188 


205 


222 


240 


w* 


85 


107 


128 


150 


171 


192 


214 


235 


257 


278 


300 



For Simply Transmitting Power and Short Counters. 
Formula : H.P. = d a B -*- 30. 



£ <w 


Number of Revolutions per Minute. 


c3 O j2 
























5 w 


100 


125 


150 


175 


200 


233 


267 


300 


333 


367 


400 


Ins. 


H.P. 


H.P. 


H.P. 


H.P. 


H.P. 


HP. 


H.P. 


H.P. 


H.P. 


H.P. 


H.P. 


VA 


6.5 


8.1 


9.7 


11.3 


13 


15.2 


17.4 


19.5 


21.7 


23.9 


26 


W& 


8.5 


10.7 


12.8 


15 


17 


19.8 


22.7 


25.5 


28.4 


31 


34 


114 


11.2 


14 


16.8 


19.6 


22.5 


26 


30 


33 


37 


41 


45 


Ws 


14.2 


17.7 


21.2 


24.8 


28.4 


33 


38 


42 


47 


52 


57 


1% 


18 


22 


27 


31 


35 


41 


47 


53 


59 


65 


71 


1% 


22 


27 


33 


38 


44 


51 


58 


65 


72 


79 


87 


2 


26 


33 


40 


46 


53 


62 


71 


80 


88 


97 


106 


2^ 


32 


40 


47 


55 


63 


73 


84 


95 


105 


116 


127 


2J4 


38 


47 


57 


66 


76 


89 


101 


114 


127 


139 


152 


m 


44 


55 


66 


77 


88 


103 


118 


133 


148 


163 


178 


2^2 


52 


65 


78 


91 


104 


121 


138 


155 


172 


190 


207 


m 


69 


84 


99 


113 


138 


181 


184 


207 


231 


254 


277 


3 


90 


112 


135 


157 


180 


210 


240 


270 


300 


330 


360 



Speed of Shafting.— Machine shops 120 to 180 

Wood-working 250 to 300 

Cotton and woollen mills 300 to 400 

There are in some factories lines 1000 ft. long, the power being applied at 
the middle. 

Hollow Shafts.— Let d be the diameter of a solid shaft, and d ± d 2 the 
external and internal diameters of a hollow shaft of the same material. 

Then the shafts will be of equal torsional strength when d 3 = — — 

di 
A 10-inch hollow shaft with internal diameter of 4 inches will weigh 16$ less 
than a solid 10-inch shaft, b at its strength will be only 2.56$ less. If the hole 
were increased to 5 inches diameter the weight would be 25# less than that 
of the solid shaft, and the strength 4.25$ less. 

Table lor laying Out Shafting.— The table on the opposite page 
Cfrom the Stevens Indicator, April, 1892) is used by Wm. Sellers & Co. to 
facilitate the laying out of shafting. 

The wood-cuts at the head of this table show the position of the hangers 
and position of couplings, either for the case of extension in both directions 
from a central head-shaft or extension in one direction from that head-shaft, 



872 



TABLE FOR LAYING OUT SHAFTING. 



II f 



v be 

c o 



•saqoui 



•satpui 






sut 'xog JO 'Sui 
-xeaa jo mSua'fi 



tOt-QOfflOi-iMM^CQOCNTPOCDO^ 



c? 






f P.w -S 60 , 

53 2«|o" 



S^o^+ 



>is^:«+ 



Jos ^g^ 
^ cilia 












c*c-§^ M 






Oi COCO CO 

C4C^O?COff 



o« w <?4 cj <n CO 



H ■* 10 SO 00 CT. 



3SS ^ $ 

aocsoeseoTtico^oc 
— ■ r-« 04 oi ot <?> ^ C4 <y 






c+3 m M a£,i, 
~'S 60 cS c *« 
P.\-S- c-C t, 

C^iS-o^^g 
1-1 d<u anbo 



PROPORTIONS OF PULLEYS. £73 



PULLEYS. 

Proportions of Pulleys. (See also Fly-wheels, pages 820 to 823.)— 
>t u = number of arms, D = diameter of pulley, S = thickness of belt. / = 
hiekness of rim at edge, T = thickness in middle, B = width of rim, /3 = 
viclrh of belt, h = breadth of arm at hub, h 1 = breadth of arm at rim, e = 
hickness of arm at hub e x = thickness of arm at rim, c = amount of crown- 
ng; dimensions in inches. 

Unwin. Reuleaux. 

5 = widthof rim..., 9/8 ((8 f 0.4) ,,,9/8/3 to 5/4/3 

t = thickness at edge of rim . 
T= " " middle of i-in: 



m . 75" + . 005D -[ (tl j/^ ^y^ ' ) 

2t + c .." 

For single , / BD 

belts = .6337 j/ ^ YaT B D_ 

For double \ / BD 4 4 20?i 



\ = breadth of arm at hub.. 



\ For double . / - 
/ belts = .798y 



li= " " " " rim %h O.SJi 

: thickness of arm at hub. 0.4ft. 0.5/i 

" " " rim 0.47i x 0.5/i x 

number of arms, for a} „ , BD ,,/► D \ 

single set, V'"; 3 + T 5 V A 5X 2b) 

t i™~fi, nf v,„k i not ^ ess than 2.5S, \ Bfor sin. -arm pulleys. 

L = length of hub -j is of fcen % ^ j-^ u ^^^.^ ,. 

M— thickness of metal in hub h to %h 

crowning of pulley 1/245 



The number of arms is really arbitrary, and may be altered if necessary. 
(Unwin.) 

Pulleys with two or three sets of arms may be considered as two or three 
separate pulleys combined in one, except that the proportions of the arms 
should be 0.8 or 0.7 time that of single-arm pulleys. (Reuleaux.) 

Example.— Dimensions of a pulley 60" diam., 16" face, for double belt y^' 
thick. 

Solution by n h h x e e^ t T L M c 

Unwin 9 3.79 2.53 1.52 1.01 .65 1.97 10.7 3.8 .67 

Reuleaux .... 4 5.0 4.0 2.5 2.0 1.25 16 5 

The following proportions are given in an article in the Amer. Machinist, 
authority not stated: 

h = .0625Z) + .5 in., h ± = .041) + 3125 in., e = .025D -j- .2 in., e x = .016D -f- 
125 in. 

These give for the above example: h — 4.25 in., /i t = 2.71 in., e = 1.7 in., 
'i = 1.09 in. The section of the arms in all cases is taken as elliptical. 

The following solution for breadth of arm is proposed by the author: 
Assume a belt pull of 45 lbs. per inch of width of a single belt, that the 
whole strain is taken in equal proportions on one half of the arms, and that 
the arm is a beam loaded at one end and fixed at the other. We have the 

formula for a beam of elliptical section fP = .0982 — j— , in which P = the 

load, R = the modulus of rupture of the cast iron, b = breadth, d = depth, 
and I = length of the beam, and/ = factor of safety. Assume a modulus 
of rupture of 36.000 lbs., a factor of safety of 10, and an additional allow- 
ance for safety in taking / = i^ the diameter of the pulley instead of y%D 

:ss the radius of the hub. 

Take d — h, the breadth of the arm at the hub, and b = e = 0.4/i, the 
^ ■ , TtT „ , ^„ „„ 45B nnn B 3535 X 0.4/i 3 

thickness. We then have fP = 10 x ^ = 900- = — tk , whence 

n^-2 n y%D 

3 /900BD " 3 /BD ,_. , . ,. „ ^ 

h — 4/ n ^r- = -633 i/ — , which is practically the same as the value 
y 6o6d)l y n 

reached by Unwin from a different set of assumptions. 



8U 



PULLEYS. 



Convexity of Pulleys.— Authorities differ. Morin gives a rise equal 
to 1/10 of the face; Molesworth, 1/24; others from y H to 1/96. Seott A. 
Smith says the crown should not be over y 8 inch for a 24-inch face. Pulleys 
for shifting belts should be " straight, 11 that is, without crowning. 

CONE OR STEP PULLEYS. 

To find the diameters for the several steps of a pair of cone-pulleys: 
1. Crossed Belts. — Let D and d be the diameters of two pulleys con- 
nected by a crossed belt, L = the distance between their centres, and /3 = 
the angle either half of the belt makes with a line joining the centres of the 

■n- •*■« 

pulleys : then total length of belt z 



/3 = angle whose sine is - 



2L 



i/--(^). 



The length of 



the belt is constant when D + d is constant; that is, in a pair of step- 
pulleys the belt tension will be uniform when the sum of the diameters of 
each opposite pair of steps is constant. Crossed belts are seldom used for 
cone-pulleys, on account of the friction between the rubbing parts of the 
belt. 

To design a pair of tapering speed-cones, so that the belt may fit 
equally tight in all positions : When the belt is crossed, use a pair of equal 
and similar cones tapering opposite ways. 

2. Open Belts. — When the belt is uncrossed, use a pair of equal and 
similar conoids tapering opposite ways, and bulging in the middle, accord- 
ing to the following formula: Let L denote the distance between the axes 
of the conoids; R the radius of the larger end of each; r the radius of the 
smaller end; then the radius in the middle, r„, is found as follows: 



B-\-r , (R - 



r) i 



6.28£ 



(Rankine.) 



If D = the diameter of equal steps of a pair of cone-pulleys, D and d — 
the diameters of unequal opposite steps, and L = distance between the 

D±d . (D - d^ 
axes, D = -^- + j^qZ- 

If a series of differences of radii of the steps, R — r, be assumed, then 
. R + r _ (R-r)* 

2 ° 6.28L 



for each pair of steps - 



-, and the radii of each may be 



computed from their half sum and half difference, as follows : 
l + r R- r . . _ R-\-r _ R-r 

2 + 2 ' ? ~ 2 2 * 



R-- 



A. J. Frith (Trans. A. S. M. E., x. 298) shows the following application of 
Rankine's method: If we had a set of cones to design, the extreme diame- 
ters of which, including thickness of belt, were 40" and 10", and the ratio 
desired 4, 3, 2, and 1, we would make a table as follows, L being 100": 



Trial 
Sum of 
D + d. 


Ratio. 


Trial Diameters. 


Values of 

(D - d)2 

12.56Z. 


Amount 

to be 
Added. 


Corrected Values. 


D 


d 


D 


d 


50 
50 
50 
50 


4 
3 

2 

1 


40 
37.5 
33.333 
25 


10 
12.5 

16.666 
25 


.7165 
.4975 
.2212 
.0000 


.0000 
.2190 
.4953 
.7165 


40 

37.2190 
33.8286 
25.7165 


10 

12.2190 
17.8286 
25.7165 



The above formulae are approximate, and they do not give satisfactory 
results when the difference of diameters of opposite steps is large and when 
the axes of the pulleys are near together, giving a large belt-angle. The 
following more accurate solution of the problem is given by C. A. Smith 
(Trans. A. S. M. E., x. 269) (Fig 152): 

Lay off the centre distance Cor EF, and draw the circles D x and d t equal 
to the first pair of pulleys, which are always previously determined by 
known conditions. Draw HI tangent to the circles Dj and d x . From B, 
midway between E and F, erect the perpendicular BG, making the length 



COKE Oil STEP PULLEYS. 



875 



BG = .314C. With G as a centre, draw a circle tangent to HI. Generally 
this circle will be outside of the belt-line, as in the cut, but when is short 
and the first pulleys D x and d x are large, it will fall on the inside of the belt- 
line. The belt-line of any other pair of pulleys must be tangent to the cir- 
cle G; hence any line, as JK or LM, drawn tangent to the circle G, will give 







G 






I /' 


f 






L 


\ 9? 


I 




J 


> i r"~' 


=~. \ s 


si / 




^^-<__J4 


/ 










_X— ^S^> 


= 2 5*^ 




>\)i 


/ i 


-K= \L---r^\ 


r^y 


7^--^ 


p\\ 


K [^ff 


E J 1 ' 


i / 


f f 


F ^~^^Vi-M 


\ V ^ 


J J i 


B 1 


i \ 


Jd z J J 


\ <*K_ 


y i 


















\ 


/ 








DjN,. 


/ 










-~" 



















Fig. 152. 

the diameters D 2 , d 2 or D 3 , d 3 of the pulleys drawn tangent to these lines 
from the centres E and F. 

The above method is to be used when the belt-angle A does not exceed 
18°. When it is between 18° and 30° a slight modification is made. In that 
case, in addition to the point G, locate another point in on the line BG .298 C 
nbove B. Draw a tangent line to the circle G, making an angle of 18? to the 
Hue of centres EF, and from the point m draw an arc tangent to this tan- 
gent line. All belt-lines with angles greater than 18° are tangent to this arc. 
The following is the summary of Mr. Smith's mathematical method: 

A M angle in degrees between the centre line and the belt of any pair of 

pulleys; 
a = .314 for belt-angles less than 18°, and .298 for angles between 18° 
and 30°; 
B° ==. an angle depending on the velocity ratio; 
C = the centre distance of the two puileys; 
D, d &r diameters of the larger and smaller of the pair of pulleys; 
E° — an angle depending on B u ; 

L = .the length of the belt when drawn tight around the pulleys; 
r — D -i- d, or the velocity ratio (larger divided by smaller). 



2C 



(3) Sin E° = sin 5° (cos A - 



r + 1 
4- d\ 



4aC 



(4) A = B° — E° when sin E° is positive ; = B° 4- E° when sin E° is negative ; 



(6) D = rd; 

(7) L = 2Ccos A -f .01745d[180 + (?■ 



- 1)(90 + ^)]. 



Equation (1) is used only once for any pair of cones to obtain the constant 
cos A, by the aid of tables of sines and cosines, for use in equation (3). 



876 



BELTING. 

Theory of Belts and Bands.— A pulley is driven by a belt by 
means of the friction between the surfaces in contact. Let 1\ be the tension 
on the driving side of the belt, '1\ the tension on the loose side; then S, — r l\, 
— T 2 , is the total friction between the band and the pulley, which is equal to 
the tractive or driving force. Let/ = the coefficient of friction, the ratio 
of the length of the arc of contact to the length of the radius, a — the angle 
of the arc of contact in degrees, e = the base of the Naperian logarithms 
= 2.71828, m = the modulus of the common logarithms = 0.434295. The 
following formulae are derived by calculus (Rankine's Mach'y & Millwork, 
p. 351 ; Carpenter's Exper. Eng'g, p. 173): 



T 2 ~ 


*>*; 


T* = 


T, 

e f0 


; t, 


- T, 


T, 

= r, - — - 

1 e/» 


= 7\(i 


- e~f e ). 


T t - 


r 2 = 


2\(1 - 


- e 


-/») = 


3*i(l- 


- io- /0m ) = 


tjjx- 


1Q -. 00758/a, 


T, 


= 10 


.00758/a. 


Ti = 


r2Xlo .00758/a. 


1C 


Ti 


2 7 2 


.00758/a • 



If the arc of contact between the band and the pulley expressed in turns 
and fractions of a turn = n, = 2n-?i; ef 9 = io 2 - 7288 /"; that is, e? 9 is the 
natural number corresponding to the common logarithm 2.7288/n. 

The value of the coefficient of friction / depends on the state and material 
of the rubbing surfaces. For leather belts on iron pulleys, Morin found 
f = .56 when dry, .36 when wet. .23 when greasy, and .15 when oily. In calcu- 
lating the proper mean tension for a belt, the smallest value, / — .15, is 
to be taken if there is a probability of the belt becoming wet with oil. The 
experiments of Henry R. Towne and Robert Briggs, however (Jour. Frank. 
Inst., 1868). show that such a state of lubrication is not of ordinary occur- 
rence; and that in designing machinery we may in most cases safely take 
f = 0.42. Reuleanx takes/ = 0.25. The following table shows the values of 
the coefficient 2.7288/, by which n is multiplied in the last equation, corre- 
sponding to different values of f; also the corresponding values of various 
ratios among the forces, when the arc of contact is half a circumference : 





f= 0.15 
2.7588/ = 0.41 


0.25 
0.68 


0.42 
0.15 


0.56 
1.53 


Lete 


= it and n = %, then 










T, -s- T« = 1 . 603 

^ ■+■&= 2.66 

2\ + Ti+-2S = 2.16 


2.188 
1 84 
1.34 


3.758 
1.86 

0.86 


5.821 
1.21 

0.71 



In ordinary practice it is usual to assume T a = 2S; Ti = 2S; T, -f T 7 -j- 
2S = 1.5. This corresponds to/ = 0.22 nearly. 

For a wire rope on cast iron / maybe taken as 0.15 nearly: and if the 
groove of the pulley is bottomed with gutta percha, 0.^5. (Kankine ) 

Centrifugal Tension of Belts.— When a belt or band runs at a 
high velocity, centrifugal force produces a tension in addition to that exist- 
ing when the belt is at rest or moving at a low velocity. This centrifugal 
tension diminishes the effective driving force. 

Rankine says : If an endless band, of any figure whatsoever, runs at a 
given speed, the centrifugal force produces a uniform tension at each cross- 
section of the band, equal to the weight of a piece of the band whose length 
is twice the height from which a heavy body must fall, in order to acquire 
the velocity of the band. (See Cooper on Belting, p. 101.) 

If To = centrifugal tension; 

V — velocity in feet per second ; 
q = acceleration due to gravity = 32.2; 
W = weight of a piece of the belt 1 ft. long and 1 sq. in. sectional area,— 

Leather weighing 56 lbs. per cubic foot gives W = 56 -+- 144 = .388. 
WV* .388F2 ntnTro 



2101 




30 lbs., 


then 


wd X 


rpm. 



BELTING PRACTICE. 87? 

Belting Practice. Handy Formulae for Belting. — Since 
in the practical application of the above formulae the value of the coefficient 
of friction must be assumed, its actual value varying within wide limits (\b% 
to 135$), and since the values of T^ and T 2 aiso are fixed arbitrarily, it is cus- 
tomary in practice to substitute for these theoretical formulae more simple 
empir.cal formulae and rules, some of which are given below. 

Let d = diam. of pulley in inches; rrd = circumference; 

V — velocity of belt in ft. per second; v = vel. in ft. per minute; 

a — angle of the arc of contact; 

L - length of arc of contact in feet = ndn ■+- (12 x 360) ; 

F — tractive force per square inch of sectional area of belt; 

w — width in inches; t — thickness; 

S = tractive force per inch of width = F-i- t ; 
rpm. = revs, per minute; rps. = revs, per second = rpm. -5- 60. 

v = -— X rpm. ; = .2618d X rpm. 
Horse-power, H.P. = §£ = ^ = ^F = Mm*** X rp m . 

If F = working tension per square inch = 275 lbs., and t = 7/32 inch, S = 
60 lbs. nearly, then 

H.P. = j™ = .mvw = .00047Qwd x rpm. I wd X rpm " 
550 

If F = 180 lbs. per square inch, and t = 1/6 inch, S = 

H.P. =^jk= -055Vw = .000838wd X rpm. = '"" 2™*"' • • ( 2 ) 

If the working strain is 60 lbs. per inch of width, a belt 1 inch wide travel- 
ling 550 ft. per minute will transmit 1 horse-power. If the working strain is 
30 lbs. per inch of width, a belt 1 inch wide, travelling 1100 ft. per minute, 
will transmit 1 horse-power. Numerous rules are given by different writers 
on belting which vary between these extremes. A rule commonly used is : 
1 inch wide travelling 1000 ft. per min. = I.H.P. 

H.P.=^ = .06^=.000882»dXrpm.==^ |jp. . . (3) 

This corresponds to a working strain of 33 lbs. per inch of width. ^ 

Many writers give as safe practice for single belts in good condition a 
working tension of 45 lbs. per inch of width. This gives 

H.P. =S" = .0818FW = .000357wd X rpm. = wd *Jl pm . . (4) 
loo <BUU 

For double belts of average thickness, some writers say that the trans- 
mitting efficiency is to that of single belts as 10 to 7, which would give 

H.P. of double belts =S = 4169Fto.= .00051u;d X rpm. = wd * r Q Pm ' ■ (5) 

Other authorities, however, make the transmitting-power of double belts 
twice that of single belts, on the assumption that the thickness of a double- 
belt is twice that of a single belt. 

Rules for hoise-power of belts are sometimes based on the number of 
square feet of surface of the belt which pass over the pulley in a minute. 
Sq. ft. per min. = wv -=- 12. The above formulae translated into this form 
give: 

(1) For S = 60 lbs. per inch wide ; H.P. = 46 sq. ft. per minute. 

(2) " S = 30 " " " H.P. = 92 

(3) " S = 33 " " " H.P. = 83 " " 

(4) " 8 = 45 " " " H.P. = 61 

(5) " S = 64.3" " " H.P. =43 " " (double belt). 



878 



The above formulae are all based on the supposition that the arc of con- 
tact is 180° For other arcs, the transmitting power is approximately pro- 
portional to the ratio of the degrees of arc to 180°. 
Some rules base the horse-power on the length of the arc of contact in 
■nda , XT „ Srw Siv ird a 

feet. Since L = ^^m and H " R = 33000 = sWo* 12 X rpm - X TSO' We 

Sw 
obtain by substitution H.P. =-— — xiX rpm., and the five formulae then 

take the following form for the several values pf S: 

wL X rpm . wL X rpm. w L X rpm. t«iX_rpm 

H.P = 55E (1), ^ (2), rjrjr (6) , — (4), 



275 
H.P. (double belt) = 



550 
wL X rpm. 
257 



(5). 



None of the handy formulae take into consideration the centrifugal ten- 
sion of belts at high velocities. When the velocity is over 3000 ft. per min- 
ute the effect of this tension becomes appreciable, and it should be taken 
account of as in Mr. Nagle's formula, which is given below. 

Horse-power of a Leather Belt One Inch wide. (Nagle.) 

Formula: H.P. = CVtiv(S - .012 V*) -* 550. 











For/ = 


40, 


a — 


180° 


C = 


= .715, w = 


= 1. 








Laced Belts, S = 275. 


Riveted Belts, 8 = 400. 


i 


Thickness in inches = t. 


.So 


Thickness in inches = t. 


is. 


1/7 


1/6 


3/16 


7/32 


1/4 


5/16 


1/3 


7/32 


1/4 


5/16 


1/3 


3/8 


7/16 


1/2 


£v 


.143 


.167 


.187 


.219 


.250 


.312 


.333 


t>«K 


219 


.250 


.312 


.333 


.375 


.437 


.500 


10 


.51 


.59 


.63 


.73 


.84 


1.05 


1.18 


15 


1.69 


1.94 


2.42 


2.58 


2.91 


3.39 


3.87 


15 


.75 


.88 


1 .00 


1.16 


1.32 


1 . 66 


1.77 


20 


2.24 


2.57 


3.21 


3.42 


3 .. 85 


4.49 


5.13 


20 


1.00 


1.17 


T32 


1.54 


1.75 


2.19 




25 


2.79 




3.98 


4.25 


4.78 


5.57 


6.37 


25 


I 23 


1 43 


1.61 


1.88 


2.16 






30 


3.31 


3.79 


4 74 


5.05 


5 67 


6.62 


7.58 


30 


1 47 


1.72 


1.93 






3.22 


3.44 


35 


3.82 


4 37 


5.46 


5.83 


6 56 


7 65 


8,75 


35 


1 m 


1 97 








3.70 


3.94 


40 


4.33 


4.95 


6.19 


6.60 


7.42 


8.66 


9 90 


40 


1 90 


2.22 








4.15 


4.44 


45 


4.85 


5,49 


6.86 


7.32 


8.43 


9.70 


10.98 


45 


2. OH 


2.45 


2.75 




3.67 


4.58 




50 




6.01 


7.51 


8.02 


9 02 


10.52112.03 


50 


2.27 


2.65 






>s 




5.30 


55 


5.68 


6.50 


8.12 


8.66 


9.74 


11.36!l3.00 


55 


2 41 


2.84 


3.19 






5.32 




60 




6.:;6 


8.70 


9.28 


10 43 


12. 17(13.91 


60 


2 58 


3.01 


3.38 




4.51 


5.64 




65 




7.37 


9.22 


9.83 


11 06 


12.9014.75 


65 


2.71 


3.16 


3.55 


4.14 


4.74 




6.32 


70 




7.75 


9 69 


10.33 


11 62 


13.56jl5.50 


70 


2.8! 


3.27 






4.91 


6.14 




75 


7.09 


8.11 


10.13 


10.84 


12,16 


14. 18116. 21 


75 




3.37 


3.79 


4.42 


5.05 


6.31 


6.73 


80 


7.36 


8.41 


10 51 


11.21 


12 61 


14.71|16.81 


80 








4.50 


5.15 


6.44 




85 


7.58 




10 82 


11,55 


13 00 


15.1617.32 


85 




3.47 








6.50 




90 


7.74 


8.85 


11.06 


11 80 


13 27 


15. 48117.69 


90 


2.97 


3.47 


3.90 


4.55 


5.20 


6.50 




100 


7.96 


9.10 


11.37 


12.13 


13.65 


15.92:18.20 


The H.P. becomes a maximum 


The H.P. becomes a maximum at 


at 87.41 ft. per sec, = 5245 ft. p. min. 


105.4 ft. per sec. = 6324 ft. per min. 



In the above table the angle of sub tension, a, is taken at 1 



Should it be I 90°|100°|110 |120°| 

Multiply above values by | .65 | .70 I .75 j .79 | 



'1140° 

1.87 



160° 1 170° 1 180° 1 200° 
.94 | .97 | 1 ll.05 



A. F. Nagle's Formula (Trans. A. S. M. E., vol. ii., 1881, p. 91. 
Tables published in 1882.) 

H.,=c^(^r , ) ; 



c = i - io - - 00758/a ; 

a = degrees of belt conta st; 
f — coefficient of friction; 
w = width in inches; 



550 

= thickness in inches; 
- velocity in feet per second: 
= T x — 1\ = stress upon belt per 
square inch. 



WIDTH OF BELT FOR A GIVEX HORSE-POWER. 879 

Taking & at 275 lbs. per sq. in. for laced belts and 400 lbs. per sq in. for 
lapped and riveted belts, the formula becomes 



H.P. = 
H.P. = 



CVtw{M - .0000218F2) for laced belts; 
CFtiv(.727 - .0000218F 2 ) for riveted belts. 



Values of C = 1 - 10 -- 00758 /a (Nagi,e.) 



i!| 








Degrees 


af contact = 


a. 


* 






II "8 S 


90° 


100° 


110° 


120° 


130° 


140° 


150° 


160° 


170° 


180° 


200° 


.15 


.210 


.230 


.250 


.270 


.288 


.307 


.325 


.342 


.359 


.376 


.408 


.20 


.270 


.295 


.319 


.342 


.364 


.386 


.408 


.428 


.448 


.467 


.503 


.25 


.325 


.354 


.381 


.407 


.432 


.457 


.480 


.503 


.524 


.544 


.582 


.30 


.376 


.408 


.438 


.467 


.494 


.520 


.544 


.567 


.590 


.610 


.649 


.35 


.423 


.457 


.489 


.520 


.548 


.575 


.600 


.624 


.646 


.667 


.705 


.40 


.467 


.502 


.536 


.567 


.597 


.624 


.649 


.673 


.695 


.715 


.753 


.45 


.507 


.544 


.579 


.610 


.640 


.667 


.692 


.715 


.737 


.757 


.792 


.55 


.578 


.617 


.652 


.684 


.713 


.739 


.763 


.785 


.805 


.822 


.853 


.60 


.610 


.649 


.684 


.715 


.744 


.769 


.792 


.813 


.832 


.848 


.877 


1.00 


.792 


.825 


.853 


.877 


.897 


.913 


.927 


.937 


.947 


.956 


.969 



The following table gives a comparison of the formulae already given for 
the case of a belt one inch wide, with arc of contact 180°. 

Horse-power of a Belt One Inch wide, Arc of Contact 180°. 

Comparison op Different Formula. 



.2 6 


la 

|*2 




Form. 1 

H.P. = 

wv 

"550* 


Form. 2 

H.P. = 

wv 

iioo' 


Form. 3 

H.P. = 

wv 

1000 ' 


Form. 4 

H.P. = 

wv 

w 


Form. 5 

dbl.belt 

H.P. = 

wv 

"513" 


Nagle's Form. 
7/32"single belt 


o a 


Laced. 


Riveted 


10 
20 
30 
40 
50 
60 
70 
80 
90 
100 
110 
120 


600 
1200 
1800 
2400 
3000 
3600 
4200 
4800 
5400 
6000 
6600 
7200 


50 
100 
150 
200 
250 
300 
350 
400 
450 
500 
550 
600 


1.09 
2.18 
3.27 
4.36 
5.45 
6.55 
7.63 
8.73 
9.82 
10.91 


.35 
1.09 
1.64 
2.18 
2.73 
3.27 
3.82 
4.36 
4.91 
5.45 


.60 
1.20 
1.80 
2.40 
3.00 
3.60 
4.20 
4.80 
5.40 
6.00 


.82 
1.64 
2.46 
3.27 
4.09 
4.91 
5.73 
6.55 
7.37 
8.18 


1.17 
2.34 
3.51 
4.68 
5.85 
7.02 
8.19 
9.36 
10.53 
11.70 


.73 
1.54 
2.25 
2.90 
3.48 
3.95 
4.29 
4.50 
4.55 
4.41 
4.05 
3.49 


1.14 
2.24 
3.31 
4.33 
5.26 
6.09 
6.78 
7.36 
7.74 
7.96 
7.97 












7.75 













Width of Belt for a Given Horse-power.— The width of belt 
required for any given horse-power may be obtained by transposing the for- 
mulae for horse-power so as to give the value of w. Thus: 

550H.P. 9.17 H.P. 2101 H.P. 275H.P. 



From formula (1), w - 

From formula (2), w - 

From formula (3), io - 

From formula (4), w = 

From formula (5),* w — 
* For double belts. 



1100 H.P. 18.33 H.P. 



1000 H.P. 16.67 H.P. 



733 H.P. 12.22 H.P. 



8.56 H.P. 
V 



d X rpm. 
_ 4202 H.P. 
~ d X rpm. 
_ 3820 HP. 
~ d X rpm. 
_ 2800 H.P . 
~ d X rpm. 

1960 H.P. 



L x rpm. ' 
_ 530 H.P. 
~ LX rpm." 
500 H.P. 
L X rpm.' 
360 H.P. 
L x rpm.' 
257 H.P. 



d x rpm. L x rpm.' 



880 



BELTIKG. 



Many authorities use formula (1) for double belts and formula (2) or (3i for ' 
single belts. 

To obtain the width by Nagle's formula, w = rVN a _ oi'op^V or divide 
the given horse-power by the figure in the table corresponding to the given \ 
thickness of belt and velocity in feet per second. 

The formula to be used in any particular case is largely a 
matter of judgment. A single belt proportioned according to formula (1), 
if tightly stretched, and if the surface is in good condition, will transmit the 
horse-power calculated by the formula, but one so proportioned is objec- 
tionable, first, because it requires so great an initial tension that it is apt to 
stretch, slip, and require frequent restretching and re lacing; and second, 
because this tension will cause an undue pressure on the pulley- shaft, and 
therefore an undue loss of power by friction. To avoid these difficulties, 
formula (2), (3), or (4,) or Mr. Nagle's table, should be used; the latter espe- 
cially in cases in which the velocity exceeds 4000 ft. per min. 

Taylor's Mules for Belting. -F. W. Taylor (Trans. A. S. M. E., 
xv. 204) describes a nine years' experiment on belting in a machine-shop, 
giving results of tests of 42 belts running night and day. Some of these 
belts were run on cone pulleys and others on shifting, or fast-and-loose, pul- 
leys. The average net working load on the shifting belts wasonly 4/10 of 
that of the cone belts. 

The shifting belts varied in dimensions from 39 ft. 7 in. long, 3.5 in. wide, 
.25 in. thick, to 51 ft. 5 in. long, 6.5 in. wide, .37 in. thick. The cone belts 
varied in dimensions from 24 ft. 7 in. long, 2 in. wide, .25 in. thick, to 31 ft. 
10 in. long, 4 in. wide, .37 in. thick. 

Belt-clamps were used having spring-balances between the two pairs of 
clamps, so that the exact tension to which the belt was subjected was 
accurately weighed when the belt was first put on, and each time it was 
tightened. 

The tension under which each belt was spliced was carefully figured so as 
to place it under an initial strain — while the belt was at rest immediately 
after tightening — of 71 lbs. per inch of width of double belts. This is equiv- 
alent, in the case of 

Oak tanned and fulled belts, to 192 lbs. per sq. in. section; 
Oak tanned, not fulled belts, to 229 " " " " " 

Semi-raw-hide belts, to 253 " " " " " 

Raw-hide belts, to 284 " " " " " 

From the nine years' experiment Mr. Taylor draws a number of conclu- 
sions, some of which are given in an abridged form below. 

In using belting so as to obtain the greatest economy and the most satis- 
factory results, the following rules should be observed: 

Other Types of 
Leather Belts 
and 6- to 7-ply 
Rubber Belts. 



A double belt, having an arc of contact of 
180°, will give an effective pull on the face 

of a pulley per inch of width of belt of 

Or, a different form of same rule: 

The number of sq. ft. of double Belt passing 
around a pulley per minute required to 
transmit one horse power is 

Or : The number of lineal feet of double- 
belting 1 in. wide passing around a pulley 
per minute required to transmit one horse- 
power is 

Or : A double belt 6 in. wide, running 4000 to 
5000 ft. per min., will transmit 




90 sq. ft. 



1100 ft. 
25 H.P. 



The terms "initial tension," " effective pull," etc., are thus explained by 
Mr. Taylor : When pulleys upon which belts are tightened are at rest, both 
strands of the belt (the upper and lower) are under the same stress per in. 
of width. By " tension," " initial tension," or " tension while at rest," we 



Taylor's rules for belting. 881 

fcaean the stress per in. of width, or sq, in. of section, to which one of the 
strands of the belt is tightened, when at rest. After the belts are in motion 
and transmitting power, the stress on the slack side, or strand, of the belt 
becomes less, while that on the tight side — or the side which does the pull- 
ing—becomes greater than when the belt was at rest. By the term " total 
load " we mean the total stress per in. of width, or sq. in. of section, on the 
tight side of belt while in motion. 

The difference between the stress on the tight side of the belt and its slack 
side, while in motion, represents the effective force or pull which is trans- 
mitted from one pulley to another. By the terms "working load, 1 ' " net 
working load, 1 ' or "effective pull, 11 we mean the difference in the tension 
of the tight and slack sides of the belt per in. of width, or sq. in. section, 
while in motion, or the net effective force that is transmitted from one pul- 
ley to another per in. of width or sq. in. of section. 

The discovery of Messrs. Lewis and Bancroft (Trans. A. S. M. E., vii. 749) 
that the "sum of the tension on both sides of the belt does not remain 
constant, 11 upsets all previous theoretical belting formulae. 

The belt speed for maximum economy should be from 4000 to 4500 ft. per 
minute. 

The best distance from centre to centre of shafts is from 20 to 25 ft. 

Idler pulleys work most satisfactorily when located on the slack side of 
the belt about one quarter way from the driving-pulley. 

Belts are more durable and work more satisfactorily made narrow and 
thick, rather than wide and thin. 

It is safe and advisable to use: a double belt on a pulley 12 in. diameter or 
larger; a triple belt on a pulley 20 in. diameter or larger; a quadruple belt 
on a pulley 30 in. diameter or larger. 

As belts increase in width they should also be made thicker. 

The ends of the belt should be fastenej together by splicing and cement- 
ing, instead of lacing, wiring, or using hooks or clamps of any kind. 

A V-splice should be used on triple and quadruple belts and when idlers 
are used. Stepped splice, coated with rubber and vulcanized in place, is best 
for rubber belts. 

For double belting the rule works well of making the splice for all belts 
up to 10 in. wide, 10 in. long; from 10 in. to 18 in. wide the splice should be 
the same width as the belt, 18 in. being the greatest length of splice required 
for double belting. 

Belts should be cleaned and greased every five to six months. 

Double leather belts will last well when repeatedly tightened under a 
strain (when at rest) of 71 lbs. per in. of width, or 240 lbs per sq. in. section. 
They will not maintain this tension for any length of time, however. 

Belt-clamps having spring- balances between the two pairs of clamps 
should be used for weighing the tension of the belt accurately each time it 
is tightened. 

The stretch, durability, cost of maintenance, etc., of belts proportioned 
(A) according to the ordinary rules of a total load ot 111 lbs. per inch of 
width corresponding to an effective pull of 65 lbs. per inch of width, and (B) 
according to a more economical rule of a total load of 54 lbs., corresponding 
to an effective pull of 26 lbs. per inch of width, are found to be as follows: 

When it is impracticable to accurately weigh the tension of a belt in tight- 
ening it, it is safe to shorten a double belt one half inch for every 10 ft. of 
length for (A) and one inch for every 10 ft. for (B), if it requires tightening. 

Double leather belts, when treated with great care and run night and day 
at moderate speed, should last for 7 years (A); 18 years (B). 

The cost of all labor and materials used in the maintenance and repairs of 
double belts, added to the cost of renewals as they give out, through a term 
of years, will amount on an average per year to 37% of the original cost of 
the belts (A); 14$ or less (B). 

In figuring the total expense of beltiug, and the manufacturing cost 
chargeable to this account, by far the largest item is the time lost on the 
machines while belts are being relaced and repaired. 

The. total stretch of leather belting exceeds %% of the original length. 

The stretch during the first six months of the life of belts is 36$ of their 
entire stretch (A); \5% (B). 

A double belt will stretch 47/100 of \% of its length before requiring to be 
tightened (A); 81/100 of \% (B). 

The most important consideration in making up tables and rules for the 
use and care of belting is how to secure the minimum of interruptions to 
manufacture from this source. 



882 BELTIKG. 

The average double belt (A), when running night and day in a machine- 
shop, will cause at least 26 interruptions to manufacture during its life, or 5 
interruptions per year, but with (B) interruptions to manufacture will not 
average oftener for each belt than one in sixteen months. 

The oak-tanned and fulled belts showed themselves to be superior in all 
respects except the coefficient of friction to either the oak-tanned not fulled, 
the semi-raw-hide, or raw-hide with tanned face. 

Belts of any width can be successfully shifted backward and forward on 
tight and loose pulleys. Belts running between 5000 and COOO ft. per min. 
and driving 300 H.P. are now being daily shifted on tight and loose pulleys, 
to throw lines of shafting in and out of use. 

The best form of belt-shifter for wide belts is a pair of rollers twice the 
width of belt, either of which can be pressed onto the flat surface of the 
belt on its slack side close to the driven pulley, the axis of the roller making 
an angle of 75° with the centre line of the belt. 

Remarks on Mr. Taylor's Rules. (Trans. A. S. M. E., xv., 242.) 
—The most notable feature in Mr. Taylor's paper is the great difference be- 
tween his rules for proper proportioning of belts and those given by earlier 
writers. A very commonly used rule is, one horse-power may be transmitted 
by a single belt 1 in. wide running x ft. per min., substituting for x various 
values, according to the ideas of different engineers, ranging usually from 
550 to 1100. 

The practical mechanic of the old school is apt to swear by the figure 
600 as being thoroughly reliable, while the modern engineer is more apt to 
use the figure 1000. Mr. Taylor, however, instead of using a figure from 550 
to 1100 for a single belt, uses 950 to 1100 for double belts. If we assume that 
a double belt is twice as strong, or will carry twice as much power, as a 
single belt, then he uses a figure at least twice as large as that used in 
modern practice, and would make the cost of belting for a given shop twice 
as large as if the belting were proportioned according to the most liberal of 
the customary rules. 

This great difference is to some extent explained by the fact that the 
problem which Mr. Taylor undertakes to solve is quite a different one from 
that which is solved by the ordinary rules with their variations. The prob- 
lem of the latter generally is, " How wide a belt must be used, or how nar- 
row a belt may be used, to transmit a given horse-power ?" Mr. Taylor's 
problem is: "How wide a belt must be used so that a given horse-power 
may be transmitted with the minimum cost for belt repairs, the longest life 
to the belt, and the smallest loss and inconvenience from stopping the 
machine while the belt is being tightened or repaired ?" 

The difference between the old practical mechanic's rule of a l-in.-wide 
single belt, 600 ft. per min., transmits one horse-power, and the rule com- 
monly used by engineers, in which 1000 is substituted for 600, is due to the 
belief of the engineers, not that a horse-power could not be transmitted by 
the belt proportioned by the older rule, but that such a proportion involved 
undue strain from overtightening to prevent slipping, which strain entailed 
too much journal friction, necessitated frequent tightening, and decreased 
the length of the life of the belt. 

Mr. Taylor's rule substituting 1100 ft. per min. and doubling the belt is a 
further step, and a long one, in the same direction. Whether it will be taken 
in any case by engineers will depend upon whether they appreciate the ex- 
tent of the losses due to slippage of belts slackened by use under overstrain, 
and the loss of time in tightening and repairing belts, to such a degree as to 
induce them to allow the first cost of the belts to be doubled in order to 
avoid these losses. 

It should be noted that Mr. Taylor's experiments were made on rather 
narrow belts, used for transmitting power from shafting to machinery, and 
his conclusions may not be applicable to heavy and wide belts, such as 
engine fly-wheel belts. 

MISCELLANEOUS NOTES ON BELTING. 

Formulae are useful for proportioning belts and pulleys, but they furnish 
no means of estimating how much power a particular belt may be trans- 
mitting at any given time, any more than the size of the engine is a measure 
of the load it is actually drawing, or the known strength of a horse is a 
measure of the load on the wagon. The only reliable means of determining 
the power actually transmitted is some form of dynamometer. (See Trans. 
A. S. M. E., vol. x'ii. p. 707.) 



MISCELLANEOUS NOTES ON BELTING. 883 

If we increase the thickness, the power transmitted ought to increase in 
proportion; and for double belts we should have half the width required for 
a siugle belt under the same conditions. With large pulleys and moderate 
velocities of belt it is probable that this holds good. With small pulleys, 
however, when a double belt is used, there is not such perfect contact 
between the pulley-face and the belt, due to the rigidity of the latter, and 
more work is necessary to bend the belt-fibres than when a thinner and 
more pliable belt is used. The centrifugal force teuding to throw the belt 
from the pulley also increases with the thickness, and for these reasons the 
width of a double belt required to transmit a given horse-power when used 
with small pulleys is generally assumed not less than seven tenths the 
width of a single belt to transmit the same power. (Flather on " Dynamom- 
eters and Measurement of Power. 1 ') 

F. W. Taylor, however, finds that great pliability is objectionable, and 
favors thick belts even for small pulleys: The power consumed in bending 
the belt around the pulley he considers inappreciable. According to Ban- 
kine's formula for centrifugal tension, this tension is proportional to the 
sectional area of the belt, and hence it does not increase with increase of 
thickness when the width is decreased in the same proportion, the sectional 
area remaining constant. 

Scott A. Smith (Trans. A. S. M. E., x. 765) says: The best belts are made 
from all oak-tanned leather, and curried with the use of cod oil and tallow, 
all to be of superior quality. Such belts have continued in use thirty to 
forty years when used as simple driving-belts, driving a proper amount of 
power, and having had suitable care. The flesh side should not be run to 
the pulley-face, for the reason that the wear from contact with the pulley 
should come on the grain side, as that surface of the belt is much weaker 
in its tensile strength than the flesh side; also as the grain is hard it is more 
enduring for the wear of attrition; further, if the grain is actually worn off, 
then the belt may not suffer in its integrity from a ready tendency of the 
hard grain side to crack. 

The most intimate contact of a belt with a pulley comes, first, in the 
smoothness of a pulley -face, including freedom from ridges and hollows left 
by turning-tools; second, in the smoothness of the surface and evenness in 
the texture or body of a belt; third, in having the crown of the driving and re 
ceiving pulleys exactly alike, — as nearly so as is practicable in a commercial 
sense; fourth, in having the crown of pulleys not over \Q' for a 24" face, that 
is to say, that the pulley is not to be over J4" larger in diameter in its centre; 
fifth, in having the crown other than two planes meeting at the centre; 
sixth, the use of any material on or in a belt, in addition to those necessarily 
used in the currying process, to keep them pliable or increase their tractive 
quality, should wholly depend upon the exigencies arising in the use of 
belts: non-use is safer than over-use; seventh, with reference to the lacing 
of belts, it seems to be a good practice to cut the ends to a convex shape by 
using a former, so that there may be a nearly uniform stress on the lacing 
through the centre as compared with the edges. For a belt 10" wide, the 
centre of each end should recede 1/10". 

Lacing of Belts.— In punching a belt for lacing, use an oval punch, 
the longer diameter of the punch being parallel with the sides of the belt. 
Punch two rows of holes in each end, placed zigzag. In a 3-in. belt there 
should be four holes in each end— tv, o in each row. In a 6-inch belt, seven 
holes— four in the row nearest the end. A 10-inch belt should have nine 
holes. The edge of the holes should not come nearer than % of an inch from 
the sides, nor % of an inch from the ends of the belt. The second row should 
be at least 1% inches from the end. On wide belts these distances should 
be even a little greater. 

Begin to lace in the centre of the belt and take care to keep the ends 
exactly in line, and to lace both sides with equal tightness. The lacing 
should not be crossed on the side of the belt that runs next the pulley. In 
taking up belts, observe the same rules as putting on new ones. 

Setting a Belt on Quarter-twist.— A belt must run squarely on to 
the pulley. To connect with a belt two horizontal shafts at right angles 
with each other, say an engine-shaft near the floor with a line attached to 
the ceiling, will require a quarter-turn. First, ascertain the central point 
on the face of each pulley at the extremity of the horizontal diameter where 
the belt will leave the pulley, and then set that point on the driven pulley 
plumb over the corresponding point on the driver. This will cause the belt 
to run squarely on to each pulley, and it will leave at an angle greater or 
Jess, according to the size of the pulleys and their distance from each other. 



884 BELTING. 

In quarter-twist belts, in order that the belt may remain on the pulleys, ■ 
the central plane on each pulley must pass through the point of delivery of I 
the other pulley. This arrangement does nor, admit of reversed motion. 

To find the Length of Belt required for two given \ 
Pulleys. — When the length cannot be measured directly by a tape-line, ! 
the following approximate rule may be used : Add the diameter of the two 
pulleys together, divide the sum by 2, and multiply the quotient by 3J4, and 
add the product to twice the distauce between the centres of the shafts. 
(See accurate formula below.) 

To find the Angle of the Arc of Contact of a Belt.— Divide 
the difference between the radii of the two pulleys in inches by the distance 
between their centres, also in inches, and in a table of natural sines find the 
angle most nearly corresponding with the quotient. Multiply this angle by 
2, and add the product to 180° for the angle of contact with the larger 
pulley, or subtract it from 180° for the smaller pulley. 

Or, let R = radius of larger pulley, r = radius of smaller; 
L = distance between centres of the pulleys; 
a — angle whose sine is (R — r) ■+■ L. 

Arc of contact with smaller pulley = 180° — 2a; 
" " " " larger pulley = 180° + 2a. 

To find the Length of Belt in Contact with the Pulley.— 

For the larger pulley, multiply the angle a, found as above, by .0349, to the 
product add 3.1416, and multiply the sum by the radius of the pulley. Or 
length of belt in contact with the pulley 

= radius X O + .0349a) = radius x n(l -f jM. 

For the smaller pulley, length = radius X (7r-.0349a)= radius X ir\l - =.J • 

The above rules refer to Open Belts. The accurate formula for length 
of an open belt is, 

Length » „R(l + ^) + Tf(l - ^) + 2L cos a 

= R(n + .0349a) -f r(n - .0349a) + 2L cos a, 

in which R = radius of larger pulley, r = radius of smaller pulley, 

L = distance between centres of pulleys, and a = angle whose sine is 
(R - r) -f- L\ cos a = |"L 2 — (R — J') 2 - 
For Crossed Belts the formula is 

Length of belt = w fl(l + ^) + w(l + ^) + 2L cos p, 

= (R + r) X (7T + .03490) + 2L cos (3, 



in which |3 = angle whose sine is (R + r) -5- L ; cos $ — \ X 2 - (R + ?") 2 . 

To find the Length of Belt when Closely Rolled -The sum 

of the diameter of the roll, and of the eye in inches, x the number of turns 
made by the belt and by .1309, = length of the belt in feet 

To find the Approximate Weight of Belts —Multiply the 
length of belt, in feet, by the width in inches, and divide the product by 13 
for single, and 8 for double belt. 

Relations of the Size and Speeds of Driving and Driven 
Pulleys.— The driving pulley is called the driver, D, and the driven pulley 
the driven, d. If the number of teeth in gears is used instead of diameter, in 
these calculations, number of teeth must be substituted wherever diameter 
occurs. R — revs, per min. of driver, r = revs, per min. of driven. 

D = dr-i-R; 
Diam. of driver = diam. of driven x revs, of driven -*- revs, of driver. 

d = DR -=- r; 
Diam. of driven = diam. of driver x revs, of driver -+- revs, of driven. 

R = dr + D; 
Bevs. of driver = revs, of driven x diam. of driven -*- diam. of driver. 



MISCELLANEOUS NOTES ON BELTING. 



885 



-= DR + d; 
Revs, of driven = revs, of driver x diam. of driver -f- diam. of driven. 

Evils of Tight Belts. (Jones and Laughlins.)— Clamps with powerful 
screws are often used to put on belts with extreme tightness, and with most 
injurious strain upon the leather. They should be very judiciously used for 
horizontal belts, which should be allowed sufficient slackness to move with a 
loose undulating vibration on the returning side, as a test that they have no 
more strain imposed than is necessary simply to transmit the power. 

On this subject a New England cotton- m ill engineer of large experience, 
says: I believe that three quarters of the trouble experienced in broken pul- 
leys, hot boxes, etc., can be traced to the fault of tight belts. The enormous 
and useless pressure thus put upon pulleys must in time break them, if they 
are made in any reasonable proportions, besides wearing out the whole out- 
fit, and causing heating and consequent destruction of the bearings. Below 
are some figures showing the power it takes in average modern mills with 
first-class shafting, to drive the shafting alone : 



Mill, 

No. 


Whole 
Load, 
H.P. 


Shafting 

Horse- 
power. 


? Alone. 

Per cent 
of whole. 


Mill, 
No. 


Whole 
Load, 
H.P. 


Shafting Alone. 

Horse- Per cent 
power, of whole. 


1 
2 
3 

4 


199 
472 
486 

677 


51 

111.5 
134 
190 


25.6 
23.6 
27.5 

28.1 


5 
6 

7 
8 


759 
235 
670 
677 


172.6 
84.8 
262.9 
182 


22.7 
36.1 
39.2 
26.8 



These may be taken as a fair showing of the power that is required in 
many of our best mills to drive shafting. It is unreasonable to think that all 
that power is consumed by a legitimate amount of friction of bearings 
and belts. I know of no cause for such a loss of power but tight belts. These, 
when there are hundreds or thousands in a mill, easily multiply the friction 
on the bearings, and would account for the figures. 

Sag oi* Belts.— In the location of shafts that are to be connected with 
each other by belts, care should be taken to secure a proper distance one 
from the other. This distance should be such as to allow of a gentle sag to 
the belt when in motion. 

A general rule may be stated thus: Where narrow belts are to be run over 
small pulleys 15 feet is a good avei-age, the belt having a sag of V/» to 2 inches. 

For larger belts, working on larger pulleys, a distance of 20 to 25 feet does 
well, with a sag of 2*^ to 4 inches. 

For main belts working on very large pulleys, the distance should be 25 to 
30 feet, the belts working well with a sag of 4 to 5 inches. 

If too great a distance is attempted,the belt will have an unsteady flapping 
motion, which will destroy both the belt and machinery. 

Arrangement of Belts and Pulleys.— If possible to avoid it, con- 
nected shafts should never be placed one directly over the other, as in such 
case the belt must be kept very tight to do the work. For this purpose belts 
should be carefully selected of well-stretched leather. 

It is desirable that the angle of the belt with the floor should not exceed 
45°. It is also desirable to locate the shafting and machinery so that belts 
should run off from each shaft in opposite directions, as this arrangement 
will relieve the bearings from the friction that would result when the belts all 
pull one way on the shaft. 

In arranging the belts leading from the main line of shafting to the 
counters, those pulling in an opposite direction should be placed as near 
each other as practicable, while those pulling in the same direction should be 
separated. This can often be accomplished by changing the relative posi- 
tions of the pulleys on the counters. By this procedure much of the friction 
on the journals may be avoided. 

If possible, machinery should be so placed that the direction of the belt 
motion shall be from the top of the driving to the top of the driven pulley, 
when the sag will increase the arc of contact. 

The pulley should be a little wider than the belt required for the work. 



886 BELTING. 

The motion of driving should run with and not against the laps of the belts. 

Tightening or guide pulleys should be applied to the slack side of belts and 
near the smaller pulley. 

Jones & Laughlins, in their Useful Information, say: The diameter of the 
pulleys should be as large as can be admitted, provided they will not pro- 
duce a speed of more than 3750 feet of belt motion per minute. 

They also say: It is better to gear a mill with small pulleys and run them 
at a high velocity, than with large pulleys and to run them slower. A mill 
thus geared costs less and has a much neater appearance than with large 
heavy pulleys. 

M. Arthur Achard (Proc. Inst. M. E., Jan. 1881, p. 62) says: When the belt 
is wide a partial vacuum is formed between the belt and the pulley at a 
high velocity. The pressure is then greater than that computed from the 
tensions in the belt, and the resistance to slipping is greater. This has the 
advantage of permitting a greater power to be transmitted by a given belt, 
and of diminishing the strain on the shafting. 

On the other hand, some writers claim that the belt entraps air between 
itself and the pulley, which tends to diminish the friction, and reduce the 
tractive force. On this theory some manufacturers perforate the belt with 
numerous holes to let rhe air escape. 

Care of Belts.— Leather belts should be well protected against water 
and even loose steam and other moisture. 

Belts of coarse, loose leather will do better service in dry warm places; for 
wet or moist situations the finest and firmest leather should be used. (J. B. 
Hoyt & Co.) 

Do not allow oil to drip upon the belts. It destroys the life of the leather. 

Leather belting cannot safely stand above 110° of heat. 

Strength of Belting.— The ultimate tensile strength of belting does 
not generally enter as a factor in calculations of power transmission. 

The strength of the solid leather in belts is from 2000 to 5000 lbs. per square 
inch; at the lacings, even if well put together, only about 1000 to 1500. If 
riveted, the joint should have half the strength of the solid belt. The work- 
ing strain on the driving side is generally taken at not over one third of the 
strength of the lacing, or from one eighth to one sixteenth of the strength 
of the solid belt. Dr. Hartig found that the tension in practice varied from 
30 to 532 lbs. per square inch, averaging: 273 lbs. 

Adhesion Independent of Diameter, (Schultz Belting Co.)— 
1. The adhesion of the belt to the pulley is the same — the arc or number of 
degrees of contact, aggregate tension or weight being the same— without 
reference to width of belt or diameter of pulle3\ 

2. A belt will slip just as readily on a pulley four feet in diameter as it will 
on a pulley two feet in diameter, provided the conditions of the faces of the 
pulleys, the arc of contact, the tension, and the number of feet the belt 
travels per minute are the same in both cases. 

3. A belt of a given width, and making any given number of feet per 
minute, will transmit as much power running on pulleys two feet in diam 
eter as it will on pulleys four feet in diameter, provided the arc of contact, 
tension, and conditions of pulley faces are the same in both cases. 

4. To obtain a greater amount of power from belts the pulleys may be 
covered with leather; this will allow the belts to run very slack and give 25$ 
more durability. 

Endless Belts.— If the belts are to be endless, they should be put on 
and drawn together by " belt clamps " made for the purpose. If the belt is 
made endless at the belt factory, it should never be run on to the pulleys, lest 
the irregular strain spring the belt. Lift out one shaft, place the belt on the 
pulleys, and force the shaft back into place. 

Belt Data.— A fly-wheel at the Amoskeag Mfg. Co., Manchester, N. H., 
30 feet diameter, 110 inches face, running 61 revolutions per minute, carried 
two heavy double-leather belts 40 inches wide each, and one 24 inches wide. 
The engine indicated 1950 H.P., of which probably 1850 H.P. was transmitted 
by the belts. The belts were considered to be heavily loaded, but not over- 
taxed. 

— = 323 feet per minute for 1 H.P. per inch of width. 

Samuel Webber (Am. Mach., Feb. 22, 1894) reports a case of a belt 30 
inches wide, % inch thick, running for six years at a velocity of 3900 feet per 
minute, on to a pulley 5 feet diameter, and transmitting 556 H.P. This gives 
a velocity of 210 feet per minute for 1 H.P. per inch of width. By Mr. Nagle's 



TOOTHED-WHEEL GEARINO. 88* 

table of riveted belts this belt would be designed for 332 H.P. By Mr. Taylor's 
rule it would be used to transmit only 123 H.P. 

The above may be taken as examples of what a belt may be made to do, 
but they shou d not be used as precedents in designing. It is not stated how 
much power was lost by the journal friction due to over-tightening of these 
belts. 

Belt Dressings.— We advise, when the belt is pliable, and only dry and 
husky, the application of blood-warm tallow. This applied, and dried in by 
heat of fire or sun, will tend to keep the leather in good working condition. 
The oil of the tallow passes into the tallow of the leather, serving to soften 
it, and the stearine is left on the outside, to fill the pores and leave a smooth 
surface. The addition of resin to the tallow for belts, if used in wet or damp 
places, will be of service and help preserve their strength. Belts which have 
become hard and dry should have an application of neat's-foot or liver oil, 
mixed with a small quantity of resin. This prevents the oil from injuring the 
belt and helps to preserve it. There should not be so much resin as to leave 
the belt sticky. (J. B. Hoyt & Company.) 

Belts should not be soaked in water before oiling, and penetrating oils 
should but seldom be used, except occasionally when a belt gets very dry 
and husky from neglect. It may then be moistened a little, andfhave neat's- 
foot oil applied. Frequent applications of such oils to a new belt render the 
leather soft and flabby, thus causing it to stretch, and making it liable to 
run out of line. A composition of tallow and oil, with a little resin or bees- 
wax, is better to use. Prepared castor-oil dressing is good, and maybe 
applied with a brush or rag while the belt is running. (Alexander Bros.) 

Cement for Cloth or Leather. (Molesworth.)— 16 parts gutta- 
percha, 4 india-rubber, 2 pitch, 1 shellac, 2 linseed-oil, cut small, melted to- 
gether and well mixed. 

Rubber Belting.— The advantages claimed for rubber belting are 
perfect uniformity in width and thickness; it will endure a great degree of 
heat and cold without injury; it is also specially adapted for use in damp or 
wet places, or where exposed to the action of steam; it is very durable, and 
has great tensile strength, and when adjusted for service it has the most per- 
fect hold on the pulleys, hence is less liable to slip than leather. 

Never use animal oil or grease on rubber belts, as it will greatly injure and 
soon destroy them. 

Rubber belts will be improved, and their durability increased, by putting 
on with a painter's brush, and letting it dry, a composition made of equal 
parts of red lead, black lead, French yellow, and litharge, mixed with boiled 
linseed-oil and japan enough to make it dry quickly. The effect of this will 
be to produce a finely polished surface. If, from dust or other cause, the 
belt should slip, it should be lightly moistened on the side next the pulley 
with boiled linseed-oil. (From circulars of manufacturers.) 



GEARING-. 

TOOTHED-WHEEL GEARING. 

Pitch, Pitch-circle, etc.— If two cylinders with parallel axes are 
pressed together and one of them is rotated on its axis, it will drive the other 
by means of the friction between the surfaces. The cylinders may be con- 
sidered as a pair of spur-wheels with an infinite number of very small teeth. 
If actual teeth are formed upon the cylinders, making alternate elevations 
and depressions in the cylindrical surfaces, the distance between the axes 
remaining the same, we have a pair of gear-wheels which will drive one an- 
other by pressure upon the faces of the teeth, if the teeth are properly 
shaped. In making the teeth the cylindrical surface may entirely disap- 
pear, but the position it occupied may still be considered as a cylindrical 
surface, which is called the " pitch-surface," and its trace on the end of the 
wheel, or on a plane cutting the wheel at right angles to its axis, is called 
the " pitch-circle " or " pitch-line." The diameter of this circle is called the 
pitch-diameter, and the distance from the face of one tooth to the corre- 
sponding face of the next tooth on the same wheel, measured on an arc of 
the pitch-circle, is called the " pitch of the tooth," or the circular pitch. 

If two wheels having teeth of the same pitch are geared together so that 
their pitch-circles touch, it is a property of the pitch-circles that their diam- 
eters are proportional to the number of teeth in the wheels, and vice versa,' 



GEARING. 



thus, if one wheel is twice the diameter (measured on the pitch-circle) of the 
other, it has twice as many teeth. If the teeth are properly shaped the 
linear velocity of the two wheels are equal, and the angular velocities, or 
speeds of rotation, are inversely proportional to the number of teeth and to 
th« diameter. Thus the wheel that has twice as many teeth as the other 
will revolve just half as many times in a minute. 

The '•pitch,' 1 or distance measured on an arc of the pitch-circle from the 
face of one tooth to the face of the next, consists of two parts— the " thick- 
ness'" of the tooth and the "space" between it and the next tooth. The 
space is larger than the thickness by a small amount called the " back- 
lash," which is allowed for imperfections of workmanship. In finely cut 
gears the backlash may be almost nothing. 

The length of a tooth in the direc- 
tion of the radius of the wheel is 
called the "depth," and this is di- 
vided into two parts: First, the 
"addendum," the height of the tooth 
above the pitch line; second, the 
"dedendum," the depth below the 
pitch line, which is an amount equal 
to the addendum of the mating gear. 
The depth of the space is usually 
given a little "clearance" to allow 
for inaccuracies of workmanship, 
especially in cast gears. 

Referring to Fig. 153, pi, pi are the 
pitch-lines, al the addendum-line, rl 
the root-line or dedendum -line, cl 
the clearance-line, and b the back- 
lash. The addendum and dedendum are usually made equal to each other. 
.-.-.,, No. of teeth 3.1416 

Diametral pitch =- 




Circular pitch 



diam. of pitch-circle in inches 
diam. X 3.1416 3.1416 



circular pitch' 



No. of teeth — diametral pitch" 
Some writers use the term diametral pitch to i 
■ircular pitch 



No. of teeth 

, but the first definition is the more common and the more 
3.1416 
convenient. A wheel of 12 in. diam. at the pitch-circle, with 48 teeth is 48/12 
= 4 diametral pitch, or simply 4 pitch, The circular pitch of the same 
, . . 12 X 3.1416 r . orA 3.1416 ___. . 

wheel is ^ = .7854, or — - — = .7854 in. 

48 4 





Relation of Diametral to Circular Pitch. 




Diame- 


Circular 


Diame- 


Circular 


Circular 


Diame- 


Circular 


Diame- 


tral 


Pitch. 


tral 


Pitch. 


Pitch. 


tral 


Pitch. 


tral 


Pitch. 




Pitch. 






Pitch. 




Pitch. 


1 


3.142 in. 


11 


.286 in. 


3 


1.047 


15/16 


3.351 


m 


2.094 


12 


.262 


2fc 


1.257 


Vs 


3.590 


2 


1.571 


14 


.224 


2 


1.571 


13/16 


3.867 


2J4 


1.396 


16 


.196 


1% 


1.676 


H 


4.189 


m 


1.257 


18 


.175 


m 
m 


1.795 


11/16 


4.570 


m 


1.142 


20 


.157 


1.933 


y 8 


5.027 


3 


1.047 


22 


.143 


m 


2.094 


9/16 


5.585 


Wz 


.898 


24 


.131 


1 7/16 


2.185 


y 2 


6.283 


4 


.785 


26 


.121 


1% 


2.285 


7/16 


7.181 


5 


.628 


28 


.112 


1 5/16 


2.394 


% 


8.378 


6 


.524 


30 


.105 


m 


2.513 


5/16 


10.053 


7 


.449 


32 


.098 


1 3/16 


2.646 


X 


12.566 


8 


.393 


36 


.087 


m 


2.793 


3/16 


16.755 


9 


.349 


40 


.079 


1 1/16 


2.957 


% 


25.133 


10 


.314 


48 


.065 


1 


3.142 


1/16 


50.266 



Since circular pitch 



diam. X 3.1416 



No. of teeth ' 
which always brings out the diameter as a number with an inconvenient 



eirc. pitch X No. of t ee th 
37T416 



TOOTHED-WHEEL GEARING. 



889 



fraction if the pitch is in even inches or simple fractions of an inch. By the 
diametral-pitch system this inconvenience is avoided. The diameter may- 
be in even inches or convenient fractions, and the number of teeth is usually 
an even multiple of the number of inches in the diameter. 
Diameter of Pitch-line of Wheels from 10 to 100 Teeth 
of 1 in. Circular Pitch. 



S3 

K CD 


is 


0® 


Si .3 




h 


o - 


i.s 


ft % 


U d 


i= 


H 


ft 


H 


ft 


H 


ft 


fcH 


a 


h 


o 


H 


ft 


10 


3.183 


26 


8.276 


41 


13.051 


56 


17.825 


71 


22.600 


86 


27.375 


11 


3.501 


27 


8.594 


42 


13.369 


57 


18.144 


72 


22.918 


87 


27.693 


18 


3.820 


28 


8.913 


43 


13.687 


58 


18.462 


73 


22.236 


88 


28.011 


13 


4.138 


29 


9.231 


44 


14.006 


59 


18.781 


74 


23.555 


89 


28.329 


14 


4.456 


30 


9.549 


45 


14.324 


(JO 


19.099 


75 


23.873 


90 


28.648 


15 


4.775 


m 


9.868 


46 


14.642 


61 


19.417 


76 


24.192 


91 


28.966 


16 


5.093 


32 


10.186 


47 


14.961 


62 


19.735 


77 


24.510 


92 


29.285 


17 




















93 




18 


5.730 


84 


10.823 


49 


15.597 


64 


20.372 


79 


25.146 


94 


29.921 


19 


6.048 


35 


11.141 


50 


15.915 


65 


20.690 


SO 


25.465 


95 


30.239 


20 


6.366 


36 


11.459 


51 


16.234 


66 


21.008 


81 


25.783 


96 


30.558 


81 


6.685 


37 


11.777 


52 


16.552 


67 


21.327 


S2 


26.101 


97 


30.876 


22 


7.003 


3H 


12.096 


53 


16.870 


68 


21.645 


83 


26.419 


98 


31.194 


23 


7.321 


39 


12 414 


54 


17.189 


69 


21.963 


84 


26.738 


99 


31.512 


24 


7.639 


40 


12.732 


55 


17.507 


70 


22.282 


85 


27.056 


1 00 


31.831 


25 


7.958 























For diameter of wheels of any other pitch than 1 in., multiply the figures 
in the table by the pitch. Given the diameter and the pitch, to find the num- 
ber of teeth. Divide the diameter by the pitch, look in the table under 
diameter for the figure nearest to the quotient, and the number of teeth will 
be found opposite. 

Proportions of Teeth. Circular Pitch = 1. 



1. 



2. 


3. 


4. 


5. 


.30 


.37 


.33 


.30 


.40 


.43 




.40 


.60 


.73 


.66 




.70 


.80 


.75 


.70 


.10 


.07 






.45 


.47 


.45 


.475 


.55 


.53 


.55 


.525 


.09 


.07 


.10 


.05 




.47 


.45 


.70 



Depth of tooth above pitch-line. . 
" " below pitch-line. . 

Working depth of tooth 

Total depth of tooth 

Clearance at root 

Thickness of tooth 

Width of space 

Backlash 

Thickness of rim — 



.70 
.75 
.05 
.45 
.54 
.10 



.485 
.515 
.03 



10.* 



Depth of tooth above pitch-line.. 
" " " below pitch-line. 

Working depth of tooth 

Total depth of tooth 

Clearance at root 

Thickness of tooth 



Width of space 
Backlash 



25 to .33 
,35 to .42 



.6 to .75 



.35+.0 
.65+ '.6 



.48 to .485 
52 to .515 
04 to . 



.52+ .03'' 
.04+. 06'' 



.687 
,04 to .05 
.48 to .5 -j 

.52 to .5] 
.0 to .04 



1-^-P 
1.157-^-P 

2h-P 
2.157^-P 
.157h-P 
1.51 -^-Pto 
1.57h-P 
1.57 h-P to 
1.63 -hP 
to 6-=-P 



* In terms of diametral pitch. 

Authorities.— 1. Sir Wm. Fairbairn. 2, 3. Clark,, R. T. D.; "used by en- 
gineers in good practice.' 11 4. Molesworth. 5, 6. Coleman Sellers : 5 for 
cast, 6 for cut wheels. 7, 8. Unwin. 9, 10. Leading American manufacturers 
of cut gears. 

The Chordal Pitch (erroneously called "true pitch" by some 
authors) is the length of a sti-aight line or chord drawn from centre to 
centre of two adjacent teeth. The term is now but little used, 



890 



GEARIKG. 



Chordal pitch = diam. of pitch-circle X sine of - 



180° 



Chordal 
No. of teeth 

pitch of a wheel of 10 in. pitch diameter and 10 teeth, 10 x sin 18° = 3.0903 
in. Circular pitch of same wheel = 3.1416. Chordal pitch is used with chain 
or sprocket wheels, to conform to the pitch of the chain. 

Formulae for Determining the Dimensions of Small Gears. 

(Brown & Sharpe Mfg. Co.) 
P = diametral pitch, or the number of teeth to one inch of diameter of 
pitch- circle; 



D' — diameter of pitch circle . 

D = whole diameter 

N — number of teeth — . ... 
V = velocity 



d' = diameter of pitch-circle.. 

d = whole diameter 

n = number of teeth 

v = velocity 



[Larger 
Wheel. 



Smaller 
Wheel. 



These wheels 

run 

together. 



a = distance between the centres of the two wheels; 
b = number of teeth in both wheels; 
t = thickness of tooth or cutter on pitch-circle; 
s = addendum; 
D" — working depth of tooth; 
/ = amount added to depth of tooth for rounding the corners and for 
clearance ; 
D"-\-f — whole depth of tooth; 
n = 3.1416. 

P' = circular pitch, or the distance from the centre of one tooth to the 
centre of the next measured on the pitch-circle. 

Formulae for a single wheel: 



iV+2. DxN. 
r ~~ D~' U ~ N+2 ' 


*>*$- 


1 P' 
2s; s = — = ±-=£l 


*-'£ »T?' 




N = PD' 
N = PD 


D' D 

-2; S ~ N ~ iV+2 


r = ^. D £5+l, 


/ = -; 

J 10' 


«H-/=i0+i) 


P=^; D=D' + 


P' 


,«« 


= y 2 p'- 


Formulae for a pair of wheels: 




b = 2aP; 


n = 


PD'V 

V 


D _2a(N+2). 
b ' 


AT = — : 


v = 


PD'V, 
'n ' 


2a(n + 2): 
d- b -, 


NV 
n = ; 

V 


V = 


NV. m 
n ' 


6 
a = -2P ; 




V = 


nv m 


D'-\-d' 
Ct =— IT* 


bV 


jy = 


2a V m 
v + V ; 


2a F 



The following proportions of gear wheels are recommended by Prof. Cole* 
man Sellers. {Stevens Indicator, April, 189£.) 



TOOTHED-WHEEL GEARING. 



891 



Proportions of Gear-wheels. 





Circular 
Pitch. 


Outside of 

Pitch-line. 

PX .3 


Inside of Pitch-line. 


Width of Space. 


"3 
is* 

<v o 

Is 

5 


For Cast or 

Cut Bevels 

or for Cast 

Spurs. 

PX .4 


For Cut 
Spurs. 
PX .35 


For Cast 
Spurs or 
Bevels. 
P X .525 


For Cut 

Bevels or 

Spurs. 

PX .51 




Ya 


.075 


.100 


.088 


.131 


.128 


12 


.2618 


.079 


.105 


.092 


.137 


.134 


10 


.31416 


.094 


.126 


.11 


.165 


.16 




% 


.113 


.150 


.131 


.197 


.191 


8 


.3927 


.118 


.157 


.137 


.206 


.2 


7 


.4477 


.134 


.179 


.157 


.235 


.228 




H 


.15 


.20 


.175 


.263 


.255 


6 


.5236 


.157 


.209 


.183 


.275 


.267 




9/16 


.169 


.225 


.197 


.295 


.287 




% 


.188 


.25 


.219 


.328 


.319 


5 


.62832 


.188 


.251 


.22 


.33 


.32 




H 


.225 


.3 


.263 


.394 


.383 


4 


.7854 


.236 


.314 


.275 


.412 


.401 




Vs 


.263 


.35 


.307 


.459 


.446 




l 


.3 


.4 


.35 


.525 


.51 


3 


1.0472 


.314 


.419 


.364 


.55 


.534 




1% 


.338 


.45 


.394 


.591 


.574 


2% 


1.1424 


.343 


.457 


.40 


.6 


.583 




iM 


.375 


.5 


.438 


.656 


.638 


2^ 


1/25664 


.377 


.503 


.44 


.66 


.641 




1% 


.413 


.55 


.481 


.722 


.701 




1^ 


.45 


.6 


.525 


.788 


.765 


2 


1.5708 


.471 


.628 


.55 


.825 


.801 




m 


.525 


.7 


.613 


.919 


.893 




2 


.6 


.8 


.7 


1.05 


1.02 


1J6 


2.0944 


.628 


.838 


.733 


1.1 


1.068 




2^ 


.675 


.9 


.788 


1.181 


1.148 




.75 


1.0 


.875 


1.313 


1.275 




2% 


.825 


1.1 


.963 


1.444 


1.403 




3 


.9 


1.2 


1.05 


1.575 


1.53 


1 


3.1416 


.942 


1.257 


1.1 


1.649 


1.602 




■m 


.975 


1.3 


1.138 


1.706 


1 657 




1.05 


1.4 


1.225 


1.838 


1.785 



Thickness of rim below root = depth of tooth. 

Width, of Teeth.— The width of the faces of teeth is generally made 
from 2 to 3 times the circular pitch = from 6.28 to 9.42 divided by the diam- 
etral pitch. There is no standard rule for width. 

The following sizes aro given in a stock list of cut gears in " Gram's 
Gears:' 1 

Diametral pitch 3 4 6 8 12 16 

Face, inches 3 and 4 2^ 1% and 2 1*4 and 1 J£ % and 1 ^ and % 

The Walker Mfg. Co. give: 
Circular pitch, in.. ^ % % % 1 1V£ 2 2^ 3 4 5 6 
Face, in 1J4 1}4 \% 2 2^ 4^ 6 7^ 9 12 16 20 

Rules for Calculating; the Speed of Gears and Pulleys.— 

The relations of the size and speed of driving and driven gear wheels are 
the same as those of belt pulleys. In calculating for gears, multiply or 
divide by the diameter of the pitch-circle or by the number of teeth, as 
may be required. In calculating for pulleys, multiply or divide by their 
diameter in inches. 

If D — diam. of driving wheel, d = diam. of driven, R — revolutions per 
minute of driver. r-= revs, per min. of drven. 

R=rd~D; r = RD h- d; D = dr -*- R; d = DR h- r. 

If N = number of teeth of driver and n = number of teeth of driven, 
N = nr -s- R; n = NR^-r; R = rn -+- N; r = RN -*- n. 



892 GEARIKG. 

To find the number of revolulions of the last wheel at the end of a train 
of spur-wheels, all of which are in a line and mesh into one another, when 
the revolutions of the first wheel and the number of teeth or the diameter 
of the first and last are given: Multiply the revolutions of the first wheel by 
its number of teeih or its diameter, and divide the product by the number 
of teeth or the diameter of the last wheel. 

To find the number of teeth in each wheel for a train of spur-wheels, 
each to have a given velocity: Multiply the number of revolutions of the 
driving-wheel by its number of teeth, and divide the product by the number 
of revolutions each wheel is to make. 

To find the number of revolutions of the last wheel in a train of wheels 
and pinions, when the revolutions of the first or driver, and the diameter, 
the teeth, or the circumference of all the drivers and pinions are given: 
Multiply the diameter, the circumference, or the number of teeth of all the 
driving-wheels together, and this continued product by the number of revo- 
lutions of the first wheel, and divide this product by the continued product 
of the diameter, the circumference, or the number of teeth of all the driven 
wheels, and the quotient will be the number of revolutions of the last wheel. 

Example.— 1. A train of wheels consists of four wheels each 12 in. diameter 
of pitch-circle, and three pinions 4, 4, and 3 in. diameter. The large wheels 
are the drivers, and the first makes 36 revs, per min. Required the speed 
of the last wheel. 



36 X 12 X 12 X 12 ,__. 

4X4X3 = ^96rpm. 



2. What is the speed of the first large wheel if the pinions are the drivers, 
the 3-in. pinion being the first driver and making 36 revs, per min.? 

Sfi v S v d v A. 

■- 1 rpm. Ans. 



12 X 12 X 12 



Milling Cutters for Interchangeable Gears.— The Pratt & 
Whitney Co. make a series of cutters for cutting epicycloidal teeth. The 
number of cutters to cut from a pinion of 12 teeth to a rack is 24 for each 
pitch coarser than 10. The Brown & Sharpe Mfg. Co. make a similar series, 
and also a series for involute teeth, in which eight cutters are made for 
each pitch, as follows: 

No 1. 2. 3. 4. 5. 6. 7. 8. 

Will cut from 135 55 35 26 21 17 14 12 

to Rack 134 54 34 25 20 16 13 

FORMS OF THE TEETH. 

In order that the teeth of wheels and pinions may run together smoothly 
and with a constant relative velocity, it is necessary that their working 
faces shall be formed of certain curves called odontoids. The essential 
property of these curves is that when two teeth are in contact the common 
normal to the tooth curves at their point of contact must pass through the 
pitch-point, or point of contact of the two pitch circles. Two such curves 
are in common use — the cyloid and the involute. 

The Cycloidal Tooth.— In Fig. 154 let PL and pi be the pitch-circles 
of two gear-wheels; GCand gc are two equal generating-circles, whose radii 
should be taken as not greater than one half of the radius of the smaller 
pitch-circle. If the circle gc be rolled to the left on the larger pitch-circle 
PL, the point O will describe an epicycloid, oefgh. If the other generating- 
circle GC be rolled to the right on PIj, the point O will describe a hypocy- 
cloid oabcd. These two curves, which are tangent at 0, form the two parts 
of a tooth curve for a gear whose pitch-circle is PL. The upper part oh is 
called the face and the lower part od is called the flank, If the same circles 
be rolled on the other pitch-circle pi, they will describe the curve for a tooth 
of the gear pi, which will work properly with the tooth on PL. 

The cycloidal curves may be drawn without actually rolling the generat- 
ing-circle, as follows: On the line PL, from O, step off and mark equal dis- 
tances, as 1, 2, 3, 4, etc. From 1, 2, 3, etc., draw radial lines toward the centre 
of PL, and from 6, 7, 8, etc., draw radial lines from the same centre, but be- 
yond PL. With the radius of the generating-circle, and with centres suc- 
cessively placed on these radial lines, draw arcs of circles tangent to PL at 
1 2 3, 6 7 8, etc. With the dividers set to one of the equal divisions, as O l7 



FORMS OF THE TEETH. 



step off la and 6e; step off two such divisions on the circle from 2 to ft, and 
from 7 to/; three such divisions from 3 to c, and from 8 to a; and so on, thus 
locating the several points abcdH and efgk, and through these points draw 
the tooth curves. 

The curves for the mating tooth on the other wheel may be found in like 
manner by drawing arcs of the generating-circle tangent at equidistant 
points on the pitch-circle pi. 

The tooth curve of the face oh is limited by the addendum-line r or r^ 




Fig. 154. 7 

and that of the flank oH by the root curve R or R x . R and r represent the 
root and addendum curves for a large number of small teeth, and R^r the 
like curves for a small number of large teeth. The form or appearance of 
the tooth therefore varies according to the number of teeth, while the pitch- 
circle and the generating-circle may remain the same. 

In the cycloidal system, in order that a set of wheels of different diam- 
eters but equal pitches shall all correctly work together, it is necessary that 
the generating-circle used for the teeth of all the wheels shall be the same, 
and it should have a diameter not greater than half the diameter of the pitch- 
line of the smallest wheel of the s*t. The customary standard size of the 
generating-circle of the cycloidal system is one having a diameter equal to 
the radius of the pitch-circle of a wheel having 12 teeth. (Some gear- 
makers adopt 15 teeth.) This circle gives a radial flank to the teeth of a 
wheel having 12 teeth. A pinion of 10 or even a smaller number of teeth 
can be made, hut in that case the flanks will be undercut, and the tooth will 
not be as strong as a tooth with radial flanks. If in any case the describing 
circle be half the size of the pitch-circle, the flanks will be radial; if it be 
less, they will spread out toward the root of the tooth, giving a stronger 
form; but if greater, the flanks will curve in toward each other, whereby the 
teeth become weaker and difficult to make. 

In some cases cycloidal teeth for a pair of gears are made with the gener- 
ating-circle of each gear, having a radius equal to half the radius of its pitch- 
circle. In this case each of the gears will have radial flanks. This method 
makes a smooth working gear, but a disadvantage is that the wheels are 
not interchangeable with other wheels of the same pitch but different num- 
bers of teeth. 



894 



GEARING. 



The rack in the cycloidal system is equivalent to a wheel with an infinite 
number of teeth. The pitch is equal to the circular pitch of the mating 
gear. Both faces and flanks are cycloids formed by rolling the generating- 
circle of the mating gear-wheel on each side of the straight pitch-line of 
the rack. 




%l 



Fig. 155. 

Another method of drawing the cycloidai curves is shown in Fig. 155. It 
is known as the method of tangent arcs. The generating-circles, as before, 
are drawn with equal radii, the length of the radius being less than half the 
radius of pi, the smaller pitch-circle. Equal divisions 1, 2, 3, 4, etc., are 
marked off on the pitch-circles and divisions of the same length stepped off 
ou one of the generating-circles, as oabc, etc. From the points 1, 2, 3, 4. 5 on 
the line po, with radii successively equal to the chord distances oa. ob, or. 
od, oe, draw the five small arcs F. A line drawn through the outer edges of 
these small arcs, tangent to them all, will be the hypocycloidal curve for the 
flank of a tooth below the pitch-line pi. From the points 1, 2, 3, etc., on the 
line ol, with radii as before, draw the small arcs Q. A line tangent to these 
arcs will be the epicycloid for the face of the same tooth for which the flank 
curve has already been drawn. In the same way, from centres on the line 
P , and oL, with the same radii, the tangent arcs H and Kmay be drawn, 
which will give the tooth for the gear whose pitch-circle is PL. 
■ If the generating-ciicle had a radius just one half of the radius of pi, the 
hypocycloid F would be a straight line, and the flank of the tooth would 
have been radial. 

The Involute Tooth.— In drawing the involute tooth curve, the 
angle of obliquity, or the angle which a common tangent to the teeth, when 
they are in contact at the pitch-point, makes with a line joining the centres 
of the wheels, is first arbitrarily determined. It is customary to take it at 15°. 
The pitch-lines pi and PL being drawn in contact at O. theline of obliquity 
AB is drawn through normal to a common tangent to the tooth curves, or 
at the given angle of obliquity to a common tangent to the pitch-circles. In 



FORMS OF THE TEETH. 



895 



the cut the angle is 20°. From the centres of the pitch-circles draw circles c 
and d tangent to the line AB. These circles are called base-lines or base- 
circles, from which the involutes FandK are drawn. By laying off conven- 
ient distances 0, 1, 2, 3, which should each be less than 1/10 of the diameter 
of the base-circle, small arcs can be drawn with successively increasing- 
radii, which will form the involute. The involute extends from the points F 




Fig. 156. 

and K down to their respective base-circles, where a tangent to the invo- 
lute becomes a radius of the circle, and the remainders of the tooth curves, 
as G and H, are radial straight lines. 

In the involute system the customary standard form of tooth is one 
having an angle of obliquity of 15° (Brown and Sharpe use 14^°), an adden- 
dum of about one third the circular pitch, and a clearance of about one 
eighth of the addendum. In this system the smallest gear of a set has 12 
teeth, this being the smallest number of teeth that will gear together when 
made with this angle of obliquity. In gears with less than 30 teeth the 
points of the teeth must be slightly rounded over to avoid interference (see 
Giant's Teeth of Gears). All involute teeth of the same pitch and with the 
same angle of obliquity work smoothly together. The rack to gear with an 
involute-toothed wheel has straight faces on its teeth, which make an angle 
with the middle line of the tooth equal to the angle of obliquity, or in the 
.standard form the faces are inclined at an angle of 30° with each, other. 

To draw the teeth of a rack which is to gear with an involute wheel (Fig. 
157).— Let AB be the pitch-line of the rack and AI= I7'=the pitch. Through 




the pitch-point I draw EF at the given angle of obliquity. Draw A E and 
I'F perpendicular to EF. Through E and F draw lines EGG' and FH par- 
allel to the pitch-line. EGG' will be the addendum-line and HF the flank- 
line. From /draw IK" perpendicular to AB equal to the greatest addendum 
in the set of wheels of tho given pitch and obliquity plus an allowance for 
clearance equal to \i of the addendum. Through K, parallel to AB, draw 
the clearance-line. The fronts of the teeth are planes perpendicular lo EF. 
and the backs are planes inclined at the same angle to AB in the contrary 
direction. The outer half of the working face AE may be slightly curved. 
Mr. Grant makes it a circular arc drawn from a centre on the pitch-line 



896 



GEARING. 




Fig. 158. 



with a radius = 2.1 inches divided by the diametral pitch, or .67 in. x cir- 
cular pitch. 

To Draw an Angle of 15° without using a Protractor.— From C, on the 
line AC, with radius AC, draw 
an arc AB, and from A, with 
the same radius, cut the arc at 
B. Bisect the arc BA by draw- 
ing small arcs at D from A and B 
as centres, with the same radius, 
which must be greater than one 
half of AB. Join DC, cutting BA 
at E. The angle EC A is 30°. Bi- 
sect the arc AE in like manner, 
and the angle FCA will be 15°. 

A property of involute-toothed 
wheels is that the distance between 
the axes of a pair of gears may be 
altered to a considerable extent 
without interfering with their ac- 
tion. The backlash is therefore 
variable at will, and may be ad- 
justed by moving the wheels farther from or nearer to each other, and may 
thus be adjusted so as to be no greater than is necessary to prevent jam- 
ming of the teeth. 

The relative merits of cycloidal and involute-shaped teeth are still a sub- 
ject of dispute, but there is an increasing tendency to adopt the involute 
tooth for all purposes. 

Clark (R. T. D., p. 734) says : Involute teeth have the disadvantage of 
being too much inclined to the radial line, by which an undue pressure is 
exerted on the bearings. 

Unwin (Elements of Machine Design, 8th ed., p. 265) says : The obliquity 
of action is ordinarily alleged as a serious objection to involute wheels. Its 
importance has perhaps been overrated. 
George B. Grant (Am. Much., Dec. 26, 1885) says : 

1. The work done by the friction of an involute tooth is always less than 
the same work for any possible epicycloidal tooth. 

2. With respect to work done by friction, a change of the base from a 
gear of 12 teeth to one of 15 teeth makes an improvement for the epicycloid 
of less than one half of one per cent. 

3. For the 12-tooth system the involute has an advantage of 1 1/5 per 
cent, and for the 15-tooth system an advantage of % per cent. 

4. That a maximum improvement of about one per cent can be accom- 
plished by the adoption of any possible non -interchangeable radial flank 
tooth in preference to the 12-tooth interchangeable system. 

5. That for gears of very few teeth the involute has a decided advantage. 

6. That the common opinion among millwrights and the mechanical pub- 
lic in general in favor of the epicycloid is a prejudice that is founded on 
long-continued custom, and not on an intimate knowledge of the properties 
of that curve. 

Wilfred Lewis (Proc. Engrs. Club of Phila., vol. x., 1893) says a strong 
reaction in favor of the involute system is in progress, and he believes that 
an involute tooth of 22^° obliquity will finally supplant all other forms. 

Approximation by Circular Arcs.— Having found the form of 
the actual tooth-curve on the drawing-board, circular arcs maybe found by 
trial which will give approximations to the true curves, and these may be 




FORMS OF THE TEETH. 



897 



used in completing the drawing and the pattern of the gear-wheels. The 
root of the curve is connected to the clearance by a fillet, which should be 
as large aspossible to give increased strength to the tooth, provided it is not 
large enough to cause interference. 

Molesworth gives the following method of construction by circular arcs : 

From the radial line at the edge of the tooth on the pitch-line, lay off the 
line HKaX an angle of 75° with the radial line; on this line will be the cen- 
tres of the root AB and the point EF. The lines struck from these centres 
are shown in thick lines. Circles drawn through centres thus found will 
give the lines in which the remaining centres will be. The radius DA for 
striking the root AB is = pitch + the thickness of the tooth. The radius 
CE for striking the point of the tooth EF = the pitch. 

George B. Grant says : It is sometimes attempted .to construct the curve 
by some handy method or empirical rule, but such methods are generally 
worthless. 

Stepped Gears. -Two gears of the same pitch and diameter mounted 
side by side on the same shaft will act as a single gear. If one gear is keyed 
on the shaft so that the teeth of the two wheels are not in line, but the 
teeth of one wheel slightly in advance of the other, the two gears form a 
stepped gear. If mated with a similar stepped gear on a parallel shaft the 
number of teeth in contact will be twice as great as in an ordinary gear, 
which will increase the strength of the gear and its smoothness of action. 

Twisted Teeth.— If a great number of very thin gears were placed 
together,' one slightly in advance of the other, they would still act as a 
stepped gear. Continuing the subdivision until the 
thickness of each separate gear is infinitesimal, the 
faces of the teeth instead of being in steps take the 
form of a spiral or twisted surface, and we have a 
twisted gear. The twist may take any shape, and if it is 
in one direction for half the width of the gear and in the 
opposite direction for the other half, we have what is 
known as the herring-bone or double helical tooth. The 
obliquity of the twisted tooth if twisted in one direction 
causes an end thrust on the shaft, but if the herring- 
bone twist is used, the opposite obliquities neutralize 
each other. This form of tooth is much used in heavy 
rolling-mill practice, where great strength and resistance 
to shocks are necessary. They are frequently made of 
steel castings (Fig. 160). The angle of the tooth with a 
line parallel to the axis of the gear is usually 30°. 

Spiral Gears.— If a twisted gear has a uniform tw 
spiral gear. The line in which the pitch-surface intersects the face of the 
tooth is part of a helix drawn on the pitch-surface. A spiral wheel may be 
imade with only one helical tooth wrapped around the cylinder several 
times, in which it becomes a screw or worm. If it has two or three teeth 
so wrapped, it is a double- or triple-threaded screw or worm. A spiral -gear 
meshing into a rack is used to drive the table of some forms of planing- 
inachine. 

Worm-gearing. -When the axes of two spiral gears are at rignt 
angles, and a wheel of one, two, or three threads works with a larger wheel 
3f many threads, it becomes a worm-gear, or endless screw, the smaller 




Fig. 160. 

it becomes a 




vheel or driver being called the worm, and the larger, or driven wheel, the 
vorm-wheel. With this arrangement a high velocity ratio may be obtained 
vith a single pair of wheels. For a one-threaded wheel the velocity ratio is 



GEARING. 



the number of teeth in the worm-wheel. The worm and wheel are com- 
monly so constructed that the worm will drive the wheel, but the wheel will 
not drive the worm. 

To find the diameter of a worm-ivheel at the throat, number of teeth and 
pitch of the worm being given: Add 2 to the number of teeth, multiply the 
sum by 0.3183, and by the pitch of the worm in inches. 

To find the number of teeth, diameter at throat and pitch of worm being 

given: Divide 3.1416 times the diameter by the pitch, and subtract 2 from 

the quotient. 

In Fig. 161 ab is the diam. of the pitch-circle, cd is the diam. at the throat. 

Example.— Pitch of worm V± in., number of teeth 70, required the diam. 

at the throat. (70 f 2) X .3183 X .25 = 5.73 in. 

Teeth of Bevel- wheels. (Rankine's Machinery and Millwork.)— 
The teeth of a bevel -wheel have acting surfaces of the conical kind, gen- 
erated by the motion of a line traversing the apex of the conical pitch- 
surface, while a point in it is carried round the traces of the teeth upon a 
spherical surface described about that apex. 

The operations of drawing the traces of the teeth of bevel-wheels exactly, 
whether by involutes or by rolling curves, are in every respect analogous to 
those for drawing the traces of the teeth of spur-wheels; except that in the 
case of bevel- wheels all those operations are to be performed on the surface 
of a sphere described about the apex, instead of on a plane, substituting 
poles for centres and great circles for straight lines. 

In consideration of the practical difficulty, especially in the case of large 
wheels, of obtaining an accurate spherical surface, and of drawing upon it 
when obtained, the following approximate method, proposed originally by 
Tredgold, is generally used: 

Let O, Fig. 162, be the common apex of the pitch- cones, OBI, OB' I, of a 
pair of bevel-wheels; OC, OC, the axes of those cones; 01 their line of con- 
tact. Perpendicular to 01 draw 
AIA', cutting the axes in A, A'; 
make the outer rims of the patterns 
and of the wheels portions of the 
cones ABI, A'B'I, of which the nar- 
row zones occupied by the teeth will 
be sufficiently near for practical pur- 
poses to a spherical surface described 
about O. As the cones ABI, A'B'I 
cut the pitch -cones at right angies in 
the outer pitch- circles IB, IB', ihey 
may be called the normal cones. To 
find the traces of the teeth upon the 
normal cones, draw on a flat surface 
circular arcs, ID, ID', with the radii 
AI, A'l; those arcs will be the de- 
velopments of arcs of the pitch- 
circles IB, IB' wheu the conical sur- 
Describe the traces of teeth for the 




Fig. 162. 



faces ABI, A'B'I are spread out flat. 

developed arcs as for a pair of spur-wheels, then wrap the developed arcs 
on the normal cones, so as to make them coincide with the pitch-circles, and 
trace the teeth on the conical surfaces. 

For formulae and instructions for designing bevel-gears, and for much other 
valuable information on the subject of gearing, see " Practical Treatise on 
Gearing," and " Formulas in Gearing,' 1 published by Brown & Sharpe Mf 'g 
Co.; and "Teeth of Gears," by George B. Giant, Lexington, Mass. The 
student may also consult Rankine's Machinery and Millwork, Reuleaux's 
Constructor, and Unwin's Elements of Machine Design. See also article on 
Gearing, by C. W. MacCord in App. Cyc. Mech., vol. ii. 

Annular and Differential Gearing. (S. W. Balch., Am. Much., 
Aug. 24, 1893.)— In internal gears the sum of the diameters of the describing 
circles for faces and flanks should not exceed the difference in the pitch 
diameters of the pinion and its internal gear. The sum may be equal to this 
difference or it may be less; if it is equal, the faces of the teeth of each 
wheel will drive the faces as well as the flanks of the teeth of the other 
wheel. The teeth will therefore make contact with each other at two points 
at the same time. 

Cycloidal tooth-curves for interchangeable gears are formed with describ- 
ing circles of about % the pitch diameter of the smallest gear of the series 
To admit two such circles between the pitch-circles of the pinion and intern; 



EFFICIENCY OF GEARING. 



899 



gear the number of teeth in the internal gear should exceed the number in 
the pinion by 12 or more, if the teeth are of the customary proportions and 
curvature used in interchangeable gearing. 

Very often a less difference is desirable, and the teeth may be modified in 
several ways to make this possible. 

First. The tooth curves resulting from smaller describing circles may be 
employed. These will give teeth which are more rounding and narrower at 
their tops, and therefore not as desirable as the regular forms. 

Second. The tips of the teeth may be rounded until they clear. This is a 
cut-and-try method which aims at modifying the teeth to such outlines as 
smaller describing circles would give. 

Third. One of the describing circles may be omitted and one only used, 
which may be equal to the difference between the pitch -circles. This will 
permit the meshing of gears differing by six teeth. It will usually prove 
inexpedient to put wheels in inside gears that differ -by much less than 12 
teeth. 

If a regular diametral pitch and standard tooth forms are determined on, 
the diameter to which the internal gear-blank is to be bored is calculated by 
subtracting 2 from the number of teeth, and dividing the remainder by the 
diametral pitch. 

The tooth outlines are the match of a spur-gear of the same number of 
teeth and diametral pitch, so that the spur-gear will fit the internal gear as 
a punch fits its die, except that the teeth of each should fail to bottom in 
the tooth spaces of the other by the customary clearance of one tenth the 
thickness of the tooth. 

Internal gearing is particularly valuable when employed in differential 
action. This is a mechanical movement in which one of the wheels is 
mounted on a crank so that its centre can move in a circle about the centre 
of the other wheel. Means are added to the device which restrain the wheel 
on the crank from turning over and confine it to the revolution of the crank. 

The ratio of the number of teeth in the revolving wheel compared with 
the difference between the two will represent the ratio between the revolv- 
ing wheel and the crank-shaft by which the other is carried. The advan- 
tage in accomplishing the change of speed with such an arrangement, as 
compared with ordinary spur-gearing, lies in the almost entire absence of 
friction and consequent wear of the teeth. 

But for the limitation that the difference between the wheels must not be 
too small, the possible ratio of speed might be increased almost indefinitely, 
and one pair of differential gears made to do the service of a whole train of 
wheels. If the problem is properly worked out with bevel-gears this limita- 
tion may be completely set aside, and external and internal bevel-gears, 
differing by but a single tooth if need be, made to mesh perfectly with each 
other. 

Differential bevel-gears have been used with advantage in mowing-ma- 
chines. A description of their construction and operation is given by Mr. 
Balch in the article from which the above extracts are taken. 

EFFICIENCY OF GEARING. 

An extensive series of experiments on the efficiency of gearing, chiefly 
worm and spiral gearing, is described by Wilfred Lewis in Trans. A. S. M. E., 
vii. 273. The average results are shown in a diagram, from which the fol- 
lowing approximate average figures are taken : 

Efficiency of Spur, Spiral, and Worm Gearing. 



Gearing. 



Velocity at Pitch line in feet per mm. 



Spur pinion 

Spiral pinion , 

Spiral pinion or worm, 



.935 
.87 
.815 
.75 
.70 
.615 
.53 
.43 



.845 
.805 
.74 
.72 
.60 



.955 
.93 
.90 



.965 
.945 



.815 
.765 



900 



The experiments showed the advantage of spur-searing 1 over all other 
kinds in both durability and efficiency. The variation from the mean results 
rarely exceeded b% in either direction, so long as no cutting occurred, but 
the variation became much greater and very irregular as soon as cutting 
began. The loss of power varies with the speed, the pressure, the tempera- 
ture, and the condition of the surfaces. The excessive friction of worm and 
spiral gearing is largely due to thee nd thrust on the collars of the shaft. 
This may be considerably red uced by roller-bearings for the collars. 

When two worms with opposite spirals run in two spiral worm-gears that 
also work with each other, and the pressure on one gear is opposite that on 
the other, there is no thrust on the shaft. Even with light loads a worm 
will begin to heat and cut if run at too high a speed, the limit for safe work- 
ing being a velocity of the rubbing surfaces of 200 to 300 ft. per minute, the 
former being preferable where the gearing has to work continuously. The 
wheel teeth will keep cool, as they form part of a casting having a large 
radiating surface; but the worm itself is so small that its heat is dissipated 
slowly. Whenever the heat generated increases faster than it can be con- 
ducted and radiated away, the cutting of the worm may be expected to be- 
gin. A low efficiency for a worm-gear means more than the loss of power, 
sinc6 the power which is lost reappears as heat and may cause the rapid 
destruction of the worm. 

Unwin (Elements of Machine Design, p. 294) says : The efficiency is greater 
the less the radius of the worm. Generally the radius of the worm = 1.5 to 
3 times the pitch of the thread of the worm or the circular pitch of the 
worm-wheel. For a one-threaded worm the efficiency is only 2/5 to J4; 
for a two-threaded worm, 4/7 to 2/5; for a three-threaded worm, % to y%. 
Since so much work is wasted in friction it is not surprising that the wear 
is excessive. The following table gives the calculated efficiencies of worm- 
wheels of 1, 2, 3, and 4 threads and ratios of radius of worm to pitch of teeth 
of from 1 to 6, assuming a coefficient of friction of 0.15 : 



No. of 
Threads 



Radius of Worm -h Pitch. 



1 


m 


Wz 


m 


2 


2^ 


3 


4 


6 


.50 


.44 


.40 


.36 


.33 


.28 


.25 


.20 


.14 


.67 


.62 


.57 


.53 


.50 


.44 


.40 


.33 


.25 


.75 


.70 


.67 


.63 


.60 


.55 


.50 


.43 


.33 


.80 


.76 


.73 


.70 


.67 


.62 


.57 


.50 


.40 



STRENGTH OF GEAR-TEETH. 

The strength of gear-teeth and the horse-power that may be transmitted 
by them depend upon so many variable and uncertain factors that it is not 
surprising that the formulas and rules given by different writers show a 
wide variation. In 1879 John H. Cooper {Jour. Frank. last., July, 1879) 
found that there were then in existence about 48 well-established rules for 
horse-power and working strength, differing from each other in extreme 
cases about 500$. In 1886 Prof. Win. Harkness (Proc. A. A. A. S. 1886), 
from an examination of the bibliography of the subject, beginning in 1796, 
found that according to the constants and formulae used by various authors 
there were differences of 15 to 1 in the power which could be transmitted 
by a given pair of gearsd wheels. The various elements which enter into 
the constitution of a formula to represent the working strength of a toothed 
wheel are the following: 1. The strength of the metal, usually cast iron, which 
is an extremely variable quantity. 2. The shape of the tooth, and espec- 
ially the relation of its thickness at the root or point of least strength to the 
pitch and to the length. 3. The point at which the ioad is taken to be ap- 
plied, assumed by some authors to be at the pitch-line, by others at the 
extreme end. along the whole face, and by still others at a single outer 
corner. 4. The consideration of whether che total load is at any time re- 
ceived by a single tooth or whether it is divided between two teeth. 5. The 
influence of velocity in causing a tendency to break the teeth by shock. 6. 
The factor of safety assumed to cover all the uncertainties of the other ele- 
ments f the problem. 



STRENGTH OF GEAR-TEETH. 



901 



Prof. Harkness, as a result of his investigation, found that all the formula? 
on the subject might be expressed in one of three forms, viz.: 

Horse-power = CFpf, or CFp 2 , or CVp 2 f; 

in which C is a coefficient, V = velocity of pitch-line in feet per second, p — 
pitch in inches, and / = face of tooth in inches. 

From an examination of precedents he proposed the following formula 
for cast-iron wheels: 



H.P. : 



0.910Fjo/ 
4/1 + 0.65 V 



He found that the teeth of chronometer and watch movements were sub- 
ject to stresses four times as great as those which any engineer would dare 
to use in like proportion upon cast-iron wheels of large size. 

It appears that all of the earlier rules for the strength of teeth neglected 
the consideration of the variations in their form; the breaking strength, as 
said by Mr. Cooper, being based upon the thickness of the teeth at the pitch- 
line or circle, as if the thickness at the root of the tooth were the same in 
all cases as it is at the pitch-line. 

Wilfred Lewis (Proc. Eng'rs Club, Phila., Jan. 1893; Am. Mach., June 22, 
1893) seems to have been the first to use the form of the tooth in the con- 
struction of a working formula and table. He assumes that in well-con- 
structed machinery the load can be more properly taken as well distribuied 
across the tooth than as concentrated in one corner, but that it cannot be 
safely taken as concentrated at a maximum distance from the root less 
than the extreme end of the tooth. He assumes that the whole load is 
taken upon one tooth, and considers the tooth as a beam loaded at one end, 
and from a series of drawings of teeth of the involute, cycloidal, and radial 
flank systems, determines the point of weakest cross-section of each, and 
the ratio of the thickness at that section to the pitch. He thereby obtains 
the general formula, 

W = spfy; 

in which W is the load transmitted by the teeth, in pounds; s is the safe 
working stress of the material, taken at 8000 lbs. for cast iron, when the 
working speed is 100 ft. or less per minute; p = pitch;/ = face, in inches; 
y = a factor depending on the form of the tooth, whose value for different 
cases is given in the following table: 





Factor for Strength, y. 




Factor for Strength, y. 


No. of 
Teeth. 








No. of 
Teeth. 








Involute 
20° Obli- 
quity. 


Involute 
15° and 
Cycloidal 


Radial 
Flanks. 


Involute 
20° Obli- 
quity. 


Involute 
15° and 
Cycloidal 


Radial 
Flanks. 


12 


.078 


.067 


.052 


27 


.111 


.100 


.064 


13 


.083 


.070 


.053 


30 


.114 


102 


.065 


14 


.088 


.072 


.054 


34 


.118 


.104 


.066 


15 


.092 


.075 


.055 


38 


.122 


.107 


.067 


16 


.094 


.077 


.056 


43 


126 


.110 


.068 


17 


.096 


.080 


.057 


50 


-30 


.112 


.069 


18 


.098 


.083 


.058 


60 


134 


.114 


.070 


19 


.100 


.087 


.059 


75 


.138 


.116 


.071 


20 


.102 


.090 


.060 


100 


.142 


.118 


.072 


21 


.104 


.092 


.061 


150 


146 


.120 


.073 


23 


.106 


.094 


.062 


300 


.150 


122 


.074 


25 


.108 


.097 


.063 


Rack. 


.154 


.124 


.075 



Safe Working Stress, s, for Different Speeds. 



Speed of Teeth in 
ft. per minute. 


100 or 
less. 


soo 


300 


600 

4000 
10000 


900 

3000 
7500 


1200 


1800 

2000 
5000 


2400 




8000 
20000 


6000 
15000 


4800 
12000 


2400 
6000 


1700 


Steel 


4300 




902 GEARING. 

The values of s in the above table are given by Mr. Lewis tentatively, in 
the absence of sufficient data upon which to base more definite values, but 
they have been found to give satisfactory results in practice. 

Mr. Lewis gives the following example to illustrate the use of the tables: 

Let it be required to find the working strength of a 12- toothed pinion of 1- 

inch pitch, 2^-inch face, driving a wheel of 60 teeth at 100 feet or less per 

minute, and let the teeth be of the 20-degree involute 

form. In the formula W - spfij we have for a cast-iron 

pinion s = 8000, pf = 2 5. and y = .078; and multiplying these 

I values together, we hnve W = l;o60 pounds. For the wheel 

I we have y = .134 and W = 2680 pounds. 

' The cast-iron pinion is, therefore, the measure of 

Strength; but if a si eel pinion be substituted we have 

s = 20,000 and W= 3900 pounds, in which combination 

the wheel is the weaker, and it therefore becomes the 

measure of strength. 

For bevel-wheels Mr. Lewis gives the following-, refer- 
ring to Fig. 163: D = large diameter of bevel; d = 
small diameter of bevel; p — pitch at large diameter; 
n = actual number of teeth; / = face of bevel: N — for- 
mative number of teeth = n X secant a, or the number 
corresponding to radius R ; y = factor depending upon 
shape of teeth and formative number N ; W — working load on teeth. 

■ w = *pfv sd^d - ay or ' more simply ' w=s Pfy-f ) ' 

which gives almost identical results when d is not less than % D, as is the 
case in good practice. 

In Am. Much., June 22, 1893, Mr. Lewis gives the following formula? for 
the working strength of the three systems of gearing, which agree very 
closely with those obtained by -use of the table: 

For involute, 20° obliquity, W = spf ( AM — : ) ; 

/ 684 \ 

For involute 15°, and cyc^oidal, W = spf .124 - '—— ) ; 

For radial flank system, W= spf I .075 — • : — ) ; 

in which the factor within the parenthesis corresponds to y in the general 
formula. For the horse-power transmitted, Mr. Lewis's general formula 

T , 7 , 33,000 H. P. «.*«.* tt-o spfyv . ,. . 

W = spfy, = , may take the form H.P. = % nnn , in which v = 

velocity in feet per minute; or since v = dir X rpm. -*- 12 = ,2618d X rpm., in 
which d = diameter in inches and rpm. = revolutions per minute, 

H-P- = £& - ^tm^ = .0—^ X rpo, 

It must be borne in mind, however, that in the case of machines which 
consume power intermittently, such as punching and shearing machines, 
the gearing should be designed with reference to the maximum load W, 
which can be brought upon the teeth at any time, and not upon the average 
horse-power transmitted 

Comparison of the Harkness and. Lewis Formulas.— 
Take an average case in which the safe working strength of the material, 
s = 6000, v = 200 ft. per min., and?/ = .100, the value in Mr. Lewis's table 
for an involute tooth of 15° obliquity, or a cycloidal tooth, the number of 
teeth in the wheel being 27. 

_ spfyv _ 6000pfv X .100 
~ 33,000 ~ 33,000 

if V is taken in feet per second. 

910 Vvf 
Prof. Harkness gives H.P.= — * J . If the V in the denominator 
Vl 4- 0.65 V 



STRENGTH OF GEAR-TEETH. 



903 



oe taken at 200 -=- 60 = 8}4 feet per second, Vl + 0.65 V = \' 3.167 = 
.910 T , 



formula. This is probably as close an agreement as can be expected, since 
Prof. Harkness derived his formula from an investigation of ancient prece- 
dents and rule-of-thumb practice, largely with common cast gears, while 
Mr. Lewis's formula was derived from considerations of modern practice 
with machine moulded and cut gears. 

Mr. Lewis takes into consideration the reduction in working strength of a 
tooth due to increase in velocity by the figures in his table of the values of 
the safe working stress s for different speeds. Prof. Harkness gives expres- 
sion to the same reduction by means of the denominator of his formula, 
V 1 + 0.657". The decrease in strength as computed by this formula is 
somewhat less than that given in Mr. Lewis's table, and as the figures given 
in the table are not based on accurate data, a mean between the values given 
by the formula and the table is probably as near to the true value as may 
be obtained from our present knowledge. The following table gives the 
values for different speeds according to Mr. Lewis's table and Prof. Hark- 
ness's formula, taking for a basis a working stress s, for cast-iron 8000, and 
for steel 20,000 lbs. at speeds of 100 ft. per minute and less: 



v = speed of teeth, ft. per min. , 
V= " " ft. per. sec. 



Safe stress s, cast-iron, Lewis. . 
Relative do., .s h- 8000 



c = 1-4-f 1+0.65F.... 

Relative val. c h- .693 

s x = 8000 X (c -^ .693) 

Mean of s and s l5 cast-iron = s 

.' " -t for steel = j 

Safe stress for steel, Lewis 



100 


200 


300' 


600 


900 


1200 


1800 


m 


*% 


5 


10 


15 


20 


30 


8000 


6000 


4800 


4000 


3000 


2400 


2000 


1 


.75 


.6 


.5 


.375 


.3 


.25 


.6930 


.5621 


.4850 


.3650 


.3050 


2672 


.2208 


1 


.811 


.700 


.526 


.439 


.385 


.318 


8000 


6488 


5600 


4208 


3512 


3080 


2544 


8000 


6:200 


'5200 


4100 


3300 


2700 


2300 


20000 


15500 


13000 10300 


8100 


6800 


5700 


20000 


15000 


12000 


10000 


7500 


6000 


5000 



2400 
40 

1700 
.2125 

.1924 

2216 
2000 
4900 
4300 



Comparing the two formulas for the case of s 
speed of 100 ft. per min., we have 



: 8000, corresponding to a 



Lewis: 



H.P. = 



= 2i.24pfy, 



Harkness: H.P. = 1 -=- VT+ 0.65F X MOVpf = .695 x .91 X l%pf= 1.051/)/' 

spfyv _ spfyV _ 8000 X \%pfy 

33,000 550 550 

in which y varies according to the shape and number of the teeth. 

For radial-flank gear with 12 teeth y ^ .052; 24.24pfy = 1.260p/ : 

For 20° involute, 19 teeth, or 15° inv., 27 teeth y = .100; 24.24pfy = 2.424pf; 
For 15° involute, 300 teeth ■ y = .150; 24.24pfy = 3.636p/. 

Thus the weakest-shaped tooth, according to Mr. Lewis, will transmit 20 
per cent more horse-power than is given by Prof. Haikness's formula, in 
which the shape of the tooth is not considered, and the average-shaped 
tooth, according to Mr. Lewis, will transmit more than double the horse- 
power given by Prof. Harkness's formula. 

Comparison of Other Formulae.— Mr. Cooper, in summing up 
his examination, selected an old English rule, which Mr. Lewis considers as 
a passably correct expression of good general averages, viz. : X— 2000p/, 
X = breaking load of tooth in pounds, p — pitch, / = face. If a factor of 
safety of 10 be taken, this would give for safe working load W = 200/?/. 

George B. Grant, in his Teeth of Gears, page 33. takes the breaking load 
at 3500p/, and, with a factor of safety of 10, gives W = 350pf. 

Nystrom's Pocket-Book, 20th ed., 1891, says : " The strength and durability 
of cast-iron teeth require that they shall transmit a force of 80 lbs. per inch 
of pitch and per inch breadth of face." This is equivalent to W = 80p/, or 
only 40$ of that given by the English rule. 

F. A. Halsey (Clark's Pocket Book) gives a table calculated from the 
formula H.P. = pfd x rpm. -f- 850. 

Jones & Laughlins give H.P. = pfd X rpm. -i- 550. 

These formulas transfoi'med give W= 128p/and W ~ 218p/, respectively, 



904 GEARING. 

Unwin, on the assumption that the load acts on the corners of the teeth, 
derives a formula p = K Vw, in which K is a coefficient derived from ex- 
isting wheels, its values being : for slowly moving gearing not subject to 
much vibration or shock K = .04; in ordinary mill-gearing, running at 
greater speed and subject to considerable vibration, K — .05; and in wheels 
subjected to excessive vibration and shock, and in mortise gearing, K = .06. 
Reduced to the form W = Cpf, assuming that/ = 2p, these values of K give 
W = 262p/, 200p/, and 139p/, respectively. 

Unwin also gives the following formula, based on the assumption that the 

pressure is distributed along the edge of the tooth : p = K x a/-^ VW, 

where K x = about .0707 for iron wheels and .0848 for mortise wheels when 
the breadth of face is not less than twice the pitch. For the case of / = 2p 
and the given values of K 1 this reduces to W = 200pf and W = 139p/, 
respectively. 

Box, in his Treatise on Mill Gearing, gives H.P. = " n , in which n 

= number of revolutions per minute. This formula differs from the more 
modern formulae in making the HP. vary as p 2 /, instead of asp/, and in 
this respect it is no doubt incorrect. 

Making the H.P. vary as Vdn or as Vv , instead of directly as v, makes 
the velocity a factor of the working strength^ as in the Harkness and Lewis 

Vv 1 

formulae, the relative strength varying as - L — , or as ~Tr, which for different 

v yv 

velocities is as follows : 

Speed of teeth in ft. per min.,^ = 100 200 300 600 900 1200 1800 2400 
Relative strength = 1 .707 .574 .408 .333 .289 .236 .204 

Showing a somewhat more rapid reduction than is given by Mr. Lewis. 

For the purpose of comparing different formulas they may in general be 
reduced to either of the following forms : 

H.P. = Cpfv, H.P. = Cjpfd x rpm., W = cpf, 

in which p = pitch, / = face, d — diameter, all in inches ; v = velocity in 
feet per minute, rpm. revolutions per minute, and C, d and c coefficients. 
The formulae for transformation are as follows ; 

HP- Wv — W x d x rpm " 
" " ~ 33000 ~ 126,050 ' 

Trr 33,000 H.P. 126,050 H.P. 



v dX rpm. ' ^ ' *y Cv G x d x rpm. ~ c " 

d = .2618a; c - 33,000C; C = 3.82(7, , = ^— ; c = 126,0500,. 

33,000 

In the Lewis formula C varies with the form of the tooth and with the 
speed, and is equal to sy -v- 33,000, in which y and s are the values taken from 
the table, and c = sy. 

910 
In the Harkness formula (7 varies with the speed and is equal to~7==== 

y 1+0.65 V 
, I7U . . . , ,, .01517 

(V being in feet per second), = — - 

Vl + .Ollv. 
In the Box for mula C v aries with the pitch and also with the velocity, 

and equals 12p V f* rpm - = .02345 -4, c = 33 ,000(7 = 774 -£- 
1000v Vv \/v 

For v = 100 ft. per min. C = 77. 4p ; for v = 600 ft. per minute c =31.6p. 

In the other formulae considered C, C x , and c are constants. Reducing 
the several formulae to the form W = cpf, we have the following : 



FRICTtONAL GEARIHG. §05 

Comparison of Different Formula for Strength of Gear-teeth. 

Safe working pressure per inch pitch and per inch of face, or value of c in 
formula W = cpf: 

v = 100 ft. v = 600 ft. 
permin. per min. 
Lewis: Weak form of tooth, radial flank, 12 teeth. . . c = 416 208 

Medium tooth, inv. 15°, or cycloid, 27 teeth., c = 800 400 

Strong form of tooth, or cycloid, 300 teeth. . . c = 1200 600 

Harkness: Average tooth c = 347 184 

Box: Tooth of 1 inch pitch c= 77 A 31.6 

" " 3 inches pitch c= 232 95 

Various, in which c is independent of form and speed: Old English 
rule, c = 200; Grant, c = 350; Nystrom, c = 80; Halsey, c = 128; Jones & 
Laughlins, c = 218; Unwin, c = 262, 200, or 139, according to speed, shock, 
and vibration. 

The value given by Nystrom and those given by Box for teeth of small 
pitch are so much smaller than those given by the other authorities that they 
may be rejected as having an entirely unnecessary surplus of strength. The 
values given by Mr. Lewis seem to rest on the most logical basis, the form of 
the teeth as well as the velocity being considered ; and since they are said to 
have proven satisfactory in an extended machine practice, they may be con- 
sidered reliable for gears that are so well made that the pressure bears 
along the face of the teeth instead of upon the corners. For rough ordi- 
nary work the old English rule W = 200p/is probably as good as any, ex- 
cept that the figure 200 may be too high for weak forms of tooth and for 
high speeds. 

The formula W = 200p/is equivalent to H.P. = nfd * 6 J Vm ' = ~^-, or 

H.P. = .0015873p/d X rpm. = .006063pft\ 

Maximum Speed of Gearing.— A. Towler, Eng'p, April 19, 1889, 
p. 388, gives the maximum speeds at which it was possible under favorable 
conditions to run toothed gearing safely as follows: 

Ft. per min. 

Ordinary cast-iron wheels 1800 

Helical " " " 2400 

Mortise " " " 2400 

Ordinary cast-steel wheels 2600 

Helical " " " 3000 

Special cast-iron machine-cut wheels 3000 

Prof. Coleman Sellers (Stevens Indicator, April, 1892) recommends that 
gearing be not run over 1200 ft. per minute, to avoid great noise. The 
Walker Mfg. Co., Cleveland, O., say that 2200 ft. per min. for iron gears and 
3000 ft. for wood and iron (mortise gears) are excessive, and should be 
avoided if possible. The Corliss engine at the Philadelphia Exhibition (1876) 
had a fly-wheel 30 ft. in diameter running 35 rpm. geared into a pinion 12 ft. 
diam. The speed of the pitch-line was 3300 ft. per min. 

A Heavy Machine-cut Spur-gear was made in 1891 by the 
Walker Mfg. Co., Cleveland, O., for a diamond mine in South Africa, with 
dimensions as follows: Number of teeth, 192; pitch diameter, 30' 6.66"; face, 
30"; pitch, 6": bore, 27"; diameter of hub, 9' 2"; weight of hub, 15 tons; and 
total weight of gear, 66% tons. The rim was made in 12 segments, the joints 
of the segments being fastened with two bolts each. The spokes were bolted 
to the middle of the segments and to the hub with four bolts in each end. 

Frictional Gearing.— In frictional gearing the wheels are toothless, 
and one wheel drives the other by means of the friction between the two 
surfaces which are pressed together. They may be used where the power 
to be transmitted is not very great; when the speed is so high that toothed 
wmeels would be noisy; when the shafts require to be frequently put into 
and out of gear or to have their relative direction of motion reversed; or 
when it is desired to change the velocity-ratio while the machinery is in mo- 
tion, as in the case of disk friction-wheels for changing the feed in machine 
tools. 

Let P = the normal pressure in pounds at the line of contact by which 
two wheels are pressed together. T= tangential resistance of the driven 
wheel at the line of contact, / = the coefficient of friction. V = the velocity 
of the pitch-surface in feet per second, and H.P. = horse-power ; then 
T may be equal to or less than fP\ H.P. = TV + 550. The value of/ for 



906 



H01ST1KG. 



metal on metal may be taken at .15 to .20; for wood on metal, .25 to .30; and 
for wood on compressed paper, .20. The tangential driving force T may be 
as high as 80 lbs. per inch width of face of the driving surface, but this is ac- 
companied by great pressure and friction on the journal-bearings. 

In frictional grooved gearing circumferential wedge-shaped grooves are 
cut in the faces of two wheels in contact. If P ~ the force pressing the 
wheels together, and N= the normal pressure on all the grooves, P= N 
(sin a + / cos a), in which 2a = the inclination of the sides of the grooves, 
and the maximum tangential available force T — fN. The inclination of the 
sides of the grooves to a plane at right angles to the axis is usually 30°. 

Frictional Grooved Gearing.— A set of friction-gears for trans- 
mitting 150 H.P. is on a steam- dredge described in Proc. Inst. M. E., July, 
1888. Two grooved pinions of 54 in. diam., with 9 grooves of \% in. pitch and 
anyde of 40° cut on their face, are geared into two wheels of 127*^ in diam. 
similarly grooved. The wheels can be thrown in and out of gear by levers 
operating eccentric bushes on the large wheel shaft. The circumferential 
speed of the wheels is about 500 ft. per min. Allowing for engine-friction, 
if half the power is transmitted through each set of gears the tangential 
force at the rims is about 3960 lbs., requiring, if the angle is 40° and the co- 
efficient of friction .18, a pressure of 7524 lbs. between the wheels and 
pinion to prevent slipping. 

The wear of the wheels proving excessive, the gears were replaced by spur- 
gear wheels and brake-wheels with steel brake-bands, which arrangement 
has proven more durable thau the grooved wheels. Mr. Daniel Adamson 
states that if the frictional wheels had been run at a higher speed the results 
would have been better, and says they should run at least 30 ft. per second. 



HOISTING. 

Approximate Weight and Strength of Cordage. (Boston 
and Lockport Block Co.)— See also pages 339 to 345. 



Size in 
Circum- 
ference. 


Size in 
Diam- 
eter. 


Weight of 
100 ft. 
Manila, 
in lbs. 


Strength 

of Manila 

Rope, 

in lbs. 


Size in 
Circum- 
ference. 


Size in 
Diam- 
eter. 


Weight of 
100 ft. 
Manila, 
in lbs. 


Strength 

of Manila 

Rope, 

in lbs. 


inch. 


inch. 






inch. 


inch. 






2 


% 


13 


4,000 


m 


19/16 


72 


22,500 


m 


% 


16 


5,000 


5 


1% 


80 


25,000 


tyi 


13/16 


20 


6,250 


&a 


m 


97 


30,250 


2% 


% 


24 


7,500 


6 


2 


113 


36,000 


3 


1 


28 


9,000 


Ws 


% 


133 


42,250 


3M 


11/16 


33 


10,500 


7 


m 


153 


49,000 


H 


.m 


38 


12,250 


VA 


2^ 
2% 


184 


56,250 


m 


45 


14,000 


8 


211 


64,000 


4 


1 5/16 


51 


16,000 


8^ 


2% 


236 


72,250 


414 
4}4 


1% 


58 


18,062 


9 


3 


262 


81,000 


1« 


65 


20,250 











Working Strength of Blocks. (B. & L. Block Co.) 



Regular Mortise-blocks Single and 
Double, or Two Double Iron- 
strapped Blocks, will hoist about- 



inch. 


lbs. 


F 


250 


6 


350 


7 


600 


8 


1,200 


9 


2,000 


10 


4,000 


12 


10.000 


14 


16,000 



Wide Mortise and Extra Heavy 
Single and Double, or Two Double, 
Iron-strapped Blocks, will hoist 
about — 

inch. lbs. - 



8 


2,000 


10 


6,000 


12 


12,000 


14 


24,000 


16 


36,000 


18 


50.000 


20 


90,000 



Where a double and triple block are used together, a certain extra propor- 
tioned amount of weight can be safely hoisted, as larger hooks are used. 



PROP RTIONS OF HOOKS. 



907 



Comparative Efficiency in Chain- blocks both in 
Hoisting and. Lowering. 

(Tests by Prof. R. H. Thurston, Hoisting, March, 1892.) 





Work op 


EOISTING. 




Work op Lowering. 






Load of 2000 lbs. 


Load of 2000 lbs 


, lowered 7 ft. 


in each case. 




Exclusive of Factor of Time. 


Indus 
Tir 




& 


u 

Si 


11 

Si 

Is a 




o 

>> 

'5 
o 


lve ot 
ne. 


O 

s 

c 


O c6 




£=3 


fell 

Us 


a 




3 


rt 
£ 


s 


tf 


> 


S-3 


a 


H 




l 


20.50 


79.50 


1.00 


32.50 


8.00 


227. 


1,816 


1.00 


0.75 


1.000 


2 


6S.00 


32.00 


.40 


62.44 


14.00 


436. 


6,104 


3.33 


1 20 


.186 


3 


69.00 


31.00 


.39 


30 00 


92.30 


196. 


18,090 


10.00 


1.50 


.050 


4 


71.20 


28.80 


.36 


28.00 


92.60 


168. 


15,556 


8.60 


2.50 


.035 


5 


73.96 


26.04 


.33 


48. 0C 


73.30 


17.5 


1,282 


0.71 


2 80 


.380 


6 


75.66 


24.34 


.31 


53. 0C 


56.60 


370 


20,942 


11.60 


1.80 


.036 


7 


77.00 


23.00 


.29 


44. 3C 


55.00 


310. 


17,050 


9.4C 


2.75 


.029 


8 


81.03 


18.97 


.24 


61.00 


48.50 


426. 


20,000 


11.60 


3.75 


.018 



No. 1 was Weston's triplex block; No. 3, Weston's differential; No. 4, 
Weston's imported. The others were from different makers, whose names 
are not given. All the blocks were of one-ton capacity. 

Proportions of Hooks.— The following formulae are given by 
Henry R. Towne, in his Treatise on Cranes, as a result of an extensive 
experimental and mathematical investi- 
gation. Thev apply to hooks of capaci- 
ties from 250 lbs. to 20,000 lbs. Each size 
of hook is made from some commercial 
size of round iron. The basis in each 
case is, therefore, the size of iron of 
which the hook is to be made, indicated 
by A in the diagram. The dimension D 
is arbitrarily assumed. The other di- 
mensions, as given by the formulae, are 
those which, while preserving a proper 
bearing-face on the interior of the hook 
for the ropes or chains which may be 
passed through it, give the greatest re- 
sistance to spreading and to ultimate 
rupture, which the amount of material 
in the original bar admits of. The sym- 
bol A is used to indicate the nominal ca- 
pacity of the hook in tons of 2000 lbs. 
The formulae which determine the lines 




ot the other parts ot the hooks ot the j 
several sizes are as follows, the measure- F 
ments being all expressed in inches: IG 


164. 


D = .5 A -f 1.25 
E = .64 A + 1.60 
F = .33 A -f .85 


G = .75 D. H= 1.08.4. 

O = .363 A + .66 i"= 1.334 

Q = .64 A + 1.60 J= 1.204 

K = 1.134 


L = 1.054 
M = .504 
N= .80B - .16 
U= .8664 


The dimensions A are necessarily based upon the ordinary merchant sizes 
of round iron. The sizes which it has been found best to select are the 
following: 


Capacity of hook: 


1 1^ 2 3 4 5 6 


8 10 tons. 


Dimension A: 
% 11/J6 H 


11/36 I14 \% 1% 2 2% 2y % 


2V8 ®A in. 



908 HOISTING. 

Experiment has shown that hooks made according to the above formulae 
will give waj r first by opening of the jaw, which, however, will not occur 
except with a load much in excess of the nominal capacity of the hook. 
This yielding of the hook when overloaded becomes a source of safety, as it 
constitutes a signal of danger which cannot easily be overlooked, and which 
must proceed to a considerable length before rupture will occur and the 
load be dropped. 

POWER OF HOISTING-ENGINES. 

Horse-power required to raise a Load at a Given 

„ .. „ ,, Gross weight in lbs . . -> . ^ , . 

Speed. — H.P. = 33000 '""' X s P eed in ft - P er min - To tn,s add 

25$ to 50% for friction, contingencies, etc. The gross weight includes the 
weight of cage, rope, etc. In a shaft with two cages balancing each other 
use the net load + weight of one rope, instead of the gross weight. 

To find the load ivhich a given pair of engines ivill start.— Let A = area 
of cylinder in square inches, or total area of both cylinders, if there are two; 
P = mean effective pressure in cylinder in lbs. per sq. in.; S= stroke of 
cylinder in inches; C = circumference of hoisting-drum in inches; L = load 
lifted by hoisting- rope in lbs. ; F — friction, expressed as a diminution of 

the load. Then £ = ^1^- P. 

An example in ColVy Engr., July, 1891, is a pair of hoisting-engines 24" X 
40", drum 12 ft. diam., average steam-pre-sure in cylinder = 59.5 lbs.; A = 
904.8; P- 59.5; S = 40; C- 452.4. Theoretical load, not allowing for friction, 
AP2S -h C = 9589 lbs. The actual load that could just be lifted on trial was 7988 
lbs., making friction loss F — 1601 lbs., or 20 + per cent of the actual load 
lifted, or 1G$£% of the theoretical load. 

The above rule takes no account of the resistance due to inertia of the 
load, but for all ordinary cases in which the acceleration of speed of the 
cage is moderate, it is covered by the allowance for friction, etc. The re- 
sistance due to inertia is equal to the force required to give the load the 
velocity acquired in a given time, or, as shown in Mechanics, equal to the 

product of the mass by the acceleration, or R — — — , in which R — resist- 
ance in lbs. due to inertia; W — weight of load in lbs. ; V = maximum veloc- 
ity in feet per second; T — time in seconds taken to acquire the velocity V\ 
g = 32.16. 

Effect of Slack Rope upon Strain in Hoisting.— A. series of 
tests with a dynamometer are published by the Trenton Iron Co., which 
show that a dangerous extra strain may be caused by a few inches of slack 
rope. In one case the cage and full tubs weighed 11,300 lbs. ; the strain when 
the load was lifted gently was 11,525 lbs.; with 3 in. of slack chain it was 
19.0-25 lbs , with 6 in. slack 25.750 lbs., and with 9 in. slack 27,950 lbs. 

Limit of Depth for Hoisting.— Taking the weight of a cast-steel 
nbisting-rope of \% inches diameter at 2 lbs. per running foot, and its break- 
ing strength at 84,000 lbs., it should, theoretically, sustain itself until 42,000 
feet long before breaking from its own weight. But taking the usual factor 
of safety of 7, then the safe working length of such a rope would be only 
6000 feet. If a weight of 3 tons is now hung to the rope, which is equivalent 
to that of a cage of moderate capacity with its loaded cars, the maximum 
length at which such a rope could be used, with the factor of safety of 7, is 
3000 feet, or 

2x -f 6000 = Mi^°_. .-.x = 3000 feet. 

This limit may be greatly increased by using special steel rope of higher 
strength, by using a smaller factor of safety, and by using taper ropes. 
(See paper by H. A. Wheeler, Trans. A. I. M. E., xix. 107.) 

Large Hoisting Records.— At a colliery in North Derbyshire dur- 
ing the first week in June, 1890, 6309 tons were raised from a depth of 509 
yards, the time of winding being from 7 a.m. to 3.30 p.m. 

At two other Derbyshire pits, 170 and 140 yards in depth, the speed of 
winding and changing has been brought to such perfection that tubs are 
drawn and changed three times in one minute. (Proc. Inst, M, E., 1890.) 



POWER OF HOISTING-ENGINES. 909 

At the Nottingham Colliery near Wilkesbarre, Pa., in Oct. 1891, 70,152 tons 
were shipped in 24.15 days, the average hoist per day being 1318 mine cars. 

The depth of hoist was 470 feet, and all coal came from one opening. The 
engines were fast motion, 22 x 48 inches, conical drums 4 feet 1 inch long. 7 
feet diameter at small end and 9 feet at large end. (Eng'g Neics, Nov. 1891.) 

Pneumatic Hoisting. (H. A. Wheeler, Trans. A. I. M. E., xix. 107.)- 
A pneumatic hoist was installed in 1876 at Epinac, France, consisting of two 
continuous air-tight iron cylinders extending from the bottom to the top of 
the shaft. Within the cylinder moved a piston from which was hung the 
cage. It was operated by exhausting the air from above the piston, the 
lower side being open to the atmosphere. Its use vas discontinued on ac- 
count of the failure of the mine. Mr. Wheeler gives a description of the sys- 
tem, but criticises it as not being equal on the whole to hoisting by steel ropes. 

Pneumatic hoisting-cylinders using compressed air have been used at 
blast-furnaces, the weighted piston counterbalancing the weight of the cage, 
and the two being connected by a wire rope passing over a pulley-sheave 
above the top of the cylinder. In the more modern furnaces steam-engine 
hoists are generally used. 

Counterbalancing of Winding-engines. (H. W. Hughes, Co- 
lumbia Coll. Qly.)— Engines running unbalanced are subject to enormous 
variations in the load; for let W ~ weight of cage and empty tubs, say 6270 
lbs.; c = weight of coal, say 4480 lbs.; r = weight of hoisting rope, say 6000 
lbs. ; %•' = weight of counterbalance rope hanging down pit, say 6000 lbs. The 
weight to be lifted will be: 

If weight of rope is unbalanced. If weight of rope is balanced. 

At beginning of lift: 

W+c + r- Woy 10,480 lbs. W + c + r - ( W+ r'), 

At middle of lift: 



Y 4480 
lbs. 



*F+c-r-|'~(V-f 0or 4480 lbs. W+c + r -+ | _(tT+|+^)' 

At end of lift: 

W+c— {W+ r) or minus 1520 lbs. W+ c + r' — (W-\- r), 

That counterbalancing materially affects the size of winding-engines is 
shown by a formula given by Mr. Robert Wilson, which is based on the fact 
that the greatest w r ork a winding-engine has to do is to get a given mass into 
a certain velocity uniformly accelerated from rest, and to raise a load the 
distance passed over during the time this velocity is being obtained. 

Let W — the weight to be set in motion: one cage, coal, number of empty 
tubs on cage, one winding rope from pit head-gear to bottom, 
and one rope from banking level to bottom. 

v = greatest velocity attained, uniformly accelerated from rest; 

g = gravity = 32.2; 

t — time in seconds during which v is obtained; 

L = unbalanced load on engine; 

E — ratio of diameter of drum and crank circles; 

P = average pressure of steam in cylinders; 

N = number of cylinders; 

S — space passed over by crank-pin during time t ; 

C = %, constant to reduce angular space passed through by crank, to 
the distance passed through by the piston during the time t; 

A = area of one cylinder, without margin for friction. To this an ad- 
dition for friction, etc., of engine is to be made, varying from 10 
to 30$ of A. 

1st. Where load is balanced, 

PNSC. 

2d. Where load is unbalanced: 

The formula is the same, with the addition of another term to allow for 
the variation in the lengths of the ascending and descending ropes. In this 
case 



910 HOISTING. 

hi — reduced length of rope in t attached to ascending cage; 
h 2 — increased length of rope in t attached to descending cage; 
w — weight of rope per foot in pounds. Then 

'=["( %->+:-K*t)-.*^*> 

PNSC. 

Applying the ahove formula when designing new engines, Mr. Wilson 
found that 30 inches diameter of cylinders would produce equal results, when 
balanced, to those of the 36-inch cylinder in use, the latter being unbal- 
anced. 

Counterbalancing may be employed in the following methods : 

(a) Tapering Rope.—A.t the initial stage the tapering rope enables us to 
wind from greater depths than is possible with ropes of uniform section. 
The thickness of such a rope at any point should only be such as to safely 
bear the load on it at that point. 

With tapering ropes we obtain a smaller difference between the initial and 
final load, but the difference is still considerable, and for perfect equaliza- 
tion of the load we must rely on some other resource. The theory of taper 
ropes is to obtain a rope of uniform strength, thinner at the cage end where 
the weight is least, and thicker at the drum end where it is greatest; 

(b) The Counterpoise System consists of a heavy chain working up and 
down a staple pit, the motion being obtained by means of a special small 
drum placed on the same axis as the winding drum. It is so arranged that 
the chain hangs in full length down the staple pit at the commencement of 
the winding; in the centre of the run the whole of the chain rests on the 
bottom of the pit, and, finally, at the end of the winding the counterpoise 
has been rewound upon the small drum, and is in the same condition as it 
was at the commencement. 

(c) Loaded-wagon System. — A plan, formerly much employed, was to 
have a loaded wagon running on a short incline in place of this heavy chain; 
the rope actuating this wagon being connected in the same manner as t.ie 
above to a subsidiary drum. The incline was constructed steep at the com- 
mencement, the inclination gradually decreasing to nothing. At the begin- 
ning of a wind the wagon was at the top of the incline, and during a portion 
of the run gradually passed down it till, at the meet of cages, no [pull was 
exerted on the engine— the wagon by this time being at the bottom. In the 
latter part of the wind the resistance was all against the engine, owing to 
its having to pull the wagon up the incline, and this resistance increased 
from nothing at the meet of cages to its greatest quantity at the conclusion 
of the lift. 

(d) The Endless-rope System is preferable to all others, if there is suffi- 
cient sump room and the shaft is free from tubes, cross timbers, and other 
impediments. It consists in placing beneath the cages a tail rope, similar 
in diameter to the winding rope, and, after conveying this down the pit, it is 
attached beneath the other cage. 

(e) Fiat Ropes Coiling on Reels —This means of winding allows of a cer- 
tain equalization, for the radius of the coil of fascending rope continues to 
increase, while that of the descending one continues to diminish. Conse- 
quently, as the resistance decreases in the ascending load the leverage 
increases, and as the power increases in the other, the leverage diminishes. 
The variation in the leverage is a constant quantity, and is equal to the 
thickness of the rope where it is wound on the drum. 

By the above means a remarkable uniformity in the load may be ob- 
tained, the only objection being the use of flat ropes, which weigh heavier 
and only last about two thirds ihe time of round ones. 

(/) Conical Drums.— Results analogous to the preceding may be obtained 
by using round ropes coiling on conical drums, which may either be smooth, 
with the successive coils lying side by side, or they may be provided with a 
spiral groove. The objection to these forms is. tliat, perfect equalization is 
not obtained with the conical drums unless the sides are very sleep, and con- 
sequently there is great risk of the rope slipping ; to obviate this, scroll 
drums were proposed. They are, however, very expensive, and the lateral 
displacement of the winding rope from the centre line of pulley becomes 
very great, owing to their necessary large width. 

(g) The Koepe System of Winding. — An iron pulley with a single circular 
groove takes the place of the ordinary drum. The winding rope passes 
from one cage, over its head-gear pulley, round the drum, and, after pass- 



CRAKES. 911 

ing over the other head-gear pulley, is connected with the second cage. The 
winding rope thus encircles about half the periphei-y of the drum in the 
same manner as a driving-belt on an ordinary pulley. There is a balance 
rope beneath the cages, passing round a pulley in the sump; the arrange- 
m nt may be likened to an endless rope, the two cages being simply points 
of attachment. 

BELT-CONVEYORS. 

Grain-elevators. — American Grain-elevators are described in a 
paper by E. Lee Heidenreich, read at the International Engineering Con- 
gress at Chicago (Trans. A. S. C. E. 1893). See also Trans. A.-S. M. E. vii, 660. 

Bands for carrying Grain. — Flexible-rubber bands are exten- 
sively used for carrying grain in and around elevators and warehouses. An 
article on the grain-storage warehouses of the Alexandria Dock, Liverpool 
(Proc. Inst. M. E., July, 1891), describes the performance of these bauds, 
aggregating three miles in length. A band 16^ inches wide, 1270 feet long, 
running 9 to 10 feet per second has a carrying capacity of 50 tons per hour. 
See also paper on Belts as Grain Conveyors, by T. W. Hugo, Trans. A. S. 
M. E . vi. 400. 

Carrying-bands or Belts are used for the purpose both of sorting 
coal and of removing impurities. These carrying-bands may be said to be 
confined to two descriptions, namely, the wire belt, which consists of an 
endless length of woven wire; and the steel-plate belt, which consists of 
two or three endless chains, carrying steel plates varying in width from 6 
inches to 14 inches. (Proc. Inst. M. E., July, 1890.) 

CRANES. 
Classification of Cranes. (Henry R. Towne, Trans. A. S. M. E., iv. 

288. Revised in Hoisting, published by The Yale & Towne Mfg. Co.) 

A Hoist is a machine for raising and lowering weights. A Crane is a 
hoist with the added capacity of moving the load in a horizontal or lateral 
direction. 

Cranes are divided into two classes, as to their motions, viz., Rotary and 
Rectilinear, and into four groups, as to their source of motive power, viz.: 

Hand. — When operated by manual power. 

Poiver. — When driven by power derived from line shafting. 

Steam, Electric. Hydraulic, or Pneumatic. — When driven by an engine or 
motor attached to the crane, and operated by steam, electricity, water, or 
air transmitted to the crane from a fixed source of supply. 

Locomotive. — When the crane is provided with its own boiler or other 
generator of power, and is self-propelling ; usually being capable of both 
rotary and rectilinear motions. 

Rotary and Rectilinear Cranes are thus subdivided : 

Rotary Cranes. 

(1) Swing-cranes. — Having rotation, but no trolley motion. 

(2) Jib-cranes. — Having rotation, and a trolley travelling on the jib. 

(3) Column-cranes. — Identical with the jib-cranes, but rotating around a 
fixed column (which usually supports a floor above). 

(4) Pillar-cranes.— Having rotation only; the pillar or column being sup- 
ported entirely from the foundation. 

(5) Pillar Jib-cranes.— Identical with the last, except in having a jib and 
trolley motion. 

(6) Derrick-cranes.— Identical with jib-cranes, except that the head of the 
mast is held in position by guy- rods, instead of by attachment to a roof or 
ceiling. 

(7) Walking-cranes.— Consisting of a pillar or jib-crane mounted on wheels 
and arranged to travel longitudinally upon one or more rails. 

(8) Locomotive-cranes.— Consisting of a pillar crane mounted on a truck, 
and provided with a steam-engine capable of propelling and rotating the 
crane, and of hoisting and lowering the load. 

Rectilinear Cranes. 

(9) Bridge-cranes. — Having a fixed bridge spanning an opening, and a 
trolley moving across the bridge. 

(10) Tram-cranes.— Consisting of a truck, or short bridge, travelling lon- 
gitudinally on overhead rails, and without trolley motion. 

(11) Travelling-cranes. — Consisting of a bridge moving longitudinally on 
overhead tracks, and a trolley moving transversely on the bridge. 






012 HOISTING. 






(12) Gantries.— Consisting of an overhead bridge, carried at each end by a 
trestle travelling on longitudinal tracks on the ground, and having a trolley 
moving transversely on the bridge. 

(13) Rotary Bridge-cranes. — Combining rotary and rectilinear movements 
and consisting of a bridge pivoted at one end to a central pier or post, 
and supported at the other end on a circular track ; provided with a trolley 
moving transversely on the bridge. 

For descriptions of these several forms of cranes see Towne's "Treatise 
on Cranes." 

Stresses in Cranes.— See Stresses in Framed Structures, p. 440, ante. 

Position of the Inclined Brace in a Jib-crane. — The most 
economical arrangement is that in which the inclined brace intersects the 
jib at a distance from the mast equal to four fifths the effective radius of 
the crane. (Hoisting.) 

A Large Travelling-crane, designed and built by the Morgan 
Engineering Co., Alliance, (J., for the 12-inch-gun shop at the Washington 
Navy Yard, is described in American Machinist, June 12, 1890. Capacity, 
150 net tons; distance between centres of inside rails, 59 ft. 6 in.; maximum 
cross travel, 44 ft. 2 in.; effective lift, 40 ft. ; four speeds for main hoist, 1, 2, 
4, and 8 ft. per min. ; loads for these speeds, 150, 75, 37^£, and 18% tons respec- 
tively ; traversing speeds of trolley on bridge, 25 and 50 ft. per minute ; 
speeds of bridge on main track, 30 and 60 ft. per minute. Square shafts are 
employed for driving-. 

A i 50-ton Pillar-crane was erected in 1893 on Finnieston Quay, 
Glasgow. The jib is formed of two steel tubes, each 39 in. diam. and 90 ft. 
long. The radius of sweep for heavy lifts is 65 ft. The jib and its load are 
counterbalanced by a balance-box weighted with 100 tons of iron and steel 
punchings. In a test a 130-ton load was lifted at the rate of 4 ft. per minute, 
and a complete revolution made with this load in 5 minutes. Eng'g News, 
July 20, 1893. 

Compressed-air Travelling-cranes,— Compressed-air overhead 
travelling-cranes have been built by the Lane & Bodley Co., of Cincinnati. 
They are of 20 tons nominal capacity, each about 50 ft. span and 400 ft. length 
of travel, and are of the triple-motor type, a pair of simple reversing-engines 
being used for each of the necessary operations, the pair of engines for the 
bridge and the pair for the trolley travel being each 5-inch bore by 7-inch 
stroke, while the pair for hoisting is 7-inch bore by 9-inch stroke. Air is 
furnished by a compressor having steam and air cylinders each 10-in. diam. 
and 12-in. stroke, which with a boiler-pressure of about 80 pounds gives an air- 
pressure when required of somewhat over 100 pounds. The air-compressor 
is allowed to run continuously without a governor, the speed being regulated 
by the resistance of the air in a receiver. From a pipe extending from the 
receiver along one of the supporting trusses communication is continuously 
maintained with an auxiliary receiver on each traveller by means of a one- 
inch hose, the object of the auxiliary receiver being to provide a supply of 
air near the engines for immediate demands and independent of the hose 
connection, which may thus be of small dimension. Some of the advantages 
said to be possessed by this type of crane are: simplicity; absence of all mov- 
ing parts, excepting those required for a particular motion when that motion 
is in use; no danger from fire, leakage, electric shocks, or freezing; ease of 
repair; variable speeds and reversal without gearing; almost entire absence 
of noise; and moderate cost. 

Quay-cranes.— An illustrated description of several varieties of sta- 
tionary and travelling cranes, with results of experiments, is given in a 
gaper on Quay-cranes in the Port of Hamburg by Chas. Nehls, Trans. A. S. 
. E., Chicago Meeting. 1893. 

Hydraulic Cranes, Accumulators, etc.— See Hydraulic Press- 
ure Transmission, page 616, ante. 

Electric Cranes.— Travelling-cranes driven by electric motors have 
largely supplanted cranes driven by square shafts or flying-ropes. Each of 
the three motions, viz., longitudinal, traversing and hoisting, is usually ac- 
complished by a separate motor carried upon the crane. 

WIRE-ROPE HAULAGE. 

Methods for transporting coal and other products by means of wire rope, 
though varying from each other in detail, may be grouped in five classes: 
I. The Self-acting or Gravity Inclined Plane. 
II. The Simple Engine-plane. 



WIRE-ROPE HAULAGE. 013 

III. The Tail-rope System. 

IV. The Endless-rope System. 
V. The Cable Tramway. 

The following 1 brief description of these systems is abridged from a 
pamphlet on Wire-rope Haulage, by Wm. Hildenbraud, C.E., published by 
John A. Roebling's Sons Co., Trenton. N. J. 

I. The Self-acting Inclined Plane.— The motive power for the 
self-acting inclined plane is gravity; consequently this mode of transport- 
ing coal finds application only in piaces where the coal is conveyed from a 
higher to a lower point and where the plane has sufficient grade for the 
loaded descending cars to raise the empty cars to an upper level. 

At the head of the plane there is a drum, which is generally constructed 
of wood, having a diameter of seven to ten feet. It is placed high enough 
to allow men and cars to pass under it. Loaded cars coming from the pit 
are either singly or in sets of two or three switched on the track of the 
plane, and their speed in descending is regulated by a brake on the drum. 

Supporting rollers, to prevent the rope dragging on the ground, are 
generally of wood, 5 to 6 inches in diameter and 18 to 24 inches long, with 
H- t0 %-inch iron axles. The distance between the rollers varies from 15 to 
30 feet, steeper planes requiring less rollers than those with easy grades. 
Considering only the reduction of friction and what is best for the preserva- 
tion of rope, a general rule may be given to use rollers of the greatest 
possible diameter, and to place them as close as economy will permit. 

The smallest angle of inclination at which a plane can be made self-acting 
will be when the motive and resisting forces balance each other. The 
motive forces are the weights of the loaded car and of the descending rope. 
The resisting forces consist of the weight of the empty car and ascending 
rope, of the rolling and axle friction of the cars, and of the axle friction of 
the supporting rollers. The friction of the drum, stiffness of rope, and 
resistance of air may be neglected. A general rule cannot be given, because 
a change in the length of the plane or in the weight of the cars changes the 
proportion of the forces; also, because the coefficient of friction, depending 
on the condition of the road, construction of the cars, etc., is a very uncer- 
tain factor. 

For working a plane with a %-inch steel rope and lowering from one to 
four pit cars weighing empty 1400 lbs. and loaded 4000 lbs., the rise in 100 
feet necessary to make the plane self-acting will be from about 5 to 10 feet, 
decreasing as the number of cars increase, and increasing as the length of 
plane increases. 

A gravity inclined plane should be slightly concave, steeper at the top 
than at the bottom. The maximum deflection of the curve should be at an 
inclination of 45 degrees, and diminish for smaller as well as for steeper 
inclinations. 

II. The Simple Engine-plane.— The name "Engine-plane" is 
given to a plane on which a load is raised or lowered by means of a single 
wire rope and stationary steam-engine. It is a cheap and simple method of 
conveying coal underground, and therefore is applied wherever circum- 
stances permit it. 

Under ordinary conditions such as prevail in the Pennsylvania mine 
region, a train of twenty-five to thirty loaded cars will descend, with reason- 
able velocity, a straight plane 5000 feet long on a grade of 1% feet in 100, 
while it would appear that 2J4 feet in 100 is necessary for the same number 
of empty cars. For roads longer than 5000 feet, or when containing sharp 
curves, the grade should be correspondingly larger. 

III. The Tail-rope System.— Of all methods for conveying coal 
underground by wire rope, the tail-rope system has found the most applica- 
tion. It can be applied under almost any condition. The road may be 
straight or curved, level or undulating, in one continuous line or with side 
branches. In general principle a tail-rope plane is the same as an engine- 
plane worked in both directions with two ropes. One rope, called the " main 
rope," serves for drawing the set of full cars outward; the other, called 
the " tail-rope," is necessary to take back the empty set, which on a level 
or undulating road cannot return by gravity. The two drums may be 
located at the opposite ends of the road, and driven by separate engines, 
but more frequently they are on the same shaft at one end of the plane. 
In the first case each rope would require the length of the plane, but in the 
second case the tail rope must be twice as long, being led from the drum 
around a sheave at the other end of the plane and back again to its starting- 



9U HOtSTlttG. 

point. When the main rope draws a set of full cars out, the tail-rope drum 
runs loose on the shaft, and the rope, being attached to the rear car, un- 
winds itself steadily. Going in, the reverse takes place. Each drum is 
provided with a brake to check the speed of the train on a down grade and 
prevent its overrunning the forward rope. As a rule, the tail rope is 
strained less than the main rope, but in cases of heavy grades dipping out- 
ward it is possible that the strain in the former may become as large, or 
even larger, than in the latter, and in the selection of the sizes reference 
should be had to this circumstance. 

IV. The Endless-rope System.— The principal features of this 
system are as follows: 

1. The rope, as the name indicates, is endless. 

2. Motion is given to the rope by a single wheel or drum, and friction is 
obtained either by a grip-wheel or by passing the rope several times around 
the wheel. 

3. The rope must be kept constantly tight, the tension to be produced by 
artificial means. It is done in placing either the return-wheel or an extra 
tension wheel on a carriage and connecting it with a weight hanging over a 
pulley, or attaching it to a fixed post by a screw which occasionally can be 
shortened. 

4. The cars are attached to the rope by a grip or clutch, which can take 
hold at any place and let go again, starting and stopping the train at will, 
without stopping the engine or the motion of the rope. 

5. On a single-track road the rope works forward and backward, but on a 
double track it is possible to run it always in the same direction, the full 
cars going on one track and the empty cars on the other. 

This method of conveying coal, as a rule, has not found as general an in- 
troduction as the tail-rope system, probably because its efficacy is not so 
apparent and the opposing difficulties require greater mechanical skill and 
more complicated appliances. Its advantages are, first, that it requires 
one third less rope than the tail-rope system. This advantage, however, 
is partially counterbalanced by the circumstance that the extra tension in 
the rope requires a heavier size to move the same load than when a main 
and tail rope are used. The second and principal advantage is that it is 
possible to start and stop trains at will without signalling to the engineer. 
On the other hand, it is more difficult to work curves with the endless sys- 
tem, and still more so to work different branches, and the constant stretch 
of the rope under tension or its elongation under changes of temperature 
frequently causes the rope to slip on the wheel, in spite of every attention, 
causing delay in the transportation and injury to the rope. 

V. Wire-rope Tramways.— The methods of conveying products on 
a suspended rope tramway find especial application in places where a mine 
is located on one side of a river or deep ravine and the loading station on 
the other. A wire rope suspended between the two stations forms the track 
on which material in properly constructed "carriages" or "buggies' 1 is 
transported. It saves the construction of a bridge or trestlework, and is 
practical for a distance of 2000 feet without an intermediate support. 

There are two distinct classes of rope tramways: 

1. The rope is stationary, forming the track on which a bucket holding 
the material moves forward and backward, pulled by a smaller endless 
wire rope. 

2. The rope is movable, forming itself an endless line, which serves at 
the same time as supporting track and as pulling rope. 

Of these two the first method has found more general application, and is 
especially adapted for long spans, steep inclinations, and heavy loads. The 
second method is used for long distances, divided into short spans, and is 
only applicable for light loads which are to be delivered at regular intervals. 

For detailed descriptions of the several systems of wire-rope transporta- 
tion, see circulars of John A. Roebliug's Sons Co., The Trenton Iron Co., and 
other wire-rope manufacturers. See also paper on Two-rope Haulage 
Systems, by R. Van A. Norris, Trans. A. S. M. E., xii. 626. 

In the Bleichert System of wire-rope tramways, in which the track rope is 
stationary, loads of 1000 pounds each and upward are carried. While the 
average spans on a level are from 150 to 200 feet, in crossing rivers, ravines, 
etc., spans up to 1500 feet are frequently adopted. In a tramway on this 
system at Granite, Montana, the total length of the line is 9750 feet, with a 
fall of 1225 feet. The descending loads, amounting to a constant weight of 
about 11 tons, develop over 14 horse-power, which is sufficient to haul the 
empty buckets as well as about 50 tons of supplies per day up the line, and 



SUSPENSION CABLEWAYS OR CABLE HOISTS. 915 



also to run the ore crusher and elevator. It is capable of delivering 250 
tons of material in 10 hours. 

SUSPENSION CABLEWAYS OR CABLE HOISTS. 

(Trenton Iron Co.) 

In quarrying, rock-cutting, stripping, piling, dam -building, and many 
other operations where it is necessary to hoist and convey large individual 
loads economically, it frequently happens that the application of a system 
of derricks is impracticable, by reason of the limited area of their efficiency 
and the room which they occupy. 

To meet such conditions cable-hoists are adapted, as they can be efficiently 
operated In clear spans up to 1500 feet, and in lifting individual loads up to 
15 tons. Two types are made — one in which the hoisting and conveying are 
done by separate running ropes, and the other applicable only to inclines, 
in which the carriage descends by gravity, and but one running rope is re- 
quired. The moving of the carriage in the former is effected by means of 
an endless rope, and these are commonly known as "endless -rope" cable- 
hoists to distinguish them from the latter, which are termed "inclined" 
cable-hoists. 

The general arrangement of the endless-rope cable-hoists consists of a 
main cable passing over towers, A frames or masts, as may be most conve- 
nient, and anchored firmly to the ground at each end, the requisite tension 
in the cable being maintained by a turnbuckle at one anchorage. 

Upon this cable travels the carriage, which is moved back and forth over 
the line by means of the endless rope. The hoisting is done by a separate 
rope, both ropes being operated by an engine specially designed for the 
purpose, which may be located at either end of the line, and is constructed 
in such a way that the hoisting-rope is coiled up or paid out automatically 
as the carriage is moved in ai.d out. Loads may be picked up or discharged 
at any point along the line. Where sufficient inclination can be obtained in 
the main cable for the carriage to descend by gravity, and the loading and 
unloading is done at fixed points, the endless rope can be dispensed with. 
The carriage, which is similar in construction to the carriage used in the 
endless-rope cableways, is arrested in its descent by a stop-block, which 
may be clamped to the main cable at any desired point, the speed of the 
descending carriage being under control of a brake on the engine-drum. 
Stress In Hoisting-ropes on Inclined Planes. 
(Trenton Iron (Jo. ) 





a 


c • 




ri 


C • 




a 


G • 


£8§ 


o.2 

<D eg 

■&o.S 


oa So 

<v p,o 


cdo o 


&c.2 


23% 

III 


a>o 


&J3.S 


III 






&£ o 


3"j 




wM"o 




<1 


^|o 


ft. 






ft. 






ft. 






5 


2° 52' 


140 


55 


28° 49' 


1003 


110 


470 44/ 


1516 


10 


5° 43' 


240 


60 


30° 58' 


1067 


120 


50° 12' 


1573 


15 


8° 32' 


336 


65 


33° 02' 


1128 


130 


52° 26' 


1620 


20 


11° 10' 


432 


70 


35° 00' 


1185 


140 


54° 28' 


1663 


25 


14° 03' 


527 


75 


36° 53' 


1238 


150 


56° 19' 


1699 


30 


16° 42' 


613 


80 


38° 40' 


1287 


160 


58° 00' 


1730 


35 


19° 18' 


700 


85 


40° 22' 


1332 


170 


59° 33' 


1758 


40 


21° 49' 


782 


90 


42° 00' 


1375 


180 


60° 57' 


1782 


45 


24° 14' 


860 


95 


43° 32' 


1415 


190 


62° 15' 


1801 


50 


26° 34' 


983 


100 


45° 00' 


1450 


200 


63° 27' 


1822 



The above table is based on an allowance of 40 lbs. per ton for rolling fric- 
tion, but an additional allowance must be made for stress due to the weight 
of the rope proportional to the length of the plane. A factor of safety of 5 
to 7 should be taken. 

In hoisting the slack-rope should be taken up genliy before beginning the 
lift, otherwise a severe extra strain will be brought on the rope. 

The best rope for inclined planes is composed of six strands of seven wires 
each, laid about a hempen centre. The wires are much coarser than those 
of the 114-wire rope of the same diameter, and for this reason the 42-wire 
rope is better adapted to withstand the rough usage and surface wear 
encountered upon inclined planes. 

A DQuble-suspension Cableway, carrying loads of 26 tons, erected near 



916 



Williamsport, Pa., by the Trenton Iron Co., is described by E. G. Spilsbury 
in Trans. A, I. M. E. xx. 766. The span is 733 feet, crossing the Susquehanna 
River. Twoisteel cables, each 2 in. diam., are used. On these cables runs a 
carriage supported on four wheels and moved by an endless cable 1 inch in 
diam. The load consists of a cage carrying a railroad-car loaded with lum- 
ber, the latter weighing about 12 tons. The power is furnished by a 50-H.P. 
engine, and the trip across the river is made in about three minutes. 

A hoisting cableway on the endless-rope system, erected by the Lidger- 
wood Mfg. Co., at the Austin Dam, Texas, had a single span 1350 ft. in 
length, with main cable 2J<£ in. diam., and hoisting-rope \% in. diam. Loads 
of 7 to 8 tons were handled at a speed of 600 to 800 ft. per minute. 

Tension required to Prevent Slipping of Wire on Drum. 
(Trenton Iron Co. )— The amount of artificial tension to be applied in an 
endless rope to prevent slipping on the driving-drum depends on the char- 
acter of the drum, the condition of the rope and number of laps which it 
makes. If Tand S represent respectively the tensions in the taut and slack 
lines of the rope; W, the necessary weight to be applied to the tail-sheave; 
JS, the resistance of the cars and rope, allowing for friction ; n, the number 
of half-laps of the rope on the driving-drum; and/, the coefficient of fric- 
tion, the following relations must exist to prevent slipping: 

T= Sef" v , W=T+S, and R = T - S; 
e fnir . j 
from which we obtain W — ! — R, 

e fnir _ ! 

in which e = 2.71828, the base of the Naperian system of logarithms. 
The following are some of the values of / : 

Dry. Wet. Greasy. 

Rope on a grooved iron drum 120 .085 .070 

Rope on wood-filled sheaves 235 .170 .140 

Rope on rubber and leather filling 495 .400 .205 

e fwr _i_ j 

The values of the coefficient — , corresponding to the above values 

efmr _ j 

of /, for one up to six half-laps of the rope on the driving-drum or sheaves, 
are as follows: 



/ 


n 


- Number of Half-laps on Driving-wheel. 




1 


2 


3 


4 


5 


6 


.070 


9.130 


4.623 


3.141 


2.418 


1.999 


1.729 


.085 


7.536 


3.833 


2.629 


2.047 


1.714 


1.505 


.120 


5.345 


2.777 


1.953 


1.570 


1.358 


1.232 


.140 


4.623 


2.418 


1.729 


1.416 


1.249 


1.154 


.170 


3.833 


2.047 


1.505 


1.268 


1.149 


1.085 


.205 


3.212 


1.762 


1.338 


1.165 


1.083 


1.043 


.235 


2.831 


1.592 


1.245 


1.110 


1.051 


1.024 


.400 


1.795 


1.176 


1.047 


1.013 


1.004 


1.001 


.495 


1.538 


1.093 


1.019 


1.004 


1.001 

















The importance of keeping the rope dry is evident from these figures. 

When the rope is at rest the tension is distributed equally on the two lines 
of the rope, but when running there will be a difference in the tensions of 
the taut and slack lines equal to the resistance, and the values of T and S 
mav be readily computed from the foregoing formula?. 

Taper Ropes of Uniform Tensile Strength.— Prof. A. S. 
Herschel in The Engineer, April, 1880, p, 267, gives an elaborate mathe- 
matical investigation of the problem of making a taper hoisting-rope of 
uniform tensile strength at every point in its length. Mr. Charles D. West, 
commenting on Prof. Hersche^s paper, gives a similar solution, and derives 
therefrom the following formula, based on a breaking strain of 80,000 lbs. 
per sq. in. of the rope, core included, with a factor of safety of 10: 

F = 3680[log G - log g] ; log G = —^ + log g; 

in which F = length in fathoms, and G and g the girth in inches at any two 
sections F fathoms apart. 



WIRE-ROPE TRANSMISSION. 



917 



Example —Let it be required to find the dimensions of a steel-wire rope to 
draw 6720 lbs.— cage, trams, and coal— from a depth of 400 fathoms. 

Area of section at lower end = 6720 -*- 8000 = .84 sq. in.; therefore girth = 
3J4 in. at bottom. 



Log G = 400 -r- 3 



- log 3.25 = .11 



I + .51188 = .62057; 



therefore G = 4.174, or, say, 4 3/16 in. girth at top. 

The equations show that the true form of rope is not a regular taper or 
truncated cone, but follows a logarithmic curve, the girth rapidly increas- 
ing towards the upper end. 

Relative Effect of Various-sized Sheaves or Drums on the 
Life of Wire Ropes. 

(Thos. E. Hughes, GolVy Eng., April, 1893.) 

Cast-steel Ropes for Inclines. 

Made of 6 strands, of 7 wires each, laid around a hemp core. 





Diameters of Sheaves or Drums in feet, showing perceut- 


Diam. of 




ages of life for various diameters. 


Rope in 






inches. 


1001 


901 


801 


751 


601 


501 


251 


m 


16 


14 


12 


11 


9 


7 


4.75 


w% 


14 


12 


10 


8.5 


7 


6 


4.5 




12 


10 


8 


7.25 


6.5 


5.5 


4.25 


10 


• 8.5 


7.75 


7 


6 


5 


4 


l 


8.5 


7.75 


6.75 


6 


5 


4.5 


3.75 


% 


7.75 


7 


6.25 


5.75 


4.5 


3.75 


3.25 


& A 


7 


6.25 


5.5 


5 


4.25 


3.5 


2.75 


6 


5.25 


4.5 


4 


3.25 


3 


2.5 


¥> 


5 


4.5 


4 


3.5 


2.75 


2.25 


1.75 



The use of iron ropes for inclines has been generally abandoned, steel 
ropes being more satisfactory and economical. 

Cast-steel Hoisting-ropes. 

Made of 6 strands, of 19 wires each, laid around a hemp core. 



Diam. of 
Rope in 


Diameters of Sheaves or Drums in feet, showing percent- 




ages of life for various diameters. 


inches. 


100*. 


901 


801 


751 


60^. 


501 


25$. 


1H 


14 


12 


10 


8.5 


7 


6 


4.5 


!« 


12 


10 


8 


7 


6 


5.25 


4 25 


10 


8.5 


7.5 


6.75 


5.5 


5 


4 


M 


9 


7.5 


6.5 


6 


5 


4.5 


3.75 


i 


8 


7 


6 


5.5 


4.5 


4 


3.50 


% 


• 7.5 


6.75 


5.75 


5 


4.25 


3.5 


3 


H 


5.5 


4.5 


4 


3.75 


3.25 


3 


2.25 


% 


4.5 


4 


3.75 


3.25 


3 


2.5 


2 


H 


4 


3 


3 


2.75 


2.25 


2 


1.5 


% 


3 






2 




1.5 





WIRE-ROPE TRANSMISSION". 

The following data and formulae are taken from a paper by Wm. Hewitt, 
of the Trenton Iron Co., 1890. (See also circulars of John A. Roebling's 
Sons Co., Trenton, N. J.; "Transmission of Power by Wire Ropes, 1 ' by A. 
\V. Stahl, Van Nostrand's Science Series No. 28; and Reuleaux's Constructor.) 

The Section of Wire Rope best suited, under ordinary conditions, 
for the transmission of power is composed of 6 strands of 7 wires each, laid 
together about a hempen centre. Ropes of 12 and 19 wires to the strand are 
also used. They are more flexible, and may be applied with advantage un- 
der conditions which do not allow the use of large transmission wheels, but 
admit of high speed. They are not as well adapted to stand surface wear, 
however, on account of the smaller size of the wires. 



918 



WIRE-ROPE TRANSMISSION". 




Section 
of Rim. 



Section 
of Arm. 



Fig. 1&, 



Tlie Driving-wheels (Fig. 165) are usually of cast iron, and are 
made as light as possible consistent with the requisite strength. Various 
materials have been used for filling the bot- 
tom of the groove, such as tarred oakum, 
jute yarn, hard wood, India-rubber and leather. 
The filling which gives the best satisfaction, 
however, consists of segments of leather and 
blocks of India-rubber, soaked in tar and 
packed alternately in the groove, and then 
turned to a true surface. 

In long spans, intermediate supporting 
wheels are frequently used, and it is usually 
sufficient to support only the slack or follow 
ing side of the rope; but whatever the distance 
that the power is transmitted, the driving side 
of the rope will require a less number of sup- 
ports than the slack side. The sheaves sup- 
porting the driving side, however, should in 
all cases be of equal diameter with the driving- 
wheels. With the slack side smaller wheels 
may be used, but their diameter sbould not be less than one half that of the 
driving-sheaves. 

The system of carrying sheaves may generally be replaced to advantage 
by that of intermediate stations. The rope thus, instead of running the 
whole length of the transmission, runs only from one station to the other; and 
it is advisable to make the stations equidistant, so that a rope may be kept 
on hand, ready spliced, to put on the wheels of any span, should its rope 
give out. This method is to be preferred where there is sometimes a jerking 
motion to the rope, as it prevents sudden movements of this kind from be- 
ing transmitted over the entire line. 

Gross horse-power transmitted = iV = .0003702D 2 v 

D = diameter of rope in inches (= 9 times diameter of single wire); v = 
velocity of rope in feet per second; k = safe stress per square inch on wires 
= for iron 25,700 lbs.; E - modulus of elasticity - 28,500,000 for iron; R - 

ED 
radius of driving-wheels in inches. The term — — = the stress per square 

inch due to bending of wires around sheaves. 

Loss due to centrifugal force = N t = . 00004 24 D 2 ?; 3 : 

Loss due to journal friction of driving-wheels = N 2 = . 0000045 ( 1 6502V,, -\-ivv) ; 
" " " " " intermediate-wheels = .0000045( W -f- w)v; 

in which W = total weight of rope; w — weight of wheel and axle. 

Net horse-power transmitted, 



(h -— ") 



in which 



= N - JV, - N. 2 



: D*v [.0003675(fc - JjS)- 



.0000424?; - .D000045mw. 



For a maximum value of iVthe diameter of the wheels should be approxi- !. 
mately from 185 to 192 times the diameter of the rope, and f»r the latter j 
ratio of diameters an approximate formula for the actual horse-power 
transmitted is N = 3 0148 D 3 V, in which V — number of revolutions of 
wheels per minute. 

The proper deflections when the rope is at rest are obtained from the for- 1 
mula Deflection = .00005765 span 2 , and are as follows: 

Span in feet.. 50 100 150 200 250 300 350 400 450 
Deflection. . . \%" 7" V Z%" 2' 3%" 3' 734" 5' 2J4" 7'%" 9' 2%" 11 '81 j 

It has been found in practice that when the deflection of the rope at rest 
is less than 3 inches the transmission cannot be effected with satisfaction, 
and shafting or belting is to be preferred. This deflection corresponds to a 
span of about 54 feet. It is customary to make the under side of the rope 
the driving side. The maximum limit of span is determined by the maxi- 
mum deflection that may be given to the upper side of the rope when in 
motion. Assuming that the clearance between the upper and lower sides of 
the rope should not be less than two feet, and that the wheels are at least 10 
feet in diameter, we have a maximum deflection of the upper side of 8 feet, 
which corresponds to a span of about 370 feet. 

Much greater spans than this are practicable, in cases where the contour 
of the ground is such that the upper side of the rope may be made the 



WIRE-ROPE TRANSMISSION". 



919 



drivei', as in crossing 1 gullies or valleys, and there is nothing to interfere with 
obtaining the proper deflections. Some very long transmissions of power 
have been effected in tins way without an intervening support. There is one 
at Lockport, N. Y., for instance, with a clear span of about 1700 feet. 

In a later circular of the Trenton Iron Co. (1892) the above figures are 
somewhat modified, giving lower values for the power transmitted by a given 
rope, as follows: 

The proper ratio between the diameters of rope and sheaves is that which 
will permit the maximum working tension to be obtained without overstrain- 
ing the wires in bending. For rope of 7-wire strands this ratio is about 
1 : 150; for rope of 12-wire strands, 1 : 115; and for rope of 19-wire strands, 
1:90; which gives the following minimum diameter of sheaves, in inches, 
corresponding to maximum efficiency. 



Diam. rope, in inches. 


Ya 


5/16 


% k/i6 


M 


9/16 


% 


11/16 


% 


% 


1 


m 


7-wire strands.. 

12 '• " 

19 " " 


37 


47 
36 


56 | 66 
43 50 
34 1 39 


75 
57 

45 


84 
65 
51 


56 


103 

78 
6"2 


112 
86 
68 


ioi 

79 


115 

90 


'ioi 



Assuming the sheaves are of equal diameter, and not smaller than con- 
sistent with maximum efficiency as determined by the preceding table, the 
actual horse-power transmitted approximately equals 3.1 times the square 
of the diameter of the rope in inches multiplied by the velocity in feet per 
second. 

From this rule we deduce the following: 

Horse-power of Wire- rope Transmission, 



Velocity, in feet ( 
per second. f 


20 


30 


40 I 50 1 60 


70 


80 




; 








Diam. Rope, in 
inches. 


Horse-power Transmitted. 


1/4 


4 


6 


8 


10 


12 


14 


16 


5/16 


G 


9 


12 


15 


18 


21 


24 


3/8 


9 


13 


17 


22 


26 


31 


35 


7/16 


12 


18 


24 


30 


36 


42 


47 


1/2 


16 


23 


31 


39 


47 


54 


62 


9/16 


20 


29 


39 


49 


59 


69 


78 


5/8 


24 


36 


48 


61 


73 


85 


97 


11/16 


29 


44 


59 


73 


88 


103 


117 


3/4 


35 


52 


70 


87 


105 


122 


140 


7/8 


48 


71 


95 


119 


142 


166 


190 


1 


62 


93 


124 


155 


186 


217 


248 



The proper deflection to give the rope in order to secure the necessary 
tension is 

h = .0000695S 2 . 

h — the deflection with the rope at rest, and S = the span, both in feet. 

Durability of Wire Kopes.-At the Risdou Iron Works, San Fran- 
cisco, a steel wire rope 2*4 inches in circumference running over 10-foot 
sheaves at 5000 ft. per minute lias transmitted 40 H.P. for six years without 
renewing the rope. At the wire-mills a steel wire rope 234 i' 1 - i° circumfer- 
ence running over 8-foot sheaves has been running steadily for a period of 
three years at a velocity of 4500 ft. per minute, transmitting 80 H.P. 

In "inclined Transmissions, when the angle of inclination is 
great, the proper deflections cannot be readily determined, and the rope be- 
comes more sensitive to the ordinary variations in the deflections, so that 
tightening sheaves must be resorted to for producing the requisite tension, 
as in the case of very short spans. When the horizontal distance between 
the two wheels is less than 60 ft., or when the angle of inclination exceeds 
30 to 45 degrees, it will be found desirable to use tightening sheaves. 

Tightening 1 pulleys should be placed on the slack side of the rope. 

The Wire-rope Catenary. (From an article on Wire-rope 
Transmission, by M. Arthur Achard, Proc. Inst. M. E., Jan. 1881.)- The 
wires have to bear two distinct molecular strains : First, the tension 6' 



920 WIRE-HOPE TRANSMISSION. 

resulting from the maximum tension T necessary to transmit the motion, 

whose value in pounds per square inch is S — ^p—^-i d being the diameter 

of the wires and i their number ; second, the strain produced by flexure 

upon the pulley, which is approximately Z — E—, R being the radius of 

the pulley and E the modulus of elasticity of the metal. The approximate 
values allowed in practice for iron-wire ropes are S = 14,220 lbs. per square 
inch, and Z = 11,380 lbs. per square inch. £ -f Z should not exceed say 11 
tons (24,640 lbs.) per square inch. 

The curve in which the rope hangs is a catenary; and it is upon the form 
of the particular catenary in which it hangs, whether more or less deep, as 
well as upon its lineal weight, that the tension to which it is subjected de- 
pends. By fixing the weight of the rope and its length, the forms Avhich its 
two spans assume in common, when at rest, is determined, and consequently 
their common tension ; which latter must be such as to produce in running 
the two unequal tensions, T and t, necessary for the transmission of the 
power. The driving force = T — t. 

Moreover, the tension in either span is not the same throughout its whole 
length; it is a minimum at the lowest point of the curve and goes on increas- 
ing towards the two extremities. The calculation of the tension at the low- 
est point is very complicated if based upon the true form of the catenary; 
but by substituting a parabola for the catenary, which is allowable in almost 
all cases, the calculation becomes simple. If the two pulleys are on the 
same level, the lowest point is midway between them, and the tension at 

this point is S = ^-, p being the lineal weight, or pounds per foot, of the 

rope, I its horizontal projection, which is approximately equal to the distance 
between the centres of the pulleys, and h the deflection in the middle. The 
catenary possesses the remarkable mechanical property that the difference 
between the tensions at any two points is equal to the weight of a length of 
rope corresponding to the difference in level between the two points. The 

pi 2 
tensions therefore at the two ends will be S t = S -f ph = -=- -{-ph. By 

substituting for S] in the above equation the required values of Tand //, and 
solving it with relation to 7i, the deflections /<j and h 2 of the driving and 
trailing spans will be obtained. The de flection h , com mon to the two spans 
at rest, is given by the equation li = Vl/xh^ -f l/2/* 2 2 - If to — the sectional 
area of the iron portion of the rope, and S the unit strain which the maximum 

tension T produces on it, we h&veivS = T = %7--\-p>h 1 . Taking the sectional 

area w of the rope in square inches, and its weight p in pounds per foot run, 
the ratio w -s- p differs little from a mean value of 0.24. The safe limit of 
working tension usually assigned for iron-wire ropes is 5 = 14,220 lbs. per 
square inch. Hence xos -*- p = 0.24 x 14,220 = 3410; and we have theapprox- 

imate equation ^ — |- /i l = 3410, which is useful as giving a relation between 

the length I and deflection h t for the driving-span of a rope. In the case of 
leather, w -=- p = 2.53 approximately, and it is impossible to give <S a higher 
value than about 355 lbs. per square inch; the relation obtained would be 

— — p- /t 1 = 900, which with equal deflections would give much shorter spans. 

If the working tension S were reduced to the American limit of 185 lbs. per 
square inch for leather belts, the above figure 900 would be reduced to 470, 
which would further shorten the span one half. 

It is therefore owing to the great strength which iron-wire ropes possess 
in proportion to their weight that they admit of long spans, with a smaller 
number of supports, and consequently smaller loss of power by friction. 
They may therefore be expected to yield a high efficiency, The experiments 
of M. Ziegler on the transmission of power at Oberusel give for the mean 
efficiency of a single relay = 96.2 per cent. The efficiency of transmission 
by relays, including m intermediate stations, is approximately obtained by 

raising the efficiency of a single relay to the power of — k~- 

It often happens that the two pulleys of a single relay are at different 
levels, in which case neither span of the rope has the same tension at its 



WIRE-ROPE TRANSMISSION. 



921 



two extremities; the tension at the upper end of each exceeds that at the 
lower by the quantity pH, H being the difference in level between the two 
extremities, or, which is approximately the same, between the centres of 
the two pulleys. It is evidently the tension of the driving-span at its lower 
end which must be regulated so as to obtain the proper driving tension T 
for the transmission; so that there is a certain excess of tension at the 
upper pulley. Large diameter of pulleys tends to preserve the ropes, makes 
the effect of stiffness insignificant, and diminishes the effect of friction on 
the bearings. 
Another formula for the tension at the ends of a catenary (assuming it to 

W , 

be a parabola) is 8 X = -- y (H2O 2 + ( 2 >0 2 . m which S = the tension in lbs ; 

W = weight of the rope in lbs.; I = span, and h = deflection, in feet. 

Diameter and Weigbt of Pulleys for Wire Rope, Ordi- 
nary : 

Diameter, ft 18 14.9 12.4 7.0 

Single groove, lbs .. . 6232 5180 2425 798 

Double groove, lbs.. 8267 6988 4078 1164 

Table of Transmission of Power by "Wire Ropes. 

(J. A. Roebling's Sons Co., 1886.) 



— ? = 

tea 


1 °a 

goo 


ft . 
go 


K A 


•2u_i c 


§£•2 
£o- 




80 
100 
120 
140 

80 


23 
23 
23 
23 
23 


8 
8 


3 

fi 

4 


7 
8 
8 
8 
8 


140 
80 
100 
120 
140 


20 
19 
19 
19 
19 


100 


23 


% 


5 


9 


80 


j 20 
119 


120 


23 


% 


6 


9 


100 


j 20 

119 


140 


23 


¥a 


7 


9 


120 


J 20 
1 19 


80 


22 


7/16 


9 


9 


140 


J20 
119 


100 


22 


7/16 


11 


10 


80 


(19 

118 


120 


22 


7/16 


13 


10 


100 


(19 
118 


140 


22 


7/16 


15 


10 


120 


i 19 

118 


80 


21 


H 


14 


10 


140 


(19 

118 


100 


21 


X 


17 


12 


80 


J 18 

117 


120 


21 


% 


20 


12 


100 


(18 
117 


140 


21 


H 


23 


12 


120 


J 18 

117 
16 


80 


20 


9/16 


20 


12 


120 


100 


20 


9/16 


25 


14 


80 


1? 


120 


20 


9/16 


30 


14 


100 


{? 



go 



9/16 % 

9/16 % 

9/16 % 

9/16 % 

% 11/16 

% H/16 

% 11/16 

% 11/16 

11/16 % 

11/16 % 

11/16 % 

% 

1 IK 

1 V/8 



39 

45 

j 47 

1 48 
j 58 
1 60 
j 69 
1 73 
( 82 
1 84 
( 64 



1102 

( 112 
1119 
( 93 
1 99 
I 116 
I 124 
(140 
1 149 
173 
j 141 
1 148 
(176 
1185 



liOng-distance Transmissions. (From Circular of the Trenton 
Iron Co., 1892.)— In very long: transmissions of power the conditions do not 
always admit of obtaining the proper tensions required in the ordinary sys- 
tem, or "flying transmission of power," as it is termed. In other words, to 
obtain the proper conditions, it would necessitate numerous and expensive 
intermediate stations. In case, for instance, it is desired to utilize the power 
of a turbine to drive a factory, say a mile away, the best method is to em- 
ploy a larger rope than would ordinarily be used, running jt at a moderate 



922 EOPE-DRIVIKG. 

speed. The rope may be in one continuous length, supported, at intervals 
of about 100 ft., on sheaves of comparatively small diameter, since the 
greater rigidity of these ropes preserves them from undue bending strains. 
Where sharp angles occur in the line, however, sheaves must be used of a 
size corresponding to the safe limit of tension due to bending. The rope is 
run under a high working tension, far in excess of what in the ordinary 
system would cause the rope to slip on the sheaves. The working tension 
may be four or five times as great as the tension in the slack portion of the 
rope, and in order to prevent slipping, the rope is wrapped several times 
about grooved drums, or a series of sheaves at each end of the line. To 
provide for the slack due to the stretch of the rope, one of the sheaves is 
placed on a slide worked by long-threaded bolts, or, better still, on a car- 
riage provided with counterweights, which runs back and forth on a track. 
The latter preserves a uniform tension in the slack portion of the rope, 
which is very important. 

Wire-rope tramways are practically transmissions of power of this kind, 
in which the load, however, instead of being concentrated at one terminal, 
is distributed uniformly over the entire line. Cable railways are also trans- 
missions of this class. The amount of horse-power transmitted is given by 
the formula 

N =[4. 7552)2 - .000006 (W+ g + gj]v; 

in which D = diameter of the rope in inches; v = velocity in ft. per second; 
W = weight of the rope; g — weight of the terminal sheaves and axles, and 
g<z = weight of the intermediate sheaves and axles. 



ROPE-DRIVING. 

The transmission of power by cotton or manila ropes promises to become 
a formidable competitor with gearing and leather belting for use where the 
amount of power is large, or the distance between the power and the work 
is comparatively great. The following is condensed from a paper by Charles 
W. Hunt, Trans. A. S. M. E., vol. xii. p. 230: 

But few accurate data are available, on account of the long period re- 
quired in each experiment, a rope lasting from three to six years. In many 
of the early applications so great a strain was put upon the rope that the 
wear was rapid, and success only came when the work required of the rope 
was greatly reduced. The strain upon the rope has been decreased until it 
is approximately known what it should be to secure reasonable durability. 
Installations which have been successful, as well as those in which the wear 
of the rope was destructive, indicate that 200 lbs. on a rope one inch in diam- 
eter is a safe and economical working strain. When the strain is materially 
increased, the wear is rapid. 

In the following equations 

C = circumference of rope in inches; g = gravity: 

D — sag of the rope in inches; H— horse-power; 

F — centrifugal force in pounds; L — distance between pulleys in ft. 

P — pounds per foot cf rope; iv — working strain in pounds; 

B = force in pounds doing useful work; 

S — strain in pounds on the rope at the pulley; 

T — tension in pounds of driving side of the rope; 

t = tension in pounds on slack side of the rope; 

v = velocity of the rope in feet per second; 
W = ultimate breaking strain in pounds. 

W = 720C 2 ; P = .32C 2 ; to = 20C 2 . 

This makes the normal working strain equal to 1/36 of the breaking 
strength, and about 1/25 of the strength at the splice. The actual strains are 
ordinarily much greater, owing to the vibrations in running, as well as from 
imperfectly adjusted tension mechanism. 

For this investigation we assume that the strain on the driving side of a 
rope is equal to 200 lbs. on a rope one inch in diameter, and an equivalent 
strain for other sizes, and that the rope is in motion at various velocities of 
from 10 to 140 ft. per second. 

The centrifugal force of the rope in running oyer the pulley will reduce 



ROPE-DMVltfG. 923 

the amount of force available for the transmission of power. The centrifu- 
gal force F = Pv" 2 h- g. 

At a speed of about 80 ft. per second, the centrifugal force increases faster 
than the power from increased velocity of the rope, and at about 140 ft. per 
second equals the assumed allowable tension of the rope. Computing this 
force at various speeds and then subtracting it from the assumed maximum 
tension, we have the force available for the transmission of power. The 
whole of this force cannot be used, because a certain amount of tension on 
the slack side of the rope is needed to give adhesion .to the pulley. What 
tension should be given to the rope lor this purpose is uncertain, as there 
are no experiments which give accurate data. It is known from considerable 
experience that when the rope runs in a groove whose sides are inclined 
toward each other at an angle of 45° there is sufficient adhesion when the 
ratio of the tensions T-=- t — 2. 

For the present purpose, T can be divided into three parts: 1. Tension 
doing useful work; 2. Tension from centrifugal force; 3. Tension to balance 
the strain for adhesion. 

The tension t can be divided into two parts: 1. Tension for adhesion ; 
2. Tension from centrifugal force. 

It is evident, however, that the tension required to do a given work should 
not be materially exceeded during the life of the rope. 

There are two methods of putting ropes on the pulleys; one in which the 
ropes are single and spliced on, being made very taut at first, and less so as 
the rope lengthens, stretching until it slips, when it is respliced. The other 
method is to wind a single rope over the pulley as many turns as needed to 
obtain the necessary horse power and put a tension pulley to give the neces- 
sary adhesion and also take up the wear. The tension t required to trans- 
mit the normal horse-power for tiie ordinary speeds and sizes of rope is com- 
puted by formula (1). below. The total tension Ton the driving side of the 
rope is assumed to be the same at all speeds. The centrifugal force, as well 
as an amount equal to the tension for adhesion on the slack side of the rope, 
must be taken from the total tension T to ascertain the amount of force 
available for the transmission of power. 

It is assumed that the tension on the slack side necessary for giving 
adhesion is equal to one half the force doing useful work on the driving side 

of the rope; hence the force for useful work is R — — — ^ ; and the ten- 
sion on the slack side to give the required adhesion is ^$(T — F). Hence 

t=^l + F. ....(.) 

The sum of the tensions !Tand t is not the same at different speeds, as the 
equation (1) indicates. 

As F varies as the square of the velocity, there is, with an increasing 
speed of the rope, a decreasing useful force, and an increasing total tension, 
t, on the slack side. 

With these assumptions of allowable strains the horse-power will be 

2v(T-F) 
H= 3X550 &) 

Transmission ropes are usually from 1 to 1% inches in diameter. A com- 
putation of the horse-power for four sizes at various speeds and under 
ordinary conditions, based on a maximum strain equivalent to 200 lbs. for a 
rope one inch in diameter, is given in Fig. 166. The horse-power of other 
sizes is readily obtained from these. The maximum power is transmitted, 
under the assumed conditions, at a speed of about 80 feet per second. 

The wear of the rope is both internal and external; the internal is caused 
by the movement of the fibres on each other, under pressure in bending 
over the sheaves, and the external is caused by the slipping and the wedg- 
ing in the grooves of the pulley. Both of these causes of wear are, within 
the limits of ordinary practice, assumed to be directly proportional to the 
speed. Hence, if we assume the coefficient of the wear to be k, the wear 
will be lev, in which the wear increases directly as the velocity, but the 
horse-power that can be transmitted, as equation (2) shows, will not vary at 
the same rate. 

The rope is supposed to have the strain T constant at all speeds on the 
driving side, and in direct proportion to the area of the cross-section ; hence 



1 



924 



ROPE-DllIVttfG. 



, 



the catenary of the driving side is not affected by the speed or by the diam- 
eter of the rope. 

The deflection of the rope between the pulleys on the slack side varies 
with each change of the load or change of the speed, as the tension equation 
(1) indicates. 

The deflection of the rope is computed for the assumed value of T and t 



42 
40 
33 






























^ 






























42 
40 


ROPE DRIVIN 








































Horse Power c 
rope at vario 


fmanilla 










































3b 


is speeds. 








































3b 






















































































































u 












<" 
















































<sy 






„e 


/ 






































: 














■§■/ 




$ 




















































s 


A 






















































« 




ki 










«■ 


































"IS 










fjf 


®l 








^ 


-^ 






















\ 




\ 








18 








r 








JV L 




























s 




\ 
















ly 




-i 


- 


































S 










14 
12 
10 
6 
6 








> 




































. 




. 


V 


















^ 


irrt 


[>« 




























> 




s 


\ 








- 


f/ 






o 






fcP 




































v 






6 
4 
2 




































\ 


s 


\ 






















































\ 


\ 


\\ 
























































X 






2 






















































^ 


1 




3 


1 





2 





3 





4 





5 





6 





7 





8 





E 





1 


jO 


1 





120 


130 


140 





Velocity of Drving Rope in feet per second. 

Fig. 166. 

PL 2 
by the parabolic formula S - -^=r + PD, S being the assumed strain T on 

the driving side, and t, calculated by equation (1), on the slack side. The 
tension t varies with the speed. 

Horse-power of Transmission Rope at Various Speeds. 

Computed from formula (2), given above. 



CW 


43 <* CO 


©■■ 


Speed of the Rope in feet per minute. 


CD °. Kc 


.23 q3 




a «8 s — 


1500 


2000 


2500 


3000 


3500 


4000 


4500 


5000 


6000 


7000 


8000 


U, 


1.45 


1.9 


2.3 


2.7 


3 


3.2 


3.4 


3.4 


3.1 


2.2 





20 




2.3 


3.2 


3.6 


4.2 


4.6 


5.0 


5.8 


5.3 


4.9 


3.4 





24 


3.3 


4.3 


5.2 


5.8 


6.7 


7.2 


7.7 


7.7 


7.1 


4.9 





30 


4 


4.5 


5.9 


7.0 


8.2 


9.1 


9.8 


10.8 


10.7 


9.3 


6.9 





36 


i'° 


5.8 


7.7 


9.2 


10.7 


11.9 


12 8 


13.6 


13.7 


12.5 


8.8 





42 


8$ 


9.2 


12.1 


14.3 


16.8 


18. 6 


20.0 


21.2 


21.4 


19.5 


13.8 





54 


13.1 


17.4 


20.7 


23 1 


26.8 


28.8 


30.6 


30.8 


28.2 


19.8 





60 


V^i 


18 


23.7 


28.2 


32.8 


36.4 


39.2 


41.5 


41.8 


37.4 


27.6 





72 


2 


23.2 30.8 


36.8 


42.8 


47.6 


51.2 


54.4 


54 8 


50 


35.2 





84 



The following notes are from the circular of the C. W. Hunt Co., New 
York : 

For a temporary installation, when the rope is not to be long in use, it 
might be advisable to increase the work to double that given in the table. 

For convenience in estimating the necessary clearance on the driving and 
on the slack sides, we insert a table showing the sag of the rope at different 
speeds when transmitting the horse-power given in the preceding table. 
When at rest the sag is Dot the same as when running, being greater on the 
driving and less on the slack sides of the rope. The sag of the driving side 
when transmitting the normal horse-power is the same no matter what size 
of rope is used or what the speed driven at, because the assumption is that 
the strain on the rope shall be the same at all speeds when transmitting the 



SAG OF THE ROPE BETWEEN PULLEYS. 



925 



assumed horse-power, but on the slack side the strains, and consequently 
the sag, vary with the speed of the rope and also with the horse -power. 
The table gives the sag for three speeds. If the actual sag is less than given 
in the table, the rope is strained more than the work requires. 

This table is only approximate, and is exact only when the rope is running 
at its normal speed, transmitting its full load and strained to the assumed 
amount. All of these conditions are varying in actual work, and the table 
must be used as a guide ouly. 

Sag of tlie Rope between Pulleys. 



Distance 


Driving Side. 


between 




Pulleys 




in feet. 


All Speeds. 


40 


feet 4 inches 


60 


" 10 " 


80 


1 " 5 " 


100 


2 " " 


120 


2 " 11 " 


140 


3 " 10 " 



160 



Is 



1 



Slack Side of Rope. 



80 ft. per 


sec. 


60 ft. per 


sec. 


40 ft 


per sec. 


Ofeet 7 inches 

1 " 5 " 

2 " 4 " 

5 " 3 " 
7 " 2 " 
9 " 3 " 


Ofeet 9 inches 
1 " 8 " 
1 " 10 " 
4 " 5 " 
6 " 3 " 
8 " 9 " 
11 " 3 " 


Ofeet 11 inches 
1 " 11 " 
3 " 3 " 
5 " 2 " 
7 « 4 « 
9 " 9 " 
14 " " 



The size of the pulleys has an important effect on the wear of the rope — 
the larger the sheaves, the less the fibres of the rope slide on each other, and 
consequently there is less internal wear of the rope. The pulleys should not 
be less than' forty times the diameter of the rope for economical wear, and 
as much larger as it is possible to make them. This rule applies also to the 
idle and tension pulleys as well as to the main driving-pulley. 

The angle of the sides of the grooves in which the rope runs varies, with 
different engineers, from 45° to 60°. It is very important that the sides of 
these grooves should be carefully polished, as the fibres of the rope rubbing 
on the metal as it comes from the lathe tools will gradually break fibre by 
fibre, and so give the rope a short life. It is also necessary to carefully avoid 
all sand or blow holes, as they will cut the rope out with surprising rapidity. 

Much depends also upon the arrangement of the rope on the pulleys, es- 
pecially where a tension weight is used. Experience shows that the 
increased wear on the rope from bending the rope first in one direction and 
then in the other is similar to that of wire rope. At mines where two cages 
are used, one being hoisted and one lowered by the same engine doing the 
same work, the wire ropes, cut from the same coil, are usually arranged so 
that one rope is bent continuously in one direction and the other rope is bent 
first in one direction and then in the other, in winding on the drum of the 
engine. The rope having the opposite bends wears much more rapidly than 
the other, lasting about three quarters as long as its mate. This difference 
in wear shows in mauila rope, both in transmission of power and in coal- 
hoisting. The pulleys should be arranged, as far as possible, to bend the 
rope in one direction. 

The wear of the rope is independent of the distance apart of the shafts, 
since the wear takes place only on the pulleys; hence in transmitting power 
any distance within the limits of rope-driving, the life of the rope will be 
the same whether the distance is small or great, but the first cost will be in 
proportion to the distance. 

Tension on the Slack Part op the Rope. 



Speed of 


Diameter of the Rope and Pounds Tension on 


the Slack Rope. 


Rope, in feet 
per second. 
















H 


Vs 


Va 


% 


1 


m 


M 


1% 


2 


20 


10 


27 


40 


54 


71 


110 


162 


216 


283 


30 


14 


29 


42 


56 


74 


115 


170 


226 


296 


40 


15 


31 


4?. 


60 


79 


123 


181 


240 


315 


50 


16 


33 


49 


65 


85 


132 


195 


259 


339 


60 


18 


36 


53 


71 


93 


145 


214 


285 


373 


70 


19 


39 


59 


78 


101 


158 


236 


310 


406 


80 


21 


43 


64 


85 


111 


173 


255 


340 


445 


90 


24 


48 


70 


93 


122 


190 


279 


372 


487 



926 



ROPE-DRIVING. 



For large amounts of power it is common to use a number of ropes lying 
side by side in grooves, each spliced separately. For lighter drives some 
engineers use one rope wrapped as many times around the pulleys as is 
necessary to get the horse-power required, with a tension pulley to take up 
the slack as the rope wears when first put in use. The weight put upon this 
tension pulley should be carefully adjusted, as the overstraining of the rope 
from this cause is one of the most common errors in rope driving. We 
therefore give a table showing the proper strain on the rope for the various 
sizes, from which the tension weight to transmit the horse-power in the 
tables is easily deduced. This strain can be still further reduced if the 
horse-power transmitted is usually less than the nominal work which the 
rope was proportioned to do, or if the angle of groove in the pulleys is 
acute. 

Diameter of Pulleys and Weight of Rope. 



Diameter of 


Smallest Diameter 


Length of Rope to 


Approximate 


Rope, 


of Pulleys, in 


allow for Splicing, 


Weight, in lbs. per 


in iuches. 


inches. 


in feet. 


foot of rope. 


Vi 


20 


6 


.12 


% 


24 


6 


.18 


H 


30 


7 


.24 


% 


36 


8 


.32 


1 


42 


9 


.49 


VA 


54 


10 


,60 


m 


60 


12 


.83 


m 


72 


13 


1.10 


2 


84 


14 


1.40 



1W 


1%" 


.98 


W 


2" 


.72 


.844 


1.125 


1.3 


44 


38 


33 


28 


25 


145 


170 


193 


228 


256 


430 


500 


600 


675 


780 


43 


50 


60 


67 


7S 


242 


280 


347 


380 


446 


34 


41 


49 


54 


63 



With a given velocity of the driving-rope, the weight of rope required for 
transmitting a given horse-power is the same, no matter what size rope is 
adopted. The smaller rope will require more parts, but the weight will be 
the same. 

Miscellaneous Notes on Rope-driving.— W. H. Booth commu- 
nicates to the Amer. Machinist the following data from English practice with 
cotton ropes. The calculated figures are based on a total allowable tension 
on a 1%-inch rope of 600 lbs., and an initial tension of 1/10 the total allowed 
stress, which corresponds fairly with practice. 

Diameter of rope 1*4" Wd' 

Weight per foot, lbs 5 .6 

Centrifugal tension = F 2 divided by 64 53 
" for V= 80 ft. per sec, lbs. 100 121 

Total tension allowable 300 360 

Initial tension 30 36 

Net working tension at 80 ft. velocity 170 203 
Horse-power per rope " " 24 28 

The most usual practice in Lancashire is summed up roughly in the fol- 
lowing figures: 154-inch cotton ropes at 5000 ft. per minute velocity = 50 H. P. 
per rope. The most common sizes of rope now used are 1% and \% in. The 
maximum horse-power for a given rope is obtained at about 80 to 83 feet 
per second. Above that speed the power is reduced by centrifugal tension. 
At a speed of 2500 ft. per minute four ropes will do about the same work as 
three at 5000 ft. per min. 

Cotton ropes do not require much lubrication in the sense that it is re- 
quired by ropes made of the rough fibre of manila hemp. Merely a slight 
surface dressing is all that is required. For small ropes, common in spin- 
ning machinery, from J^ to §4 inch diameter, it is the custom to prevent the 
fluffing of the ropes on the surface by a light application of a mixture of 
black-lead and molasses,— but only enough should be used to lay the fibres,— 
put upon one of the pulleys in a series of light dabs. 

Reuleaux's Constructor gives as the " specific capacity " of hemp rope in 
actual practice, that is, the horse-power transmitted per square inch of 
cross-section for each foot of linear velocity per minute, .004 to .002, the 
cross-section being taken as that due to the full outside diameter of the 
rope. For a 1%-in. rope, with a cross section of 2.405' q. in., at a velocity of 
5000 ft. per min., this gives a horse-power of from 24 to 48, as against 41.8 
by Mr. Hunt's table and 49 by Mr. Booth's. 



MISCELLANEOUS NOTES ON ROPE-DRIVING. 927 

Reuleaux gives formulae for calculating sources of loss in hemp-rope 
transmission due to(l) journal friction, (2) stiffness of ropes, and (3) creep 
of ropes. The constants in these formulae are, however, uncertain from 
lack of experimental data. He calculates an average case giving loss of 
power due to journal friction = 4%, to stiffness 7.8$. and to creep 5%, or 16.8$ 
in all, and says this is not to be considered higher than the actual loss. 

T. Spencer Miller (Eug'g News, Dec. 6, 1890) says: In England hemp and 
mauila ropes have been largely superseded by ropes of cotton; and I am 
satisfied that one reason for this is that dry manila ropes wear out too fast, 
while lubricated ropes give too low a coefficient of friction. The angle of 
45° for the groove lias been in use for 33 years, having been first introduced 
by Jas. Combe in Belfast, Ireland; but if we are to use tallow-laid or other 
lubricated ropes, we should certainly use a sharper angle in the groove, 
especially in the American system, which employs a continuous rope with 
many wraps. 

Mr. Hunt's formula, Tension of driving side of rope -s- tension of slack 
side of rope = 2, implies a coefficient of friction of only .10. But I have 
obtained a coefficient of friction of .26, and have found one authority giving 
.28. Reuleaux advises for single-line transmission 30° angle of groove. 
Ramsbottom, an English engineer, and Yale & Towne use a 30° groove in 
transmission-wheels of travelling-cranes, and I hope to see the best Ameri- 
can practice use 30° or 35° as a standard groove angle. The work done in 
pulling out a greasy mauila rope from a 30° groove is not worth considera- 
tion, although we hear a great deal about the loss of power on this account. 
I am strongly in favor of using the continuous-rope system, and also of 
using smaller ropes than are recommended in Mr. Hunt's paper. 

The most perfect small transmission I have ever seen (about 20 H.P.) em- 
ploys 5/1(5- in. manila rope on wheels 30 in. in diameter, using a tension car- 
riage. Rather than use large ropes I think it wiser to replace small ones 
oftener, for by so doing a great gain may be made in efficiency, thus saving 
fuel. 

A large majority of failures in the continuous-rope plan have occurred 
where the driving and driven sheaves were of widely different diameters, as 
for example, driving dynamos, or driving a line-shaft from an engine fly- 
wheel. As ordinarily installed the ropes will not pull alike, and by calcula- 
tion or by experiment we may find one rope pulling twice or three times as 
much as the others on the sheave. 

An installation designed by the writer employs an engine-driving sheave 
about three times the diameter of the driven sheave. To equalize the pull 
on the different ropes the grooves of the large driving-sheaves were made 
with an angle of 30° and those of the small sheaves with an angle of 45°. 
This change of groove angle has entirely remedied the unequal pulling com- 
plained of. 

It has been observed that in sheaves of the same diameter, by the use of a 
proper tension weight, the ropes may all pull alike; while, where the sheaves 
are of unequal diameter, the pull is unequal. The only difference of condi- 
tions in the two cases lies in the different arc of contact of the rope on the 
two sheaves, which leads to a greater frictional hold of the rope on the large 
sheave. To equalize the frictional hold on the two sheaves we may sharpen 
the angle of the small sheave or increase the angle of the large sheave. 

The Walker Mfg. Co. adopts a curved form of groove instead of one with 
straight sides inclined to each other at 45°. The curves are concave to the 
rope. The rope rests on the sides of the groove in driving and driven pul- 
leys. In idler pulleys the rope rests on the bottom of the groove, which is 
semicircular. The Walker Mfg. Co. also uses a '•' differential " drum for heavy 
rope drives, in which the grooves are contained each in a separate ring 
which is free to slide on the turned surface of the drum in case one rope 
pulls more than another. 

A heavy rope-drive on the separate, or English, rope system is described 
and illustrated in Power, April, 1892. It is in use at the India Mill at Darwen, 
England. This mill was originally driven by gears, but did not prove success- 
ful, and rope-driving was resorted to. The 85.000 spindles and preparation 
are driven by a 2000 horse-power tandem compound engine, with cylinders 
23 and 44 inches in diameter and 72- inch stroke, running at 54 revolutions 
per minute. The fly-wheel is 30 feet in diameter, weighs 65 tons, and is 
arranged with 30 grooves for 1%-inch ropes. These ropes lead off to receiv- 
ing-pulleys upon the several floors, so that each floor receives its power direct 
from the fly-wheel. The speed of the ropes is 50b9 feet per minute, and five 
7-foot receivers are used, the number of ropes upon each being proportioned^ 



928 



FRICTIOK AND LUBRICATION". 



to the amount of power required upon the several floors. Lambeth cotton 
ropes are used. 



FRICTION AND LUBRICATION. 



Friction is defined by Rankine as that force which acts between two 
bodies at their surface of contact so as to resist their sliding on each other, 
and which depends on the force with which the bodies are pressed together. 

Coefficient of Friction.— The ratio of the force required to slide a 
body along a horizontal plane surface to the weight of the body is called the 
coefficient of friction. It is equivalent to the tangent of the angle of repose, 
which is the angle of inclination to the horizontal of an inclined plane on 
which the body will just overcome its tendency to slide. The angle is usually 
denoted by 0, and the coefficient by /. / = tan 6. 

Friction of Rest and of Motion.— The force required to start a 
body sliding is called the friction of rest, and the force required to continue 
its sliding: after having: started is called the friction of motion. 

Rolling Friction is the force required to roll a cylindrical or spheri- 
cal body on a plane or on a curved surface. It depends on the nature of the 
surfaces and on the force with which they are pressed together, but is 
essentially different from ordinary, or sliding, friction. 

Friction of Solids.— Rennie's experiments (I8x!9) on friction of solids, 
usually unlubricated and dry, led to the following conclusions: 

1. The laws of sliding friction differ with the character of the bodies 
rubbing together. 

2. The friction of fibrous material is increased by increased extent of 
surface and by time of contact, and is diminished by pressure and speed. 

3. With wood, metal, and stones, within the limit of abrasion, friction 
varies only with the pressure, and is independent of the extent of surface, 
time of contact and velocity. 

4. The limit of abrasion is determined by the hardness of the softer of the 
two rubbing parts. 

5. Friction is greatest with soft and least with hard materials. 

6. The friction of lubricated surfaces is determined by the nature of the 
lubricant rather than by that of the solids themselves. ' 

Friction of Rest. (Rennie.) 



Pressure, 
lbs. 


Values of /. 










per square 
inch. 


Wrought iron on 


Wrought on 


Steel on 


Brass on 


Wrought Iron. 


Cast Iron. 


Cast Iron. 


Cast Iron. 


187 


.25 


.28 


.30 


.23 


224 


.27 


.29 


.33 


.22 


336 


.31 


.33 


.35 


.21 


448 


.38 


.37 


.35 


.21 


560 


.41 


.37 


.36 


.23 


672 


Abraded 


.38 


.40 


.23 


784 




Abraded 


Abraded 


.23 



Law of Unlubricated Friction.— A. M. Wellington, Eng'g Neivs, 
April 7, 1888, states that the most important and the best determined of all 
the laws of unlubricated friction may be thus expressed: 

The coefficient of unlubricated friction decreases materially with velocity, 
is very much greater at minute velocities of -f, falls very rapidly with 
minute increases of such velocities, and continues to fall much less rapidly 
with higher velocities up to a certain varying point, following closely the 
laws which obtain with lubricated friction. 

Friction of Steel Tires Sliding on Steel Rails. (Westing- 
house & Gal ton.) 



Speed, miles per hour 

Coefficient of friction 

Adhesion, lbs. per ton (2240 lbs.) 



10 
0.110 
246 



15 



25 



.051 



45 50 
.047 .040 
114 90 



929 



Rolling Friction is a consequence of the irregularities of form and 
the roughness of surface of bodies rolling one over the other. Its laws 
are not yet definitely established in consequence of the uncertainty which 
exists in' experiment as to how much of the resistance is due to roughness of 
surface, how much to original and permanent irregularity of form, and how 
much to distortion under the load. (Thurston.) 

Coefficients of Rolling Friction.— If R = resistance applied at 
the circumference of the wheel, W — total weight, r = radius of the wheel, 
and / = a coefficient, R = fW -¥■ r. f is very variable. Coulomb gives .06 
for wood, .005 for metal, where W is in pounds and r in feet. Tredgold 
made the value of /for iron on iron .002. 

For wagons on soft soil Morin found/ — .065, and on hard smooth roads 
.02. 

A Committee of the Society of Arts (Clark, R. T. D.) reported a loaded 
omnibus to exhibit a resistance on various loads as below: 

Pavement Speed per hour. Coefficient. Resistance. 

Granite. 2.87 miles. .007 17.41 per ton. 

Asphalt 3.56 " .0121 27.14 

Wood 3.34 " .0185 41.60 

Macadam, gravelled 3.45 " .0199 44.48 

granite, new. . 3.51 " .0451 101.09 

Thurston gives the value of /for ordinary railroads, .003, well-laid railroad 
track, .002; best possible railroad track, .001. 

The few experiments that have been made upon the coefficients of rolling 
friction, apart from axle friction, are too incomplete to serve as a basis for 
practical rules. (Trautwine). 

Laws of Fluid Friction.— For all fluids, whether liquid or gaseous, 
the resistance is (1) independent of the pressure between the masses in 
contact; (2) directly proportional to the area of rubbing-surface; (3) pro- 
portional to the square of the relative velocity at moderate and high speeds, 
and to the velocity nearly at low speeds; (4) independent of the nature of 
the surfaces of the solid against which the stream may flow, but dependent 
to some extent upon their degree of roughness; (5) proportional to the den- 
sity of the fluid, and related in some way to its viscosity. (Thurston.) 

The Friction of Lubricated Surfaces approximates to that of solid fric- 
tion as the journal is run dry, and to that of fluid friction as it is flooded 
with oil. 

Angles of Repose and Coefficients of Friction of Build- 
in ir Materials. (From Rankine's Applied Mechanics.) 



Dry masonry and brickwork 
Masonry and brickwork with 

damp mortar 

Timber on stone 

Iron on stone . . 

Timber on timber 

• " " metals 

Metals on metals 

Masonry on dry clay 

" " moist clay 

Earth on earth 

" " dry sand, clay, 

and mixed earth. 

Earth on earth, damp clay 

" " " wet clay 

" " " shingle and 
gravel 



35° to 16%° 
26^° to 11^° 
31° to \1%° 

14° to sy 2 ° 

27° 

18M° 

14° to 45° 

21° to 37° 
45° 



> to 48° 



/ = tan 0. 



about .4 
.7 to .3 
.5 to .2 
.6 to .2 
.25 to .15 
.51 
.33 
.25 to 1.0 

.38 to .75 
1.0 



.81 



1.67 to 1.4 

1.35 

2.5 

1.43 to 3.3 

2 to 5 

1.67 to 5 

4 to 6.67 

1.96 

3. 

4tol 

2.63 to 1.33 

1 

3.23 

1.23 to 0.9 



Friction of Uf otion.— The following is a table of the angle of repose 
. the coefficient of friction / = tan 0, and its reciprocal, 1 -=-/, for the ma- 
terials of mechanism— condensed from the tables of General Morin (1831), 
and other sources, as given by Rankine; 



930 



FRICTION AND LUBRICATION". 




Surfaces. 



Wood on wood, dry .... 
" " " soaped.. 

Metals on oak, dry 

" " " wet 

" " " soapy.. . 

'• " elm, dry 

Hemp on oak, dry 

" " " wet 

Leather on oak ... 

" " metals, dry.. 
" " " wet.. 
" " " greasy 
" oily... 
Metals on metals, dry. .. 
" " " wet... 
Smooth surfaces, occa- 
sionally greased 

Smooth surfaces, con- 
tinuously greased 

Smooth surfaces, best 

results 

Bronze or lignum vitee, 
constantly wet 



14° to 26^° 
uy 2 ° to 2° 
2G\i° to 31° 
13^° to 14° 

1114° 

11^° to 14° 

28° 

18^° 
15° to 19^° 

20° 
13° 

&A° 

8^° to 11° 

16^>° 



m° to 2° 



.25 to . 

.2 to . 

5 to 

.24 to . 
.2 

.2 to A 
.53 
.33 

.27 to . 
.56 
.36 
.23 



.07 to . 

.05 
.03 to .( 

.05? 



Coefficients of Friction of Journals. (Morin.) 



Cast iron on cast iron. , 
Cast iron on bronze.. . . 



Cast iron on lignum-vitse . 

Wrought iron on cast iron ) 

" " " bronze.. \ 

Iron on lignum vitse \ 



Bronze on bronze. . 



Oil, lard tallow. 
Unctuous and wet. 
Oil, lard, tallow. 
Unctuous and wet. 
Oil, lard. 

Oil, lard, tallow. 

Oil, lard. 
Unctuous. 
Olive-oil. 
Lard. 



Intermittent. Continuous, 



.07 to .08 

.14 
.07 to .08 

.16 

.07 to .08 
.11 
.19 
.10 
.09 



.03 to .054 
.03 to .054 

.09 
.03 to .054 



Prof. Thurston says concerning the above figures that much better results 
are probably obtained in good practice with ordinary machinery. Those 
here given are so greatly modified byvariations of speed, pressure, and tem- 
perature, that they cannot be taken as correct for general purposes. 

Average Coefficients of Friction. Journal of cast iron in bronze 
bearing; velocity 120 feet per minute; temperature 70° F.; intermittent 
feed through an oil-hole. (Thurston on Friction and Lost Work.) 



Oils. 


Pressures, pounds per square inch. 


8 


16 


32 


48 


Sperm, lard, neat's-foot,etc. 
Olive, cotton -seed, rape, etc. 
Cod and menhaden. . .... 

Mineral lubricating-oils. . . . 


.159 to .250 
.160 " .283 
.248 " .278 
.154 " .261 


.138 to .192 
.107 " .245 
.124 " .167 
.145 " .233 


.086 to .141 
.101 " .168 
.097 " .102 
.086 " .178 


.077 to .144 
.079 " .131 
081 " .122 
.091 " .222 



With fine steel journals running in bronze bearings and continuous lubri- 
cation, coefficients far below those above given are obtained. Thus with 
sperm-oil the coefficient with 50 lbs. per square inch pressure was .0034; with 
200 lbs., .0051; with 300 lbs., ,0057, 



FRICTION". 931 

For very low pressures, as in spindles, the coefficients are much higher. 
Thus Mr, Woodbury found, at a temperature of 100° and a velocity of 600 
feet per minute, 

Pressures, lbs. per sq. iu 1 2 3 4 5 

Coefficient 38 .27 .22 .18 .17 

These high coefficients, however, and the great decrease in the coefficient 
at increased pressures are limited as a practical matter only to the smaller 
pressures which exist especially in spinning machinery, where the pressure 
is so light and the film of oil so thick that the viscosity of the oil is an import- 
ant part of the total frictional resistance. 

Experiments on Friction of a Journal Lubricated by an 
Oil-bath (reported by the Committee on Friction, Froc. Inst. M. E., 
Nov. 1883) show that the absolute friction, that is, the absolute tangential 
force per square inch of bearing, required to resist the tendency of the brass 
to go round with the journal, is nearly a constant under all loads, within or- 
dinary working limits. Most certainly it does not increase in direct propor- 
tion to the load, as it should do according to the ordinary theory of solid 
friction. The results of these experiments seem to show that the friction of 
a perfectly lubricated journal follows the laws of liquid friction much more 
closely than those of solid friction. They show that under these circum- 
stances the friction is nearly independent of the pressure per square inch, 
and that it increases with the velocity, though at a rate not nearly so rapid 
as the square of the velocity. 

The experiments on friction at different temperatures indicate a great 
diminution in the friction as the temperature rises. Thus in the case of 
lard-oil, taking a speed of 450 revolutions per minute, the coefficient of fric- 
tion at a temperature of 120° is only one third of what it was at a tempera- 
ture of 60. 

The journal was of steel, 4 inches diameter and 6 inches long, and a gun- 
metal brass, embracing somewhat less than half the circumference of the 
journal, rested on its upeer side, on which the load was applied. When the 
bottom of the journal was immersed in oil, and the oil therefore carried 
under the brass by rotation of the journal, the greatest load carried with 
rape-oil was 573 lbs. per square inch, and with mineral oil 625 lbs. 

In experiments with ordinary lubrication, the oil being fed in at the cen- 
tre of the top of the brass, and a distributing groove being cut in the brass 
parallel to the axis of the journal, the bearing would not run cool with only 
100 lbs. per square inch, the oil being 1 pressed out from the bearing-surface 
and through the oil-hole, instead of being carried in by it. On introducing 
the oil at the sides through two parallel grooves, the lubrication appeared 
to be satisfactory, but the bearing seized with 380 lbs. per square inch. 

When the oil. was introduced through two oil-holes, one near each end of 
the brass, and each connected with a curved groove, the brass refused to 
take its oil or run cool, and seized with a load of only 200 lbs. per square 
inch. 

With an oil-pad under the journal feeding rape-oil, the bearing fairly car- 
ried 551 lbs. Mr. Tower's conclusion from these experiments is that the 
friction depends on the quantity and uniformity of distribution of the oil, 
and may be anything between the oil-bath results and seizing, according to 
the perfection or imperfection of the lubrication. The lubrication may be 
very small, giving a coefficient of 1/100; but it appeared as though it could 
not be diminished and the friction increased much beyond this point with- 
out imminent risk of heating and seizing. The oil-bath probably represents 
the most perfect lubrication possible, and the limit beyond which friction 
cannot be reduced by lubrication ; and the experiments show that with speeds 
of from 100 to 200 feet per minute, by properly proportioning the bearing- 
surface to the load, it is possible to reduce the coefficient of friction to as low 
as 1/1000. A coefficient of 1/1500 is easily attainable, and probably is fre- 
quently attained, in ordinary engine-bearings in which the direction of the 
force is rapidly alternating and the oil given an opportunity to get between 
the surfaces, while the duration of the force in one direction is not sufficient 
to allow time for the oil film to be squeezed out. 

Observations on the behavior of the apparatus gave reason to believe that 

with perfect lubrication the speed of minimum friction was from 100 to 150 

feet per minute, and that this speed of minimum friction tends to be higher 

■I, with an increase of load, and also with less perfect lubrication. By the 

I speed of minimum friction is meant that speed in approaching which from 

rest the friction diminishes, and above which the friction increases. 



932 



FRICTION AND LUBRICATION. 



Coefficients of Friction of Journal with Oil-bath.— Ab- 
stract of results of Tower's experiments on friction (Proc. Inst. M. E., Nov. 
1683). Journal, 4 in. diam., 6 in. long; temperature, 90° F. 



Lubricant in Bath. 



Nominal Load, in pounds per square inch. 



625 



520 415 310 205 153 i 100 



Coefficients of Friction. 



Lard-oil : 

157 ft. per miu.. 

471 " 
Mineral grease : 

157 ft. per min.. 

471 ' 



Sperm-oil : 

157 ft. per min 

471 " " 

Rape-oil : 

157 ft. per min 

471 kl " 

Mineral-oil : 

157 ft. per min 

471 " " .., 

Rape-oilfed by syphon lubricator: 

157 ft. per miu 

314 " " 

Rape-oil, pad under journal: 

157 ft. per min 

314 " " 



.001 
.002 



(573 lb.) 
001 



.0009 
.0017 



.0014 
.0022 



.001 
.0015 



.0012 
.0018 



0009 
0016 



,0012 
.002 



.0014 
.0029 


.0020 
.0042 


.0022 
.004 


.0034 
.0066 


.0011 
.0019 


.0016 
.0027 


.0008 
.0016 


.0014 

.0024 


.0014 
.0024 


.0021 
.0035 


.0056 
.0068 


.0098 
.0077 


.0099 
.0099 


.0105 
.0078 



.0042 
.009 



.0076 
.0151 



.003 
.0064 



.004 
.007 



.004 
.007 



.0125 
.0152 



.0099 
.0133 



Comparative friction of different lubricants under same circumstances, 
temperature 90°, oil-bath: 

Sperm-oil 100 per cent. I Lard 135 per cent. 

Rape-oil 106 " Olive-oil 135 

Mineral oil 129 " | Mineral grease 217 " 

Coefficients of Friction of Motion and of Rest of a 
Journal.— A cast-iron journal in steel boxes, tested by Prof. Thurston at 
a speed of rubbing of 150 feet per minute, with lard and with sperm oil, 
gave the following: 



Pressures per sq. in., lbs 50 

Coeff., with sperm 013 

" lard 02 

The coefficients at starting were: 



100 



250 
.005 
.0085 



With sperm . . 
With lard. . . . 



.07 
.07 



500 
.004 
.0053 

.15 
.10 



750 
.0043 
.0066 

.185 
.12 



1000 
.009 
.0125 



The coefficient at a speed of 150 feet per minute decreases with increase 
of pressure until 500 lbs. per sq. in. is reached ; above this it increases. The 
coefficient at rest or at starting increases with the pressure throughout the 
ranee of the tests. 

Value of Anti-friction Metals. (Denton.)— The various white 
metals available for lining brasses do not afford coefficients of friction 
lower than can be obtained with bare brass, but they are less liable to 
"overheating," because of the superiority of such material over bronze in 
ability to permit of abrasion or crushing, without excessive increase of 
friction. 

Thurston (Friction and Lost Work) says that gun-bronze, Bnbbitt, and 
other soft white alloys have substantially the same friction ; in other words, 
the friction is determined by the nature of the unguent and not by that of 
the rubbing-surfaces, when the latter are in good order. The soft metals 
run at higher temperatures than the bronze. This, however, does not nec- 
essarilv indicate a serious defect, but simply deficient conductivity. The 
value of the white alloys for bearings lies mainly in their ready reduction 
to a smooth surface after any local or general injury by alteration of either 
surface or form. 



morin's laws of frictioh. 933 

Cast-iron for Bearings. (Joshua Rose.)— Cast iron appears to be an 
exct-piion to the general rule, that the harder the metal the greater the 
resistance to wear, because cast iron is softer in its texture and easier to 
cut with steel tools than steel or wrought iron, but in some situations it is 
far more durable than hardened steel; thus when surrounded by steam it 
will wear better than will any other metal. Thus, for instance, experience 
has demonstrated that piston-rings of cast iron will wear smoother, better, 
and equally as long as those of steel, and longer than those of either 
wrought iron or brass, whether the cylinder in which it works be composed 
of brass, steel, wrought iron, or cast iron; the latter being the more note- 
worthy, since two surfaces of the same metal do not, as a rule, wear or 
work well together. So also slide-valves of brass are not found to wear so 
long or so smoothly as those of cast iron, let the metal of which the seating 
is composed be whatever it may; while, on the other hand, a cast iron slide- 
valve will wear longer of itself and cause less wear to its seat, if the latter 
is of cast iron, than if of steel, wrought iron, or brass. 

Friction of Metals under Steam-pressure.— The friction of 
brass upon iron under steam-pressure is double that of iron upon iron. 
(G. H. Babcock, Trans. A. S. M. E., i. 151.) 

Morin's "Laws of Friction."— 1. The friction between two bodies 
is directly proportioned to the pressure; i.e., the coefficient is constant for 
all pressures. 

2. The coefficient and amount of friction, pressure being the same, is in- 
dependent of the areas in contact. 

3 The coefficient of friction is independent of velocity, although static 
friction (friction of rest) is greater than the friction of motion. 

Eng'g News, April 7, 1888, comments on these "laws" as follows : From 
1831 till about 1876 there was no attempt worth speaking of to enlarge our 
knowledge of the laws of friction, which during all that period was assumed 
to be complete, although it was really worse than nothing, since it was for 
the most part wholly false. In the year first mentioned Morin began a se- 
ries of experiments which extended over two or three years, and which 
resulted in the enunciation of these three " fundamental laws of friction," 
no one of which is even approximately true. 

For fifty years these laws were accepted as axiomatic, and were quoted as 
such without question in every scientific work published during that whole 
period. Now that they are so thoroughly discredited it has been attempted 
to explain away their defects on the ground that they cover only a very lim- 
ited range of pressures, areas, velocities, etc., and that Morin himself only 
announced them as true within the range of his conditions. It is now clearly 
established that there are no limits or conditions within which any one of 
them even approximates to exactitude, and that there are many conditions 
under which they lead to the wildest kind of error, while many of the con- 
stants were as inaccurate as the laws. For example, in Morin's "' Table of 
Coefficients of Moving Friction of Smooth Plane Surfaces, perfectly lubri- 
cated," which may be found in hundreds of text-books now in use. the coeffi- 
cient of wrought iron on brass is given as .075 to .103, which would make the 
rolling friction of railway trains 15 to 20 lbs. per ton instead of the 3 to 6 lbs. 
which it actually is. 

General Morin, in a letter to the Secretary of the Institution of Mechanical 
Engineers, dated March 15, 1879, writes as follows concerninghis experiments 
on friction made more than forty years before: " The results furnished by my 
experiments as to the relaiions between pressure, surface, and speed on the 
one hand, and sliding friction on the other, have always been regarded by 
myself, not as mathematical laws, but as close approximations to the truth, 
within the limits of the data of the experiment? themselves. The same holds, 
in my opinion, for many other laws of practical mechanics, such as those of 
rolling resistance, fluid resistance, etc." 

Prof. J. E. Denton (Stevens Indicator, July, 1890) says: It has been gen- 
erally assumed that friction between lubricated surfaces follows the simple 
law that the amount of the friction is some fixed fraction of the pressure be- 
tween the surfaces, such fraction being independent of the intensity of the 
pressure per square inch and the velocity of rubbing, between certain limits 
of practice, and that the fixed fraction referred to is represented by the co- 
efficients of friction given by the experiments of Morin or obtained from ex- 
perimental data which represent conditions of practical lubrication, such as 
those given in Webber's Manual of Power. 

By the experiments of Thurston, Woodbury, Tower, etc., however, it 
appears that the friction between lubricated metallic surfaces, such as ma- 



934 FRICTIOH AND LUBRICATION. 

chine bearings, is not directly proportional to the pressure, is not indepen- 
dent of the speed, and that the coefficients of Morin and Webber are about 
tenfold too great for modern journals. 

Prof. Denton offers an explanation of this apparent contradiction of au- 
thorities by showing, with laboratory testing machine data, that Moriu's 
laws hold for bearings lubricated by a restricted feed of lubricant, such as 
is afforded by tne oil-cups common to machinery; whereas the modern ex- 
periments have been made with a surplus feed or superabundance of lubri- 
cant, such as is provided only in railroad -car journals, and a few special 
cases of practice. 

That the low coefficients of friction obtained under the latter conditions 
are realized in the case of car journals, is proved by the fact that the tem- 
perature of car-boxes remains at 100° at high velocities; and experiment shows 
that this temperature is consistent only with a coefficient of friction of a 
fraction of one per cent. Deductions from experiments on train resistance 
also indicate the same low degree of friction. But these low co-efficients do 
not account for the internal friction of steam-engines as well as do the co- 
efficients of Morin and Webber. 

In American Machinist, Oct. 23, 1890, Prof. Denton says: Morin's measure- 
ment of friction of lubricated journals did not extend to light pressures. 
They apply only to the conditions of general shafting and engine work. 

He clearly understood that there was a frictional resistance, due solely to 
the viscosity of the oil, and that therefore, for very light pressures, the laws 
which he enunciated did not prevail. 

He applied his dynamometers to ordinary shaft-journals without special 
preparation of the rubbmg-surfaces, and without resorting to artificial 
methods of supplying the oil. 

Later experimenters have with few exceptions devoted themselves exclu- 
sively to the measurement of resistance practically due to viscosity alone. 
They have eliminated the resistance to which Morin confined his measure- 
ments, namely, the friction due to such contact of the rubbing-surfaces as 
prevail with a very thin film of lubricant between comparatively rough sur- 
faces. 

Prof. Denton also says (Trans. A. S. M. E., x. 518): " I do not believe there 
is a particle of proof in any investigation of friction ever made, that Morin's 
laws do not hold for ordinary practical oil-cups or restricted rates of feed." 

Laws of Friction of well-lubricated Journals.— John 
Goodman (Trans. Inst. C. E. 1886, Eiufg News, Apr. 7 and 14, 1888), review- 
ing the results obtained from the testing-machines of Thurston, Tower, and 
Stroudley, arrives at the following laws: 

Laws of Friction: Well- lubricated Surfaces. 
(Oil-bath.) 

1. The coefficient of friction with the surfaces efficiently lubricated is from 
1/6 to 1/10 that for dry or scantily lubricated surfaces. 

2. The coefficient of friction for moderate pressures and speeds varies ap- 
proximately inversely as the normal pressure; the frictional resistance va- 
ries as the area in contact, the normal pressure remaining constant. 

3. At very low journal speeds the coefficient of friction is abnormally 
high; but as the speed of sliding increases from about 10 to 100 ft. per min , 
the friction diminishes, and again rises when that speed is exceeded, varying 
approximately as the square root of the speed. 

4. The coefficient of friction varies approximately inversely as the temper- 
ature, within certain limits, namely, just before abrasion takes place. 

The evidence upon which these laws are based is taken from various mod- 
ern experiments. That relating to Law 1 is derived from the " First Report 
on Friction Experiments, 1 ' by Mr. Beauchamp Tower. , 



Method of Lubrication. 


Coefficient of 
Friction. 


Comparative 
Friction. 


Oil-bath 


.00139 
.0098 
.0090 


1 00 


Siphon lubricator 

Pad under journal 


7.06 
6.48 



With a load of 293 lbs. per sq. in. and a journal speed of 314 ft. per min. 
Mr. Tower found the coefficient of friction to be .0016 with an oil- bath, and 



LAWS OF FRICTION". 



935 



.0097, or six times as much, with a pad. The very low coefficients ob- 
tained by Mr. Tower will be accounted for by Law 2, as he found that the 
frictional resistance per square inch under varying loads is nearly constant, 
as below: 

Load in lbs. per sq. in 529 468 415 363 310 258 205 153 100 

Frictional resist, persq. in. .416 .514 .498 .472 .464 .438 .43 .458 .45 

The frictional resistance per square inch is the product of the coefficient 
of friction into the load per square inch on horizontal sections of the brass. 
Hence, if this product be a constant, the one factor must vary inversely as 
the other, or a high load will give a low coefficient, and vice versa. 

For ordinary lubrication, the coefficient is more constant under varying 
loads; the frictional resistance then varies directly as the load, as shown by 
Mr. Tower in Table VIII of his report (Proc. Inst. M. E. 1883). 

With respect to Law 3, A. M. Wellington (Trans. A. S. C. E. 1884), in ex- 
periments on journals revolving at very low velocities, found that the friction 
was then very great, and nearly constant under varying conditions of the 
lubrication, load, and temperature. But as the speed increased the friction 
fell slowly and regularly, and again returned to the original amount when 
the velocity was reduced to the same rate. This is shown in the following 
table: 
Speed, feet per minute: 

0+ 2.16 3.33 4.86 8.82 21.42 35.37 53.01 89.28 106.02 
Coefficient of friction : 

.118 .094 .070 .069 .055 .047 .040 .035 .030 .026 

It was also found by Prof. Kimball that when the journal velocity was in- 
creased from 6 to 110 ft. per minute, the friction was reduced 70%; in another 
case the friction was reduced 67% when the velocity was increased from 1 to 
100 ft. per minute; but after that point was reached the coefficient varied 
approximately with the square root of the velocity. 

The following results were obtained by Mr. Tower: 



Feet per minute. .. 


209 


262 


314 


366 


419 


471 


Nominal Load 
per sq. in. 


Coeff . of friction . . 


.0010 
.0013 
.0014 


.0012 
.0014 
.0015 


.0013 
.0015 
.0017 


.0014 
.0017 
.0019 


.0015 
.0018 
.0021 


.0017 
.002 
.0024 


520 lbs. 
468 " 

415 " 



The variation of friction with temperature is approximately in the inverse 
ratio, Law 4. Take, for example, Mr. Tower's results, at 262 ft. per minute: 



Temp. F. 


110° 


100° 


90° 


80° 


70° 


60° 


Observed 

Calculated 


.0044 
.00451 


.0051 
.00518 


.006 
.00608 


.0073 
.00733 


.0092 
.00964 


.0119 
.01252 



This law does not hold good for pad or siphon lubrication, as then the co- 
efficient of friction diminishes more rapidly for given increments of tem- 
perature, but on a gradually decreasing scale, until the normal temperature 
has been reached; this normal temperature increases directly as the load 
per sq in. This is shown in the following table taken from Mr. Stroudley's 
experiments with a pad of rape oil: 



Temp. F 


105° 


110° 


115° 


120° 


125° 


130° 


135° 1 140° 


145° 




.022 


.0180 


.0160 


.0140 
0020 


.0125 
.0015 


.0115 
.0010 


.0110 
.0005 


.0106 
.0004 


.0102 


Decrease of coeff. . 


.0040 


.0020 


.0002 



In the Galton-Westinghouse experiments it was found that with velocities 
below 100 ft. per min., and with low pressures, the frictional resistance 
varied directly as the normal pressure; but when a velocity of 100 ft. per 
min. was exceeded, the coefficient of friction greatly diminished; from the 
same experiments Prof. Kennedy found that the coefficient of friction for 
high pressures was sensibly less than for low. 

Allowable Pressures on Bearing-surfaces. (Proc. Inst. M. E., 
May, 1888.)— The Committee on Friction experimented with a steel ring of 



936 FRICTION" AND LUBRICATION. 

rectangular section, pressed between two cast-iron disks, the annular bear- 
ing-surfaces of which were covered with gun-metal, and were 12 in. inside 
diameter and 14 in. outside. The two disks were rotated together, and the 
steel ring was prevented from rotating by means of a lever, the holding 
force of which was measured. When oiled through grooves cut in each face 
of the ring and tested at from 50 to 130 revs, per min., it was found that a 
pressure of 75 lbs. per sq. in. of bearing-surface was as much as it would 
bear safely at the highest speed without seizing, although it carried 90 lbs. 
per sq. in. at the lowest speed. The coefficient of friction is also much 
higher than for a cylindrical bearing, and the friction follows the law of the 
friction of solids much more nearly than that of liquids. This is doubtless 
due to the much less perfect lubrication applicable to this form of bearing 
compared with a cylindrical one. The coefficient of friction appears to be 
about the same with the same load at all speeds, or, in other words, to be 
independent of the speed; but it seems to diminish somewhat as the load is 
increased, and may be stated approximately as 1/20 at 15 lbs. per sq. in., 
diminishing to 1/30 at 75 lbs. per sq. in. 

The high coefficients of friction are explained by the difficulty of lubricat- 
ing a collar-bearing. It is similar to the slide-block of an engine, which can 
carry only about one tenth the load per sq. in. that can be carried by the 
crank-pins. 

In experiments on cylindrical journals it has been shown that when a 
cylindrical journal was lubricated from the side on which the pressure bore, 
100 lbs. per sq. in. was the limit of pressure that it would carry; but when it 
came to be lubricated on the lower side and was allowed to drag the oil in 
with it, 600 lbs. per sq. in. was reached with impunity; and if the 600 lbs. per 
sq. in., which was reckoned upon the full diameter of the bearing, came to 
be reckoned on the sixth part of the circle that was taking the greater pro- 
portion of the load, it followed that the pressure upon that part of the circle 
amounted to about 1200 lbs. per sq. in. 

In connection with these experiments Mr. Wicksteed states that in drill- 
ing-machines the pressure on the collars is frequently as high as 336 lbs. per 
sq. in., but the speed of rubbing in this case is lower than it was in any of 
the experiments of the Research Committee. In machines working very 
slowly and intermittently, as in testing-machines, very much higher pres- 
sures are admissible. 

Mr. Adamson mentions the case of a heavy upright shaft carried upon a 
small footstep-bearing, where a weight of at least 20 tons w,as carried on a 
shaft of 5 in. diameter, or, say, 20 sq. in. area, giving a pressure of 1 ton per 
sq. in. The speed was 190 to 200 revs, per min. It was necessary to force the 
oil under the bearing by means of a pump. For heavy horizontal shafts, 
such as a fly-wheel shaft, carrying 100 tons on two journals, his practice for 
getting oil into the bearings was to flatten the journal along one side 
throughout its whole length to the extent of about an eighth of an inch in 
width for each inch in diameter up to 8 in. diameter; above that size rather 
less flat in proportion to the diameter. At first sight it appeared alarming 
to get a continuous flat place coming round in every revolution of a heavily 
loaded shaft; yet it earned the oil effectually into the bearing, which ran 
much better in consequence than a truly cylindrical journal without a flat 
side. 

In thrust-bearings on torpedo-boats Mr. Thornycroft allows a pressure of 
never more than 50 lbs. per sq. in. 

Prof. Thurston (Friction and Lost Work, p. 240) says 7000 to 9000 lbs. 
pressure per square inch is reached on the slow-working and rarely moved 
pivots of swing bridges. 

Mr. Tower says (Proc. Inst M. E., Jan. 1884): In eccentric-pins of punch- 
ing and shearing-machines very high pressures are sometimes used without 
seizing. In addition to the alternation in the direction, the pressure is ap- 

Elied for only a very short space of time in these machines, so that the oil 
as no time to be squeezed out. 

In the discussion on Mr. Tower's paper (Proc. Inst. M. E. 1885) it was 
stated that it is well known from practical experience that with a constant 
load on an ordinary journal it is difficult and almost impossible to have more 
than 200 ibs. per square inch, otherwise the bearing would get hot and the 
oil go out of it; but when the motion was reciprocating, so that the load was 
alternately relieved from the journal, as with crank-pins and similar jour- 
nals, much higher loads might be applied than even 700 or 800 lbs. per square 
inch. 



FRICTION OF CAR-JOURNAL BRASSES. 937 

Mr. Goodman (Proc. Inst. C. E. 1886) found that the total frictional re- 
sistance is materially reduced by diminishing the width of the brass. 

The lubrication is most efficient in reducing the friction when the brass 
subtends an angle of from 1J0° to 60°. The film is probably at its best be- 
tween the angles 80° and 110°. 

In the case of a brass of a railway axle-bearing where an oil-groove is cut 
along its crown and an oil-hole is drilled through the top of lhe brass into it, 
the wear is invariably on the off side, which is probably due to the oil escap- 
ing as soon as it reaches the crown of the brass, and so leaving the off side 
almost dry, where the wear consequently ensues. 

In railway axles the brass wears always on the forward side. The same ob- 
servation has been made in marine engine .journals, which always wear in 
exactly the reverse way to what they might be expected. Mr. Stroudley 
thinks this peculiarity is due to a film of lubricant being drawn in from the un- 
der side of the journal to the aft part of the brass, which effectually lubri- 
cates and prevents wear on that side; and that when the lubricant reaches 
the forward side of the brass it is so attenuated down to a wedge shape that 
there is insufficient lubrication, and greater wear consequently follows. 

Prof. J. E, Denton (Am. Alack., Oct. 30, 1890) says: Regarding the pres- 
sure to wnich oil is subjected in railroad car-service, it is probably more severe 
than in any other class of practice. Car brasses, when used bare, are so im- 
perfectly fitted to the journal, that during the early stages of their use the 
area of bearing may be but about one square inch. In this case the pressure 
per square inch is upwards of 6000 lbs. But at the slowest speeds of freight 
service the wear of a brass is so rapid that, within about thirty minutes the 
area is either increased to about three inches, and is thereby able to relieve 
the oil so that the latter can successfully prevent overheating of the journal, 
or else overheating takes place with any oil. and measures of relief must be 
taken which eliminate the question of differences of lubricating power 
among the different lubricants available. A brass which has been run about 
fifty miles under 5000 lbs. load may have extended the area of bearing-surface 
to about three square inches. The pressure is then about 1700 lbs. per square 
inch. It may be assumed that this is an average minimum area for car-ser- 
vice where no violent and unmanageable overheating has occurred during the 
use of a brass for a short time. This area will very slowly increase with any 
lubricant. 

C. J. Field (Poirer, Feb. 1893) says: One of the most vital points of an en- 
gine for electrical service is that of main bearings. They should have a sur- 
face velocity of not exceeding 350 feet per minute, with a mean bearing- 
f>ressure per square inch of projected area of journal of not more than 80 
bs. This is considerably within the safe limit of cool performance and easy 
operation. If the bearings are designed in this way, it would admit the use 
of grease on all the main wearing-surface, which in a large type of engines 
for this class of work we think advisable. 

Oil-pressure in a Bearing.— Mr. Beauchamp Tower (Proc. Inst. 
M. E , Jan. 1885) made experiments with a brass bearing 4 inches diameter 
by 6 inches Ions:, to determine the pressure of the oil between the brass and 
the journal. The bearing was half immersed in oil, and had a total load of 
8008 lbs. upon it. The journal rotated 150 revolutions per minute. The 
pressure of the oil was determined by drilling small holes in the bearing at 
different points and connecting them by tubes to a Bourdon gauge. It was 
found that the pressure varied from 310 to 625 lbs. per square inch, the great- 
est pressure being a little to the " off " side of the centre line of the top of 
the bearing, in the direction of motion of the journal. The sum of the up- 
ward force exerted by these pressures for the whole lubricated area was 
nearly equal to the total pressure on the bearing. The speed was reduced 
from 150 to 20 revolutions, but the oil-pressure remained the same, showing 
that the brass was as completely oil-borne at the lower speed as at the 
higher. The following was the observed friction at the lower speed: 

Nominal load, lbs. per square inch . . . 443 333 211 89 
Coefficient of friction 00132 .00168 .00247 .0044 

The nominal load per square inch is the total load divided by the product of 
the diameter and length of the journal. At the same low speed of 20 revo- 
lutions per minute it was increased to 676 lbs. per square inch without any 
signs of heating or seizing. 

Friction of Car-journal Brasses. (J. E. Denton, Trans. A. S. M. 
E , xii. 405.) — A new brass dressed with an emery-wheel, loaded with 5000 lbs., 
may have an actual bearing-surface on the journal, as shown by the polish 



938 FKICTIOH AND LUBttlCATiOtf. 

of a portion of the surface, of only 1 square inch. With this pressure of 5000 
lbs. per square inch, the coefficient of friction may be 6%, and the brass may 
be overheated, scarred and cut but, on the contrary, it may wear down evenly 
to a smooth bearing, giving a highly polished area of contact of 3 square 
inches, or more, inside of two hours of running, gradually decreasing the 
pressure per square inch of contact, and a coefficient of friction of less than 
0.5%. A reciprocating motion in the direction of the axis is of importance 
in reducing the friction. With such polished surfaces any oil will lubricate, 
and the coefficient of friction then depends on the viscosity of the oil. With 
a pressure of 1000 lbs per square inch, revolutions from 170 to 320 per minute, 
and temperatures of 75° to 113° F. with both sperm and parraffine oils, a co- 
efficient of as low as 0.11$ has been obtained, the oil being fed continuously 
by a pad. 

Experiments on Overheating of Bearings.— Hot Boxes. 
(Denton.)— Tests with car brasses loaded from 1100 to 4500 lbs. per square 
inch gave 7 cases of overheating out of 32 trials. The tests show how purely 
a matter of chance is the overheating, as a brass which ran hot at 5000 lbs. 
load on one day would run cool on a later date at the same or higher pres- 
sure. The explanation of this apparently arbitrary difference of behavior is 
that the accidental variations of the smoothness of the surfaces, almost in- 
finitesimal in their magnitude, cause variations of friction which are always 
tending to produce overheating, and it is solely a matter of chance when 
these tendencies preponderate over tbe lubricating influence of the oil. 
There is no appreciable advantage shown by sperm-oil, when there is no ten- 
dency to overheat— that is, paraffine can lubricate under the highest pres- 
sures which occur, as well as sperm, when the surfaces are within the condi- 
tions affording the minimum coefficients of friction. 

Sperm and other oils of high heat-resisting qualities, like vegetable oil and 
petroleum cylinder stocks, only differ from the more volatile lubricants, 
like paraffine, in their ability to reduce the chances of the continual acci- 
dental infinitesimal abrasion producing overheating. 

The effect of emery or other gritty substance in reducing overheating of a 
bearing is thus explained : 

The effect of the emery upon the surfaces of the bearings is to cover the 
latter with a series of parallel grooves, and apparently after such grooves 
are made the presence of the emery does not practically increase the friction 
over the amount of the latter when pure oil only is between the surfaces. 
Tbe infinite number of grooves constitute a very perfect means of insuring 
a uniform oil supply at every point of the bearings. As long as grooves in 
the journal match with those in the brasses the friction appears to amount 
to only about 10% to 15% of the pressure. But if a smooth journal is placed 
between a set of brasses which are grooved, and pressure be applied, the 
journal crushes the grooves and becomes brazed or coated with brass, and 
then the coefficient of friction becomes upward of 40$. If then emery is 
applied, the friction is made very much less by its presence, because the 
grooves are made to match each other, and a uniform oil supply prevails at 
every point of the bearings, whereas before the application or the emery 
many spots of the latter receive no oil between them. 

Moment of Friction and Work of Friction of Sliding- 
surfaces, etc. 

Moment of Fric- Energy lost by Friction 
tion, inch-lbs. in ft. -lbs. per miu. 

Flat surfaces fWS 

Shafts and journals YzfWd .2618/TTdn 

Flat pivots YsfWr A745fWrn 

Collar-bearing %fW *\ ~ *\ .1745/PTn *" 2 ,, ~ *'*' 

Conical pivot % fWr cosec a A745fWrn cosec a 

Conical 3ournal %fW r r sec a A745fWrn sec a 

Truncated-cone pivot HfW 3 ' 2 T r * . l?45/T T r2 T Tx 

v aj r 2 sin a r 2 sin a 

Hemispherical pivot fWr .2618/PFr 

Tractrix, or Schiele's " anti- 
friction " pivot fWr .2618/PTr. 



PIVOT-BEARINGS. 939 

In the above / — coefficient of friction; 

W = weight on journal or pivot in pounds; 
r = radius, d = diameter, in inches; 
S = space in feet through which sliding takes place; 
r 2 = outer radius, r 3 = inner radius; 
n = number of revolutions per minute; 

a = the half-angle of the cone, i.e., the angle of the slope 
with the axis. 

To obtain the horse-power, divide the quantities in the last column by 

fWdn 
33,090. Horse-power absorbed by friction of a shaft = o Kn - n - 

The formula for energy lost by shafts and journals is approximately true 
for loosely fitted bearings. Prof. Thurston shows that the correct formula 
varies according to the character of fit of the bearing; thus for loosely 
fitted journals, if U = the energy lost, 

2fTrr Tir . , , .2618/ Wdn . . „ 
U = — Wn inch-pounds = foot-lbs. 

Vi + P V\ +P 

For perfectly fitted journals U = 2MfnrWn inch-lbs. = . 3325/ Wdn, ft.-lbs. 

For a bearing in which the journal is so grasped as to give a uniform 
pressure throughout, U — fit^rWn inch-lbs. — AU2 fWdn, ft.-lbs. 

Resistance of railway trains and wagons due to friction of trains: 

f X 2240 
Pull on draw-bar = - — ^— pounds per gross ton, 

in which R is the ratio of the radius of the wheel to the radius of journal. 

A cylindrical journal, perfectly fitted into a bearing, and carrying a total 
'oad. distributes the pressure due to this load unequally on the bearing, the 
maximum pressure being at the extremity of the vertical radius, while at 
the extremities of the horizontal diameter the pressure is zero. At any 
point of the bearing-surface at the extremity of a radius which makes an 
angle 6 with the vertical radius the normal pressure is proportional to cos 6. 
If p — normal pressure on a unit of surface, to — total load on a unit of 
length of the journal, and r = radius of journal, 

. . K „ to COR 9 

w cos = l.olrp, p = < „ — . 
1 .oir 

PIVOT-BEARINGS. 

The Scliiele Curve.— W. H. Harrison, in a letter to the Am. Machin- 
ist. 1891, says the Schiele curve is not as good a form for a bearing as the 
segment of a sphere. He says: A mill-stone weighing a ton frequently 
bears its whole weight upon the flat end of a hard-steel pivot V/Q' diameter, 
or one square inch area of bearing; but to carry a weight of 3000 lbs. he 
advises an end bearing about 4 inches diameter, made in the form of a seg- 
ment of a sphere about ^ inch in height. The die or fixed bearing should 
be dished to fit the pivot. This form gives a chance for the bearing to 
adjust itself, which it does not have when made flat, or when made with the 
Schiele curve. If a side bearing is necessary it can be arranged farther up 
the shaf t. The pivot and die should be of steel, hardened; cross-gutters 
should be in the die to allow oil to flow, and a central oil-hole should be 
made in the shaft. 

The advantage claimed for the Schiele bearing is that the pressure is uni- 
formly distributed over its surface, and that it therefore wears uniformly. 
Wilfred Lewis (Am. Mach., April 19, 1894) says that its merits as a thrust- 
bearing have been vastly overestimated; that the term '•anti-friction'" 
applied to it is a misnomer, since its friction is greater-, than that of a flat 
step or collar of the same diameter. He advises that flat thrust-bearings 
should always be annular in form, having an inside diameter one half of 
the external diameter 

Friction of a Flat Pivot-bearing.— The Research Committee 
on Friction (Proc. Inst. M. E. 1891) experimented on a step-bearing, flat- 
ended, 3 in. diam., the oil being forced into the bearing through a hole in 
its centre and distributed through two radial grooves, insuring thorough 
lubrication, The step was of steel and the bearing of manganese-bronze. 



940 FMCTIOK AND LUBRICATION. 

At revolutions per min 50 128 194 290 353 

The coefficient of friction varied j .0181 .0053 .0051 .0044 .0053 

between I and .0221 .0113 .0102 .0178 .0167 

With a white-metal bearing at 128 revolutions the coefficient of friction 
was a little larger than with the manganese-bronze. At the higher speeds 
the coefficient of friction was less, owing to the more perfect lubrication, as 
shown by the more rapid circulation of the oil. At 128 revolutions the 
bronze bearing heated and seized on one occasion with a load of 260 pounds 
and on another occasion with 300 pounds per square inch. The white-metal 
bearing under similar conditions heated and seized with a load of 240 
pounds per square inch. The steel footstep on manganese-bronze was after- 
wards tried, lubricating with three and with four radial grooves; but the 
friction was from one and a half times to twice as great as with only the two 
grooves. (See also Allowable Pressures, page 936.) 

Mercury-hath. Pivot.— A nearly frictionless step-bearing may be 
obtained by floating the bearing with its superincumbent weight upon mer- 
cury. Such an apparatus is used in the lighthouses of La Heve, Havre. It 
is thus described in Eng'g, July 14, 1893, p. 41: 

The optical apparatus, weighing about 1 ton, rests on a circular cast-iron 
table, which is supported by a vertical shaft of wrought iron 2.36 in. 
diameter. 

This is kept in position at the top by a bronze ring and outer iron support, 
and at the bottom in the same way, while it rotates on a removable steel 
pivot resting in a steel socket, which is fitted to the base of the support. To 
the vertical shaft there is rigidly fixed a floating cast-iron ring 17.1 in. diam- 
eter and 11.8 in. in depth, which is plunged into and rotates in a mercury 
bath contained in a fixed outer drum or tank, the clearance between the 
vertical surfaces of the drum and ring being only 0.2 in., so as to reduce as 
much as possible the volume of mercury (about 220 lbs.), while the horizon- 
tal clearance at the bottom is 0.4 in. 

BALL-BEARINGS, FRICTION ROLLERS, ETC. 

A. H. Tyler (Encfg, Oct. 20, 1893, p. 483), after experiments and com- 
parison with experiments of others arrives at the following conclusions: 

That each ball must have two points of contact only. 

The balls and race must be of glass hardness, and of absolute truth. 

The balls should be of the largest possible diameter which the space at 
disposal will admit of. 

Any one ball should be capable of carrying the total load upon the bearing. 

Two rows of balls are always sufficient. 

A ball-bearing requires no oil, and has no tendency to heat unless over- 
loaded. 

Until the crushing strength of the balls is being neared, the frictional re- 
sistance is proportional to the load. 

The frictional resistance is inversely proportional to the diameter of the 
balls, but in what exact proportion Mr. Tyler is unable to say. Probably it 
varies with the square. 

The resistance is independent of the number of balls and of the speed. 

No rubbing action will take place between the balls, and devices to guard 
against it are unnecessary, and usually injurious. 

The above will show that the ball-bearing is most suitable for high speeds 
and light loads. On the spindles of wood-carving machines some make as 
much as 30.000 revolutions per minute. They run perfectly cool, and never 
have any oil upon them. For heavy loads the balls should not be less than 
two thirds the diameter of the shaft, and are better if made equal to it. 

Ball-hearings have not been found satisfactory for thrust-blocks, for 
the reason apparently that the tables crowd together. Better results have 
been obtained from coned rollers. A combined system of rollers and balls 
is described in Encfo. Oct. 6, 1893, p. 429. 

Friction-rollers. —If a journal instead of revolving on ordinary 
bearings be supported on friction-rollers the force required to make the jour- 
nal revolve will be reduced in nearly the same proportion that the diameter 
of the axles of the rollers is less than the diameter of the rollers themselves. 
In experiments by A. M. Wellington with a journal 3Vs in. diam. supported 
on rollers 8 in. diam., whose axles were Yji in. diam., the friction in starting 
from rest was J4 the friction of an ordinary 3J^-in. bearing, but at a car 
speed of 10 miles per hour it was % that of the ordinary bearing. The ratio 
of the diam. of the axle to diam. of roller was 1%: 8, or as 1 to 4.6, 



FRICTION OF STEAM-ENGINES. 941 

Bearings for Very High Rotative Speeds. (Proc. Inst. M. E., 
Oec. 1888, p. 48-'.) — Iu the Parsons steam-turbine, which has a speed of as 
high as 18,000 i ev. per min., as it is impossible to secure absolute accuracy 
of balance, the bearings are of special construction so as to allow of a certain 
very small amount of lateral freedom. For this purpose the bearing is sur- 
rouudel by two sets of steel washers 1/16 inch thick and of different diam- 
eters, the larger fitting close in the casing and about 1/32 inch clear of the 
bearing, and the smaller fitting close on the bearing and about 1/32 inch 
clear of the casing. These are arranged alternately, and are pressed 
together by a spiral spring. Consequently any lateral movement of the 
bearing causes them to slide mutually against one another, and by their 
friction to check or damp any vibrations that may be set up in the spindle. 
The tendency of the spindle is then to rotate about its axis of mass, or prin- 
cipal axis as it is called; and the bearings are thereby relieved from exces- 
sive pressure, and the machine from undue vibration. The finding of the 
centre of gyration, or rather allowing the turbine itself to find its own 
centre of gyration, is a well-known device in other branches of mechanics: 
as in the instance of the centrifugal hydro-extractor, where a mass very 
much out of balance is allowed to find its own centre of gyration ; the faster 
it ran the more steadily did it revolve and the less was the vibration. An- 
other illustration is to be found in the spindles of spinning machinery, 
which run at about 10,000 or 11.000 revolutions per minute: they are made 
of hardened and tempered steel, and although of very small dimensions, the 
outside diameter of the largest portion or driving whorl being perhaps not 
more than 1J4 in., it is found impracticable to run them at that speed in 
what might be called a hard-and-fast bearing. They are therefore run with 
some elastic substance surrounding the bearing, such as steel springs, hemp, 
or cork. Any elastic substance is sufficient to absorb the vibration, and 
permit of absolutely steady running. 

FRICTION OF STEAM-ENGINES. 
Distribution of the Friction of Engines.— Prof . Thurston in 
Irs " Friction and Lost Work," gives the following: 

1. 

Main bearings. , 47.0 

Piston and rod 32.9 

Crank-pin 6.8 

Cross-head and wrist-pin 5.4 

Valve and rod 2.5 

Eccentric strap. - 5.3 

Link and eccentric 

Total 

100.0 100.0 100.0 

No. 1, Straight-line, 6" X 12", balanced valve; No. 2, Straight-line, 6" X 12", 

unbalanced valve; No. 3, 7" X 10", Lansing traction locomotive valve-gear. 

Prof. Thurston's tests on a number of different styles of engines indicate 

that the friction of any engine is practically constant under all loads. 

(Trans. A. S. M. E., viii. 86; ix. 74.) 

In a Straight-line engine, 8" X 14", I.H.P. from 7.41 to 57.54, the friction H. 
P. varied irregularly between 1.97 and 4.02, the variation being independent 
of the load. With 50 H.P. on the brake the I.H.P. was only 52.6, the friction 
being only 2.6 H.P., or about 5%. 

In a compound condensing-engine, tested from to 102.6 brake H.P., gave 
I.H.P. from 14.92 to 117.8 H.P., the friction H.P. varying only from 14.92 to 
17.42. At the maximum load the friction was 15.2 H.P., or 12.9^. 

The friction increases with increase of the boiler-pressure from 30 to 70 
lbs., and then becomes constant. The friction generally increases with in- 
crease of speed, but there are exceptions to this rule. 

Prof. Denton (Stevens Indicator, July, 1890), comparing the calculated 
friction of a number of engines with the friction as determined by measure- 
ment, finds that iu one case, a 75-ton ammonia ice-machine, the friction of 
the compressor, 17J^ H.P., is accounted for by a coefficient of friction of 7}?&% 
on all the external bearings, allowing 6$ of the entire friction of the machine 
for the friction of pistons, stuffing-boxes, and valves. In the case of the 
Pawtucket pumping-engine, estimating the friction of the external bearings 
with a coefficient of friction of 6% and that of the pistons, valves, and stuff- 
ing-boxes as in the case of the ice-machine, we have the total friction 
disiributed as follows : 



2. 


3. 


35.4 


35.0 


25.0 


21.0 


5.1| 
4.1 f 


13.0 


26.4/ 
4.0) 


22.0 




9.01 



942 FRICTION AND LUBRICATION. 

Horse- Per cent 

power, of Whole. 

Crank-pins and effect of piston-thrust on main shaft.. 0.71 11.4 

Weight of fly-wheel and main shaft 1.95 32.4 

Steam-valves 0.23 3.7 

Eccentric 0.07 1.2 

Pistons 0.43 7.2 

Stuffing-boxes, six altogether 0.72 11.3 

Air-pump 2.10 32 . 8 

Total friction of engine with load 6.21 100.0 

Total friction per cent of indicated power ... 4.27 

The friction of this engine, though very low in proportion to the indicated 
power, is satisfactorily accounted for by Morin's law used with a coefficient 
of friction of 5%. In both cases the main items of friction are those due to i 
the weight of the fly-wheel and main shaft and to the piston-thrust on 
crank-pins and main-shaft bearings. In the ice-machine the latter items 
are the larger owing to the extra crank pin to work the pumps, while I 
in the Pawtucket engine the former preponderates, as the crank-thrusts are 
partly absorbed by the pump-pistons, and only the surplus effect acts on 
the crank -shaft. 

Prof. Denton describes in Trans. A. S. M. E., x. 392, an apparatus by 
which he measured the friction of a piston packing-ring. When the parts 
of the piston were thoroughly devoid of lubricant, the coefficient of friction 
was found to be about 7*4%; with an oil-feed of one drop in two minutes the 
coefficient was about 5$; with one drop per minute it was about 3%. These 
rates of feed gave unsatisfactory lubrication, the piston groaning at the 
ends of the stroke when run slowly, and the flow of oil left upon the surfaces 
was found by analysis to contain about 50$ of iron. A feed of two drops per 
minute reduced the coefficient of friction to about 1%, and gave practically 
perfect lubrication, the oil retaining its natural color and purity. 

LUBRICATION. 

Measurement of the Durability of Lubricants. (J. E. Den- 
ton, Trans. A. S. M. E., xi. 1013.)— Practical differences of durability of lubri- 
cants depend not on any differences of inherent ability to resist being "worn 
out" by rubbing, but upon the rate at which they flow through and away 
from the bearing-surfaces. The conditions which control this flow are so 
delicate in their influence that aU attempts thus far made to measure dura- 
bility of lubricants may be said to have failed to make distinctions of lubri- 
cating value having any practical significance. In some kinds of service the 
limit to the consumption of oil depends upon the extent to which dust or other 
refuse becomes mixed with ir, as in railroad-car lubrication and in the case 
of agricultural machinery. The economy of one oil over another, so far as 
the quality used is concerned— that is, so far as durability is concerned— is 
simply proportional to the rate at which it can insinuate itself into and flow 
out of minute orifices or cracks. Oils will differ in their ability to do this, 
first, in proportion to their viscosity, and, second, in proportion to the ca- 
pillary properties which they may possess by virtue of the particular ingre- 
dients' used in their composition. Where the thickness of film between rub- 
bing-surfaces must be so great that large amounts of oil pass through 
beaiings in a given time, and the surroundings are such as to permit oil to 
be fed at high temperatures or applied by a method not requiring a perfect 
fluidity, it is probable that the least amount of oil will be used when the vis- 
cosity is as great as in the petroleum cylinder stocks. When, however, the 
oil must flow freely at ordinary temperatures and the feed of oil is 
restricted, as in the case of crank-pin bearings, it is not practicable to feed 
such heavy oils in a satisfactory manner. Oils of less viscosity or of a 
fluidity approximating to lard-oil must then be used. 

Relative Value of Lubricants. (J. E.Denton, Am. Much., Oct. 30, 
1890.)— The three elements which determine the value of a lubricant are the 
cost due to consumption of lubricants, the cost spent for coal to overcome 
the fiictional resistance caused by use of the lubricant, and the cost due to 
the metallic wear on the journal and the brasses. In cotton-mills the cost 
of the power is alone to be considered; in rolling-mills and marine engines 
the cost of the quantity of lubricant used is the only important factor; but 
in railroads not only do both these elements enter ihe problem as tangible 



LUBRICATION. 943 

factors, but the cost of the wearing away of the metallic parts enters in ad- 
dition, and furthermore, the latter is the'greatest element of cost in the case. 
The Qualifications of a Good. Lubricant, as laid down by 
W. H. Bailey, in Proc. Inst. C. E., vol. xlv., p. '672, are: 1. Sufficient body 
to keep the surfaces free from contact under maximum pressure, 2. The 
greatest possible fluidity consistent with the foregoing condition. 3. The 
lowest possible coefficient of friction, which in bath lubrication would be for 
fluid friction approximately. 4. The greatest capacity for storing and 
carrying away heat. 5. A high temperature of decomposition. 6. Power 
to resist oxidation or the action of the atmosphere. 7. Freedom from cor- 
rosive action on the metals upon which used. 

Best Lubricants for Different Purposes. (Thurston.) 

Low temperatures, as in rock-drills j T .,. ^ ; „ /i ,.„ 1 i„u„i„„ f; „„ ^;i„ 
driven by compressed air: } Ll S ht mineral lubncating-oils. 

Very great pressures, slow speed... -j G Stt' nt s s oapstone ' and other Solid 

Heavy pressures, with slow speed. . . -j T * e re £ s ™' a ' nd lard ' tallow ' and other 

Heavy pressures and high speed. . . . { Sp £Xus'. ca8tor_oi1 ' and heavy min " 

Light pressures and high speed -j ^X^-seed* petroleum ' ° live ' rape ' 

Ordinarv machinerv i Lard-oil, tallow-oil, heavy mineral oils, 

Ordinary machineiy -, ftnd the heavier vegetable oils. 

Steam -cylinders Heavy mineral oils, lard, tallow. 

Watches and other delicate media ( Clarified sperm, neat's-foot, porpoise, 
w n a S s ana otner aellcate mecna-; oliv6) and ljght mineral lubricating 
nifem - ( oils. 

For mixture with mineral oils, sperm is best: lard is much used; olive and 
cotton-seed are good. 

Amount of Oil needed to Run an Engine.— The Vacuum Oil 
Co. in 1892, in response to an inquiry as to cost of oil to run a 1000-H.P. 
Corliss engine, wrote: The cost of running two engines of equal size of the 
same make is not always the same. Therefore while we could furnish 
figures showing what it is costing some of our customers having Corliss 
engines of 1000 H.P., we could only give a general idea, which in itself 
might be considerably out of the way as to the probable cost of cylinder- 
and engine-oils per year for a particular engine. Such an engine ought to 
run readily on less than 8 drops of 600 W oil per minute. If 3000 drops are 
figured to the quart, and 8 drops used per minute, it would take about 
two and one half barrels (52.5 gallons) of 600 W cylinder-oil, at 65 cents per 
gallon, or about $85 for cylinder-oil per year, running 6 days a week and 10 
hours a day. Engine-oil would be even more difficult to guess at what the 
cost would be, because it would depend upon the number of cups required 
on the engine, which varies somewhat according to the style of the engine. 
It would doubtless be safe, however, to calculate at the outside that not 
more than twice as much engine-oil would be required as of cylinder-oil. 

The Vacuum Oil Co. in 1892 published the following results of practice 
with " 600 W " cylinder-oil: 

Corliss compound engine J 20 and 33 X 48; 83 revs - P er min - ; 1 dro P of oil 
oornss compound engine, ( per mjn tQ 1 drop in twQ minutes 

" triple exp. " 20, 33, and 46 X 48; 1 drop every 2 minutes. 

Porter Allen " -I 20 and 36 X 36; 143 revs - V ev min -' 2 drops of oil 

( per min., reduced afterwards to 1 dropper min. 
tj-m u J 15 X 25 X 16; 240 revs, per min.; 1 drop every 4 

cau 1 minutes. 

Results of tests on ocean-steamers communicated to the author by Prof. 
Denton in 1892 gave: for 1200-H.P. marine engine, 5 to 6 English gallons (6 to 
7.2 U. S. gals.) of engine-oil per 24 hours for external lubrication; and for a 
1500-H.P. marine engine, triple expansion, running 75 revs, per min., 6 to 7 
English gals, per 24 hours. The cylinder-oil consumption is exceedingly 
variable, — from 1 to 4 gals, per day on different engines, including cylinder- 
oil used to swab the piston-rods. 

Quantity of Oil used on a Locomotive Crank-pin.— Prof. 
Denton. Trans. A. S. M. E., xi. 1020, says: A very economical case of practical 
oi'-consiunption is when a locomotive main crank-pin consumes about six 



944 FRICTION AND LUBRICATIOH. 

cubic inches of oil in a thousand miles of service. This is equivalent to a 
consumption of one milligram to seventy square inches of surface rubbed 
over. 

The Examination of Lubricating-oils. (Prof. Thos. B. Still- 
man, Stevens Indicator, July, 1890.) — The generally accepted conditions of 
a good lubricant are as follows: 

1. " Body " enough to prevent the surfaces, to which it is applied, from 
coming in contact with each other. (Viscosity.) 

2. Freedom from corrosive acid, either of mineral or animal origin. 

3. As fluid as possible consistent with " body." 

4. A minimum coefficient of friction. 

5. High "flash 1 ' and burning points. 

6. Freedom from all materials liable to produce oxidation or " gumming." 
The examinations to be made to verify the above are both chemical and 

mechanical, and are usually arranged in the following order : 

1. Identification of the oil, whether a simple mineral oil, or animal oil, or 
a mixture. 2. Density. 3. Viscosity. 4. Flash-point. 5. Burning -point. 
6. Acidity. 7. Coefficient of friction. 8. Cold test. 

Detailed directions for making all of the above tests are given in Prof. 
St ill man's article. 

Weights of Oil per Gallon.— The following are approximately the 
weights per gallon of different kinds of oil (Penn. R. R. Specifications): 

Lard-oil, tallow -oil, neat's-foot oil, bone-oil, colza-oil, mustard-seed oil, 
rape-seed oil, paraffine-oil, 500° fire-test oil, engine-oil, and cylinder lubricant, 
7$4 pounds per gallon. 

Well-oil and passenger-car oil, 7.4 pounds per gallon; navy sperm-oil, 7.2 
pounds per gallon; signal -oil, 7.1 pounds per gallon; 300° burning-oil, 6.9 
pounds per gallon; and 150° burning-oil, 6.6 pounds per gallon. 

Penna. R. R. Specifications for Petroleum Products. 
1889. — Five different grades of petroleum products will be used. 

The materials desired under this specification are the products of the dis- 
tillation and refining of petroleum unmixed with any other substances. 

150° Fire-test Oil.— This grade of oil will not be accepted if sample (1) is 
not "water-white" in color; (2) flashes below 130° Fahrenheit; (3) burns 
below 151° Fahrenheit; (4) is cloudy or shipment has cloudy barrels when 
received, from the presence of glue or suspended matter; (5) becomes 
opaque or shows cloud when the sample has been 10 minutes at a temper- 
ature of 0° Fahrenheit. 

The flashing and burning points are determined by heating the oil in an 
open vessel, not less than 12° per minute, and applying the test flame every 
7°, beginning at 123° Fahrenheit. The cold test may be conveniently made 
by having an ounce of the oil, in a four-ounce sample bottle, with a ther- 
mometer suspended in the oil, and exposing this to a freezing mixture of 
ice and salt. It is advisable to stir with the thermometer while the oil is 
cooling. The oil must remain transparent in the freezing mixture ten 
minutes after it has cooled to zero. 

300° Fire-test Oil.— This grade of oil will not be accepted if sample (1) is 
not "water white " in color; (2) flashes below 249° Fahrenheit; (3) burns 
below 298° Fahrenheit; (4) is cloudy or shipment has cloudy barrels when 
received, from the presence of glue or suspended matter; (5) becomes 
opaque or shows cloud when the sample has been 10 minutes at a temper- 
ature of 32° Fahrenheit. 

The flashing and burning points are determined the same as for 150° fire- 
test oil. except that the oil is heated 15° per minute, test-flame being applied 
first at 212° Fahrenheit. The cold test is made the same as above, except 
that ice and water are used. 

Paraffine-oil. — This grade of oil will not be accepted if the sample (1) is 
other than pale-lemon color; (2) flashes below 249° Fahrenheit; (3) shows 
viscosity less than 40 seconds or more than 65 seconds when tested as 
described under " Well Oil " at 100° Fahrenheit throughout the year; (4) has 
gravity at 60° Fahrenheit, below 24° Baume, or above 29° Baume; (5) from 
October 1st to May 1st has a cold test above 10° Fahrenheit. 

The flashing-point is determined same as for 300° fire-test oil. The cold 
test is determined as follows: A couple of ounces of oil is put in a four-ounce 
sample bottle, and a thermometer placed in it. The oil is then frozen, a 
freezing mixture of ice and salt being used if necessary. When the oil has 
become hard, the bottle is removed from the freezing mixture and the 
frozen oil allowed to soften, being stirred and thoroughly mixed at the same 
time by means of the thermometer, until the mass will run from one end of 



SOLID LUBRICANTS. 945 

the bottle to the other. The reading of the thermometer when this is the 
case is regarded as the cold test of the oil. 

Well Oil.— This grade of oil will not be accepted if the sample (1) flashes, 
from May 1st to October 1st, below 249° Fahrenheit, or from October 1st to 
May 1st below 200° Fahrenheit; (2) has a gravity, at 60° Fahrenheit, below 
28° Bauine. or above 30°; (3) from October 1st to May 1st has a cold test 
above 10° Fahrenheit; (4) shows any precipitation in 10 minutes when 5 
cubic centimetres are mixed with 95 cubic centimetres of 88° gasoline; (5) 
shows a viscosity less than 55 seconds, or more than 100 seconds, when tested 
as described below. From October 1st to May 1st the test must be made 
at 100° Fahrenheit, and from May 1st to October 1st at 110° Fahrenheit. 

For summer oil the flashing-point is determined the same as for paraffine- 
oil; and for winter oil the same, except that the test-flame is applied first 
at 193° Fahrenheit. The cold test is made the same as for parafflne-oil. 

The precipitation test is to exclude tarry and suspended matter. It is 
easiest made by putting 5 cubic centimetres of the oil in a 100-cubic -cen- 
timetre graduate, then filling to the mark with gasoline, and thoroughly 
shaking. 

The viscosity test is made as follows: A 100 cubic -centimetre pipette of the 
long bulb form is regraduated to hold just 100 cubic centimetres to the bottom 
of the bulb. The size of the aperture at the bottom is then made such that 
100 cubic centimetres of water at 100° Fahrenheit will run out the pipette 
down to the bottom of the bulb in 34 seconds. Pipettes with bulbs varying 
from \% inches to \]4, inches in diameter outside, and about 4]4 inches long 
give almost exactly the same results, provided the aperture at the bottom 
is the proper size. The pipette being obtained, the oil sample is heated to 
the required temperature, care being taken to have it uniformly heated, and 
then is drawn up into tbe pipette to the proper marK. The time occupied 
by the oil in running out, down to the bottom of the bulb, gives the test 
figures. 

500° Fire-test Oil— This grade of oil will not be accepted if sample (1) 
flashes below 415° Fahrenheit; (2) shows precipitation with gasoline when 
tested as described for well -oil. 

The flashing-point is determined the same as for well-oil, except that the 
test flame is applied first at 438° Fahrenheit. 

SOLID LUBRICANTS. 

Graphite in a condition of powder and used as a solid lubricant, so 
called, to distinguish it from a liquid lubricant, has been found to do well 
where the latter has failed. 

Rennie, in 1829, says : " Graphite lessened friction in all cases where it 
was used." General Morin, at a later date, concluded from experiments 
that it could be used with advantage under heavy pressures; and Prof. 
Thurston found it well adapted for use under both light and heavy pressures 
when mixed with certain oils. It is especially valuable to prevent abrasion 
and cutting under heavy loads and at low velocities. 

Soapstone, also called talc and steatite, in the form of powder and 
mixed with oil or fat, is sometimes used as a lubricant. Graphite or soap- 
stone, mixed with soap, is used on surfaces of wood working against either 
iron or wood. 

Fifore-grapliite.— A new self-lubricating bearing known as fibre- 
graphite is described by John H. Cooper in Trans. A. S. M. E., xiii. 374, as 
the invention of P. H. Holmes, of Gardiner, Me. This bearing material is 
composed of selected natural graphite, which has been finely divided and 
freed from foreign and gritty matter, to which is added wood-fibre or other 
growth mixed in water in various proportions, according to the purpose to 
be served, and then solidified by pressure in specially prepared moulds ; 
after removal from which the bearings are first thoroughly dried, then satu- 
rated with a drying oil. and finally subjected to a current of hot, dry air for 
the purpose of oxidizing the oil, and hardening the mass. When finished, 
they may be " machined " to size or shape with the same facility and means 
employed on metals. 

Metaline is a solid compound, usually containing graphite, made in the 
form of small cylinders which are fitted permanently into holes drilled in 
the surface of the bearing. The bearing thus fitted runs without any other 
lubrication. 



946 THE FOUHDRY. 



THE FOUNDRY. 

CXJPOL.A PRACTICE. 

The following notes, with the accompanying table, are taken from an 
article by Simpson Bolland in American Machinist, June 30, 1892. The table 
shows heights, depth of bottom, quantity of fuel on bed, proportion of fuel 
and iron in charges, diameter of main blast-pipes, number of tuyeres, blast - 
pressure, sizes of blowers and power of engines, and melting capacity per 
hour, of cupolas from 24 inches to 84 inches in diameter. 

Capacity of Cupola.— The accompanying table will be of service in deter- 
mining the capacity of cupola needed for the production of a given quantity 
of iron in a specified time. 

First, ascertain the amount of iron which is likely to be needed at each 
cast, and the length of time which can be devoted profitably to its disposal; 
and supposing that two hours is all that can be spared for that purpose, and 
that ten tons is the amount which must be melted, find in the column, Melt- 
ing Capacity per hour in Pounds, the nearest figure to five tons per hour, 
which is found to be 10,760 pounds per hour, opposite to which in the column 
Diameter of Cupolas, Inside Lining, will be found 48 inches ; this will be the 
size of cupola required to furnish ten tons of molten iron in two hours. 

Or suppose that the heats were likely to average 6 tons, with an occasional 
increase up to ten, then it might not be thought wise to incur the extra ex- 
pense consequent on working a 48-inch cupola, in which case, by following 
the directions given, it will be found that a 40-inch cupola would answer the 
purpose for 6 tons, but would require an additional hour's time for melting 
whenever the 10-ton heat came along. 

; The quotations in the table are not supposed to be all that can be melted 
in the hour by some of the very best cupolas, but are simply the amounts 
which a common cupola under ordinary circumstances may be expected to 
melt in the time specified. 

Height of Cupola.— By height of cupola is meant the distance from the 
base to the bottom side of the charging hole. 

Depth of Bottom of Cupola.— Depth of bottom is the distance from the 
sand-bed, after it has been formed at the bottom of the cupola, up to the 
under side of the tuyeres. 

All the amounts for fuel are based upon a bottom of 10 inches deep, and 
any departure from this depth must be met by a corresponding change in 
the quantity of fuel used on the bed ; more in proportion as the depth is 
increased, and less when it is made shallower. 

Amount of Fuel Required on the Bed. — The column " Amount of Fuel re- 
quired on Bed. in Pounds" is based on the supposition that the cupola is a 
straight one all through, and that the bottom is 10 inches deep. If the bot- 
tom be more, as in those of the Colliau type, then additional fuel will be 
needed. 

The amounts being given in pounds, answer for both coal and coke, for, 
should coal be used, it would reach about 15 inches above the tuyeres ; the 
same weight of coke would bring it up to about 22 inches above the tuyeres, 
which is a reliable amount to stock with. 

First Charge of Iron. — The amounts given in this column of the table are 
safe figures to work upon in every instance, yet it will always be in order, 
after proving the ability of the bed to carry the load quoted, to make a slow 
and gradual increase of the load until it is fully demonstrated just how much 
burden the bed will carry. 

Succeeding Charges of Fuel and Iron. — In the columns relating to succeed- 
ing charges of fuel and iron, it will be seen that the highest proportions are 
not favored, for the simple reason that successful melting with any greater 
proportion of iron to fuel is not the rule, but, rather, the exception. When- 
ever we see that iron has been melted in prime condition in the proportion 
of 12 pounds of iron to one of fuel, we may reasonably expect that the talent, 
material, and cupola have all been up to the highest degree of excellence. 

Diameter of Main Blast -pipe. —The table gives the diameters of main 
blast-pipes for all cupolas from 24 to 84 inches diameter. The sizes given 
opposite each cupola are of sufficient area for all lengths up to 100 feet. 



CUPOLA PRACTICE. 



947 





43d Xipud 


_ 


1,500 

2,000 
2,500 
3,000 
3,500 
4,000 
4,820 
5,640 
6,460 
7,550 
8,640 
9,730 
10,760 
11,790 
12.820 
13.850 
14,880 
15,910 
16,940 
18,340 
19,770 
21,200 
22,630 
24,060 
26,070 
27,980 
29,890 
31.800 
33,710 
35,620 
37.530 




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241 

282 
325 
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150 
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175 
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2!6 
2.6 
3.3 
3.3 
3.7 
5. 
5. 
5.8 
5.8 
6.8 
6.8 
8. 

10.7 
10.7 
12.2 
12.2 
13.7 
13.7 
15.4 
15.4 
17.1 
19. 
19. 
23.9 
23.9 
26. 
26. 
28. 
31. 


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1,206 
1,584 
1,962 
2,340 
2,718 
3,096 
3,474 
3,852 
4,230 
4,608 
4,986 
5,364 
5,742 
6,120 
6,498 
6,876 
7,254 
7,632 
8.110 
8,388 
8,766 
9,144 
9,522 
9,900 
10,278 
10,656 
11,134 
11,412 
11,790 




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92 

134 

176 

218 

260, 

302 

344 

386 

428 

470 

512 

554 

596 

638 

680 

722 

764 

806 

848 

890 

932 

974 

1,016 

1,058 

1,100 

1,142 

1,184 

1,226 

1,268 

1,310 




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1,170 
1,440 
1,710 
1,980 
2,250 
2.520 
2,790 
3,060 
3,330 
3,600 
3,870 
4,140 
4,410 
4,680 
4,950 
5,220 
5,490 
5,760 
6.030 
6,300 
6,570 
6,840 
7,110 
7,380 
7,650 
7,920 
8,190 
8,460 
8,730 
9,000 




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948 THE FOUNDRY. 

Tuyeres for Cupola. — Two columns are devoted to the number and sizes of 
tuyeres requisite for the successful working of each cupola ; one gives the 
number of pipes 6 inches diameter, and the other gives the number and 
dimensions of rectangular tuyeres which are their equivalent in area. 

From these two columns any other arrangement or disposition of tuyeres 
may be made, which shall answer in their totality to the areas given in the 
table. 

When cupolas exceed 60 inches in diameter, the increase in diameter 
should begin somewhere above the tuyeres. This method is necessary in all 
common cupolas above 60 inches, because it is not possible to force the blast 
to the middle of the stock, effectively, at any greater diameter. 

On no consideration must the tuyere area be reduced; thus, an 84-inch 
cupola must have tuyere area equal to 31 pipes 6 inches diameter, or 16 flat 
tuyeres 16 inches by 13J^ inches. 

if it is found that the given number of flat tuyeres exceed in circumference 
that of the diminished part of the cupola, they can be shortened, allowing 
the decreased length to be added to the depth, or they may be built in on 
end; by so doing, we arrive at a modified form of the Blakeney cupola. 

Another important point in this connection is to arrange the tuyeres in 
such a manner as will concentrate the fire at the melting-point into the 
smallest possible compass, so that the metal in fusion will have less space 
to traverse while exposed to the oxidizing influence of the blast. 

To accomplish this, recourse has been had to the placing of additional 
rows of tuyeres in some instances— the "Stewart rapid cupola' 1 having 
three rows, and the "Colliau cupola furnace" having two rows, of tuyeres. 

Blast -pressure. — Experiments show that about 30,000 cubic feet of air are 
consumed in melting a ton of iron, which would weigh about 2400 pounds, 
or more than both iron and fuel. When the proper quantity of air is sup- 
plied, the combustion of the fuel is perfect, and carbonic-acid gas is the 
result. When the supply of air is insufficient, the combustion is imperfect, 
and carbonic-oxide gas is the result. The amount of heat evolved in these 
two cases is as 15 to 4^ showing a loss of over two thirds of the heat by im- 
perfect combustion. 

It is not always true that we obtain the most rapid melting when we are 
forcing into the cupola the largest quantity of air. Some time is required 
to elevate the temperature of the air supplied to the point that it will enter 
into combustion. If more air than this is supplied, it rapidly absorbs heat, 
reduces the temperature, and retards combustion, and the fire in the cupola 
may be extinguished with too much blast. 

Slag in Cupolas.— A certain amount of slag is necessary to protect the 
molten iron which has fallen to the bottom from the action'of the blast ; if 
it was not there, the iron would suffer from decarbonization. 

When slag from any cause forms in too great abundance, it should be led 
away by inserting a hole a little below the tuyeres, through which it will 
find its way as the iron rises in the bottom. 

In the event of clean iron and fuel, slag seldom forms to any appreciable 
extent in small heats ; this renders any preparation for its withdrawal un- 
necessary, but when the cupola is to be taxed to its utmost capacity it is 
then incumbent on the melter to flux the charges all through the heat, car- 
rying it away in the manner directed. 

The best flux for this purpose is the chips from a white marble yard. 
About 6 pounds to the ton of iron will give good results when all is clean. 

When fuel is bad, or iron is dirty, or both together, it becomes imperative 
that the slag be kept running all the time. 

Fuel for Cupolas.— -The best fuel for melting iron is coke, because it re- 
quires less blast, makes hotter iron, and melts faster than coal. When coal 
must be used, care should be exercised in its selection. All anthracites 
which are bright, black, hard, and free from slate, will melt iron admirably. 
The size of the coal used affects the melting to an appreciable extent, and, 
for the best results, small eupolas should be charged with the size called 
l 'egg," a still larger grade for medium-sized cupolas, and what is called 
" lump " will answer for all large cupolas, when care is taken to pack it 
carefully oti the charges. 

Charging a Cupola.— Ohas. A. Smith (Am. Mach , Feb. 12, 1891) gives 
the following: A 28-in. cupola should have from 300 to 400 pounds of coke 
on bottom bed; a 36-in. cupola, 700 to 800 pounds; a 48-in. cupola, 1500 lbs.; 
and a 60-in. cupola should have one ton of fuel on bottom bed. To every 
pound of fuel on the bed, three, and sometimes four pounds of metal can be 
added with safety, if the cupola has proper blast; in after-charges, to every 



CUPOLA PKACTICE. 



949 



pound of fuel add 8 to 10 pounds of metal; any well-constructed cupola will 
stand ten. 

F. P. WolcottM»u. Mach., Mar. 5, 1891) gives the following as the practice 
of the Colwell Iron-works, Carteret, N. J.: " We melt daily from twenty to 
forty tons of iron, with an average of 11.2 pounds of iron to one of fuel. In 
a 36-in. cupola seven to nine pounds is good melting, but in a cupola that 
lines up 48 to 60 inches, anything less than nine pounds shows a defect in 
arrangement of tuyeres or strength of blast, or in charging up." 

"The Moulder's Text-book, 1 ' by Thos. D. West, gives forty-six reports in 
tabular form of cupola practice in thirty States, reaching from Maine to 
Oregon. 

Cupola Charges in Stove-foundries. (Iron Age, April 14, 1892.) 
No two cupolas are charged exactly the same. The amount of fuel on the 
bed or between the charges differs, while varying amounts of iron are used 
in the charges. Below will be found charging-lists from some of the prom- 
inent stove-foundries in the country : 

lbs. 



A— Bed of fuel, coke 1,500 

First charge of iron 5,000 

All other charges of iron . . 1,000 
First and second charges 
of coke, each 200 



Four next charges of coke, 

each 150 

Six next charges of coke, each 120 
Nineteen next charges of coke, 
each 100 



Thus for a melt of 18 tons there would be 5120 lbs. of coke used, giving a 
ratio of 7 to 1. Increase the amount of iron melted to 24 tons, and a ratio of 
8 pounds of iron to 1 of coal is obtained. 



lbs. 

-Bed of fuel, coke 1,600 

First charge of iron 1,800 

First charge of fuel 150 

All other charges of iron, 
each 1,000 



Second and third charges of 

fuel 130 

All other charges of fuel, each 100 



For an 18- ton melt 5060 lbs. of coke would be necessary, giving a ratio of 
7.1 lbs. of iron to 1 pound of coke. 



lbs. 

-Bed of fuel, coke 1,600 

First charge of iron 4,000 

First and second charges 
of coke . . 200 



All other charges of iron . . 
All other charges of coke . 



lbs. 

2,000 
150 



In a melt of 18 tons 4100 lbs. of coke would be used, or a ratio of 8.5 to 1. 
lbs. I lbs. 

D— Bed of fuel, coke 1,800 | All charges of coke, each 200 

First charge of iron 5,600 | All other charges of iron 2,900 

In a melt of 18 tons, 3900 lbs. of fuel would be used, giving a ratio of 9.4 
pounds of iron to 1 of coke. Very high, indeed, for stove-plate. 

lbs. 



All other charges of iron, each 2,000 
All other charges of coal, each 175 



E— Bed of fuel, coal 1,900 

First charge of iron 5,000 

First charge of coal 200 

In a melt of 18 tons 4700 lbs. of coal would be used, giving a ratio of 7.7 
lbs. of iron to 1 lb. of coal. 

These are sufficient to demonstrate the varying practices existing among 
different stove-foundries. In all these places the iron was proper for stove- 
plate purposes, and apparently there was little or no difference in the kind 
of work in the sand at the different foundries. 

Results of Increased ©riving. (Erie City Iron-works, 1891.)— 
May— Dec. 1890: 60-in. cupola, 100 tons clean castings a week, melting 8 tons 
per hour; iron per pound of fuel, 7^ lbs. ; percent weight of good castings to 
iron charged, 75%. Jan. -May, 1891 : Increased rate of melting to 1 \}4 tons per 
hour; iron per lb. fuel, 9^; per cent weight of good castings, 75; one week, 
1334 tons per hour, 10.3 lbs. iron per lb. fuel; per cent weight of good cast- 
ings, 75.3. The increase was made by putting in an additional row of tuyeres 
and using stronger blast, 14 ounces. Coke was used as fuel. (W. O. Webber, 
Trans. A. S. M. E. xii. 1045.) 



95 


) 








THE FOUKDLiY. 








I 


Buffalo Steel Pressure-Mowers. 


t 
Speeds and Capacities t 




as applied to Cupolas. 






1 


03 . 

5.S 

leg 


o 


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Ceo 


sw 03 
O ft 


i 


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a 

6 

03 


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o £ 




o 

3 



6 


11 


03 


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4 


20 


4793 


1545 


412 


1.0 


9 


5095 


164? 


438 


1.3 


6 


5 


25 


8 


3911 


2321 


619 


1.2 


10 


4509 


2600 


694 


2.2 


8 


6 


30 


8 


3156 


3093 


825 


2.05 


10 


3974 


3671 


926 


3.1 


11 


7 


35 


8 


3092 


4218 


1125 


3.1 


10 


3476 


4777 


1274 


4.25 ! 


14 


8 


40 


8 


2702 


5425 


1444 


3.9 


10 


3034 


6082 


1622 


5.52 


IS 


9 


45 


10 


2617 


7818 


2085 


7.1 


12 


2916 


8598 


2293 


9.36 


26 


10 


55 


10 


2139 


11295 


3012 


10.2 


12 


2353 


1237>« 


3301 


12. 


46 


11 


73 


12 


1639 


21978 


5861 


23.9 


14 


1777 


23838 


6357 


30.3 


68 


12 


88 


12 


1639 


32395 


8636 


35.2 


14 


1777 


35190 


9384 


43.7 



In the table are given two different speeds and pressures for each size of 
blower, and the quantity of iron that may be melted, per hour, with each. 
In all cases it is recommended to use the lowest pressure of blast that will do 
the work. Run up to the speed given for that pressure, and regulate quan- 
tity of air by the blast-gate. The tuyere area should be at least one ninth 
of the area of cupola in square inches, with not less than four tuyeres at 
equal distances around cupola, so as to equalize the blast throughout. Va- 
riations in temperature affect the working of cupolas materially, hot 
weather requiring increase in volume of air. 
(For tables of the Sturtevant blower see pages 519 and 520.) 
Loss in Melting Iron in Cupolas.— G. O. Vair, Am. Mach., 
March 5, 1891, gives a record of a 45-in. Colliau cupola as follows: 

Ratio of fuel to iron, 1 to 7.42. 

Good castings 21,314 lbs. 

New scrap 3,005 " 

Millings 200 " 

Loss of metal 1,481 " 

Amount melted 26,000 lbs. 

Loss of metal, 5.69$. Ratio of loss, 1 to 17.55. 

Use of Softeners in Foundry Practice. (W. Graham, Iron Age, 
June 27, 18S9.)— In the foundry the problem is to have the right proportions 
of combined and graphitic carbon in the resulting casting; this is done by 
getting the proper proportion of silicon. The variations in the proportions 
of silicon afford a reliable and inexpensive means of producing a cast iron 
of any required mechanical character which is possible with the material 
employed. In this way, by mixing suitable irons in the right proportions, 
a required grade of casting can be made more cheaply than by using irons 
in which the necessary proportions are already found. 

If a strong machine casting were required, it would be necessary to keep 
the phosphorus, sulphur, and manganese within certain limits. Professor 
Turner found that cast iron which possessed the maximum of the desired 
qualities contained, graphite, 2.59$; silicon, 1.42$; phosphorus, 0.39$; sul- 
phur, 0.06$; manganese, 0.58$. 

A strong casting could not be made if there was much increase in the 
amount of phosphorus, sulphur, or manganese. Irons of the above percent- 
ages of phosphorus, sulphur, and manganese would be most suitable for this 
purpose, but they could be of different grades, having different percentages 
of silicon, combined and graphitic carbon. Thus hard irons, mottled and 
white irons, and even steel scrap, all containing low percentages of silicon 
and high percentages of combined carbon, could be employed if an iron 
having a large amount of silicon were mixed with them in sufficient amount. 
This would bring the silicon to the proper proportion and would cause the 
combined carbon to be forced into the graphitic state, and the resulting 



SHRINKAGE OP CASTINGS. 



951 



easting would be soft. High-silicon irons used in this way are called " soft- 
eners.' 1 
The following are typical analyses of softeners: 





Ferro-silicon. 


Softeners, American. 


Scotch 
Irons, No. 1. 




Foreign. 


American. 


Well- 
ston. 


Globe 


Belle- 
fonte. 


Eg- 
linton 


Colt- 
ness. 


Silicon 

Combined C. 
Graphitic C 
Manganese . . 
Pliosphorus. . 
Sulphur 


10.55 
1.84 
0.52 
3.86 
0.04 
0.03 


9.80 
0.09 
1.12 
1.95 
0.21 
0.04 


12.08 
0.06 
1.52 
0.76 
0.48 

Trace 


10.34 
0.07 
1.92 
0.52 
0.45 

Trace 


6.67 

2^57 

0^50 
Trace 


5.89 
0.30 
2.85 
1.00 
1.10 
0.02 


3 to 6 
0.25 
3. 

0.53 
0.35 
0.03 


2.15 
(1.21 
3.76 
2.80 
0.62 
0.03 


2.59 

' V.70* 
0.85 
0.01 



(For other analyses, see pages 371 to 373.) 

Ferro -silicons contain a low percentage of total carbon and a high per- 
centage of combined carbon. Carbon is the most important constituent of 
cast iron, and there should be about 3.4$ total carbon present. By adding 
ferro-silicon which contains only 2% of carbon the amount of carbon in the 
resulting mixture is lessened. 

Mr. Keep found that more silicon is lost during the remelting of pig of 
over 10% silicon than in remelting pig iron of lower percentages of silicon. 
He also points out the possible disadvantage of using ferro-silicons contain- 
ing as high a percentage of combined carbon as 0.70$ to overcome the bad 
effects of combined carbon in other irons. 

The Scotch irons generally contain much more phosphorus than is desired 
in irons to be employed in making the strongest castings. It is a mistake to 
mix with strong low-phosphorus irons an iron that would increase the 
amount of phosphorus for the sake of adding softening qualities, when soft- 
ness can be produced by mixing irons of the same low phosphorus. 

(For further discussion of the influence of silicon see page 365.) 

Shrinkage of Casting's. — The allowance necessary for shrinkage 
varies for different kinds of metal, and the different conditions under which 
they are cast. For castings where the thickness runs about one inch, cast 
under ordinary conditions, the following allowance can be made: 



For cast-iron, % inch per foot. 
" brass, 3/16 " " " 
" steel, % " " " 
" mal. iron, % " " " 



For zinc, 5/16 inch per foot. 

" tin, 1/12 " " " 

" aluminum, 3/16 " " " 
" Britannia, 1/32 " " " 



Thicker castings, under the same conditions, will shrink less, and thinner 
ones more, than this standard. The quality of the material and the manner 
of moulding and cooling will also make a difference. 

Numerous experiments by W. J. Keep (see Trans. A. S. M. E., vol. xvi.) 
showed that the shrinkage of cast iron of a given section decreases as the 
percentage of silicon increases, while for a given percentage of silicon the 
shrinkage decreases as the section is increased. Mr. Keep gives the follow- 
ing table showing the approximate relation of shrinkage to size and per- 
centage of silicon: 





Sectional Area of Casting. 


Percentage 














of 


Yz" n 


1" D 


1" X 2" 


2" D 


3" □ 


4" D 


Silicon. 
















Shrinkage in Decimals of an inch per foot of Length. 


1. 


.183 


.158 


.146 


.130 


.113 


.102 


1.5 


.171 


.145 


.133 


.117 


.098 


.087 


2. 


.159 


.133 


.121 


.104 


.085 


.074 


2.5 


.147 


.121 


.108 


.092 


.073 


.060 


3. 


.135 


.108 


.095 


.077 


.059 


.045 


3.5 


.123 


.095 


.082 


.065 


.046 


.032 



952 



THE FOUNDRY. 



Mr. Keep also gives the following " approximate key for regulating foun- 
dry mixtures" so as to produce a shrinkage of Y & in. per ft. in castings of 
different sections: 

Size of casting \i 1 2 3 4 in. sq. 

Silicon required, per cent 3.25 2.75 2.25 1.75 1.25 percent. 

Shrinkage of a ^-iu. test-bar. .125 .135 .145 .155 .165 in. per ft. 

"Weight of Castings determined from Weight of Pattern. 

(Rose's Pattern-maker's Assistant.) 



A Pattern weighing One Pound, 
made of— 



Will weigh when cast in 



*« Zinc. Copper. gJJ« »•* 



Mahogany — Nassau 

Honduras 
" Spanish. 

Pine, red ... 

" white.. 

" yellow 



10.7 
12.9 
8.5 
12.5 
16.7 
14.1 



10.4 
12.7 
8.2 
12.1 
16.1 
13.6 



lbs. 
12.8 
15.3 
10.1 
14.9 
19.8 
16.7 



lbs. 
12.2 



14.2 
19.0 
16.0 



lbs. 
12.5 



14.6 
19.5 
16.5 



Moulding Sand. (From a paper on "The Mechanical Treatment of 
Moulding Sand." by Walter Bagshaw, Proc. Inst. M. E. 1891.)— The chemical 
composition of sand will affect the nature of the casting, no matter what 
treatment it undergoes. Stated generally, good sand is composed of 94 parts 
silica, 5 parts alumina, and traces of magnesia and oxide of iron. Sand con- 
taining much of the metallic oxides, and especially lime, is to be avoided. 
Geographical position is the chief factor governing the selection of sand; 
and whether weak or strong, its deficiencies are made up for by the skill of 
the moulder. For this reason the same sand is often used for both heavy and 
light castings, the proportion of coal varying according to the nature of the 
casting. A common mixture of facing-sand consists of six parts by weight 
of old sand, four of new sand, and one of coal-dust. Floor-sand requires 
only half the above proportions of new sand and coal-dust to renew it. Ger- 
man founders adopt one part by measure of new sand to two of old sand; 
to which is added coal-dust in the proportion of one tenth of the bulk for 
large castings, and one twentieth for small castings. A few founders mix 
street-sweepings with the coal in order to get porosity when the metal in 
the mould is likely to be a long time before setting. Plumbago is effective in 
preventing destruction of the sand; but owing to its refractory nature, it 
must not be dusted on in such quantities as to close the pores and prevent 
free exit of the gases. Powdered French chalk, soapstone, and other sub- 
stances are sometimes used for facing the mould; but next to plumbago, oak 
charcoal takes the best place, notwithstanding its liability to float occasion- 
ally and give a rough casting. 

For the treatment of sand in the moulding-shop the most primitive method 
is that of hand-riddling and treading. Here the materials are roughly pro- 
portioned by volume, and riddled over an iron plate in a flat heap, where 
the mixture is trodden into a cake by stamping with the feet; it is turned 
over with the shovel, and the process repeated. Tough sand can be obtained 
in this manner, its toughness being usually tested by squeezing a handful 
into a ball and then breaking it; but the process is slow and tedious. Other 
things being equal, the chief characteristics of a good moulding-sand are 
toughness and porosity, qualities that depend on the manner of mixing as 
well as on uniform ramming. 

Toughness of Sand.— In order to test the relative toughness, sand 
mixed in various ways was pressed under a uniform load into bars 1 in. sq. 
and about 12 in. long, and each bar was made to project further and 
further over the edge of a table until its end broke off by its own weight. 
Old sand from the shop floor had very irregular cohesion, breaking at all 
lengths of projections from ^ in. to 1J^ in. New sand in its natural state 
held together until an overhang of 2% in. was reached. A mixture of old 
sand, new sand, and coal-dust 
Mixed under rollers broke at 2 to 2*4 in. of overhang. 

" in the centrifugal machine " " 2 " 2J4 " " " 

through a riddle " " 1% " 2^ " " " 



SPEED OF CUTTING-TOOLS IN LATHES, ETC. 



953 



Showing as a mean of the tests only slight differences between the last 
three methods, but in favor of machine-work. In many instances the frac- 
tures were so uneven that minute measurements were not taken. 

Dimensions of Foundry ladles.— The following table gives the 
dimens ons. inside the lining, of ladles from 25 lbs. to 16 tons capacity. All 
the ladles are supposed to have straight sides. (Am. Mach., Aug. 4, 1892.) 



Capacity. 



Diam. 


Depth. 


in. 


in. 


54 


56 


52 


53 


49 


50 


46 


48 


43 


44 


39 


40 


34 


35 


31 


32 


27 


28 


24^ 


25 


22 


22 



Capacity. 



Diam. 



Depth. 



i tons 

14 " 

12 " 

10 " 

8 " 

6 " 

4 " 

3 " 

2 " 

1 " 



%ton .., 
Y2 " ... 
H " ... 

300 pound: 

250 

200 

150 

100 

75 

50 

35 



10% 
10 



^y 2 



13^ 



10V6 



THE MACHINE-SHOP. 



SPEED OF CUTTING-TOOLS IN LATHES, MILLING 

MACHINES, ETC. 

Relation of diameter of rotating tool or piece, number of revolutions, 
and cutting-speed : 
Let d = diam. of rotating piece in inches, n = No. of revs, per min.; 
S = speed of circumference in feet per minute; 



3.82S 



12 



Approximate rule : No. of revs, per min. — 4 X speed in ft. per min. -r- 
diam. in inches. 

Speed of Cut-for Lathes and Planers. (Prof. Coleman Sellers, 
Stevens'" Indicator, April, 1892.)— Brass may be turned at high speed like 
wood. 

Bronze.— A speed of 18 feet per minute can be used with the soft alloys- 
say 8 to 1, while for hard mixtures a slow speed is required— say 6 feet per 
minute. 

Wrought Iron can be turned at 40 feet per minute, but planing-machines 
that are used for both cast and forged iron are operated at 18 feet per 
minute. 

Machinery Steel. — Ordinary, 14 feet per minute; car-axles, etc., 9 feet per 
minute. 

Wheel Tires.— 6 feet per minute; the tool stands well, but many prefer 
to run faster, say 8 to 10 feet, and grind the tool more frequently. 

Lathes. — The speeds obtainable by means of the cone-pulley and the back 
gearing are in geometrical progression from the slowest to the fastest. In 
a well-proportioned machine the speeds hold the same relation through nil 
the steps. Many lathes have the same speed on the slowest of the cone and 
the fastest of the back-gear speeds. 

The Speed of Counter-shaft of the lathe is determined by an assumption 
of a slow speed with the back gear, say 6 feet per minute, on the largest 
diameter that the lathe will swing. 

Example. — A 30-inch lathe will swing 30 inches =, say, 90 inches. circumfer- 
ence = 7' 6" ; the lowest triple gear should give a speed of 5 or 6 per minute. 

In turning or planing, if the cutting-speed exceed 30 ft. per minute, so 
much heat will be produced that the temper will be drawn from the tool. 
The speed of cutting is also governed by the thickness of the shaving, and 
by the hardness and tenacity of the metal which is being cut; for instance, 
in cutting mild steel, with a traverse of % in. per revolution or stroke, and 
with a shaving about % in. thick, the speed of cutting must be reduced to 
about 8 ft. per minute. A good average cutting-speed for wrought or e$s$ 



954 



THE MACHINE-SHOP. 



iron is 20 ft. per minute, whether for the lathe, planing, shaping, or slotting 
machine. (Proc. Inst. M. E., April, 1883, p. 248.) 

Table of Cutting-speeds. 



Feet per minute. 



Revolutions per minute. 



76.4 


152.8 


229.2 


305 6 


382.0 


50.9 


101.9 


153.8 


203.7 


254.6 


38.2 


76.4 


114.6 


152.8 


191.0 


30.6 


61.1 


91.7 


122.2 


152.8 


25.5 


50.9 


76.4 


101.8 


127.3 


21 .8 


43.7 


65.5 


87.3 


109.1 


19.1 


38.2 


57.3 


76.4 


95.5 


17.0 


34.0 


50.9 


67.9 


84.9 


15.3 


30.6 


45.8 


61.1 


76.4 


13 9 


27.8 


41.7 


55.6 


69.5 


12.7 


25.5 


38.2 


50.9 


63.6 


10.9 


21.8 


32.7 


43.7 


54.6 


9.6 


19.1 


28.7 


38.2 


47.8 


8.5 


17.0 


25.5 


34.0 


42.5 


7.6 


15.3 


22 9 


30.6 


38.2 


6.9 


13.9 


20.8 


27.8 


34.7 


6.4 


12.7 


19.1 


25.5 


31.8 


5.5 


10.9 


16.4 


21.8 


27.3 


4.8 


9.6 


14.3 


19.1 


23.9 


4.2 


8.5 


12.7 


17.0 


21.2 


3.8 


7.6 


11.5 


15.3 


19.1 


3.5 


6.9 


10.4 


13.9 


17.4 


3.2 


6.4 


9.5 


12.7 


15.9 


2.7 


5.5 


8.2 


10.9 


13.6 


2.4 


4.8 


7.2 


9.6 


11.9 


2.1 


4.2 


6.4 


8.5 


10.6 


1.9 


. 3.8 


5.7 


7.6 


9.6 


1.7 


3.5 


5.2 


6.9 


8.7 


1.6 


3.2 


4.8 


6.4 


8.0 


1.5 


2.9 


4.4 


5.9 


7.3 


1.4 


2.7 


4.1 


5.5 


6.8 


1.3 


2.5 


3.8 


5.1 


6.4 


1.2 


2.4 


3.6 


4.8 


6.0 


1.1 


2.1 


3.2 


4.2 


5.3 


1.0 


1.9 


2.9 


3.8 


4.8 


.9 


1.7 


2.6 


3.5 


4.3 


.8 


1.6 


2.4 


3.2 


4.0 


.7 


1.5 


2.2 


2.9 


3.7 


7 


1.4 


2.0 


2.7 


3.4 


.6 


1.3 


1.9 


2.5 


3.2 


.5 


1.1 


1.6 


2.1 


2.7 


.5 


.9 


1.4 


1.8 


2.3 


.4 


.8 


1.2 


1.6 


2.0 


.4 


.7 


1.1 


1.4 


1.8 


.3 


.6 


1.0 


1.3 


1.6 



458.4 
305.6 
229.2 
183.4 
152.8 
13i 

114.6 
101.8 
91. 
83.3 
76.4 
65.5 
57.3 
50.9 
45 
41 



19.1 
16.4 
14.3 
12.7 
11.5 
10.4 
9.5 



5. 

5.2 

4.8 

4.4 

4.1 

3.8 

3.: 

2.' 

2.4 

2.1 

1.9 



534.8 


611.2 


687.6 


350.5 


407.4 


458.3 


267.4 


305.6 


343 8 


213.9 


244.5 


275.0 


178.2 


203.7 


229.1 


152.8 


174.6 


196.4 


133.7 


152.8 


171.9 


118.8 


135.8 


152.8 


106.9 


122.2 


137.5 


97.2 


111.1 


125.0 


89.1 


101.8 


114.5 


76.4 


87.3 


98.2 


66.9 


76.4 


86.0 


59.4 


67.9 


76.4 


53.5 


61.1 


68.8 


48.6 


55.6 


62.5 


44.6 


50.9 


57.3 


38.2 


43.7 


49.1 


33.4 


38 2 


43.0 


29.7 


34.0 


38.2 


26.7 


30.6 


34.4 


24.3 


27.8 


31.2 


22.3 


25.5 


28.6 


19.1 


21.8 


24.6 


16.7 


19.1 


21.5 


14.8 


17.0 


19.1 


13.3 


15.3 


17.2 


12.2 


13.9 


15.6 


11.1 


12.7 


14.3 


10.3 


11.8 


13.2 


9.5 


10.9 


12.3 


8.9 


10.2 


11.5 


8.4 


9.5 


10.7 


7.4 


8.5 


9.5 


6.7 


7.6 


8.6 


6.1 


6.9 


7.8 


5.6 


6.4 


7.2 


5.1 


5.9 


6.6 


4.8 


5.5 


6.1 


4.5 


5.1 


5 7 


3.7 


4.2 


4.8 


3.2 


3.6 


4.1 


2.8 


3.2 


3.6 


2.5 


2.8 


3.2 


2.2 


2.5 


2.9 



764.0 
509.3 
382.0 
305.6 
254.6 
218.3 
191.0 
169.7 
152.8 
138.9 
127.2 
109.2 
95.5 
84.9 
76.4 
69.5 
63.7 
54.6 
47.8 
42.5 
38.1 
34.7 
31.8 
27.3 
23.9 
21.2 
19.1 
17.4 
15.9 
14.7 
13.6 
12.7 
11.9 
10.6 



Speed of Cutting with Turret Lathes.— Jones & Lamson Ma- 
chine (Jo. give the following cutting-speeds for use with their flat turret 
lathe: 

Ft. per minute. 

( Tool steel and taper on tubing 10 

Threading < Machinery , 15 

| Very soft steel 20 

Tiirninp- Cut wnich reduces the stock to )4 of its original diam. . 20 
hi .o iiiTiPPv J Cut wll 'eh reduces the stock to % of its original diam. . 25 
sreli i Cut which reduces the stock to % of its original diam. . 30 to 35 
(. Cut which reduces the stock to 15/16 of its original diam. 40 to 45 
Turning very soft machinery steel, light cut and cool worj?. , , 50 to 6Q 



GEARING OF LATHES. 955 

Forms of Metal-cutting Tools.— " Hutte" the German Engi- 
neers' Pocket-book, gives the following cutting-angles for using least power: 
Top Rake. Angle of Cutting-edge. 

Wrought iron 3° 51° 

Cast iron 4° 51° 

Bronze 4° 66° 

The American Machinist comments on these figures as follows : We are 
not able to give the best nor even the generally used angles for tools, 
because these vary so much to suit different circumstances, such as degree 
of hardness of the metal being cut, quality of steel of which the tool is 
made, depth of cut, kind of finish desired, etc. The angles that cut with 
the least expenditure of power are easily determined by a few experiments, 
but the best angles must be determined by good judgment, guided by expe- 
rience. In nearly all cases, however, we think the best practical angles are 
greater than those given. 

For illustrations and descriptions of various forms of cutting-tools, see 
articles on Lathe Tools in App. Cyc. App. Mech., vol. ii., and in Modern 
Mechanism. 

Cold Chisels.— Angle of cutting- faces (Joshua Rose): For cast steel, 
about 65 degrees; for gun-metal or brass, about 50 degrees; for copper and 
soft metals, about 30 to 35 degrees. 

Rule for Gearing Lathes for Screw-cutting. (Garvin Ma- 
chine (Jo.) — Read from Hie lathe index the number of threads per inch cut 
by equal gears, and multiply it by any number that will give for a product 
a grear on the index; put this gear upon the stud, then multiply the number 
of threads per inch to be cut by the same number, and put the resulting gear 
upon the screw. 

Example.— To cut 11^ threads per inch. We find on the index that 48 into 
48 cuts 6 threads per inch, then 6 X 4 = 24, gear on stud, and 11' j X 4 = 46, 
gear on screw. Any multiplier may be used so long as the products include 
gears that belong with the lathe. For instance, instead of 4 as a multiplier 
we may use 6. Thus, 6 X 6 = 36, gear upon stud, and llj^ x 6 = 69, gear 
upon screw. 

Rules for Calculating Simple and Compound Gearing 
where there is no Index. {Am Mach.)—lt the lathe is simple- 
geared, and the stud runs at the same speed as the spindle, select some gear 
for the screw, and multiply its number of teeth by the number of threads 
per inch in the lead-screw, and divide this result by the number of threads 
per inch to be cut. This will give the "number of teeth in the gear for the 
stud. If this result is a fractional number, or a number which is not among 
the gears on hand, then try some other gear for the screw. Or, select the 
gear for the stud first, then multiply its number of teeth by the number of 
threads per inch to be cut, and divide by the number of threads per inch on 
the lead-screw. This will give the number of teeth for the gear on the 
screw. If the lathe is compound, select at random all the driving-gears, 
multiply the numbers of their teeth together, and this product by the num- 
ber of threads to be cut. Then select at random all the driven gears except 
one; multiply the numbers of their teeth together, and this product by the 
number of threads per inch in the lead-screw. Now divide the first result by 
the second, to obtain the number of teeth in the remaining driven gear. Or, 
select at random all the driven gears. Multiply the numbers of their teeth 
together, and this product by the number of threads per inch in the lead- 
screw. Then select at random all the driving-gears except one. Multiply 
the numbers of their teeth together, and this result by the number of threads 
per inch of the screw to be cut. Divide the first result by the last, to obtain 
the number of teeth in the remaining driver. When the gears on the com- 
pounding stud are fast together, and cannot be changed, then the driven one 
has usually twice as many teeth as the other, or driver, in which case in the 
calculations consider the lead-screw to haze twice as many threads per inch 
as it actually nas ; and then ignore the compounding entirely. Some lathes 
are so constructed that the stud on which the first driver is placed revolves 
only half as fast as the spindle. This can be ignored in the calculations by 
doubling the number of threads of the lead-screw. If both the last condi- 
tions are present ignore them in the calculations by multiplying the number 
of threads per inch in the lead-screw by four. If the thread to be cut is a 
fractional one, or if the pitch of the lead-screw is fractional, or if both are 
fractional, then reduce the fractions to a common denominator, and use 
the numerators of these fractions as if they equalled the pitch of the screw 



950 



THE MACHINE-SHOP. 



to be cut, and of the lead-screw, respectively. Then use that part of the rule 
given above which applies to the lathe in question. For instance, suppose 
it is desired to cut a thread of 25/32-inch pitch, and the lead-screw has 4 
threads per inch Then the pitch of the lead-screw will be J4 inch, which is 
equal to 8/32 inch. We now have two fraction, 25/32 and 8/32, and the two 
screws will be in the proportion of 25 to 8, and the gears can be figured by 
the above rule, assuming the number of threads to be cut to be 8 per inch, 
and those on the lead-screw to be 25 per inch. But this latter number may 
be further modified by conditions named above, such as a reduced speed of 
the stud, or fixed compound gears. In the instance given, if the lead-screw 
had been 2^ threads per inch, then its pitch being 4/10 inch, we have the 
fractions 4/10 and 25/32, which, reduced to a common denominator, are 
64/160 and 125/160, and the gears will be the same as if the lead-screw had 125 
threads per inch, and the screw to be cut 64 threads per inch. 

On this subject consult also " Formulas in Gearing," published by Brown 
& Sharpe Mfg. Co.. and Jamieson's Applied Mechanics. 

Change-gears for Screw-cutting Lathes. — There is a lack of 
uniformity among lathe-builders as to the change-gears provided for screw- 
cutting. W. R. Macdonald, in Am. Mach., April 7, 1892, proposes the follow- 
ing series, by which 33 whole threads (not fractional) maybe cut by changes 
of only nine gears: 



70 
110 



130 



Spindle. 



40 50 60 70 



4 4/5 
7 1/5 
9 3/5 



14 2/5 
16 4/5 
26 2/5 
28 4/5 
31 1/5 



5 1/7 

6 6/7 
8 4/7 

10 2/*" 



18 
20 4/7 
2.2 3/~ 



110 120 130 



2 2/11 

3 3/11 

4 4/11 

5 5/11 

6 6/11 

7 7/11 

13'i/ii 

14 2/11 



13 



1 11/13 

2 10/13 

3 9/13 

4 8/13 

5 7/13 

6 6/13 

10 2/13 

11 1/13 



Whole Threads. 



2 


11 


22 


3 


12 


24 


4 


13 


26 


5 


14 


28 


6 


15 


30 


7 


16 


33 


8 


IK 


36 


9 


20 


39 


10 


21 


42 



Ten gears are sufficient to cut all the usual threads, with the exception of 
perhaps ll^j, the standard pipe-thread; in ordinary practice any fractional 
thread between 11 and 12 will be near enough for the customary short pipe- 
thread; if not, the addition of a single gear will give it. 

In this table the pitch of the lead-screw is 12, and it may be objected to as 
too fine for the purpose. This may be rectified by making the real pitch 6 
or any other desirable pitch, and establishing the proper ratio between the 
lathe spindle and the gear-stud. 

Metric Screw-threads may be cut on lathes with inch-divided lead- 
ing-screws, by the use of a change-wheel with 127 teeth; for 127 millimetres 
equal 5 inches (126 X .03937 = 4.99999 in.). 

Rule for Setting the Taper in a Lathe. (Am. Mach.)— No 
rule can be given which will produce esact results, owing to the fact that 
the centres enter the work an indefinite distance. If it were not for this cir- 
cumstance the following would be an exact rule, and it is an approximation 
as it is. To find the distance to set the centre over: Divide the difference in 
the diameters of the large and small end of the taper by 2, and multiply this 
quotient by the ratio which the total length of the shaft bears to the length 
of the tapered portion. Example: Suppose a shaft three feet long is to have 
a taper turned on the end one foot long, the large end of the taper being two 

inches and the small end one inch diameter. — — - X - = \% inches. 

Electric Drilling-machines -Speed of Drilling Holes in 
Steel Plates. (Proc. Inst. M. E., Aug. 1887, p. 329.;— In drilling holes in 
the shell of the S.S. "Albania," after a very small amount of practice the 
men working the machines drilled the %-inch holes in the shell with great 
rapidity, doing the work at the rate of one hole every 69 seconds, inclusive of 
the time occupied in altering the position of the machines by means of differ- 
ential pulley-blocks, which were not conveniently arranged as slings for 
this purpose. Repeated trials of these drilling-machines have also shown 
that, when using electrical energy in both holding-on magnets and motor 



MILLING-CUTTERS. 



95? 



amounting to about % H.P., they have drilled holes of 1 inch diameter 
through l^j inch thickness of solid wrought iron, or through \% inch of mild 
steel in two plates of 13/16 inch each, taking exactly 1% minutes for each 
hole. 

Speed of Twist-drills.— The cutting-speeds and rates of feed recom- 
mended by the Morse Twist -drill and Machine Company are given in the 
following table. 

Revolutions per minute for drills 1/16 in. to 2 in. diam., as usually applied: 



Diameter 


Speed 


Speed 


Speed 


Diameter 


Speed 


Speed 


Speed 


of 


for 


for 


for 


of 


for 


for 


for 


Drills. 


Steel. 


Iron. 


Brass. 


Drills. 


Steel. 


Iron. 


Brass. 


inch. 








inch. 








1/16 


940 


1280 


1560 


1 1/16 


54 


75 


95 


% 


460 


660 


785 


v& 


52 


70 


90 


3/16 


SIC 


420 


540 


1 3/16 


49 


66 


85 


x 4 


230 


320 


400 


VA 


46 


62 


80 


5/16 


190 


260 


320 


1 5/16 


44 


60 


75 


% 


150 


220 


260 


Ws 


42 


58 


72 


7/16 


130 


185 


230 


1 7/16 


40 


56 


69 


H 


115 


160 


200 


1H 


39 


54 


66 


9/16 


100 


140 


180 


1 9/16 


37 


51 


63 


% 


95 


130 


160 


1% 


36 


49 


60 


11/16 


85 


115 


145 


1 11/16 


34 


47 


58 


H 


75 


105 


130 


m 


33 


45 


56 


1-3/16 


70 


100 


120 


1 13/16 


32 


43 


54 


% 


65 


90 


115 


1% 


31 


41 


52 


15/16 


62 


85 


110 


1 15/16 


30 


40 


51 


1 


58 


80 


100 


2 


29 


39 


49 



To drill one inch in soft cast iron will usually require: For %-in. drill, 125 
revolutions; for J^-in. drill, 120 revolutions; for %-m.. drill, 100 revolutions; 
for 1-in. drill, 95 revolutions. 

The rates of feed for twist drills are thus given by the same company: 
Diameter of drill 1/16 y± ' % \£ % 1 \\& 



Revs, per inch depth of hole. 125 125 120 to 140 1 inch feed per min. 
MILLING-CrTTEBS. 

George Addy, (Proc. Inst. M. E., Oct. 1890, p. 537), gives the following: 
Analyses of Steel.— The following are analyses of milling-cutter 

blanks, made from best quality crucible cast steel and from self-hardening 

" Ivanhoe V steel : 



Carbon 

Silicon 

Phosphorus 

Manganese 

Sulphur 

Tungsten , 

Iron, by difference . . 



Crucible Cast Steel, 
per cent. 

1.2 

0.112 

0.018 
.. . 0.36 
0.02 



Ivanhoe Steel, 
per cent. 
1.67 
0.252 
0.051 
2.557 
0.01 
4.65 
90.81 



100.000 100.000 

The first analysis is of a cutter 14 in. diam., 1 in. wide, which gave very 
good service at a cutting-speed of 60 ft. per min. Large milling-cutters are 
sometimes built up, the cutting-edges only being of tool steel. A cutter 22 in. 
diam. by 5)4 in. wide has been made in this way, the teeth being clamped 
between two cast-iron flanges. Mr. Addy recommends for this form of 
tooth one with a cutting-angle of 70°, the face of the tooth being set 10° back 
of a radial line on the cutter, the clearance -angle being thus 10°. At the 
Clarence Iron-works, Leeds, the face of the tooth is set 10° back of the radial 
line for cutting wrought iron and 20° for steel. 

Pitch of Teeth.— For obtaining a suitable pitch of teeth for milling- 
cutters of various diameters there exists no standard rule, the pitch being 
usually decided in an arbitrary manner, according to individual taste. 



958 THE MACEIKE-SHOP. 

For estimating the pitch of teeth in a cutter of any diameter from 4 in. to 15 
in., Mr. Addy has worked out the following rule, which he has found capa- 
ble of giving good results in practice: 

Pitch in inches = 4/(diam. in inches x 8) X 0.0625 = .177 Vdiam. 

J. M. Gray gives a rule for pitch as follows: The number of teeth in a 
milling-cutter ought to be 100 times the pitch in inches; that is, if there 
were 27 teeth, the pitch ought to be 0.27 in. The rules are practically the 
same, for if d = diam., n = No. of teeth, p = pitch, c = circumference, c = 

pn; d = ^ = — ^ = 31.83p2; p = \/MUd = .177 Vd; No. of teeth, n, = 

3.14d-=-p. 

Number of Teeth in Mills or Cutters. (Joshua Rose.)— The teeth 
of cutters must obviously be spaced wide enough apart to admit of the emery- 
wheel grinding one tooth without touching the next one, and the front faces 
of the teeth are always made in the plane of a line radiating from the axis of 
the cut'.er. In cutters up to 3 in. in diam. it is good practice to provide 8 
teetli per in. of diam., while in cutters above that diameter the spacing 
may be coarser, as follows: 

Diameter of cutter, 6 in. ; number of teeth in cutter, 40 

7 k ' " " " " " 45 

8 " " " " " " 50 

Speed of Cutters.— The cutting speed for milling was originally fixed 
very low; but experience has shown that with the improvements now in 
use it may with advantage be considerably increased, especially with cutters 
of large diameter. The following are recommended as safe speeds for cut- 
ters of 6 in. and upwards, provided there is not any great depth of material 
to cut away: 

Steel. Wrought iron. Cast iron. Brass. 

Feet per minute 36 48 60 120 

Feed, inch per min. .. J^ 1 \% 2% 

Should it be desired to remove any large quantity of material, the same 
cutting-speeds are still recommended, but with a finer feed. A simple rjle 
for cutting-speed is: Number of revolutions per minute which the outer 
spindle should make when working on cast iron = 240, divided by the diam- 
eter of the cutter in inches. 

Speed of Milling-cutters. (Proc. Inst. M. E., April, 1883, p. 248.)— 
The cutting-speed which can be employed in milling is much greater than 
that which can be used in any of the ordinary operations of turning in the 
lathe, or of planing, shaping, or slotting. A milling-cutter with a plentiful 
supply of oil, or soap and water, can be run at from 80 to 100 ft. per min., 
when cutting wrought iron. The same metal can only bs turned in a lathe, 
with a tool-holder having a good cutter, at the rate of 30 ft. per min., or at 
about one third the speed of milling. A milling-cutter will cut cast steel at 
the rate of 25 to 30 ft. per min. 

The following extracts are taken from an article on speed and feed of 
milling-cutters in Eng'g, Oct. 22, 1891: Milling-cutters are successfully em- 
ployed on cast iron at a speed of 250 ft. per min. ; on wrought iron at from 
80 ft. to 100 ft. per min. The latter materials need a copious supply of good 
lubricant, such as oil or soapy water. These rates of speed are not ap- 
proached by other tools. The usual cutting-speeds on the lathe, planing, 
shaping, and slotting machines rarely exceed about one third of those given 
above, and frequently average about a fifth, the time lost in back strokes not 
being reckoned. 

The feed in the direction of cutting is said by one writer to vary, in ordi- 
nary work, from 40 to 70 revs, of a 4-in. cutter per in. of feed. It must always 
to an extent depend on the character of the work done, but the above gives 
shavings of extreme thinness. For example, the circumference of a 4-in. 
cutter being, say, 12^ in., and having, say, 60 teeth, the advance corre- 
sponding to the passage of one cutting-tooth over the surface, in the coarser 
of the above-named feed-motions, is 1/40 X 1/60 = 1/2100 in.; the finer feed 
gives an advance for each tooth of onlv 1/70 X 1/60 = 1/4200 in. Such fine 
feeds as these are used only for light finishing cuts, and the same author- 
ity recommends, also for fini-hing, a cutter about 9 in. in circumference, or 
nearly 3 in. in diameter, which should be run at about 60 revs, per min. to 
cut tough wrought steel, 120 for ordinary cast iron, about 80 for wrought 



MILLING-MACHINES. 959 

iron, and from 140 to 160 for the various qualtities of gun-metal and brass. 
With cutters smaller or larger the rates of revolution are increased or 
diminished to accord with the following table, which gives these rates of 
cutting-speeds and shows the lineal speed of the cutting-edge: 

Steel. Wrought Iron. Cast Iron. Gun-metal. Brass. 
Feet per minute... 45 60 90 105 120 

These speeds are intended for very light finishing cuts, and they must be 
reduced to about one half for heavy cutting. 

The following results have been found to be the highest that could be at- 
tained in ordinary workshop routine, having due consideration to economy 
and the time taken to change and grind the cutters when they become dull: 
Wrought iron— 36 ft. to 40 ft. per min.; depth of cut. 1 in.; feed, % in. per 
min. Soft mild steel — About 30 ft. per min.; depth of cut, J4 m -\ feed, % 
in. per min. Tough gun-metal— 80 ft. per min. ; depth of cut, % in. ; feed, % 
in. per min. Cast-iron gear-wheels— 26J^ ft. per min.; depth of cut, J^ in.; 
feed, % in. per min. Hard, close-grained cast iron— 30 ft. per min.; depth 
of cijt, 2J^ in.; feed, 5/16 in. per min. Gun-metal joints, 53 ft. per min.; 
depth of cut, l%in.; feed, % in. per nun. Steel-bars— 21 ft. per min.; depth 
of cut, 1/32 in.; feed, % in. per min. 

A stepped milling-cutter, 4 in. in diam. and 12 in. wide, tested under two 
conditions of speed in the same machine, gave the following results: The 
cutter in both instances was worked up to its maximum speed before it gave 
way, the object being to ascertain definitely the relative amount of work 
done by a high speed and a light feed, as compared with a low speed and a 
heavy cut. The machine was used single-geared and double-geared, and in 
both cases the width of cut was lOJ^ in. 

Single-gear, 42 ft. per min.; 5/16 in. depth of cut; feed, 1.3 in. per min. =: 
4.16 cu. in. per min. Double-gear, 19 ft. per min.; %in. depth of cut; feed, 
% in. per rain. = 2.40 cu. in. per min. 

Extreme Results witli Milling-machines. — Horace L. 
Arnold (Am. Mack., Dec. 28, 1893) gives the following results in flat-surface 
milling, obtained in a Pratt & Whitney milling-machine : The mills for the 
flat cut were 5" diam., 12 teeth, 40 to 50 revs, and 4%" feed per min. One 
single cut was run over this piece at a feed of 9" per min., but the mills 
showed plaiuly at the end that this rate was greater than they could endure. 
At 50 revs, for these mills the figures are as follows, with 4%" feed: Surface 
speed, 64 ft., nearly; feed per tooth, 0. 00812": cuts per inch, 123. And with 
9" feed per min.: Surface speed, 64 ft. per min.; feed per tooth, 0.015"; cuts 
per inch, 66%. 

At a feed of 4%" per min. the mills stood up well in this job of cast-iron 
surfacing, while with a 9" feed they required grinding after surfacing one 
piece; in other words, it did not damage the mill-teeth to do this job with 
123 cuts per in. of surface finished, but they would not endure 66% cuts per 
inch. In this cast-iron milling the surface speed of the mills does not seem 
to be the factor of mill destruction: it is the increase of feed per tooth that 
prohibits increased production of finished surface. This is precisely the re- 
verse of the action of single-pointed lathe and planer tools in general: with 
such tools there is a surface-speed limit which cannot be economically ex- 
ceeded for dry cuts, and so long as this surface-speed limit is not reached, 
the cut per tooth or feed can be made anything up to the limit of the driv- 
ing power of the lathe or planer, or to the safe strain on the work itself, 
which can in many cases be easily broken by a too great feed. 

In wrought metal extreme figures were obtained in one experiment made 
in cutting key ways 5/16" wide by %" deep in a bank of 8 shafts 1J4" diam. 
at once, on a Pratt & Whitney No. 3 column milling-machine. The 8 mills 
were successfully operated with 45 ft. surface speed and 19^ in. per min. 
feed; the cutters were 5" diam., with 28 teeth, giving the following figures, 
in steel: Surface speed, 45 ft. per min.; feed per tooth. 0.02024"; cuts per 
inch, 50, nearly. Fed with the revolution of mill. Flooded with oil, that is, 
a large stream of oil running constantly over each mill. Face of tooth 
radial. The resulting keyway was described as having a heavy wave or 
cutter-mark in the bottom, and it was said to have shown no signs of being 
heavy work on the cutters or on the machine. As a result of the experiment 
it was decided for economical steady work to run at 17 revs., with a feed of 
4" per min., flooded cut, work fed with mill revolution, giving the following 
figures: Surface speed, 22!4 ft- V er niin.; feed per tooth, 0.0084"; cuts per 
inch, 119, 



960 THE MACHINE-SHOP. 

An experiment in milling a wrought iron connecting-rod of a locomotive 
on a Pratt & Whitney double-head milling-machine is described in the Iron 
Age, Aug. 27, 1891. The amount of metal removed at one cut measured 3)4 
in. wide by 1 3/16 in. deep in the groove, and across the top % in. deep by 4% 
in. wide. This represented a section of nearly Ay% sq. in. This was done at 
the rate of 1% in. per min. Nearly 8 cu. in. of metal were cut up into chips 
every minute. The surface left by the cutter was very perfect. The cutter 
moved in a direction contrary to that of ordinary practice; that is, it cut 
down from the upper surface instead of up from the bottom. 

Milling "with" or "against" tlie Feed.— Tests made with 
the Brown & Sharpe No. 5 milling-machine (described by H. L. Arnold, in 
Am. Mach., Oct. 18, 1894) to determine the relative advantage of running 
the milling-cutter with or against the feed—" with the feed " meaning that 
the teeth of the cutter strike on the top surface or "scale" of cast-iron 
work in process of being milled, and "against the feed " meaning that the 
teeth begin to cut in the clean, newly cut surface of the work and cut up- 
wards toward the scale — showed a decided advantage in favor of running 
the cutter against the feed. The result is directly opposite to that obtained 
in tests of a Pratt & Whitney machine, by experts of the P. & W. Cot 

In the tests with the Brown & Sharpe machine the cutters used were 6 
inches face by 414 an( i 3 inches diameter respectively, 15 teeth in each mill, 
42 revolutions per minute in each case, or nearly 50 feet per minute surface 
speed for the 4J^-inch and 33 feet per minute for the 3-inch mill. The revo- 
lution marks were 6 to the inch, giving a feed of 7 inches per minute, and a 
cut per tooth of .011". When the machine was forced to the limit of its 
driving the depth of cut was 11/32 inch when the cutter ran in the " old " 
way, or against the feed, and only J4 inch when it ran in the " new " way, 
or with the feed. The endurance of the milling-cutters was much greater 
when they were run in the " old " way. 

Spiral Milling-cutters.— There is no rule for finding the angle of 
the spiral; from 10° to 15° is usually considered sufficient; if much greater 
the end thrust on the spindle will be increased to an extent not desirable for 
some machines. 

Milling-cu tters with Inserted Teeth.— When it is required to 
use milling-cutters of a greater diameter than about 8 in., it is preferable to 
insert the teeth in a disk or head, so as to avoid the expense of making 
solid cutters and the difficulty of hardening them, not merely because of 
the risk of breakage in hardening them, but also on account of the difficulty 
in obtaining a uniform degree of hardness or temper. 

Milling - machine versus Planer. — For comparative data of 
work done by each see paper by J. J. Grunt, Trans. A. S. M. E., ix. 259. He 
says : The advantages of the milling machine over the planer are many, 
among which are the following : Exact duplication of work; rapidity of pro- 
duction—the cutting being continuous; cost of production, as several 
machines can be operated by one workman, and he not a skilled mechanic; 
and cost of tools for producing a given amount of work. 

POWER REQUIRED FOR MACHINE TOOL.S. 

Resistance Overcome in Cutting Metal. (Trans. A. S. M. E., 
viii. 308.) — Some experiments made at the works of William Sellers & Co. 
showed that the resistance in cutting steel in a lathe would vary from 
180,000 to 700,000 pounds per square inch of section removed, while for 
cast iron the resistance is about one third as much. The power required to 
remove a given amount of metal depends on the shape of the cut and on 
the shape and the sharpness of the tool used. If the cut is nearly square in 
section, the power required is a minimum; if wide and thin, a maximum. 
The dulness of a tool affects hut little the power required for a heavy cut. 

Heavy Work on a Planer.— Win. Sellers & Co. write as follows 
to the American Machinist : The 120'' planer table is geared to run 18 ft. per 
minute under cut, and 72 feet per minute on the return, which is equivalent, 
without allowance for time lost in reversing, to continuous cut of 14.4 feet 
per minute. Assuming the work to be 28 feet long, we may take 14 feet as 
the continuous cutting speed per minute, the .8 of a foot being much more 
than sufficient to cover time loss in reversing and feeding. The machine 
carries four tools. At %" feed per tool, the surface planed per hour would 
be 35 square feet. The section of metal cut at %" depth would be .75" X 
,125" X 4 — -375 square inch, which would require approximately 30,000 lbs, 



POWER REQUIRED FOR MACHINE TOOLS. 



961 



pressure to remove it. The weight of metal removed per hour would be 
14 X 12 X .375 X .26 x 60 = 1082.8 lbs. Our earlier form of 36" planer has 
removed with one tool on %" cut on work 200 lbs. of metal per hour, and 
the 120" machine has more than five times its capacity. The total pulling 
power of the planer is 45,000 !bs. 

Horse-power Required to Run Lathes. (J. J. Flather, Am. 
Mach., April 23, 1891.)— The power required to do useful work varies with 
the depth and breadth of chip, with the shape of tool, and with the nature 
and density of metal operated upon; and the power required to run a ma- 
chine empty is often a variable quantity. 

For instance, when the machine is new, and the working pai'ts have not 
become worn or fitted to each other as they will be after running a few 
months, the power required will be greater than will be the case after the 
x-unning parts have become better fitted. 

Another cause of variation of the power absorbed is the driving-belt; a 
tight belt will increase the friction, hence to obtain the greatest efficiency 
of a machine we should use wide belts, and run them just tight enough to 
prevent slip. The belts should also be soft and pliable, otherwise power is 
consumed in bending them to the curvature of the pulleys. 

A third cause is the variation of journal-friction, due to slacking up or 
tightening the cap-screws, and also the end-thrust bearing screw. 

Hartig's investigations show that it requires less total power to turn off a 
given weight of metal in a given time than it does to plane off the same 
amount; and also that the power is less for large than for small diameters. 

The following table gives the actual horse-power required to drive a lathe 
empty at varying numbers of revolutions of main spindle. 

HORSE-POWER FOR SMALL LATHES. 



Without Back Gears. I With Back Gears. 



Revs, of 
Spindle 
per min. 



132.72 

219.08 
365.00 



47.4 

125.0 



54.6 
82.2 



H.P. 

required 
to drive 
empty. 



.145 
.197 
.310 



.159 
.259 



.206 
.339 
.455 



.210 
.326 



Revs, ot 
Spindle 
per min. 



24.33 

38.42 



4.84 
12.8 
19.2 



6.61 
14.8 



H.P. 

required 
to drive 
empty. 



.126 
.141 
.274 



.132 
.187 
.230 



.157 
.206 
.249 



.035 
.063 

.087 



20" Fitchburg lathe. 



Smallla the (13^">, Chem- 
nitz. Germany. New 
machine. 



17^" lathe do. New 
machine. 



If H.P. = horse-power necessary to drive lathe empty, and N= number 
of revolutions per minute, then the equation for average small lathes is 
H.P.o = 0.095 -f 0.0012IV. 

For the power necessary to drive the lathes empty when the back gears 
are in, an average equation for lathes under 20" swing is 

H.P.o = 0.10 + 0.006iV. 

The larger lathes vary so much in construction and detail that no general 
rule can be obtained which will give, even approximately, the power re- 
quired to run them, and although the average formula shows that at least 
0.095 horse-power is needed to start the small lathes, there are many Amer- 
ican lathes under 20" swing working on a consumption of less than ,05 
horse-power, 



962 



THE MACHINE-SHOP. 



The amount of power required to remove metal in a machine is determin- 
able within more accurate limits. 

Referring to Dr. Hartig's researches, H.P.! = CW, where C is a constant, 
and Wthe weight of chips removed per hour. 

Average values of C are .030 for cast-iron, .032 for wrought-iron, .047 for 
steel. 

The size of lathe, and, therefore, the diameter of work, has no apparent 
effect on the cutting power. If the lathe be heavy, the cut can be increased, 
and consequently the weight of chips increased, but the value of C appears 
to be about the same for a given metal through several vaiwing sizes of 
lathes. 

Horse-power required to remove Cast Iron in a 20-inch Lathe. 
(J. J. Hobart.) 





m 


d 


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22 


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15 


3 


17 


4 


2 


5 


4 


fi 


1 


7 


1 



Side tool 

Diamond . . . 
Round nose . 
Left - hand round 

nose 

Square -faced tool 

y^' broad . 



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37.90 


.125 


.015 


.342 


13.30 


30.50 


.125 


.015 


.218 


10.70 


42.61 


.125 


.015 


.352 


14.95 


26.29 


.125 


.015 


.237 


9.22 


25.82 


.015 


.125 


.255 


9.06 


25.27 


.048 


.048 


.200 


10.89 


25.64 


.125 


.015 


.246 


8.99 



a . 

> 



.025 
.020 



.028 
.018 
.027 



The above table shows that an average of .26 horse-power is required to 
turn off 10 pounds of cast-iron per hour, from which we obtain the average 
value of the constant = .024. 

Most of the cuts were taken so that the metal would be reduced \i" in 
diameter; with a broad surface cut and a coarse feed, as in No. 5, the power 
required per pound of chips removed in a given time was a maximum; the 
least power per unit of weight removed being required when the chip was 
square, as in No. 6. 

Horse-power required to remove Metal in a 29-inch Lathe. 
(R. H. Smith.) 



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Metal. 


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Cast iron 


12.7 


.05 


.046 


.105 


5.49 


.019 


4 


Cast iron 


11.1 


.135 


.046 


.217 


12.96 


.017 


2 


Cast iron 


12.85 


.04 


.038 


.098 


3.66 


.027 


4 


Wrought iron 


9.6 


.03 


.046 


.059 


2.49 


.023 


4 


Wrought iron 


9.1 


.06 


.046 


.138 


4.72 


.029 


4 


Wrought iron 


7.9 


.14 


.046 


.186 


9.56 


.019 


2 


Wrought iron 


9.35 


.045 


.038 


.092 


2.99 


.031 


4 


Steel 


6.00 


.02 


.046 


.043 


1.03 


.042 


4 


Steel 


5.8 


.04 


.046 


.085 


2.00 


.042 


4 


Steel 


5.1 


.06 


.046 


.108 


2.64 


.040 



POWER REQUIRED FOR MACHINE TOOLS. 963 

The small values of C, .017 and .019, obtained for cast iron are probably 
due to two reasons : the iron was soft and of fine quality, known as pulley 
metal, requiring less power to cut; and, as Prof. Smith remarks, a lower 
cutting-speed also takes less horse-power. 

Hardness of metals and forms of tools vary, otherwise the amount of 
chips turned out per hour per horse-power would be practically constant, the 
higher cutting-speeds decreasing but slightly the visible work done. 

Taking into account these variations, the weight of metal removed per 
hour, multiplied by a certain constant, is equal to the power necessary to do 
the work. 

This constant, according to the above tests, is as follows : 

Cast Iron. Wrought Iron. Steel. 

Hartig 030 .032 .047 

Smith 023 .028 .042 

Hobart ... .024 

Average 026 .030 .044 

The power necessary to run the lathe empty will vary from about .05 to .3 
H.P., which should be ascertained and added to the useful horse-power, to 
obtain the total power expended. 

Power used by Machine-tools. (R. E. Dinsmore, from the Elec- 
trical World.) 

1. Shop shafting 2 3/16" X 180 ft. at 160 revs., carrying 26 pulleys 

from 6" diam. to 36", and running 20 Idle machine belts 1 .32 H.P. 

2. Lodge-Davis upright back-geared drill-press with table, 28" 
swing, drilling %" hole in cast iron, with a feed of 1 in. per 

minute 0.78 H.P. 

3. Morse twist-drill grinder No. 2, carrying 26" wheels at 3200 

revs , 0.29 H.P. 

4. Pease planer 30" X 36", table 6 ft., planing cast iron, cut J4" 

deep, planing 6 sq. in. per minute, at 9 reversals ... 1 .06 H.P. 

5. Shaping-machine 22" stroke, cutting steel die, 6" stroke, %" 

deep, shaping at rate of 1.7 square inch per minute 0.37 H.P. 

6. Engine-lathe 17" swing, turning steel shaft 2%" diam., cut 3/16 

deep, feeding 7.92 inch per minute 0.43 H.P. 

7. Engine-lathe 21" swing, boring cast-iron hole 5" diam., cut 3/16 

diam., feeding 0.3" per minute 0.23 H.P. 

8. Sturtevant No. 2, monogram blower at 1800 revs, per minute, 

no piping . 0.8 H.P. 

9. Heavy planer 28" X 28" X 14 ft. bed, stroke 8", cutting steel, 

22 reversals per minute 3.2 H.P. 

The table on the next page compiled from various sources, principally 
from Hartig's researches, by Prof. J. J. Flather {Am. Mach., April 12, 1894), 
may be used as a guide in estimating the power required to run a given 
machine; but it must be understood that these values, although determined 
by dynamometric measurements for the individual machines designated, 
are not necessarily representative, as the power required to drive a machine 
itself is dependent largely on its particular design and construction. The 
character of the work to be done may also affect the power required to 
operate; thus a machine to be used exclusively for brass work may be 
speeded from 10$ to 15$ higher than if it were to be used for iron work of 
similar size, and the power required will be proportionately greater. 

Where power is to be transmitted to the machines by means of shafting 
and countershafts, an additional amount, varying from 30$ to 50$ of the total 
power absorbed by the machines, will be necessary to overcome the friction 
of the shafting. 

Horse-power required to drive Shafting.— Samuel Webber, 
in his " Manual of Power " gives among numerous tables of power required 
to drive textile machinery, a table of results of tests of shafting. A line of 
2%" shafting, 342 ft. long, weighing 4098 lbs., with pulleys weighing 5331 lbs., 
or a total of 9429 lbs., supported on 47 bearings, 216 revolutions per minute, 
required 1.858 H.P. to drive it. This gives a coefficient of friction of 5.52$. 
In seventeen tests the coefficient ranged from 3.34$ to 11.4$, averaging 
5.73$. 



964 THE MACHINE-SHOP. 



Horse-power Required to Drive Machinery. 



Name of Machine. 


Observed Horse-power. 


Total 
Work. 


Running Light. 


Small screw-cutting lathe 13J^" swing, B, G. 

Screw-cutting lathe 17M>", B. G 


0.41 

0.867 

0.47 

0.462 

0.53 

0.91 

0.16 
0.24 
0.63 
1.14 
0.24 
84 
1.47 
0.62 
0.41 
1.33 
1.24 
0.53 
0.67 
1.08 
0.28 
0.44 
0.95 
0.28 
0.66 

0.18 
0.28 

93 
1.52 

7.12 

4.41 
0.79 
4.12 
2.70 
4.24 
3.03 
4.63 
5.00 
3.20 
6.91 
3.23 
5.64 
0.96 
0.49 

3.68 
2.11 
2.73. 
2.25 
2.00 
2.45 

1.55 
3.11 
0.56 


0.18;0.15*-0.34t 
0.207; 0.16-0.466 
0.12; 0.12 to 0.31 
0.05; 0.03 to 0.33 
0.187; 0.12to0.66 
0.37; 0.39 to 0.81 
0.23 to 3.40 
0.086 to 0.26 
0.07; 0.07 to 0.12 
0.21 ; 0.01 to 0.47 
0.26; 0.15 to 0.73 
0.12; 0.12 to 0.40 

0.27 

0.60 

0.39 
0.15; 0.15 to 0.43 

0.62 

0.62 
0.44;0.1*-0.44t 
0.30; 0.12*-0.80t 

0.46 
0.09; 0.05 to 0.25 
0.22; 0.15 to 0.65 
0.57; 0.43 to 0.94 
0.01; 0.003-0.13 
0.26; 0.26 to 0.55 

0.10 

0.11 

0.12; 0.10-0.12*; 

0.10to0.25t 

37 
0.67 

1.00 
0.16 
0.61 
.54 
3.35 

1 42 
1.25 

O.74J-0.17§ 
1.45 
4.18 
0.70 
1.16 
0.19 
0.34 

1.67; 0.65 to 2.0 
1.42 
0.61 
2.17 
1.30 
2.00 

0.32 
0.24 
0.40' 


Screw-cutting lathe 26", B. G 

Lathe, 80" face plate, will swing 108", T. G 


Large facing lathe, will swing 68", T. G 

Wheel lathe 60" swing 


Small shaper, Richards (9^" X 22") 


Shaper (15" stroke Gould & Eberhardt). .. 

Large shaper, Richards (29" X 91") 

Crank planer (capacity 23" X 27" X 28J^" stroke). . 
Planer (capacity 36' x 36" X 11 feet) 


Small drill press 

Upright slot drilling mach. (will drill 2}4" diam.).. . . 


Large drill press 




Radial drill press 


Slotter (91^" stroke) ... . 


Slotter (15" stroke) 

Universal milling mach (Brown & Sharpe No. 1).... 

Milling machine (13" cutter-head, 12 cutters) 

Small head traversing milling machine (cutter-head 


Gear cutter will cut 20" diameter 

Horizontal boring machine for iron, 22^" swing 


Large plate shears— knives 28" long, 3" stroke 

Large punch press, over-reach 28", 3" stroke, \y^' 

stock can be punched 

Small punch and shear comb'd, 7%" knives, 1}^" str. 
Circular saw for hot iron (30^" diameter of saw). . . 
Plate-bending rolls, diam. of rolls 13", length 9^ ft. 
Wood planer 13}^" (rotary knives, 2 hor'l 2 vert. . . . 
Wood planer 24" (rotary knives) 






Circular saw for wood (23" diameter of saw) 

Circular saw for wood (35" diameter of saw) 




Hor*l wood-boring and mortising machine, drill 4" 

diam., mortise 8J^ deep X HJ^" long 

Tenon and mortising machine 

Tenon and mortising machine . 

Tenon and mortising machine 

Edge-molder and shaper. (Vertical spindle) 

Wood-molding mach. (cap. 714 X 2y»). Hor. spindle 

680 ft. per minute 

Emery wheel HJ/£" diameter x M"- Saw grinder. . 



* With back gears. + Without back gears. % For 
side cutters. B. G., back-geared. T. G., triple-gear 



surface cutters. §With I 
ed. 



ABRASIVE PROCESSES. 



965 



Horse-power consumed in Machine-shops.— How much 
power is required to drive ordinary machine-tools? and h'ow many men can 
be employed per horse-power'? are questions which it ismipossible to answer 
by any fixed rule. The power varies greatly according to the conditions in 
each shop. The following table given by J. J. Flather in his work on Dyna- 
mometers gives an idea of the variation in several large works. The percen- 
tage of the total power required to drive the shafting varies from 15 to 80, 
and the number of men employed per total H.P. varies from 0.62 to 6.04. 

Horse-power; Friction; Men Employed. 



Name of Firm. 



Lane & Bodley 

J. A. Fay & Co 

Union Tron Works 

Frontier Iron & Brass W'ks 

Taylor Mfg. Co 

Baldwin Loco. Works 

W. Sellers & Co. (one de- 
partment) 

Pond Machine Tool Co — 

Pratt & Whitney Co 

Brown & Sharpe Co 

Yale&TowneCo 

Ferracute Machine Co 

T. B. Wood's Sons , 

Bridgeport Forge Co 

Singer Mfg. Co 

Howe Mfg. Co. 

Worcester Mach. Screw Co 
Hartford " " 
Nicholson File Co 





Horse-power. 




o 






<u 


o< 
















































Kind 
of 




T3 . 




T3 . 

SB 




Sec 


Work. 




^£! 


— " 


-u£ 


o 






9> <A 




c a 










- ^ 


O.J3 










?V2 


§<"• 


««2 


fl 


o 




H 


tt 


« 




3 
SB 


o 


E. &W. W. 


58 








132 


2.27 


W. W. 


100 


15 


85 


15 


300 


3.00 


E.,M. M. 


400 


95 


305 


28 


1600 


4.00 


M. E., etc. 


25 


8 


17 


32 


150 


6.00 


E. 


95 








230 


2.42 


L. 


2500 


2000 


500 


80 


4100 


1.64 


H. M. 


102 


41 


61 


40 


300 


2.93 


M.T. 


180 
120 
230 


75 


105 


41 


432 
725 
900 


2.40 
6.04 
3.91 


C. &L. 


135 


67 


68 


49 


700 


5.11 


P. &D. 


35 


11 


24 


31 


90 


2.57 


P. &S. 


12 








30 


2.50 


H. F. 


150 


75 


75 


50 


130 


.86 


S. M. 


1300 
350 








3500 
1500 


2.69 

4.28 


M.S. 


41 








80 


2.00 




40C 


100 


300 


25 


250 


0.62 


F. 


350 








400 


1.14 




346.4 






38.6$ 


818.3 


2.96 






3.53 
5.24 



4.87 
4.11 



10.25 
'5 



Abbreviations: E., engine; W.W., wood-working machinery; M. M., min- 
ing machinery; M. E., marine engines; L., locomotives; H. M., heavy ma- 
chinery; M. T., machine tools; C. & L., cranes and locks; P. & D., presses 
and dies; P. & S., pulleys and shafting; H. F., heavy forgings; S. M., sewing- 
machines; M. S., machine-screws: F., files. 

J. T. Henthorn states (Trans. A. S. M. E., vi. 462) that in print-mills which 
he examined the friction of the shafting and engine was in 7 cases below 
20$ and in 35 cases between 20$ and 30%, in 11 cases from 30$ to 35$ and in 2 
cases above 35$, the average being 25.9$. Mr. Barrus in eight cotton-mills 
found the range to be between 18$ and 25.7$, the average being 22$. Mr. 
Flather believes that for shops using heavy machinery the percentage of 
power required to drive the shafting will average from 40$ to 50$ of the total 
power expended. This presupposes that under the head of shafting are 
included elevators, fans, and blowers. 

ABRASIVE PROCESSES. 

Abrasive cutting is performed by means of stones, sand, emery, glass, 
corundum, carborundum, crocus, rouge, chilled globules of iron, and in some 
cases by soft, friable iron alone. (See paper by John Richards, read before 
the Technical Society of the Pacific Coast, Am. Mach. t Aug. 20, 1891, and 
Eng. & M. Jour., July 25 and Aug. 15, 1891.) 



966 THE MACHINE-SHOP. 

The ki Cold Saw."— For sawing any section of iron while cold the 
cold saw is sometimes used. This consists simply of a plain soft steel or 
iron disk without teeth, about 42 inches diameter and 3/16 inch thick. The 
velocity of the circumference is about 15,000 feet per minute. One of these 
saws will saw through an ordinary steel rail cold in about one minute. In 
this saw the steel or iron is ground off by the friction of the disk, and is not 
cut as with the teeth of an ordinary saw. It has generally been found more 
profitable, however, to saw iron with disks or band-saws fitted with cutting- 
teeth, which run at moderate speeds, and cut the metal as do the teeth of a 
milling-cutter. 

Reese's Fu sing-disk.— Reese's fusing-disk is an application of the 
cold saw to cutting iron or steel in the form of bars, tubes, cylinders, etc., 
in which the piece to be cut is made to revolve at a slower rate of speed 
than the saw. By this means only a small surface of the bar to be cut is 
presented at a time to the circumference of the saw. The saw is about the 
same size as the cold saw above described, and is rotated at a A'elociiy of 
about 25,000 feet per minute. The heat generated by the friction of this saw 
againsfr the small surface of the bar rotated against it is so great that the 
particles of iron or steel in the bar are actually fused, and the " sawdust " 
welds as it falls into a solid mass. This disk will cut either cast iron, wrought 
iron, or steel. It will cut a bar of steel 1% inch diameter in one minute, in- 
cluding the time of setting it in the machine, the bar being rotated about 
200 turns per minute. 

Cutting "Stone with Wire.— A plan of cutting stone by means of a 
wire cord bas been tried m Europe. While retaining sand as the cutting 
agent, M. Paulin Gay, of Marseilles, has succeeded in applying it by mechan- 
ical means, and as continuously as formerly the sand-blast and band-saw, 
with both of which appliances his system — that of the " helicoidal wire 
cord ''—has considerable analogy. An engine puts in motion a continuous 
wire cord (varying from five to seven thirty-seconds of an inch in diameter, 
according to the work), composed of three mild-steel wires twisted at a cer- 
tain pitch, that is found to give the best results in practice, at a speed of 
from 15 to 17 feet per second. 

The Sand-blast.— In the sand-blast, invented by B. F. Tilghman, of 
Philadelphia, and first exhibited at the American Institute Fair, New York, 
in 1871, common sand, powdered quartz, emery, or any sharp cutting mate- 
rial is blown by a jet of air or steam on glass, metal, or other comparatively 
brittle substance, by which means the latter is cut, drilled, or engraved. 
To protect those portions of the surface which it is desired shall not be 
abraded it is only necessary to cover them with a soft or tough material, 
such as lead, rubber, leather, paper, wax, or rubber-paint. (See description 
in App. Cyc. Meeh.; also U. S. report of Vienna Exhibition, 1873, vol. iii. 316.) 
A "jet of sand " impelled by steam of moderate pressure, or even by the 
blast of an ordinary fan, depolishes glass in a few seconds; wood is cut quite 
rapidly; and metals are given the so-called "frosted''' surface with great 
rapidity. With a jet issuing from under 300 pounds pressure, a hole was 
cue through a piece of corundrum 1J,£ inches thick in 25 minutes. 

The sand-blast has been applied to the cleaning of metal castings and 
sheet metal, the graining of zinc plates for lithographic purposes, the frost- 
ing of silverware, the cutting of figures on stone and glass, and the cutting 
of devices on monuments or tombstones, the recutting of files, etc. The 
time required to sharpen a worn-out 14-inch bastard file is about four 
minutes. About one pint of sand, passed through a No. 120 sieve, and four 
horse-power of 60-lb. steam are required for the operation. For cleaning 
castings compressed air at from 8 to 10 pounds pressure per square inch is 
employed. Chilled-iron globules instead of quartz or flint-sand are used 
with good results, both as to speed of working and cost of material, when 
the operation can be carried on under proper conditions. With the expen- 
diture of 2 horse-power in compressing air, 2 square feet of ordinary 
scale on the surface of steel and iron piates can be removed per minute. 
The surface thus prepared is ready for tinning, galvanizing, plating, bronz- 
ing, painting, etc. By continuing the operation the hard skin on the surface 
of castings, which is so destructive to the cutting edges of milling and 
other tools, can be removed. Small castings are placed in a sort of slowly 
rotating barrel, open at one or both ends, through which the blast is 
directed downward against them as they tumble over and over. No portion 
of the surface escapes the action of the sand. Plain cored work, such as 
valve-bodies, can be cleaned perfectly both inside and out. 100 lbs. of cast- 
ings can be cleaned in from 10 to 15 minutes with a blast created by 2 horse- 



EMERY-WHEELS AND GRINDSTONES. 



967 



power. The same weight of small forgings and stampings can be scaled in 
from 20 to 30 minutes.— Iron Age, March 8, 1894. 

EMERY-WHEELS AND GRINDSTONES. 

The Selection of Emery-wheels.— A pamphlet entitled " Emery- 
wheels, their Selection and Use," published by the Brown & Sharpe Mfg. 
Co., after calling attention to the fact that too much should not be expected 
of one wheel, and commenting upon the importance of selecting the proper 
wheel for the work to be done, says : 

Wheels are numbered from coarse to fine; that is, a wheel made of No. 
10 emery is coarser than one made of No. 100. Within certain limits, and 
other things being equal, a coarse wheel is less liable to change the tem- 
perature of the work and less liable to glaze than a fine wheel. As a rule, 
the harder the stock the coarser the wheel required to produce a given 
finish. For example, coarser wheels are required to produce a given sur- 
face upon hardened steel than upon soft steel, while finer wheels are re- 
quired to produce this surface upon brass or copper than upon either 
hardened or soft steel. 

Wheels are graded from soft to hard, and the grade is denoted by the 
letters of the alphabet, A denoting the softest grade. A wheel is soft or 
hard chiefly on account of the amount and character of the material com- 
bined in its manufacture with emery or corundum. But other character- 
istics being equal, a wheel that is composed of fine emery is more compact 
and harder than one made of coarser emery. For instance, a wheel of No. 
100 emery, grade B, will be harder than one of No. 60 emery, same grade. 

The softness of a wheel is generally its most important characteristic. A 
soft wheel is less apt to cause a change of temperature in the work, or to 
become glazed, than a harder one. It is best for grinding hardened steel, 
cast-iron, brass, copper, and rubber, while a harder or more compact wheel 
is better for grinding soft steel and wrought iron. As a rule, other things 
being equal, the harder the stock the softer the wheel required to produce 
a given finish. 

Generally speaking, a wheel should be softer as the surface in contact 
with the work is increased. For example, a wheel 1/16-inch face should be 
harder than one J^-inch face. If a wheel is hard and heats or chatters, it 
can often be made somewhat more effective by turning off a part of its 
cutting surface; but it should be clearly understood that while this will 
sometimes prevent a hard wheel from heating or chattering the work, such 
a wheel will not prove as economical as one of the full width and proper 
grade, for it should be borne in mind that the grade should always bear the 
proper relation to the width. (See the pamphlet referred to for other in- 
formation. See also lecture by T. Dunkin Paret. Pres't of The Tanite Co., 
on Emery-wheels. Jour. Frank. Inst., March, 1890.) 

Speed of Emery-wheels.— The following speeds are recommended 
by different make 



oa 


Revolutions per minute. 


»"1 


Revolutions 


per minute. 




P 


6 


sP 


8* 


IS 


d 


So 


bS 






83 


a a v 

8* 




n 




s 


a3 a <D 


si* 


1 


19,000 








10 


1 950 


2.160 


2,200 


2,200 


i% 


12,500 


14,400 




12,000 


12 


1,600 


1,800 


1,800 


1.850 


2 


9.500 


10,800 




10,000 


14 


1,400 


1,570 


1,600 


1,600 


Wo, 


7,600 


8,640 




8,500 


16 


1,200 


1,350 


1,400 


1,400 


3 


6,400 


7,200 


7,400 


7,400 


IS 


1.050 


1,222 


1,250 


1,250 


4 


4,800 


5,400 


5,400 


5,450 


20 


950 


1,080 


1,100 


1,100 


5 


3,800 


4,320 


4,400 


4,400 


22 


875 


1,000 


1,000 


1,000 


6 


3,200 


3,600 


3,600 


3,600 


24 


800 


917 


925 


925 


7 


2,700 


3,080 


3,200 


3,150 


2fi 


750 




600 


825 


8 


2.400 


2,700 


2,700 


2.750 


30 


675 


733 


500 


735 


9 


2,150 


2,400 


2.400 


2,450 


36 


550 


611 


400 


550 



"We advise the regular speed of 5500 feet per minute." (Detroit Emery- 
wheel Co.) 
"Experience has demonstrated that there is no advantage in running 



968 



THE MACHINE-SHOP. 



16 


" 20 


24 


" 30 


afi 


" 40 


46 


" 60 


70 


44 80 


90 


" 100 


120 


F and FF 



solid emery-wheels at a higher rate than 5500 feet per minute peripheral 
speed." (Springfield E. W. Mfg. Co.) 

" Although there is no exactly defined limit at which a wheel must be run 
to render it effective, experience has demonstrated that, taking into account 
safety, durability, and liability to heat, 5500 feet per minute at the periphery 
gives the best results. All first-class wheels have the number of revolutions 
necessary to give this rate marked on their labels, and a column of figures 
in the price-list gives a corresponding rate. Above this speed all wheels 
are unsafe. If run much below it they wear away rapidly in proportion to 
what they accomplish." (Northampton E. W. Co.) 

Grades of Emery.— The numbers representing the grades of emery 
run from 8 to 120, and the degree cf smoothness of surface they leave may 
be compared to that left by files as follows: 

8 and 10 represent the cut of a wood rasp. 

" " " a coarse rough file. 
" " " an ordinary rough file. 
" " " a bastard file. 
" " " a second-cut file. 
" " " a smooth " 

" " " a superfine " 
" " " a dead-smooth file. 
Speed of Polishing-wheels. 

Wood covered with leather, about 7000 ft. per minute 

"• " " a hair brush, about 2500 revs, for largest 

" " 1]4" to 8" diam., hair 1" to 1J4" long, ab. 4500 " "smallest 

Walrus-hide wheels, about 8000 ft. per minute 

Rag-wheels, 4 to 8 in. diameter, about 7000 " " " 

Safe Speeds for Grindstones and Emery-wheels.— G. D. 
Hiscox (Iron Age, April 7, 1892), by an application of the formula for centrif- 
ugal force in fly-wheels (see Fly-wheels), obtains the figures for strains in 
grindstones and emery-wheels which are given in the tables below. His 
formulas are: 

Stress per sq. in. of section of a grindstone = (.7071 D X iV) 2 x .0000795 
" " " " " " an emery-wheel = (.7071Z) X N)* X 00010226 
D = diameter in feet, N = revolutions per minute. 

He takes the weight of sandstone at .078 lb. per cubic inch, and that of an 
emery-wheel at 0.1 lb. per cubic inch; Ohio stone weighs about .081 lb. and 
Huron stone about .089 lb. per cubic inch. The Ohio stone will bear a speed 
at the periphery of 2500 to 3000 ft. per min., which latter should never be 
exceeded. The Huron stone can be trusted up to 4000 ft., when properly 
clamped between flanges and not excessively wedged in setting. Apart 
from the speed of grindstones as a cause of bursting, probably the majority 
of accidents have really been caused by wedging them on the shaft and over 
wedging to true them. The holes being square, the excessive driving of 
wedges to true the stones starts cracks in the corners that eventually run 
out until the centrifugal strain becomes greater than the tenacity of the 
remaining solid stone. Hence the necessity of great caution in the use of 
wedges, as well as the holding of large quick-running stones between large 
flanges and leather washers. 

Strains in Grindstones. 

Limit of Velocity and Approximate Actual Strain per Square Inch oi 

Sectional Area for Grindstones op Medium Tensile Strength. 



Diam- 


Revolutions per minute. 


eter. 


100 


150 


200 


250 


300 


350 


400 


feet. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


2 


1.58 


3.57 


6.35 


9.93 


14.30 


18.36 


25.42 


2^ 


2.47 


5.57 


9.88 


15.49 


22.29 


28.64 


39.75 


3 


3.57 


8.04 


14.28 


22.34 


32.16 






3^ 


4.86 


10.93 


19.44 


30.38 








4 


6.35 


14.30 


27.37 










V4 


8.04 


18.08 


32.16 










5 


9.93 


22.34 




Approximate breaking strain ter 


6 


14.30 


32.17 




times the strain for size opposite 
the bottom figure in each column. 


7 


19.44 







EMERY-WHEELS AND GRINDSTONES. 



969 



The figures at the bottom of columns designate the limit of velocity (in 
revolutions per minute), at the head of the columns for stones of the diam- 
eter in the first column opposite the designating: figure. 

A general rule of safety for any size grindstone that has a compact and 
strong grain is to limit the peripheral velocity to 47 feet per second. 

There is a large variation in the listed speeds of emery-wheels by different 
makers — 4000 as a. minimum and 5600 maximum feet per minute, while 
others claim a maximum speed of 10,000 feet per minute as the safe speed 
of their best emery-wheels. Rim wheels and iron centre wheels are special- 
ties that require the maker's guarantee and assignment of speed. 

Strains in Emery-wheels. 

Actual Strain per Square Inch of Section in Emery-wheels at the 

Velocities at Iiead of Columns for Sizes in First Column. 



p 








Revolutions per minute. 








600 


800 


1000 


1200 


1400 


1600 


1800 


2000 


2200 


2400 


2600 


4 












22.67 
51.13 
90.71 
141.90 


27.43 
61.86 
109.76 
171.71 


32.64 
73.62 
130.62 


38.31 


6 
















86 40 


8 

m 


'18.40 
24.80 
32.57 
41.41 
50.98 
61.81 
73.62 
86.36 
115.04 
165.64 


"S3! 72 

43.90 
57.65 
73.62 
90.23 
109.41 
130.88 
152.85 


22.67 
35.47 
51.12 
68.70 
90.24 
115.03 
141.22 
171.23 


32 65 
51.08 
73.62 
99.21 
130.31 
165.65 


44.45 
69.51 
100.21 
134.65 
177.80 


58.05 
90.81 
130.88 
175.60 


73.47 
114.94 
165.65 


153.30 


12 








Diam 




16 






Revs, per 


18 








mm. 
















in. 


2800 




22 












3000 


24 

26 
























4 
6 
8 


44.43 

100.21 
177.80 


51.12 


30 














115.03 


36 





































Joshua Rose (Modern Machine-shop Practice) says: The average speed of 
grindstones in workshops may be given as follows: 

Circumferential Speed of Stone. 

For grinding machinists' tools, about 900 feet per minute. 

" " carpenters' " " 600 " " " 

The speeds of stones for file-grinding, and other similar rapid grinding is 
thus given in the " Grinders' List." 

Diam. ft 8 7% 7 6^ 6 5J^ 5 4^ 4 3^ 3 

Revs, per min. 135 144 154 166 180 196 216 240 270 308 360 
The following table, from the Mechanical World, is for the diameter of 
stones and the number of revolutions they should run per minute (not to be 
exceeded), with the diameter of change of shift-pulleys required, varying 
each shift or change 2% inches, 214 inches, or 2 inches in diameter for each 
reduction of 6 inches in the diameter of the stone. 







Shift of Pulleys, in inches. 


Diameter 


Revolutions 






of Stone. 


per minute. 












2y 2 


m 


2 


ft. in. 










8 


135 


40 


36 


32 


7 6 


144 


37^ 


3334 


30 


7 


154 


35 


31^ 


28 


6 6 


166 


32^ 


29^ 


26 


6 


180 


30 




24 


5 6 


196 


27^ 


24% 


22 


5 


216 


25 


22V£ 


20 


. 4 6 


240 


22^ 


20^ 


18 


4 


270 


20 


18 


16 


3 6 


308 


W% 


15% 


14 


3 


360 


15 


13^ 


12 


1 


2 


3 


4 


5 



970 



THE MACHINE-SHOP. 



Columns 3, 4, and 5 are given to show that if we start an 8-foot stone with, 
say, a countershaft pulley driving a 40-inch pulley on the grindstone spindle, 
and the stone makes the right number (135) of revolutions per minute, the 
reduction in the diameter of the pulley on the grinding-stone spindle, when 
the stone has been reduced 6 inches in diameter, will require to be also re- 
duced 2L£ inches in diameter, or to shift from 40 inches to 37*4 inches, and so 
on similarly for columns 4 and 5. Any other suitable dimensions of pulley 
may be used for the stone when eight feet in diameter, but the number of 
inches in each shift named, in order to be correct, will have to be propor- 
tional to the numbers of revolutions the stone should run, as given in column 
2 of the table. 

Varieties of Grindstones. 

(Joshua Rose.) 
For Grinding Machinists' Tools. 



Name of Stone. 



Nova Scotia, > 

Bay Chaleur (New | 

Brunswick), f 

Liverpool or Melling. 



Kind of Grit. Texture of Stone. Color of Stone, 



All kinds, from 
finest to coarsest 

Medium to finest 

Medium to fine 



All kinds, from 
hardest to softest 
Soft and sharp 
Soft, with sharp 



Blue or yellowish 

gi-ay 
Uniformly light 

blue 
Reddish: 



For Wood-working Tools. 



Wickersley 

Liverpool or Melling. 

Bay Chaleur (New ( 

Brunswick)/ f 

Huron. Michigan . .. 



Medium to fine 
Medium to fine \ 

Medium to finest 
Fine 



Very soft 
Soft, with sharp 
grit 

Soft and sharp 

Soft and sharp 



Grayish yellow- 
Reddish 

Uniform light blue 
Uniform light blue 



For Grinding Broad Surfaces, as Saws or Iron Plates. 



Newcastle Coarse to med'm The hard ones Yellow 

Independence Coarse Hard to medium Grayish white 

Massillon Coarse Hard to medium Yellowish white 



TAP DRILLS. 

Taps for Machine-screws. (The Pratt & Whitney Co.) 



Approx. 






Approx. 








Diameter, 


Wire 


No. of Threads 


Diameter, 


Wire 


No. of Threa 


ds 


fractions 


Gauge. 


to inch. 


fractions 


Gauge. 


to inch. 




of an inch. 






of an inch. 










No. 1 


60,72 




No. 13 


20,24 






2 


48, 56, 64 


H 


14 


16, 18, 20, 22 


24 




3 


40, 48, 56 




15 


18, 20, 24 




7/64 


4 


32, 36, 40 


17/64 


16 


16, 18, 20, 22 






5 


30, 32, 36, 40 


9/32 


18 


16, 18, 20 




9/64 


6 


30, 32, 36, 40 




19 


16, 18, 20 






7 


24, 30, 32 


5/16 


20 


16, 18, 20 




5/32 


8 


24, 30. 32, 36, 40 




22 


16, 18 






9 


24, 28, 30, 32 


% 


24 


14, 16, 18 




3/16 


10 


20, 22, £4, 30, 32 




26 


16 






11 


22, 24 




28 


16 




7/32 


12 


20, 22, 24 




30 


16 





The Morse Twist Drill and Machine Co. gives the following table showing 
the different sizes of drills that should be used when a full thread is to be 
tapped iu a hole. The sizes given are practically correct. 



TAP DRILLS. 



971 



? T-i CO O? «C 



SW(00«M©N(OM CO «£ 



„ m £J S3 ., 

T-l CO T-Kr-KlO lOMrtOlCSlCClOrH e^M t- r-l r-Ki-H oa 1-1 ic}\OJ 



.^Eftff* . 



O* O* 01 OJ O? OltOOl OJtOOJ «©« 
OJtOOJ OJtOCO COtOCO CO «5 CO CO i-l CO COt-iCO CO-— 'CO 

•ao\^o^sji\\i-i noico \m sfijt- " OS nsoi-i i-i co \-rm co t- ^ci o ^~ 

H\iO CO 8- --i\05 lO i-i ctfsi-i fr- 1-1 ,-Ki-i 03 i-i uSsOJ i-i O* CCKOJ i-l C> t>SO* i-i CO 


















CO 

■C5~ 


■OJ 


• (M 




:»: 
















• ;^ 


: ;«; 


■ -Ol • 


: :§ : : i 



02 . 

lis 




OJO?> T* "* -<* OJ tO 0? 

— — eo ^ — 

" ~" N ^oi:u 



oo • -0000 • •■>#■<# • • oj o* • • ojoj 

KM ■ -T-Irt • -MH ■ ■«!-! • -HH 

ooaoQOco«o<oto®MKiTfT}iHH(:»5}MiH . -oo 

<o(OffltortiTfirtiT)<:i«««oOrtiHooooo)0>0!aoo(Kooa 



OJ tO OJ O? tO 0? O) tO 0! O* 5D OJ «tDM O? tO O* 

NtfCO — i CO \$CC0 i-i CO \C!TO — CO VXCO rt CO \tCC —i CO \XCO i-i CO 0* SO c? 

ri osmi cot- m t-cscj m-ico m co £- anon \\\ 

« i-i i-i i-i ti 8!n« OJi-iW OJi-iCO i-i i-l CO 



972 



THE MACHINE-SHOP. 



. TAPER BOL.TS, PINS, REAMERS, ETC. 

Taper Bolts for Locomotives,- Bolt-threads, American stan- 
dard, except stay-bolts and boiler-studs, V threads, 12 per inch; valves, 
cocks, and plugs, V threads, 14 per inch, and J^-inch taper per 1 inch. 
Standard bolt taper 1/16 inch per foot. 

Taper Reamers.— The Pratt & Whitney Co. makes standard taper 
reamers for locomotive work taper 1/16 inch per foot from y% inch diam. ; 
7 in. length of flute to 1% inch diam.; 16 in. length of flute, diameters ad- 
vancing by 16ths. P. & W. Co.'s standard taper pin reamers taper J4 in. 
per foot, are made in 14 sizes of diameters, 0.135 to 1.009 in.; length of flute 
1 5/16 in. to 12 in. 



Dimensions of the Pratt & Whitney Company's 
Standard-taper Socket. 



Reamers for Morse 



No. 


Diameter 

Small End, 

inches. 


Diameter 

Large End, 

inches. 


Gauge 
Diam.da'ge 
end, inches 


Gauge 
L'ngth, 
inches. 


Length 
Flute, 
inches. 


Total 
L'ngth. 


Taper 
per foot, 
inches. 


1 
2 
3 

4 
5 
6 


0.374 
0.574 
0.783 
1.027 
1.484 
2.117 


0.525 
0.749 
0.982 
1.283 
1.796 
2.566 


0.481 
0.699 
0.950 
1.232 
1.746 
2.500 


2^ 
2Y 2 
3 5/16 
4 
5 


3 

4 
5 
6 


5J4 

m/ 2 


0.605 
0.600 
0.605 
0.615 
0.625 
0.634 



Standard Steel Taper-pins, 

the Pratt & Whitney Co.: 
Number: 

12 3 4 5 

Diameter large end: 

.156 .172 .193 .219 .250 
Approximate fractional sizes: 

5/32 11/64 3/16 7/32 M 
Lengths from 

H H % % % 
To* 1 114 ^A 1M 2 

Diameter small end of standard taper-reamer : + 
.125 .146 .162 .183 .208 .240 .279 



The following sizes are made by 



289 .341 .409 .492 .591 



19/64 11/32 13/32 






3M 



1 



.331 






5K 



1M 

6 



.581 



Standard Steel Mandrels. (The Pratt & Whitney Co.)— These 
mandrels are made of tool-steel, hardened, and ground true on their cen- 
tres. The ends are of a form best adapted to resist injury likely to be 
caused by driving. They are slightly taper. Sizes, % in. diameter by 3% 
in. long to 3 in. diam. by 14^ in. long, diameters advancing by 16ths. 

PUNCHES AND DIES, PRESSES, ETC. 

Clearance between Pnncli and Die. —For computing the amount 
of clearance that a die should have, or, in other words, the difference in 
size between die and punch, the general rule is to make the diameter of 
die-hole equal to the diameter of the punch, plus 2/10 the thickness of the 
plate. Or, D -■ d X .2f, in which D = diameter of die-hole, d = diameter of 
punch, and t = thickness of plate. For very thick plates some mechanics 
prefer to make the die-hole a little smaller than called for by the above rule. 
For ordinary boiler-work the die is made from 1/10 to 3/10 of the thickness 
of the plate larger than the diameter of the punch; and some boiler-makers 
advocate making the punch fit the die accurately. For punching nuts, the 
punch fits in the die. (Am. Machinist.) 

Kennedy's Spiral Punch. (The Pratt & Whitney Co.)— B. Martell, 
Chief Surveyor of Lloyd's Register, reported tests of Kennedy's spiral 
punches in which a %-inch spiral punch penetrated a %-ineh plate at a pres- 
sure of 22 to 25 tons, while a flat punch required 33 to 35 tons. Steel boiler- 
plates punched with a flat punch gave an average tensile strength of 58,579 

* Taken \%' from extreme end, each size overlaps smaller one about y%" . 
Taper J4" to the foot. + Lengths vary by J4" each size, 



FORCING AND SHRINKING FITS. 973 

lbs. per square inch, and an elongation in two inches across the hole of 5.2$, 
while plates punched with a spiral punch gave 63,929 lbs., and 10.6$ elonga- 
tion. 

The spiral shear form is not recommended for punches for use in metal of 
a thickness greater than the diameter of the punch. This form is of great- 
est benefit when the thickness of metal worked is less than two thirds the 
diameter of punch. 

Size of Blanks used in the Drawing-press. Oberlin Smith 
(Jour. Frank. Inst., Nov. 1886) gives three methods of finding the size of 
blanks. The first is a tentative method, and consists simply in a series of 
experiments with various blanks, until the proper one is found. This is for 
use mainly in complicated cases, and when the cutting portions of the die 
and punch can be finally sized after the other work is done. The second 
method is by weighing the sample piece, and then, knowing the weight of 
the sheet metal per square inch, computing the diameter of a piece having 
the required area to equal the. sample i n weight. The third method is by 
computation, and the formula is x = 4/d 2 + 4dh for sharp-cornered cup, 
where x — diameter of blank, d — diameter of cup, h = height of cup. For 
round-cornered cup where the corner is s mall, say r adius of corner less than 
J4 height of cup, the formula is x — ( Vd' 2 + 4dh) — r, about; r being the 
radius of the corner. This is based upon the assumption that the thickness 
of the metal is not to be altered by the drawing- operation. 

Pressure attainable by the Use of the Drop-press. (R. H. 
Thurston, Trans. A. S. 31. E., v. 53.)— A set of copper cylinders was prepared, 
of pure Lake Superior copper; they were subjected to the action of presses 
of different weights and of different heights of fall. Companion specimens 
of copper were compressed to exactly the same amount, and measures were 
obtained of the loads producing compression, and of the amount of work 
done in producing the compression by the drop. Comparing one with the 
other it was found that the work done with the hammer was 90% of the work 
which should have been done with perfect efficiency. That is to say, 90% of 
the work done in the testing-machine was equal to that due the weight of 
the drop falling the given distance. 

„...-,, - , Weight of drop X fall X efficiency 

Formula: Mean pressure in pounds = - : -. 

compression. 

For pressures per square inch, divide by the mean area opposed to crush- 
ing action during the operation. 

Flow of Metals. (David Townsend, Jour. Frank. Inst., March, 1878.) 
—In punching holes ?/16 inch diameter through iron blocks 1% inches thick, 
it was found that the core punched out was only 1 1/16 inch thick, and its 
volume was only about 32% of the volume of the hole. Therefore, 68% of the 
metal displaced by punching the hole flowed into the block itself, increasing 
its dimensions. 

FORCING AND SHRINKING FITS. 

Forcing Fits of Pins and Axles by Hydraulic Pressure. 

—A 4-iuch axle is turned .015 inch diameter larger than the hole into which 
it is to be fitted. They are pressed on by a pressure of 30 to 35 tons. (Lec- 
ture by Coleman Sellers, 1872.) 

For forcing the crank-pin into a locomotive driving-wheel, when the pin- 
hole is perfectly true and smooth, the pin should be pressed in with a pres- 
sure of 6 tons for every inch of diameter of the wheel fit. When the hole is 
not perfectly true, which may be the result of shrinking the tire on the 
wheel centre after the hole for the crank-pin has been bored, or if the hole is 
not perfectly smooth, the pressure may have to be increased to 9 tons for 
every inch of diameter of the wheel-fit. (Am. Machinist.) 

Shrinkage Fits.— In 1886 the American Railway Master Mechanics' 
Association recommended the following shrinkage allowances for tires of 
standard locomotives. The tires are uniformly heated by gas-flames, slipped 
over the cast-iron centres, and allowed to cool. The centres are turned to a 
diameter equal to the inside diameter of the tire plus the shrinkage allow- 
ance: 

Diameter of tire, in 38 44 50 56 62 66 

Shrinkage allowance, in... .040 .047 .053 .060 .066 .070 

This shrinkage allowance is approximately 1/80 inch per foot, or 1/960. A 
common allowance is 1/1000. Taking the modulus of elasticity of steel at 



974 THE MACHINE-SHOP. 

30,000,000, thestrain caused by shrinkage would be 30,000 lbs. per square inch, 
which is well within the elastic limit of machinery steel. 

SCREWS, SCREW-THREADS, ETC.* 

Efficiency of a Screw.— Let a = angle of the thread, that is, the 
angle whose tangent is the pitch of the screw divided by the circumference 
of a circle whose diameter is the mean of the diameters at the top and 
bottom of the thread. Then for a square thread 

-n ~. • 1 — / tan a 

Efficiency = , , , y — , 

1 + / cotan a 

in which / is the coefficient of friction. (For demonstration, see Cotterill and 
Slade. Applied Mechanics, p. 146.) Since cotan = 1 -=-tan, we may substitute 
for cotan a the reciprocal of the tangent, or if p = pitch, and'c = mean cir- 
cumference of the screw, 

Efficiency = ■. 

Example.— Efficiency of square-threaded screws of y 2 in. pitch. 
Diameter at bottom of thread, in 1 2 3 4 

" top " '« » ... iy 2 2y 2 sy 2 4y 2 

Mean circumference " " "....3.927 7.069 10.21 13.35 

Cotangent a = c -s- p =7.854 14.14 20.42 26.70 

Tangent a = p -4- c '. = .1273 .0661 .0490 .0?75 

Efficiency if /= .10 =55.3* 41.2* 32.7* 27.2* 

'«;/=. 15 = 45* 31.7* 24.4* 19.9* 

The efficiency thus increases with the steepness of the pitch. 

The above formulae and examples are for square-threaded screws, and 
consider the friction of the screw-thread only, and not the friction of the 
oollar or step by which end thrust is resisted* and which further reduces the 
efficiency. The efficiency is also further reduced by giving an inclination to 
the side of the thread, as in the V-threaded screw. For discussion of this 
subject, see paper bv Wilfred Lewis, Jour. Frank. Inst. 1880; also Trans. 
A. S. M. E., vol. xii. 784. 

Efficiency of Screw-bolts.— Mr. Lewis gives the following approx- 
imate formula for ordinary screw-bolts (V threads, with collars): p = 
pitch of screw, d = outside diameier of screw, F = force applied at circum- 
ference to lift a unit of weight, E = efficiency of screw. For an average 
case, in which the coefficient of friction may be assumed at .15, 



f = p±A E = -^ 

3d ' p + d 

For bolts of the dimensions given above, J^-in. pitch, and outside diam- 
eters 1^2, 2J^, 3^, and 4% in., the efficiencies according to this formula 
would be, respectively, .25, .167, .125, and .10. 

James McBride (Trans. A. S. M. E.. xii. 781) describes an experiment with 
an ordinary 2-in. screw-bolt, with a V thread, 4]4 threads per inch, raising 
a weight of 7500 lbs., the force being applied by turning the nut. Of the 
power applied 89.8* was absorbed by friction of the nut on its supporting 
washer and of the threads of the bolt in the nut. The nut was not faced, 
and had the flat side to the washer. 

Prof. Ball in his " Experimental Mechanics " says: "Experiments showed 
in two cases respectively about % and % of the power was lost. 11 

Trautwine says: "In practice the friction of the screw (which under 
heavy loads becomes very great) make the theoretical calculations of but 
little value.' 1 

Weisbach says: " The efficiency is from 19* to 30*." 

Efficiency of a Differential Screw.- A correspondent of the 
American Machinist describes an experiment with a differential screw- 
punch, consisting of an outer screw 2 in. diam., 3 threads per in., aud an 
inner screw 1% in. diam., "&/% threads per inch. The pitch of the outer screw 

* For U. S. Standard Screw-threads, see page 204. 



KEYS. 975 

being ^ in. and that of the inner screw 2/7 in., the punch would ad- 
vance in one revolution y% — 2/7 = 1/21 in. Experiments were made to de- 
termine the force required to punch an 11/16-in. hole in iron J4 in. thick, the 
force being applied at the end of a lever arm of 47% in. The leverage would 
be 47% X 2tt X 21 = (5300. The mean force applied at the end of the lever 
was 95 lbs., and the force at the punch, if there was no friction, would be 
6300 X 95 = 598,500 lbs. The force required to punch the iron, assuming a 
shearing resistance of 50,000 lbs. per sq. in., would be 50,000 x 11/16 x tt X 
V± = 27,000 lbs., and the efficiency of the punch would be 27,000 -=- 598,500 = 
only 4.5^. With the larger screw only used as a punch the mean force at 
the end of the lever was only 82 lbs. The leverage in this case was 47% X 
%ir X 3 = 900, the total force referred to the punch, including friction, 900 X 
82 =3 73,800, and the efficiency 27,000 -=- 73,800 = SQ.7%. The screws were of 
tool-steel, well fitted, and lubricated with lard-oi! and plumbago. 

Powell's New Screw-thread.— A. M. Powell (Am. Mach., Jan. 24, 
1895) has designed a new screw-thread to replace the square form of thread, 
giving the advantages of greater ease in making fits, and provision for " take 
up " in case of wear. The dimensions are the same as those of square- 
thread screws, with the exception that the sides of the thread, instead of 
being perpendicular to the axis of the screw, are inclined 14^° to such per- 
pendicular; ihat is, the two sides of a thread are inclined 29° to each other. 
The formulae for dimensions of the thread are the following: Depth of 
thread = ^ -=- pitch; width of top of thread = width of space at bottom = 
.3707 -=- pitch; thickness at root of thread = width of space at top = .6293 -r- 
pitch. The term pitch is the number of threads to the inch. 

PROPORTIONING PARTS OF MACHINES IN A SERIES 
OF SIZES. 

(Stevens Indicator, April, 1892.) 

The following method was used by Coleman Sellers while at William Sellers 
& Co.'s to get the proportions of the parts of machines, based upon the 
size obtained in building a large machine and a small one to any series of 
machines. This formula is used in getting up the proportion-book and ar- 
ranging the set of proportions from which any machine can be constructed 
of intermediate size between the largest and smallest of the series. 

Rule to Establish Construction Formulae.— Take difference 
between the nominal sizes of the largest and the smallest machines that 
have been designed of the same construction. Take also the difference be- 
tween the sizes of similar parts on the largest and smallest machines se- 
lected. Divide the latter by the former, and the result obtained will be a 
" factor, 1 '' which, multiplied by the nominal capacity of the intermediate 
machine, and increased or diminished by a constant " increment," will give 
the size of the part required. To find the " increment :" Multiply the nomi- 
nal capacity of some known size by the factor obtained, and subtract the 
result from the size of the part belonging to the machine of nominal ca- 
pacity selected. 

Example.— Suppose the size of a part of a 72-in. machine is 3 in., and the 
corresponding part of a 42-in. machine is 1%, or 1.875 in.: then 72 — 42 — 
30, and 3 in. - 1% in. = W s in. = 1.125. 1.125 -+- 30 = .0375 = the " factor," 
and .0375 X 42 = 1.575. Then 1.875 - 1.575 = .3 = the "increment'' 1 to be 
added. Let D = nominal capacity; then the formula will read: x = 
D X .0375 + .3. 

Proof: 42 X .0375 -4- .3 = 1.875, or \% the size of one of the selected parts. 

Some prefer the formula: aD + c — a\ in which D = nominal capacity in 
inches or in pounds, c is a constant increment, a is the factor, and x — the 
part to be found. 

KEYS. 

Sizes of Keys for Mill-gearing. (Trans. A. S. M. E., xiii, 229.)— E. 
G. Parkhurst's rule : Width of key = Y 8 diam. of shaft, depth = 1/9 diam. of 
Shaft; taper % in. to (he foot. 

Custom in Michigan saw-mills : Keys of square section, side = \\ diam. of 
shaft, or as nearly as may lie in even sixteenths of an inch. 

J. T. Hawkins's rule : Width = % diam. of hole; depth of side abutment 
in shaft = % diam. of hole. 

W. S. Huson's rule : J^-inch key for 1 to 1*4 in. shafts, 5/16 key for 1J4 to 
1^4 in. shafts, % in. key for \y% to 1% in. shafts, and so on. Taper % in. to 
the foot. Total thickness at large end of splice, 4/5 width of key. 



976 THE MACHItf E-SHOP. 

Unwin (Elements of Machine Design) gives : Width = y^d -f % in. Thick- 
ness = %d -\- % in., in which d = diam. of shaft in inches. When wheels or 
pulleys transmitting only a small amount of power are keyed on large shafts, 
he says, these dimensions are excessive. In that case, if H.P. = horse- 
power transmitted by the wheel or pulley, N = revs, per min, P = force 
acting at the circumference, in lbs., and R = radius of pulley in inches, take 



3/IOO HJ\ 3/pjg 

1 = y n or y 630 ■ 



Prof. Coleman Sellers (Stevens Indicator, April, 1892) gives the following : 
The size of keys, both for shafting and for machine tools, are the propor- 
tions adopted by William Sellers & Co., and rigidly adhered to during a pe- 
riod of nearly forty years. Their practice in making keys and fitting them 
is, that the keys shall always bind tight sidewise, but not top and bottom; 
that is, not necessarily touch either at the bottom of the key-seat in the 
shaft or touch the top of the slot cut in the gear-wheel that is fastened to 
the shaft ; but in practice keys used in this manner depend upon the fit of 
the wheel upon the shaft being a forcing fit, or a fit that is so tight as to re^ 
quire screw-pressure to put the wheel in place upon the shaft. 

Size of Keys for Shafting. 

Diameter of Shaft, in. Size of Key, in. 

1M 1 7/16 1 11/16 5/16 -x % 

115/16 2 3/16 7/16 x J^ 

2 7/16 9/16 x % 

2 11/16 2 15/16 3 3/16 3 7/16 ll/16x% 

3 15/16 4 7/16 4 15/16 13/16 x % 

6 7/16 5 15/16 6 7/16 15/16x1 

6 15/16 7 7/16 7 15/16 8 7/16 8 15/16.. 1 1/16x1}$ 
Length of key-seat for coupling = 1J4 x nominal diameter of shaft. 

Size of Keys for Machine Tools. 



Diam. of Shaft, in. Size of Key, 
sq. in. 

4 to 5 7/16 13/16 

5^ to 6 15/16 15/16 

7 to 8 15/16 1 1/16 

9 to 10 15/16 1 3/16 

11 to 12 15/16 1 5/16 

13 to 14 15/16 1 7/16 



Diam. of Shaft, in. Siz f n ° f s * ey ' 

15/16 and under % 

1 to 13/16 3/16 

1M tol 7/16 34 

\y 2 tol 11/16 5/16 

1% to 2 3/16 7/16 

2M to 2 11/16 9/16 

2% to 3 15/16 11/16 

John Richards, in an article in Cassier's Magaz me, writes as follows: There 
are two kinds or system of keys, both proper and necessary, but widely dif- 
ferent in nature. 1. The common fastening key, usually made in width one 
fourth of the shaft's diameter, and the depth five eighths to one third the 
width. These keys are tapered and fit on all sides, or, as it is commonly de- 
scribed, " bear all over." They perform the double function in most cases 
of driving or transmitting and fastening the keyed-on member against 
movement endwise on the shaft. Such keys, when properly made, drive 
as a strut, diagonally from corner to corner. 

2. The other kind or class of keys are not tapered and fit on their sides 
only, a slight clearance being left on the back to insure against wedge action 
or radial strain. These keys drive by shearing strain. 

For fixed work where there is no sliding movement such keys are com- 
monly made of square section, the sides only being planed, so the depth is 
more than the width by so much as is cut away in finishing or fitting. 

For sliding bearings, as in the case of drilling-machine spindles, the depth 
should be increased, and in cases where there is heavy strain there should 
be two keys or feathers instead of one. 

The following tables are taken from proportions adopted in practical use. 

Flat keys, as in the first table, are employed for fixed work when the 
parts are to be held not only against torsional strain, but also against move- 
ment endwise ; and in case of heavy strain the strut principle being the 
strongest and most secure against movement when there is strain each way, 
as in the case of engine cranks and first movers generally. The objections 



HOLDING-POWER OF KEYS AND SET-SCREWS. 977 



to the sj'Stem for general use are, straining the work out of truth, the care 
and expense required in fitting, aud destroying the evidence of good or bad 
fitting of the keyed joint. When a wheel or other part is fastened with a 
tapering key of this kind there is no means of knowing whether the work is 
well fitted or not. For this reason such keys are not employed by machine- 
tool-makers, and in the case of accurate work of any kind, indeed, cannot 
be, because of the wedging strain, and also the difficulty of inspecting com- 
pleted work. 

I. Dimensions of Flat Keys, in Inches. 



Diam. of shaft ... 
Breadth of keys 
Depth of keys 



1 


U;, 


m m 


2 


•>u 


3 


3^> 


4 


5 


6 


7 


J 4 


5/16 


%7/16 


to 




n 


% 


1 


m 


m 


1L, 


o/32 


y/io 


M9/82 


;>/l6 


*8 


v/ie 


y s 


% 


11/16 


13/16 


58 



7 8 



II. Dimensions op Square Keys, in Inches. 



Diam. of shaft 

Breadth of keys.. 
Depth of keys 



1 


7/32 


m 


m 


2 


2Yo 


3 


3Yo 


5/32 


9/32 


11/32 


13/32 


15/32 


17/32 


9/16 


3/16 


Ya 


5/16 


% 


7/16 


to 


9/16 


Va 



11/16 



III. Dimensions of Sliding Feather-keys, in Inches. 



Diam. of shaft 

Breadth of keys. 
Depth of keys : . . 



m 


m 


\% 


2 


21/4 
% 


2Yo 


3 


VA 


4 


Va 


Vi 


5/16 


5/16 


% 


Vo 


9/16 


9/16 


% 


% 


7/16 


7/16 


to 


to 


Va 


u 


u 



4y 2 



P. Pryibil furnishes the following table of dimensions to the Am. Machin- 
ist. He says : On special heavy work and very short hubs we put in two 
keys in one shaft 90° apart. With special long hubs, where we cannot use 
keys with noses, the keys should be thicker than the standard. 



Diameter of Shafts, 


Width, 


Thick- 


Diameter of Shafts, 


Width, 


Thick- 


inches. 


inches. 


ness, in. 


inches. 


inches. 


ness.m. 


% tol 1/16 
\y 8 to 1 5/16 


3/16 


3/16 


3 7/16 to 3 11/16 


Va 


Va 


5/16 


Ya 


3 15/16 to 4 3/16 


1 


11/16 


1 7/16 tol 11/16 


Va 


5/16 


4 7/16 to 4 11/16 


15* 


u 


1 15/16 to 2 3/16 




% 


4% to 5% 


m 


15/16 


2 7/16 to 2 11/16 


Va 


to 


5% to 6% 


m 


1 


2 15/16 to 3 3/16 


H 


9/16 


W% to 7% 


m 


m 



Keys longer than 10 inches, say 14 to 16", 1/16" thicker; keys longer than 
10 inches, say 18 to 20", %" thicker; and so on. Special short hubs to have 
two keys. 

For description of the Woodruff system of keying, see circular of the 
Pratt & Whitney Co. ; also Modern Mechanism, page 455. 

HOIiDING-POWER OF KEYS AND SET-SCREWS. 

Tests of the Holding-power of Set-screws in Pulleys. 

(G. Lanza, Trans. A. S. M. E., x. 230.)— These tests were made by using a 
pulley fastened to the shaft by two set-screws with the shaft keyed to the 
holders; then the load required at the rim of the pulley to cause it to slip 
was determined, and this being multiplied by the number 6.037 (obtained by 
adding to the radius of the pulley one-half the diameter of the wire rope, 
and dividing the sum by twice the radius of the shaft, since there were two 
set-screws in action at a time) gives the holding-power of the set-screws. 
The set-screws used were of wrought-iron, % of an inch in diameter, and ten 
threads to the inch; the shaft used was of steel and rather hard, the set- 
screws making but little impression upon it. They were set up with a 
force of 75 lbs. at the end of a ten-inch monkey-wrench. The set-screws 
used were of four kinds, marked respectively A, B, C, and D. The results 
were as follows : 



978 DYNAMOMETERS. 

A, ends perfectly flat, 9/16-in. diameter, 1412 to 2294 lbs. ; average 2064. 

B, radius of rounded ends about y% inch, 2747 " 3079 " " 2912. 

C, " " " " " J4 " 1902 " 3079 " " 2573. 
D ends cup-shaped and case-hardened, 1962 " 2958 " " 2470. 

Remarks. — A. The set-screws were not entirely normal to the shaft ; hence 
they bore less in the earlier ti ials, before they had become flattened by 
wear. 

B. The ends of these set-screws, after the first two trials, were found to 
be flattened, the flattened area having a diameter of about J4 inch. 

G. The ends were found, after the first two trials, to be flattened, as in B. 

D, The first test held well because the edges were sharp, then the holding- 
power fell off till they had become flattened in a manner similar to B, when 
the holding-power increased again. 

Tests of the Holding-power of Keys. (Lanza.)— The load 
was applied as in the tests of set-screws, the shaft being firmly keyed to the 
holders. The load required at the rim of the pulley to shear the keys was 
determined, and this, multiplied by a suitable constant, determined in a sim- 
ilar way to that used in the case of set-screws, gives us the shearing strength 
per square inch of the keys. 

The keys tested were of eight kinds, denoted, respectively, by the letters 
A, B, C, D, E, F, G and H, and the results were as follows : A, B, D and F, 
each 4 tests; E, 3 tests ; C, G, and H, each 2 tests. 

A, Norway iron, 2" x A" X 15/32", 40,184 to 47,760 lbs.; average, 42,726. 

B, refined iron, 2" X Va" X 15/32", 36,482 " 39,254; " 38,059. 

C, tool steel, 1" X M" X 15/32", 91,344 & 100,056. 

D, machinery steel, 2" x Va" X 15/32", 64,630 to 70.186; " 66,875. 

E, Norway iron, \y s " x %" X 7/16", 36,850 " 37,222; " 37,036. 

F, cast-iron, 2" X M" X 15/32", 30,278 " 36,944; ■ " 33,034. 

G, cast-iron, \y B " X %" X 7/16", 37,222 & 38,700. 
H, cast-iron, 1" X W X 7/16", 29,814 & 38,978. 

In A and B some crushing took place before shearing. In E, the keys be- 
ing only 7/16 in. deep, tipped slightly in the key- way. In H, in the first test, 
there was a defect in the key-way of the pulley. 



DYNAMOMETERS. 

Dynamometers are instruments used for measuring power. They are of 
several classes, as : 1. Traction dynamometers, used for determining the 
power required to pull a car or other vehicle, or a plough or harrow. 
2. Brake or absorption dynamometers, in which the power of a rotating 
shaft or wheel is absorbed or converted into heat by the friction of a brake; 
and, 3. Transmission dynamometers, in which the power in a rotating shaft 
is measured during its transmission through a belt or other connection to 
another shaft, without being absorbed. 

Traction Dynamometers generally contain two principal parts: 
(1) A spring or series of springs, through which the pull is exerted, the exten- 
sion of the spring measuring the amount of the pulling force; and (2) a paper- 
covered drum, rotated either at a uniform speed by clockwork, or at a speed 
proportional to the speed of the traction, through gearing, on which the ex- 
tension of the spring is registered by a pencil. From the average height of 
the diagram drawn by the pencil above the zero-line the average pulling 
force in pounds is obtained, and this multiplied by the distance traversed, 
in feet, gives the work done, in foot-pounds. The product divided by the 
time in minutes and by 33,000 gives the horse-power. 

The Prohy brake is the typical form of absorption dynamometer. 
(See Fig. 167, from Flather on Dynamometers and the Measurement of 
Power.) 

Primarily this consists of a lever connected to a revolving shaft or pulley 
in such a manner that the friction induced between the surfaces in contact 
will tend to rotate the arm in the direction in which the shaft revolves. This 
rotation is counterbalanced by weights P, hung in the scale-pan at the end 
of the lever. In order to measure the power for a given number of revolu- 
tions of pulley, we add weights to the scale-pan and screw up on bolts bb, 
until the friction induced balances the weights and the lever is maintained 



THE ALDEN ABSORPTION-DYNAMOMETER. 9?9 

in its horizontal position while" the revolutions of shaft per minute remain 
constant. 

For small powers the beam is generally omitted— the friction being mea- 
sured by weighting a band or strap thrown over the pulley. Ropes or cords 
are often used for the same purpose. 

Instead of hanging weights in a scale-pan, as in Fig. 16T, the friction may be 
weighed on a platform-scale; in this . 

case, the direction of rotation being j; ,_ ml, 

the same, the lever-arm will be on the i 1 *™ -3B^ 

opposite side of the shaft. | /~* \ \ 

In a modification of this brake, the pnf -i-at -t $• 

brake-wheel is keyed to the shaft, ^— / I A 

and its rim is provided with inner ^ eJ / \ 

flanges which form an annular trough / \ 

for the retention of water to keep the /AxAx 

pulley from heating. A small stream J MpP\\ 

of water constantly discharges into 

the trough and revolves with the p IG J57 

pulley— the centrifugal force of the 

particles of water overcoming the action of gravity; a waste-pipe with its 
end flattened is so placed in the trough that it acts as a scoop, and removes 
all surplus water. The brake consists of a flexible strap to which are fitted 
blocks of wood forming the rubbing-surface; the ends of the strap are con- 
nected by an adjustable bolt-clamp, by means of which any desired tension 
may be obtained. 

The horse-power or work of the shaft is determined from the following: 

Let W — work of shaft, equals power absorbed, per minute; 

P = unbalanced pressure or weight in pounds, acting on lever-arm 

at distance L; 
L = length of lever-arm in feet from centre of shaft; 
V = velocity of a point in feet per minute at distance L, if arm were 

allowed to rotate at the speed of the shaft; 
N == number of revolutions per minute; 
H.P. = horse-power. 

Then will W = PV = 2nLNP. 

Since H.P. = PV +- 33,000, we have H.P. = 2irLNP -4- 33,000. 

If L = — , we obtain H.P. = -tj™-. 33 -h 2n is practically 5 ft. 3 in., a value 

often used in practice for the length of arm. 

If the rubbing-surface be too small, the resulting friction will show great 
irregularity— probably on account of insufficient lubrication— the jaws be- 
ing allowed to seize the pulley, thus producing shocks and sudden vibra- 
tions of the lever-arm. 

Soft woods, such as bass, plane-tree, beech, poplar, or maple are all to be 
preferred to the harder w r oods for brake-blocks. The rubbing-surface should 
be well lubricated with a heavy grease. 

The Alden Absorption-dynamometer. (G. I. Alden, Trans. 
A. S. M. E., vol. xi. 958; also xii, 700 and xiii. 429.)— This dynamometer is a 
friction-brake, w r hich is capable in quite moderate sizes of absorbing large 
powers with unusual steadiness and complete regulation. A smooth cast- 
iron disk is keyed on the rotating shaft. This is enclosed in a cast-iron 
shell, formed of two disks and a ring at their circumference, which is free 
to revolve on the shaft. To the interior of each of the sides of the shell is 
fitted a copper plate, enclosing between itself and the side a water-tight 
space. Water under pressure from the city pipes is admitted into each of 
these spaces, forcing the copper plate 'against the central disk. The 
chamber enclosing the disk is filled with oil. To the outer shell is fixed a 
weighted arm, which resists the tendency of the shell to rotate with the 
shaft, caused by the friction of the plates against the central disk. Four 
brakes of this type, 56 in. diam., were used in testing the experimental 
locomotive at Purdue University (Trans. A. S. M. E., xiii. 429). Each was 
designed for a maximum moment of 10,500 foot-pounds with a w r ater-press- 
ure of 40 lbs. per sq. in. 

The area in effective contact with the copper plates on either side is rep- 
resented by an annular surface having its outer radius equal to 28 inches, 
and its inner radius equal to 10 inches. The apparent coefficient of friction 
between the plates and the disk was 3}4%. 



980 



DYNAMOMETERS. 



W. W. Beaumont (Proc. Inst. C. E. 1889) has deduced a formula by means 
of which the relative capacity of brakes can be compared, judging from the 
amount of horse-power ascertained by their use. 

If W = width of rubbing-surface on brake-wheel in inches; V— vel. of 
point on circum. of wheel in feet per minute; K — coefficient; then 

K = WV -f- H.P. 

Capacity of Friction-brakes.— Prof. Flather obtains the values 
of K given in the last column of the subjoined table : 



21 

19 

20 

40 

33 
150 

24 
180 
475 
125 | 
250 f 

401 
125 f 



i 


Brake- 


• 




pulley. 


< 


« . 




.- 


a* 


c % 


5 © 


A 


*£ 




h^ 


bD 


o a 


d a 


a 


P3 


Ed 


Q 


J 


150 


7 


5 


33" 


148.5 


7 


5 


33.38" 


146 


7 


5 


32.19" 


180 


10.5 


5 


32" 


150 


10.5 


5 


32" 


150 


10 


9 




142 


12 


6 


38.31" 


100 


24 


5 


126.1" 


76.2 


24 


7 


191" 


290 1 
250 f 


24 


4 


63" 


322 | 
290 f 


13 


4 


27%" 



Design of Brake. 



Royal Ag. Soc, compensating 

McLaren, compensating 

" water-cooled and comp 

Garrett, " " " 

Schoenheyder, water-cooled 

Balk 

Gately & Kletsch, water-cooled . . . 
Webber, water-cooled — 

Westinghouse, water-cooled 



741 
749 

282 



465 
847 



The above calculations for eleven brakes give values of K varying from 
84 7 to 1385 for actual horse-powers tested, the average being K = 655. 

Instead of assuming an average coefficient, Prof. Flather proposes the 
following : 

Water-cooled brake, non-compensating, K = 400; W — 400 H.P. -4- V. 

Water-cooled brake, compensating, K = 750; W = 750 H.P. -?- V. 

Non-cooling brake, with or without compensating device, K = 900; 
W = 900 H.P. -s- V. 

Transmission Dynamometers are of various forms, as the 
Batchelder dynamometer, in which the power is transmitted through a 
" train-arm " of bevel gearing, with its modifications, as the one described 
by the author in Trans. A. I. M. E., viii. 177, and the one described by 
Samuel Webber in Trans. A. S. M. E., x. 514: belt dynamometers, as the 
Tatham; the Van Winkle dynamometer, in which the power is transmitted 
from a revolving shaft to another in line with it, the two almost touching, 
through the medium of coiled springs fastened to arms or disks keyed to 
the shafts; the Brackett and the Webb cradle dynamometers, used for 
measuring the power required to run dynamo-electric machines. Descrip- 
tions of the four last named are given in Flather on Dynamometers. 

Much information on various forms of dynamometers will be found in 
Trans. A. S. M. E., vol. vii. to xv., inclusive, indexed under Dynamometers. 



OPERATIONS OF A REFRIGERATING-MACHINE. 981 



ICE-MAKING OR REFRIGERATING- MACHINES. 



References.— An elaborate discussion of the thermodynamic theory of 
the action of the various fluids used in the production of cold was published by 
M. Ledoux in the ..4717! ales des Mines, and translated in Van NostrantVs Maga- 
zine in 1879. This work, revised and additions made in the light of recent ex- 
perience by Professors Denton, Jacobus, and Riesenberger, was reprinted in 
1892. (Van Nostrand's Science Series, No. 46.) The work is largely mathe- 
matical, but it also contains much information of immediate practical value, 
from which some of the matter given below is taken. Other references are 
Wood's Thermodynamics, Chap. V., and numerous papers by Professors 
Wood, Denton, Jacobus, and Linde in Trans. A. S. M. E., vols. x. to xiv. ; 
Johnson's Cyclopaedia, article on Refrigeratiug-machines; also Eng'g. June 
18, July 2 and 9, 1886; April 1, 1887; June 15, 1888; July 31. Aug. 28, 1889; Sept. 
11 and Dec. 4, 1891 ; May 6 and July 8, 1892. For properties of Ammonia and 
Sulphur Dioxide, see papers by Professors Wood and Jacobus, Trans. A. S. 
M. E., vols. x. and xii. 

For illustrated articles describing refrigerating-machines, see Am. Mach., 
May 29 and June 26, 1890, and Mfrs. Record, Oct. 7, 1892; also catalogues of 
builders, as Frick & Co., Waynesboro, Pa. ; De La Vergne Refrigerating-ma- 
chine Co , New York; and others. 

Operations of a Refrigerating-machine.— Apparatus designed 
for refrigerating is based upon the following series of operations: 

Compress a gas or vapor by means of some external force, then relieve it 
of its heat so as to diminish its volume; next, cause this compressed gas or 
vapor to expand so as to produce mechanical work, and thus lower its tem- 
perature. The absorption of heat at this stage by the gas, in resuming its 
original condition, constitutes the refrigerating effect of the apparatus. 

A refrigerating-machine is a heat-engine reversed. 

From this similarity between heat-motors and freezing-machines it results 
that all the equations deduced from the mechanical theory of heat to deter- 
mine the performance of the first, apply equally to the second. 

The efficiency depends upon the difference between the extremes of tem- 
perature. 

The useful effect of a refrigerating-machine depends upon the ratio 
between the heat-units eliminated and the work expended in compressing 
and expanding. 

This result is independent of the nature of the body employed. 

Unlike the heat-motors, the freezing-machine possesses the greatest effi- 
ciency when the range of temperature is small, and when the final tempera- 
ture is elevated. 

If the temperatures are the same, there is no theoretical advantage in em- 
ploying a gas rather than a vapor in order to produce cold. 

The choice of the intermediate body would be determined by practical 
considerations based on the physical characteristics of the body, such as the 
greater or less facility for manipulating it, the extreme pressures required 
for the best effects, etc. 

Air offers the double advantage that it is everywhere obtainable, and that 
we can vary at will the higher pressures, independent of the temperature of 
the refrigerant. But to produce a given useful effect the apparatus must 
be of larger dimensions than that required by liquefiable vapors. 

The maximum pressure is determined by the temperature of the con- 
denser and the nature of the volatile liquid: this pressure is often very high. 

When a change of volume of a saturated vapor is made under constant ' 
pressure, the temperature remains constant. The addition or subtraction of 
heat, which produces the change of volume, is represented by an increase or 
a diminution of the quantity of liquid mixed with the vapor. 

On the other hand, when vapors, even if saturated, are no longer in con- 
tact with their liquids, and receive an addition of heat either through com- 
pression by a mechanical force, or from some external source of heat, they 
comport themselves nearly in the same way as permanent gases, and be- 
come superheated. 

It results from this property, that refrigerating-machines using a liquefi- 
able gas wiil afford results differing according to the method of working, 



982 ICE-MAKING OR REFRIGERATING MACHINES. 



and depending upon the state of the gas, whether it remains constantly sat- 
urated, or is superheated during a part of the cycle of working. 

The temperature of the condenser is determined by local conditions. The 
interior will exceed by 9° to 18° the temperature of the water furnished to 
the exterior. This latter will vary from about 52° F., the temperature of 
water from considerable depth below the surface, to about 95° F., the tem- 
perature of surface-water in hot climates. The volatile liquid employed in 
the machine ought not at this temperature to have a tension above that 
which can be readilv managed by the apparatus. 

On the other hand, if the tension of the gas at the minimum temperature 
is too low, it becomes necessary to give to the compression-cylinder large 
dimensions, in order that the weight of vapor compressed by a single stroke 
of the piston shall be sufficient to produce a notably useful effect. 

These two conditions, to which may be added others, such as those de- 
pending upon the greater or less facility of obtaining the liquid, upon the 
dangers incurred in its use, either from its inflammability or unhealthful- 
ness, and finally upon its action upon the metals, limit the choice to a small 
number of substances. 

The gases or vapors generally available are: sulphuric ether, sulphurous 
oxide, ammonia, methylic ether, and carbonic acid. 

The following table, derived from Regnault, shows the tensions of the 
vapors of these substances at different temperatures between — 22° and -f- 
104°. 

Pressures and Boiling-points of Liquids available for 
Use in Refrigerating-machines. 



Temp, of 
Ebullition. 


Tension of Vapor, in lbs. per sq. in., above Zero. 


Deg. 
Fahr. 


Sul- 
phuric 
Ether. 


Sulphur 
Dioxide. 


Ammonia. 


Methylic 
Ether. 


Carbonic 
Acid. 


Pictet 
Fluid. 


- 40 






10.22 
13.23 
16.95 
21.51 
27.04 
33.67 
41.58 
50.91 
61.85 
74.55 
89.21 
105.99 
125.08 
146.64 
170.83 
197.83 
227.76 








- 31 












— 22 

- 13 




5.56 
7.23 
9.27 
11.76 
14.75 
18.31 
22.53 
27.48 
33.26 
39.93 
47.62 
56.39 
66.37 
77.64 
90.32 


11.15 
13.85 
17.06 
20.84 
25.27 
30.41 
36.34 
43.13 
50.84 
59 56 
69.35 
80.28 
92.41 


"25l!e" 
292.9 
340.1 
393.4 
453.4 
520.4 
594.8 
676.9 
766.9 
864.9 
971.1 
1085.6 
1207.9 
1338.2 




- 4 
5 
14 
23 
32 
41 
50 
59 
68 

86 
95 


1.30 
1.70 
2.19 
2.79 
3.55 
4.45 
5.54 
6.84 
8.38 
10.19 
12.31 
14.76 
17.59 


13.5 
16.2 
19.3 
22.9 
26.9 
31.2 
36.2 
41.7 
48.1 
55.6 
64 J. 
73.2 


104 




82.9 



The table shows that the use of ether does not readily lead to the produc- 
tion of low temperatures, because its pressure becomes then very feeble. 

Ammonia, on the contrary, is well adapted to the production of low tem- 
peratures. 

Methylic ether yields low temperatures without attaining too great pres- 
sures at the temperature of the condenser. Sulphur dioxide readily affords 
temperatures of — 14 to — 5, while its pressure is only 3 to 4 atmospheres 
at the ordinary temperature of the condenser. These latter substances then 
lend themselves conveniently for the production of cold by means of 
mechanical force. 

The "Pictet fluid" is a mixture of 97$ sulphur dioxide and 3% carbonic 
acid. At atmospheric pressure it affords a temperature 14° lower than 
sulphur dioxide. 

Carbonic acid is as yet (1895) in use but to a limited extent, but the rela- 
tively greater compactness of compressor that it requires, and its inoffensive 



THE AJvJMoKlA ABSoitMlOtf-MACHlKE. 983 

character, are leading to its recommendation for service on shipboard, where 
economy of space is important. 

Certain ammonia plants are operated with a surplus of liquid present dur- 
ng compression, so that superheating is prevented. This practice is known 
as the "cold system " of compression. 

Nothing definite is known regarding the application of methylic ether or 
of the petroleum product chymogene in practical refrigerating service. The 
inflammability of the latter and the cumbrousness of the compressor 
required are objections to its use. 

"Ice-melting Effect. "— It is agreed that the term "ice-melting 
effect 1 ' means the cold produced in an insulated bath of brine, on the as- 
sumption that each 142.2 B.T.U.* represents one pound of ice, this being the 
latent heat of fusion of ice, or the heat required to melt a pound of ice at 
32° to water at the same temperature. 

The performance of a machine, expressed in pounds or tons of " ice-melt- 
ing capacity," does not mean that the refrigei ating-machine would make 
the same amount of actual ice, but that the cold produced is equivalent to 
the effect of the melting of ice at 32° to water of the same temperature. 

In making artificial ice the water frozen is generally about 70° F. when sub- 
mitted to the refrigerating effect of a machine; second, the ice is chilled from 
12° to 20° below its freezing-point; third, there is a dissipation of cold, from 
the exposure of the brine tank and the manipulation of the ice-cans: there- 
fore the weight of actual ice made, multiplied by its latent heat of fusion, 
142.2 thermal units, represents only ahout three fourths of the cold produced 
in the brine by the refrigerating fluid per I.H.P. of the engine driving the 
compressing-pumps. Again, there is considerable fuel consumed to operate 
the brine-circulating pump, the condeusing-water and feed-pumps, and to 
reboil, or purify, the condensed steam from which the ice is frozen. This 
fuel, together with that wasted in leakage and drip water, amounts to about 
one half that required to drive the main steam-engine. Hence the pounds 
of actual ice manufactured from distilled water is just about half the equiv- 
alent of the refrigerating effect produced in the brine per indicated horse- 
power of the steam-cylinders. 

When ice is made directly from natural water by means of the "plate 
system," about half of the fuel, used with distilled water, is saved by avoid- 
ing the reboiling. and using steam expansively in a compound engine. 

Ether-machines, used in India, are said to have produced about G 
lbs. of actual ice per pound of fuel consumed. 

The ether machine is obsolete, because the density of the vapor of ether, 
at the necessary working-pressure, requires that the compressing-cylinder 
shall be about 6 times larger than for sulphur dioxide, and 17 times larger 
than for ammonia. 

Air-machines require about 1.2 times greater capacity of compress- 
ing cylinder, and are, as a whole, more cumbersome than ether machines, 
but they remain in use on ship-board. In using air the expansion must take 
place in a cylinder doing work, instead of through a simple expansion-cock 
which is used with vapor machines. The work done in the expansion-cylin- 
der is utilized in assisting the compressor. 

Ammonia Compression-machines. — "Cold " vs. "Dry " Systems 
of Compress ion. —In the "cold" system or "humid" system some of the 
ammonia entering the compression-cylinder is liquid, so that the heat de- 
veloped in the cylinder is absorbed by the liquid and the temperature of the 
ammonia thereby confined to the boiling-point due to the condenser-pres- 
sure. No jacket is therefore required about the cylinder. 

In the " dry" or " hot" system all ammonia entering the compressor is 
gaseous, and the temperature becomes by compression several hundred de- 
grees greater than the boiling-point due to the condenser-pressure. A water- 
jacket is therefore necessary to permit the cylinder to be properly lubri- 
cated. 

Relative Performance of* Ammonia Compression- and 
Absorption-machines, assuming no Water to be En- 
trained with tbe Ammonia-gas in the Condenser. (Denton 
and Jacobus, Trans. A. S. M. E., xiii.) — It is assumed in the calculation for 
both machines that 1 lb. of coal imparts 10,000 B.T.H. to the boiler. The 

* The latent heat of fusion of ice is 144 thermal units (Phil. Mag., 1871, 
xli.. 182); but it is customary to use 142. (Prof. Wood, Trans. A. S. M. E., 

xi. 834.) ■ . . , ■ , . . , • 



984 ICE-MARlHG OR REFRIGERATING MACHINES. 



condensed steam from the generator of the absorption-machine is assumed 
to be returned to the boiler at the temperature of the steam entering the 
generator. The engine of the compression-machine is assumed to exhaust 
through a feed-water heater that heats the feed-water to 21v2° F. The engine 
is assumed to consume 2034 lbs - °f water per hour per horse-power. The 
figures for the compression- machine include the effect of friction, which is 
taken at 15$ of the net work of compression. 



Condenser. 


Refrigerat- 
ing Coils. 




Pounds of Ice-melting Effect 
per lb. of Coal. 


O CD 
ft 








fa 






20 . 








o 


Com 


Dress. 


Absorption- 












Machine. 


machine.* 




£ 


.a 


£ 










cffl eS 


fn 


3 
a 


gfa 




amm. 
xhausts 
osphere 
eater, 

temp, 
ater. 


MaT-p 




2 

3 


fa 

0) 


s 


1 


"3 
8 fa 


Oh 


2| = 2 


2'B'o 


So 

.a 

a 


P. 
£> CO 


So 
<h 

'O 

a 
ft 

a 


03 

ft 

CD 

s.a 

,£) CO 


o 
i§ 
«1 

o 

S 


tap 
.S ° 


CO J 

£ a 

SO 3 

T-J O 

XI 

C 03 
a ft 


asorption-ma 
which the a 
circulating-p 
hausts into t 
erator. 


which the 
circ. pump e 
into the atm 
through a h 
yielding 212° 
to the feed-w 


^ 2 § 
III 

s a«J 
""' 1 ° 
ai tf£ 


^ 


< 


EH 


<j 


H 


t> 


P 


< 


M 


W 


61.2 


110.6 


5 


33.7 


61.2 


38.1 


71.4 


38.1 


33.5 


969 


59.0 


106.0 


5 


33.7 


59.0 


39.8 


74.6 


38.3 


33.9 


967 


59.0 


106.0 


5 


33.7 


130.0 


39.8 


74.6 


39.8 


35.1 


931 


59.0 


106.0 


-22 


16.9 


59.0 


23.4 


43.9 


36.3 


31.5 


1000 


86.0 


170.8 


5 


33.7 


86.0 


25.0 


46.9 


35.4 


28.6 


988 


86.0 


170.8 


5 


33.7 


130.0 


25.0 


46.9 


36.2 


29.2 


966 


86.0 


170.8 


-22 


16.9 


86.0 


16.5 


30.8 


33.3 


26.5 


1025 


86.0 


170.8 


-22 


16.9 


130.0 


16.5 


30.8 


34.1 


27.0 


1002 


104 


227.7 


5 


33.7 


104.0 


19.6 


36.8 


33.4 


25.1 


1002 


104.0 


227.7 


-22 


16.9 


104.0 


13.5 


25.3 


31.4 


23.4 


1041 



The Ammonia Absorption-machine comprises a generator 
which contains a concentrated solution of ammonia in water; this gener- 
ator is heated either directly by a fire, or indirectly by pipes leading from a 
steam-boiler. The condenser communicates with the upper part of the gen- 
erator by a tube; it is cooled externally by a current of cold water. The 
cooler or brine-tank is so constructed as to utilize the cold produced; the up- 
per part of it is in communication with the lower part of the condenser. 

An absorption-chamber is filled with a weak solution of ammonia; a tube 
puts this chamber in communication with the cooling-tank. 

The absorption-chamber communicates with the boiler by two tubes: one 
leads from the bottom of the generator to the top of the chamber, the other 
leads from the bottom of the chamber to the top of the generator. Upon 
the latter is mounted a pump, to force the liquid from the absorption -cham- 
ber, where the pressure is maintained at about one atmosphere, into the gen- 
erator, where the pressure is from 8 to 12 atmospheres. 

To work the apparatus the ammonia solution in the generator is first 
heated. This releases the gas from the solution, and the pressure rises. 
When it reaches the tension of the saturated gas at the temperature of the 
condenser there is a liquefaction of the gas, and also of a small amount of 
steam. By means of a cock the flow of the liquefied gas into the refrigerat- 
ing-coils contained in the cooler is regulated. It is here vaporized by ab- 
sorbing the heat from the substance placed there to be cooled. As fast as it 
is vaporized it is absorbed by the weak solution in the absorbing-chamber. 

Under the influence of the heat in the boiler the solution is unequally sat- 
urated, the stronger solution being uppermost. 

The weaker portion is conveyed by the pipe entering the top of the absorb- 
ing-chamber, the flow being regulated by a cock, while the pump sends an 
equal quantity of strong solution from the chamber back to the boiler. 

* 5$ of water entrained in the ammonia will lower the economy of the ab- 
sorption-machine about 15$ to 20$ below the figures given in the table. 



SULPHUR-DIOXIDE MACHINES. 



985 



The working of the apparatus depends upon the adjustment and regula- 
tion of the flow of the gas and liquid; by these means the pressure is varied, 
and consequently the temperature in the cooler may be controlled. 

The working is similar to that of compression-machines. .The absorption- 
chamber fills the office of aspirator, and the generator plays the part of 
compressor. 

The mechanical force producing exhaustion is here replaced by the affinity 
of water for ammonia gas; and the mechanical force required for compres- 
sion is replaced by the heat which severs this affinity and sets the gas at 
liberty. 

(For discussion of the efficiency of the absorption system, see Ledoux's 
work; paper by Prof. Linde, and discussion on the same by Prof. Jacobus, 
Trans. A. S. M. E., xiv. 1416, 1436; and papers by Denton and Jacobus, 
Trans. A. S. M. E. x. 792; xiii. 507. 

Sulphur-Dioxide Machines.— Results of theoretical calculations 
are given in a table by Ledoux showing an ice-melting capacity per 
hour per horse-power ranging from 134 to 63 lbs., and per pound of coal 
ranging from 44.7 to 21.1 lbs., as the temperature corresponding to the 
pressure of the vapor in the condenser rises from 59° to 104° F. The theo- 
retical results do not represent the actual. It is necessary to take into ac- 
count the loss occasioned by the pipes, the waste spaces in the cylinder, loss 
of time in opening of the valves, the leakage around the piston and valves, 
the reheating by the external air, and finally, when the ice is being made, 
the quantity of the ice melted in removing the blocks from their moulds. 
Manufacturers estimate that practically the sulphur-dioxide apparatus using 
water at 55° or 60° F. produces 56 lbs. of ice, or about 10,000 heat-units, per 
hour per horse-power, measured on the driving-shaft, which is about 55$ of 
the theoretical useful effect. In the commercial manufacture of ice about 
7 lbs. are produced per pound of coal. This includes the fuel used for re- 
boiling the water, which, together with that wasted by the pumps and lost 
oy radiation, amounts to a considerable portion of that used by the engine. 

Prof. Denton says concerning Ledoux's theoretical results: The figures 
given are higher than those obtained in practice, because the effect of 
superheating of the gas during admission to the cylinder is not considered. 
This superheating may cause an increase of work of about 25$. There are 
other losses due to superheating the gas at the brine-tank, and in the pipe 
leading from the brine-tank to the compressor, so that in actual practice a 
sulphur-dioxide machine, working under the conditions of an absolute 
pressure in the condenser of 56 lbs. per sq. in. and the corresponding tem- 
perature of 77° F., will give about 22 lbs. of ice-melting capacity per pound 
of coal, which is about 60$ of the theoretical amount neglecting friction, or 
70$ including friction. The following tests, selected from those made by 
Prof. Schroter on a Pictet ice-machine having a compression-cylinder 11.3 
In. bore and 24.4 in. stroke, show the relation between the theoretical and 
actual ice-melting capacity. 





Temp, in degrees Fahr. 
corresponding to 
pressure of vapor. 


Ice-melting capacity per pound of coal, 
assuming 3 lbs. per hour per H.P. 


No. of 
Test. 


Condenser. 


Suction. 


Theoretical 

friction 
included.* 


Actual. 


Per cent loss due to 
cylinder super- 
heating, or differ- 
ence between 
cols. 4 and 5. 


11 
12 
13 

14 


77.3 
76.2 
75.2 
80.6 


28.5 

14.4 

-2.5 

-15.9 


41.3 
31.2 
23.0 
16.6 


33.1 
24.1 
17.5 
10.1 


19.9 
22.8 
23.9 
39.2 



The Refrigerating Coils of a Pictet ice-machine described by 
Ledoux had 79 sq. ft. of surface for each 100,000 theoretic negative heat-units 
produced per hour. The temperature corresponding to the pressure of the 
dioxide in the coils is 10.4° F., and that of the bath "(calcium chloride solu- 
tion) in which they were immersed is 19 4°. 

* Friction taken at figure observed in the test, Avhich ranged from 23$ to 
§6$ of the work of the steam-cylinder, 



986 ICE-MAKING OR REFRIGERATING MACHINES. 



d I ^ 5^ 



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AMMONIA COMPRESSION-MACHINES. 



987 



03 

bJO 

a 
*d 
o 
Q 


•sanoq f% m ifcjioud 
-i?0 3ui%\aw-a6i jo 
uoj, J9d ajnujm .i9j 


"3 


(19) 
.89 
.92 
.94 
.96 
.98 

1.00 


63 
tf 
P 

Ifi 
63 

B 
63 

BS 

O ' 

B b 
<° 
. 63 

w B 

J £-, 

10 i 

Oi B 

5 
Oc 

go 

B& 

H 63 

££ 

.4 

M° 

H 

B Q 
B 2i 

Eh < 

°B 

is 

§^ 

s < 
<1b 

s 

03 B 

J s 

» c 
|~ 

O 63 
' K 

85 


(19) 
.97 
.99 
1.02 
1.04 
1.07 
1.09 


•^4}OBdBO Sill 
-3I9UI-90I JO UOJ, J9J 




(18) 
1290 
1320 
1350 
1380 
1410 
1440 


(18) 
1,390 
1,420 
1,470 
1,500 
1,540 
1,570 


•duidx Jo 
9Sa«a 0O8 Suuuns 

-SB '(J119lU90B[dS|Q 

1104s jj jo -4 j •na'j9 < j 


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(17) 

.2872 

.2882 

.2890 

.2898 

.2904 

.2910 


(17) 

.1611 

.1620 

.1628 

.1636 

.1643 

.1649 


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(16) 
.000223 
.000219 
.000215 
.000211 
.000206 
.000202 


(16) 
.000116 
.000114 
.000111 
.000109 
.000107 
.000105 


pie 


1" 


uotjou^ q^iAV 




(15) 
39.8 
33.4 
28.7 
25.0 
22.0 
19.6 


(15) 
23.4 
20.6 
18.4 
16.5 
14.9 
13.5 


•nor; 
-ouji ^noqijAi 


(J 


(14) 
45.8 
38.4 
33.0 
28.7 
25.3 
22.6 


(14) 
26.9 
23.7 
21.1 
18.9 
17.1 
15.5 




•uoi^ou^ q^AV 




(13) 
119.3 
100.2 
86.1 
75.0 
66.1 
58.9 


(13) 
70.2 
61.8 
55.1 
49.4 
44.7 
40.6 


•noq 
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(12) 
137.0 
115.2 
99.0 
86.2 
76.0 
67.7 


(12) 
80.7 
71.0 
63.3 
56.8 
51.4 
46.6 


.5 


•nopoijg; Saipnpui 
''d'H J9 d anon J^d 


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(11) 
16,960 
14,250 
12,240 
10,660 
9,400 
8,380 


(11) 
9,980 
8,790 
7,840 
7,030 
6,360 
5,780 


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(10) 
.00504 
.00444 
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7,410 
8,660 
9,890 
11,130 
12,360 
13.590 


(8) 

6,530 
7.280 
8,000 
8,750 
9,480 
10,200 


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(7) 

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8,600 
9,680 
10,750 
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(7) 
5,680 
6,330 
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7,610 
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8,870 


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3.95 
3.52 
3.15 
2.85 
2.59 


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(5) 
63.47 
62.31 
61.13 
59.93 
58.70 
57.45 


(5) 
32.93 
32.31 
31.69 
31.05 
30.41 
29.75 


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(4) 
71.81 
72.05 
72.26 
72.46 
72.61 
72.74 


(4) 
40.28 
40.50 
40.70 
40.90 
41.07 
41.23 


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(3) 

154.6 
179.9 
205.1 
230.3 
255.4 
280.:] 


(3) 
224.1 
252.2 

280.2 
308.3 
336.2 
364.0 


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(2) 

106.0 
125.1 
146.6 
170.8 
197.8 
227.8 


(2) 
106.0 
125.1 
146.6 
170.8 
197.8 
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988 ICE-MAKItfG OR REFRIGERATING MACHINES. 

The following is a comparison of the theoretical ice-melting capacity of an 
ammonia compression machine with that obtained in some of Prof. 
Schroter's tests on a Linde machine having a compression-cylinder 9.9-in. 
bore and 16.5 in. stroke, and also in tests by Prof. Denton on a machine 
having two single-acting compression cylinders 12 in. X 30 in.: 



No. 


Temp, in Degrees F. 

Corresponding to 
Pressure of Vapor. 


Ice-melting Capacity per lb. of Coal, 

assuming 3 lbs per hour per 

Horse-power. 


of 
Test. 


Condenser. 


Suction. 


Theoretical, 
Friction * in- 
. eluded. 


Actual. 


Per Cent 
of Loss Due to 

Cylinder 
Superheating. 


5 f 1 

£ 1 3 

6 I 4 


72.3 
70.5 
69.2 
68.5 


26.6 

14.3 

0.5 

-11.8 


50.4 
37.6 
29.4 
22.8 


40.6 
30.0 
22.0 
16.1 


19.4 
20.2 
25.2 
29.4 


§ (34 
t <36 
(3 (25 


84.2 
82.7 
84.6 


15.0 
- 3.2 

-10.8 


27.4 
21.6 

18.8 


24.2 
17.5 
14.5 


11.7 
19.0 
22.9 



Refrigerating Machines using Vapor of Water. (Ledoux.) 
— In these machines, sometimes called vacuum machines, water, at ordi- 
nary temperatures, is injected into, or placed in connection with, a chamber 
in which a strong vacuum is maintained. A portion of the water vaporizes, 
the heat to cause the vaporization being supplied from the water not vapor- 
ized, so that the latter is chilled or frozen to ice. If brine is used instead of 
pure water, its temperature may be reduced below the freezing-point of 
water. The water vapor is compressed from, say, a pressure of one tenth 
of a pound per square inch to one and one half pounds, and discharged into 
a condenser. It is then condensed and removed by means of an ordinary 
air-pump. The principle of action of such a machine is the same as that 
of volatile-vapor machines. 

A theoretical calculation for ice-making, assuming a lower temperature 
of 32° F., a pressure in the condenser of \y% lbs. per square inch, and a coal 
consumption of 3 lbs. per I.H.P. per hour, gives an ice-melting effect of 34.5 
lbs. per pound of coal, neglecting friction. Ammonia for ice-making condi- 
tions gives 40.9 lbs. The volume of the compressing cylinder is about 150 
times the theoretical volume for an ammonia machine for these conditions. 

Relative Efficiency of a Refrigerating Machine.— The effi- 
ciency of a refrigerating machine is sometimes expressed as the quotient of 
the quantity of heat received by the ammonia from the brine, that is, the 
quantity of useful work done, divided by the heat equivalent of the mechan- 
ical work done in the compressor. Thus in column 1 of the table of perform- 
ance of the 75-ton machine (page L98) the heat given by the brine to the 
ammonia per minute is 14,776 B.T.U. The horse-power of the ammonia cylin- 
der is 65.7. and its heat equivalent = 65.7 X 33,000 -*- 778 = 2786 B.T.U. Then 
14,776 -f- 2786 = 5.304, efficiency. The apparent paradox that the efficiency 
is greater than unity, which is impossible in any machine, is thus explained. 
The working fluid, as ammonia, receives heat from the brine and rejects 
heat into the condenser. (If the compressor is jacketed, a portion is rejected 
into the jacket-water.) The heat rejected into the condenser is greater than 
that received from the brine; the difference (plus or minus a small difference 
radiated to or from the atmosphere) is heat received by the ammonia from 
the compressor. The work to be done by the compressor is not the mechan- 
ical equivalent of the refrigeration of the brine, but only that necessary to 
supply the difference between the heat rejected by the ammonia into the con- 
denser and that received from the brine. If cooling water colder than the 
brine were available, the brine might transfer its heat directly into the cool- 
ing water, and there would be no need of ammonia or of a compressor; but 



* Friction taken at figures observed in the tests, which range from 14$ to 
20% of the work of the steam-cylinder. 



EFFICIENCY OF HEFMGERATlNG-MACHiHES. 989 

since such cold water is not available, the brine rejects its heat into the 
colder ammonia, and then the compressor is required to heat the ammonia 
to such a temperature that it may reject heat into the cooling water. 

The efficiency of a refrigerating plant referred to the amount of fuel 
consumed is 

of brine or other 



Ice-melting capacity | 
per pound of fuel, f 



( of temperature 



Pounds circulated per hour 
X specific heat x range 



circulating fluid. 



142.2 X pounds of fuel used per hour. 
The ice-melting capacity is expressed as follows: 



Tons (of 2000 lbs.) ) 
ice-melting ca- > 
pacity per 24 hours } 



24 X pounds 
X specific heat 
X range of temp. ) 



>■ of brine circulated per hour. 



142.2 X 2000 



The analogy between a heat-engine and a refrigerating -machine is as fol- 
lows: A steam-engine receives heat from the boiler, converts a part of it 
into mechanical work in the cylinder, and throws away the difference into 
the condenser. The ammonia in a compression refrigerating machine re- 
ceives heat from the brine-tank or cold-room, receives an additional amount 
of heat from the mechanical work done in the compression-cylinder, and 
throws away the sum into the condenser. The efficiency of the steam-engine 
= work done -f- heat received from boiler. The efficiency of the refrigerat- 
ing-machine = heat received from the brine-tank or cold-room -*- heat re- 
quired to produce the work in the compression-cylinder. In the ammonia 



Cold Water 

J i «. 




■X- 

Compressor " 


„o Brine Outlet 
V 


fr 


— *- 


Condenser 


309 c 
82° 


239°' 10° 3° 

v 64° 


Brine Tank 

Ammonia 
Coils 


Cold Room 








•*• 

Heat received 
from compression. 




1 1 »• 

Warm Water 
Heat rejected 


\X- 






14° 

Heat received 
from brine 


Inlet 



DIAGRAM OF AMMONIA COMPRESSION MACHINE. 



1 54° 




i J 



Torce Pump 
DIAGRAM OF AMMONIA ABSORPTION MACHINE. 



absorption-apparatus, the ammonia receives heat from the brine-tank and 
additional heat from the boiler or generator, and rejects the sum into the 
condenser and into the cooling water supplied to the absorber. The effi- 
ciency = heat received from the brine h- heat received from the boiler. 



§90 ICE-MAKlHG Oil ftEFElGERATING MACHINES. 

TEST-TRIALS OF REFRIGEBATING-MACHINES. 

(G. Linde, Trans. A. S. M. E., xiv. 1414.) 

The purpose of the test is to determine the ratio of consumption and pro- 
duction, so that there will have to be measured both the refrigerative effect 
and- the heat (or mechanical work) consumed, also the cooling water. The 
refrigerative effect is the product of the number of heat-units (Q) abstracted 

from the body to be cooled, and the quotient — c —, — ; in which Tc = abso- 
lute temperature at which heat is transmitted to the cooling water, and T = 
absolute temperature at which heat is taken from the body to be cooled. 

The determination of the quantity of cold will be possible with the proper 
exactness only when the machine is employed during the test to refrigerate 
a liquid; and if the cold be found from the quantity of liquid circulated per 
unit of time, from its range of refrigeration, and from its specific heat. 
Sufficient exactness cannot be obtained by the refrigeration of a current of 
circulating air, nor from the manufacture of a certain quantity of ice, nor 
from a calculation of the fluid circulating within the machine (for instauce, 
the quantity of ammonia circulated by the compressor). Thus the refrig- 
eration of brine will generally form the basis for tests making any pretension 
to accuracy. The degree of refrigeration should not be greater than neces- 
sary for allowing the range of temperature to be measured with the neces- 
sary exactness; a range of temperature of from 1° to 6° Fahr. will suffice. 

The con dense :• measurements for cooling water and its temperatures will 
be possible with sufficient accuracy only with submerged condensers. 

The measurement of the quantity of brine circulated, and of the cooling 
water, is usually effected by water-meters inserted into the conduits. If the 
necessary precautions are observed, this method is admissible. For quite 
precise tests, however, the use of two accurately gauged tanks must be ad- 
vised, which are alternately filled and emptied. 

To measure the temperatures of brine and cooling water at the entrance 
and exit of refrigerator and condenser respectively, the employment of 
specially constructed and frequently standardized thermometers is indis- 
pensable; no less important is the precaution of using at each spot simul- 
taneously two thermometers, and of changing the position of one such 
thermometer series from inlet to outlet (and vice versa) after the expiration 
of one half of the test, in order that possible errors may be compensated. 

It is important to determine the specific heat of the brine used in each 
instance for its corresponding temperature range, as small differences in the 
composition and the concentration may cause considerable variations. 

As regards the measurement of consumption, the programme will not have 
any special rules in cases where only the measurement of steam and cooling 
water is undertaken, as will be mainly the case for trials of absorption-ma- 
chines. For compression-machines the steam consumption depends both 
on the quality of the steam-engine and on that of the ref rigerating-machine, 
while it is evidently desirable to know the consumption of the former sep- 
arately from that of the latter. As a rule steam-engine and compressor are 
coupled directly together, thus rendering a direct measurement of the power 
absorbed by the refrigerating-machine impossible, and it will have to suffice 
to ascertain the indicated work both of steam-engine and compressor. By 
further measuring the work for the engine running empty, and by compar- 
ing the differences in power between steam-engine and compressor resulting 
for wide variations of condenser-pressures, the effective consumption of 
work Le for the refrigerating-machine can be found very closely. In gen- 
eral, it will suffice to use the indicated work found in the steam-cylinder, 
especially as from this observation the expenditure of heat can be directly 
determined. Ordinarily the use of the indicated work in the compressor- 
cylinder, for purposes of comparison," should be avoided; firstly, because 
there are usually certain accessory apparatus to be driven (agitators, etc.), 
belonging to the refrigerating-machine proper; and secondly, because the 
external friction would be excluded. 

Heat Balance. — We possess an important aid for checking the cor- 
rectness of the results found in each trial by forming the balance in each 
case for the heat received and rejected. Only such tests should be re- 
garded as correct beyond doubt which show a sufficient conformity in the 
heat balance. It is true that in certain instances it may not be easy to 
account fully for the transmission of heat between the several parts of the 
machine and its environment by radiation and convection, but generally 



TEMPERATURE RANGE. 991 

(particularly for compression-machines) it will be possible to obtain for the 
neat received and rejected a balance exhibiting: small discrepancies only. 

Report of Test.— Reports intended to be used for comparison with 
the figures found for other machines will therefore have to embrace at least 
the following observations : 
Refrigerator: 

Quantity of brine circulated per hour 

Brine temperature at inlet to refrigerator -.. 

Brine temperature at outlet of refrigerator t 

Specific gravity of brine (at 64° Fahr.) 

Specific heat of brine 

Heat abstracted (cold produced) Qe 

Absolute pressure in the refrigerator 

Condenser : 

Quantity of cooling water per hour 

Temperature at inlet to condenser 

Temperature at outlet of condenser t 

Heat abstracted Q x 

Absolute pressure in the condenser 

Temperature of gases entering the condenser 



COMPRESSTON-MACHINE. 

Compressor : 

Indicated work Lt 

Temperature of gases at inlet. . 

Temperature of gases at exit.. 
Steam-engine : 

Feed-water per hour 

Temperature of feed-water 

Absolute steam-pressure before 
steam-engine 

Indicated work of steam-engine 
Le 

Condensing water per hour 

Temperature of da 

Total sum of losses by radiation 

and convection ± Q 3 

Heat Balance : 

Qe + ALo = Qi ± Q z . 



Absorption-machine. 
Still : 

Steam consumed per hour 

Abs. pressure of heating steam. 
Temperature of condensed 

steam at outlet 

Heat imparted to still Q'e 

Absorber : 
Quantity of cooling water per 

hour 

Temperature at inlet ... 

Temperature at outlet 

Heat removed Q 2 

Pump for Ammonia Liquor : 
Indicated work of steam-engine 
Steam-consumption for pump.. 
Thermal equivalent for work of 

pump ALp 

Total sum of losses by radiation 

and convection ± Q 3 

Heat Balance : 

Qe + Q'e = Qi + Qz± Q 3 . 
For the calculation of efficiency and for comparison of various tests, the 
actual efficiencies must be compared with the theoretical maximum of effi- 
ciency {-ry-J max. = — — corresponding to the temperature range. 

Temperature Range. — As temperatures (T and To) at which the 
heat is abstracted in the refrigerator and imparted to the condenser, it is cor- 
rect to select the temperature of the brine leaving the refrigerator and that 
of the cooling water leaving the condenser, because it is in principle impos- 
sible to keep the refrigerator pressure higher than would correspond to the 
lowest brine temperature, or to reduce the condenser pressure below that 
corresponding to the outlet temperature of the cooling water. 

Prof. Linde shows that the maximum theoretical efficiency of a com- 
pression-machine may be expressed by the formula 
Q T 

AL ~ Tc - T ' 

in which Q = quantity of heat abstracted (cold produced) ; 

AL = thermal equivalent of the mechanical work expended; 
L = the mechanical work, and A = 1 -s- 778 ; 
T = absolute temperature of heat abstraction (refrigerator) ; 
Tc = " " " " rejection (condenser). 

If u = ratio between the heat equivalent of the mechanical work AL, and 
the quantity of heat Q' which must be imparted to the motor to produce 
the work i, then 



992 ICE-MAKING OR REFRIGERATING MACHINES. 



AL 
Q' 



and^: 



uT 



It follows that the expenditure of heat Q' necessary for the production of 
the quantity of cold Q in a compression-machine will be the smaller, the 
smaller the difference of temperature To, — T. 

Metering tlie Ammonia*— For a complete test of an ammonia re- 
frigerating-machine it is advisable to measure the quantity of ammonia cir- 
culated, as was done in the test of the 75-ton machine described by Prof. 
Denton. (Trans. A. S. M. E., xii. 326.) 

PROPERTIES OF SULPHUR DIOXIDE AND 

AMMONIA GAS. 

Ledoux's Table for Saturated Sulphur-dioxide Gas. 

Heat-units expressed in B.T.U. per pound of sulphur dioxide. 



®.2 

5 = fe 



3-9 x •]• 



Deg. F. 

-22 

-13 
- 4 



95 
104 



Lbs. 



5.56 
7.23 
9.27 
11.76 
14.74 
18.31 
22.53 
27.48 
33.25 
39.93 
47.61 
56.39 
66.36 
77.64 
90.31 



Ip2 



B.T.U. 



157.43 
158.64 
159.84 
161.03 
162.20 
163.36 
164.51 
165.65 



16790 
168.99 
170.09 
171.17 
172.24 
173.30 



-1 &i* 



-19.56 
-16.30 
-13.05 

- 9.79 

- 6.53 

- 3.27 
0.00 
3.27 
6.55 
9.83 

13.11 
16.39 
19.69 
22.98 
26.28 



B.T.U. 



176.99 
174.95 
172.89 
170.82 
168.73 
166.63 
164.51 
162.38 
160.23 
158.07 
155.89 
153.70 
151.49 
149.26 
147.02 






13.59 
13.83 
14.05 



14.84 
15.01 
15.17 
15.32 
15.46 
15.59 
15.71 
15.82 
15.91 



161.12 
158.84 
156.56 
154.27 
151.97 
149.68 
147.37 
145.06 
142.75 
140.43 
138.11 
135.78 
133.45 
131.11 






Cu. ft. 



13.17 
10.27 
8.12 
6.50 
5.25 
4.29 
3.54 
2.93 
2.45 
2.07 
1.75 
1.49 
1.27 
1.09 
.91 



> be 



.123 
.153 
.190 



.340 
.407 
.483 
.570 



1.046 



Density of Liquid Ammonia. (DAndreff, Trans. A. S. M. E., 
x. 641.) 

At temperature C -10—5 5 10 15 20 

F +14 23 32 41 50 59 68 

Density...., 6492 .6429 .6364 .6298 .6230 .6160 .6089 

These may be expressed very nearly by 

8 = 0.6364 - 0.0014*° Centigrade; 
8 = 0.6502 - 0.000777T Fahr. 

Latent Heat of Evaporation of Ammonia. (Wood, Trans. 
A. S. M. E.,x. 641.) 

he = 555.5 - 0.613T - 0.0002192 12 (in B.T.U., Fahr. deg.); 

Ledoux found he = 583.33 - 0.5499T - 0.0001173T 2 . 

For experimental values at different temperatures determined by Prof. 
Denton, see Trans. A. S. M. E., xii. 356. For calculated values, see 
vol. x. 646. 

Density of Ammonia Gas.— Theoretical, 0.5894; experimental, 
0.596. Regnault (Trans. A. S. M. E.. x. 633). 

Specific Heat of Liquid Ammonia. (Wood, Trans. A. S. M. E., 
x 645 )— The specific heat is nearly constant at different temperatures, and 
about equal to that of water, or unity. From 0° to 100° F., it is 
c = 1.096 - .0012T, nearly. 

In a later paper by Prof. Wood (Trans. A.S, M. E,, xii. 136) he givesahigher 
value, viz., c = 142136 + 0.00043ST, 



PKOPERTIES OF AMMONIA VAPOR. 



993 



Dr. Von Strombeck, in 1890, found from the mean of eight experiments, 
at a temperature about 80° F., c = 1.22876,— about % greater than that cal- 
culated from this formula. 

In Prof. Wood's Thermodynamics (edition of 1894) in addition to the above 
figures he gives the mean of six determinations by Ludeking and Starr, 0.886. 
This, says Prof. Wood, leaves the correct result in doubt, and one may con- 
sider it as unity until determined by further experiments. 

Properties of the Saturated Vapor of Ammonia. 

(Wood's Thermodynamics.) 



Temperature. 


Pressure, 
Abso' 11 *" 


Heat of 


Volume 


Volume 


Weight 
of a cu. 










Vaporiza- 
tion, ther- 
mal units. 


of Vapor 
per lb., 
cu. ft. 


of Liquid 
per lb., 
cu. ft. 


Degs. 
F. 


Abso- 
lute, F. 


Lbs.per 
sq. ft. 


Lbs.per 
sq. in. 


ft. of 

Vapor, 

lbs. 


- 40 


420.66 


1540.7 


10.69 


579.67 


24.372 


.0234 


.0410 


- 35 


425.66 


1773.6 


12.31 


576.69 


21.319 


.0236 


.0468 


- 30 


430.66 


2035.8 


14.13 


573.69 


18.697 


.0237 


.0535 


- 25 


435.66 


2329.5 


16.17 


570.68 


16.445 


.0238 


.0608 


- 20 


440.66 


2657.5 


18.45 


567.67 


14.507 


.0240 


.0689 


- 15 


445.66 


3022.5 


20.99 


564.64 


12.834 


.0242 


.0779 


- 10 


450.66 


3428.0 


23.80 


561.61 


11.384 


.0243 


.0878 


- 5 


455.66 


3877.2 


26.93 


558.56 


10.125 


.0244 


.0988 





460.66 


4373.5 


30.37 


555 . 50 


9.027 


.0246 


.1108 




h 5 


465.66 


4920.5 


34.17 


552.43 


8.069 


.0247 


.1239 




- 10 


470.66 


5522.2 


38.34 


549.35 


7.229 


.0249 


.1383 




- 15 


475.66 


6182.4 


42.93 


546.26 


6.492 


.0250 


.1544 




- 20 


480.66 


6905.3 


47.95 


543.15 


5.842 


.0252 


.1712 




- 25 


485.66 


7695.2 


53.43 


540.03 


5.269 


.0253 


.1898 




- 30 


490.66 


8556.6 


59.41 


536.92 


4.763 


.0254 


.2100 




- 35 


495.66 


9493.9 


65.93 


533.78 


4.313 


.0256 


.2319 




- 40 


500.66 


10512 


73.00 


530.63 


3.914 


.0257 


.2555 




- 45 


505.66 


11616 


80.66 


527.47 


3.559 


.0259 


.2809 




- 50 


510.66 


12811 


88.96 


524.30 


3.242 


.0261 


.3085 




- 55 


515.66 


• 14102 


97. y3 


521.12 


2.958 


.0263 


.3381 




- 60 


520.66 


15494 


107.60 


517.93 


2.704 


.0265 


.3698 




- 65 


525.66 


16993 


118.03 


514.73 


2.476 


.0266 


.4039 




- 70 


530.66 


18605 


129.21 


511.52 


2.271 


.0268 


.4403 




- 75 


535.66 


20336 


141.25 


508.29 


2.087 


.0270 


.4793 




- 80 


540.66 


22192 


154.11 


505.05 


1.920 


.0272 


.5208 




- 85 


545.66 


24178 


167.86 


501.81 


1.770 


.0273 


.5650 




- 90 


550.66 


26300 


182.8 


498.11 


1.632 


.0274 


.6128 




- 95 


555.66 


28565 


198.37 


495.29 


1.510 


.0277 


.6623 




-100 


560.66 


30980 


215.14 


492.01 


1.398 


.0279 


.7153 




-105 


565.66 


33550 


232.98 


488.72 


1.296 


.0281 


.7716 




-110 


570.66 


36284 


251.97 


485.42 


1.203 


.0283 


.8312 




-115 


575.66 


39188 


272.14 


482.41 


1.119 


.0285 


.8937 




-120 


580.66 


42267 


293.49 


478.79 


1.045 


.0287 


.9569 




-125 


585.66 


45528 


316.16 


475.45 


0.970 


.0289 


1.0309 




- iSu 


590.66 


48978 


340.42 


472.11 


0.905 


.0291 


1.1049 




-135 


595.66 


52626 


365.16 


468.75 


0.845 


.0293 


1.1834 




-140 


600.66 


56483 


392 22 


465.39 


0.791 


.0295 


1.2642 




-145 


605.66 


60550 


420.49 


462.01 


0.741 


.0297 


1.3495 




-150 


610.66 


64833 


450.20 


458.62 


0.695 


.0299 


1.4388 




-155 


615.66 


69341 


4S1.54 


455.22 


0.652 


.0302 


1.5337 


- 


-160 


620.66 


74086 


514.40 


451.81 


0.613 


.0304 


1.6343 




r- 165 


625.66 


79071 


549.04 


418.39 


0.577 


.0306 


1.7333 



Specific Heat of Ammonia Vapor at the Saturation 
Point. (Wood, Trans. A. S. M.;E , x. 644.)— For the range of temperatures 
ordinarily used in engineeering practice, the specific heat of saturated am- 
monia is negative, and the saturated vapor will condense with adiabatic ex- 
pansion, and the liquid will evaporate with the compression of the vapor, 
and when all is vaporized will superheat. 

Regnault (Rel. cles. Exp., ii. 162) gives for specific heat of ammonia-gas 
0-50836, (Wood, Trans. A. S. M. E., xii. 133,) 



994 ICE-MAKING OR REFRIGERATING MACHINES. 



Properties of Brine used to absorb Refrigerating Effect 
of Ammonia. (J. E. Denton, Trans. A. S. M. E., x. 799.)— A soluuon of I 
Liverpool salt in well-water having a specific gravity of 1.17, or a weight 
per cubic foot of 73 lbs., will not sensibly thicken or congeal at 0° Fahren- 
heit. (It is reported that brine of 1.17 gravity, made with American salt, 
begins to congeal at about 24° Fahr.) 

The mean specific heat between 39° and 16° Fahr. was found by Denton to 
be 0.805. Brine of the same specific gravity has a specific heat of 0.805 at 
65° Fahr., according to Naumann. 

Naumann's values are as follows {Lehr- und Handbuch der Tliermochemie, 
1882): 

Specific heat 791 .805* .863 .895 .931 .962 .978 

Specific gravity. 1.187 1.170 1.103 1.072 1.044 1.023 1.012 
* Interpolated. 

Chloride-of-calcium solution has been used instead of brine. Ac- 1 
cording to Naumann, a solution of 1.0255 sp. gr. has a specific heat of .957. 
A solution of 1.163 sp. gr. in the test reported in Eng'g, July 22, 1887, gave a 
specific heat of .827. 

ACTUAL PERFORMANCES OF ICE-MAKING 
MACHINES. 

The table given on page 996 is abridged from Denton, Jacobus, and Riesen- 
berger's translation of Ledoux on Ice-making Machines. The following 
shows the class and size of the machines tested, referred to by letters in the 
table, with the names of the authorities: 



Class of Machines. 



Authority. 



Dimensions of Compres- 
sion-cylinder in inches. 



A. Ammonia cold-compression.. 

B. Pictet fluid dry -compression. 

C. Bell-Coleman air 



D. Closed cycle air 

E. Ammonia dry-compression . 

F. Ammonia absorption 



\ Ren wick & 
( Jacobus. 
Denton. 



28.0 

10. 

12.0 



16.5 
24.4 
23.8 
18.0 
30.0 



Performance of a 75-ton Ammonia Compression- 
machine. (J. E. Denton, Trans. A. 8. M. E., xii, 326. )— The machine had 
two single-acting compression cylinders 12" X 30", and one Corliss steam - 
cylinder, double-acting, 18 ,/ X 36". It was rated by the manufacturers as a 
50-ton machine, but it showed 75 tons of ice-refrigerating effect per 24 hours 
during the test. 

The most probable figures of performance in eight trials are as follows : 





Ammonia 


Brine 


c iuo^ 


I'll 


3 ..; F<w 


"1 o.i 




— • 


Pressures, 


Tempera- 


O-J u 


o M 


Pi 

8 


.5 


lbs. above 


tures, 


H £p< 


~~°°u . 


Wtiter-cons 
tion, gals 
Water pei 
per ton o 
paeity. 


IMS 


H 


Atmosphere. 


Degrees F. 


©£ v 5s 

o 


Efficiency 
Ice pei 
Coal at 
Coal pe 
per H.P 


o 


6 


Con- 
densing 


Suc- 
tion. 


Inlet. 


Outlet. 


.2.2 
:■« 


1 


151 


28 


36.76 


28.86 


70.3 


22.60 


0.80 


1 


1.0 


8 


161 


27.5 


36.36 


28.45 


70.1 


22.27 


1.09 


1.0 


1.0 


7 


147 


13.0 


14.29 


2.29 


42.0 


16.27 


0.83 


1.70 


1.66 


4 


152 


8.2 


6.27 


2.03 


36.43 


14.10 


1.1 


1.93 


1.92 


6 


105 


7.6 


6.40 


-2.22 


37.20 


17.00 


2.00 


1.91 


1.88 


2 


135 


15.7 


4.62 


3.22 


27.2 


13.20 


1.25 


2.59 


2 57 



The principal results in four tests are given in the table on page 998. The 
fuel economy under different conditions of operation is shown in the fol- 
lowing table : 



PERFORMANCES OF ICE-MAKING MACHINES. 995 







Pounds of Ice-melting Effect with 


B.T.U. 


per lb. of Steam 


02 


1 


Engines— 


with Engines— 


Ph . 




Non-com- 


Compouud 








«l 


kg 


Non-con- 
densing. 


pound Con- 
densing. 


Con- 
densing. 


a 
*2 • 


.5 




© 


&_* 


si 


^ u 


a% 


■&~ 


fi'S 


8.2 


0> 

a 


§3 


o 
O 


3 


& © 


(S s 




g£ 


13 


££ 


o 


O 


oo 


150 


28 


24 


2.00 


30 


3.61 


37.5 


4.51 


393 


513 


640 


150 


7 


14 


1.69 


17.5 


2.11 


21.5 


2.58 


240 


300 


366 


105 


28 


34.5 


4.10 


43 


5.18 


54 


6.50 


591 


725 


923 


105 


7 


22 


2. C5 


27.5 


3.31 


34.5 


4.16 


376 


470 


591 



The non -condensing engine is assumed to require 25 lbs. of steam per 
horse-power per hour, the non-compound condensing 20 lbs., and the com- 
densing 16 lbs., and the boiler efficiency is assumed at 8.3 lbs. of water per 
lb. coal under working conditions. The following conclusions were derived 
from the investigation : 

1. The capacity of the machine is proportional, almost entirely, to the 
weight of ammonia circulated. This weight depends on the suction- 
pressure and the displacement of the compressor-pumps. The practical 
suction-pressures range from 7 lbs. above the atmosphere, with which a 
temperature of 0° F. can be produced, to 28 lbs. above the atmosphere, with 
which the temperatures of refrigeration are confined to about 28° F. At the 
lower pressure only about one half as much weight of ammonia can be cir- 
culated as at the upper pressure, the proportion being about in accordance 
with the ratios of the absolute pressures, 22 and 42 lbs. respectively. For each 
cubic foot of piston-displacement per minute a capacity of about one sixth 
of a ton of " refrigerating effect " per 24 hours can be produced at the lower 
pressure, and of about one third of a ton at the upper pressure. No other 
elements practically affect the capacity of a machine, provided the cooling- 
surface in the brine-tank or other space to be cooled is equal to about 
36 sq. ft. per ton of capacity at 28 lbs. back pressure. For example, a differ- 
ence of 100$ in the rate of circulation of brine, while producing a propor- 
tional difference in the range of temperature of the latter, made no practical 
difference in capacity. 

The brine-tank was 10^ X 13 X 19^ ft., and contained 8000 lineal feet of 
1-in. pipe as cooling-surface. The condensing-tank was 12 X 10 X 10 ft., and 
coutained 5000 lineal feet of 1-in. pipe as cooling-surface. 

2. The economy in coal-consumption depends mainly upon both the suc- 
tion-pressures and condeusing-pressures. Maximum economy, with a given 
type of engine, where water must be bought at average city prices, is 
obtained at 28 lbs. suction-pressure and about 150 lbs. condensing-pressure. 
Under these conditions, for a non-condensing steam-engine, consuming coal 
at the rate of 3 lbs. per hour per I.H.P. of steam-cylinders, 24 lbs. of ice- 
refrigerating effect are obtained per lb. of coal consumed. For the same 
condensing-pressure, and with 7 lbs. suction-pressure, which affords tem- 
peratures of 0° F., the possible economy falls to about 14 lbs. of " refrigerat- 
ing effect " per lb. of coal consumed. The condensing-pressure is determined 
by the amount of condensiug-water supplied to liquefy the ammonia in the 
condenser. If the latter is about 1 gallon per minute per ton of refrigerating 
effect per 24 hours, a condensing-pressm - e of 150 lbs. results, if the initial tem- 
perature of the water is about 56° F. Twenty-five per cent less water causes 
the condensing-pressure to increase to 190 lbs. The work of compression is 
thereby increased about 20$, and the resulting "economy" is reduced to 
about 18 lbs. of " ice effect " per lb. of coal at 28 lbs. suction-pressure and 
11.5 at 7 lbs. If, on the other hand, the supply of water is made 3 gallons 
per minute, the condensing-pressure may be confined to about 105 lbs. The 
work of compression is thereby reduced about 25$, and a proportional increase 
of economy results. Minor alterations of economy depend on the initial 
temperature of the condensing-water and variations of latent heat, but these 
are confined within about 5$ of the gross result, the main element of control 
being the work of compression, as affected by the back pressure and con- 
densing-pressure, or both. If the steam engine supplying the motive power 
may use a condenser to secure a vacuum, an increase of ecouomy of 25$ is 
available over the above figures, making the lbs of " ice effect" per lb. of 



996 ICE-MAKING OR REFRIGERATING MACHINES. 

coal for 150 lbs. condensing-pressure and 28 lbs. suction -pressure 30.0, and 
for 7 lbs. suction-pressure, 17.5. It is, however, impracticable to use a con- 
denser in cities where water is bought. The latter must be practically 
free of cost to be available for this purpose. In this case it may be assumed 
that water will also be available for condensing the ammonia to obtain as 
low a condensing-pressure as about 100 lbs., and the economy of the refrig- 
erating-machine becomes, for 28 lbs. back pressure, 43.0 lbs. of "ice effect " 
per lb. of coal, or for 7 lbs. back-pressure, 27.5 lbs. of ice effect per lb. 
of coal. If a compound condensing-engine can be used with a steam-con- 
sumption per hour per horse-power of 16 lbs. of water, the economy of the 
refrigeratiug-machine may be 25$ higher than the figures last named, mak- 
ing for 28 lbs. back pressure a refrigerating effect of 54.0 lbs. per lb. of coal, 
and for 7 lbs. back pressure a refrigerating effect of 34.0 lbs. per lb. of coal. 
Actual Performance of Ice-making Machines, 







1^* 
B c 9 
,0 3 t» 


emperature 
corresponding 
to Pressure, in 
degrees Fahr. 


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A 


1 


135 


55 


72 


27 


43 


37 


44.9 


17.9 


14.4 


26.2 


40.63 


30.8 


19.1 


54.8 


*» 


2 


131 1 42 


70 


14 


28 


23 


45.1 


18.0 


16.7 


19.5 


30.01 


33.5 


20.2 


53.4 


" 


3 


128, 30 


69 


1 


14 


9 


45.1 


16 8 


16.0 


13.3 


22. 03 


37.1 


25 2 


50.3 


" 


4 


126 22 


68 


-12 





- 5 


44.8 


15.5 


19.5 


9.0 


16.14 


42.9 


29.1 


44.7 


" 


5 


200, 42 


95 


14 


28 


23 


45.0 




10.5 


16.5 


19.07 


36.0 




77.0 


" 


6 


136 60 


72 


30 


44 


37 


45.2 


17.9 


10.7 


29.8 




28.5 


19.9 


56.8 


" 


7 


131 | 45 


71 


18 


28 


23 


45.1 


18.0 


12.1 


21.6 




31.3 


21 9 


56.4 


" 


8 


126 


24 


68 


- 9 





- 6 


44.7 


15.(3 


18.0 


9.9 


17.55 


41.1 




46.1 


l « 


9 


117 


41 


64 


13 


28 


23 


15.0 


16.4 


13.5 




33.77 


33 1 




50.6 


" 


10 


130 


60 


70 


31 


43 


37 


31.7 


12.0 


14.8 


19.5 


45.01 


35.2 


. 


52.0 


B 


11 


57 


21 


77 


28 


43 


37 


57.0 


21.5 


22.9 


25.6 


33.07 


39.9 




24.1 


" 


12 


56 


15 


76 


14 


28 


23 


56.8 


20.6 


22.9 


17.9 


24.11 


41.3 


24.0 


23.1 


" 


13 


55 


10 


75 


- 2 


14 


9 


57.1 


18.5 


21.0 


11.0 


17.47 


42.2 




20.4 


" 


14 


60 


7 


81 


-16 





- 6 


57.6 


15.7 


25.7 


5.7 


10.14 


54.5 


i 


16.8 


" 


15 


91 


15 


104 


14 


28 


23 


59 3 


27.2 


16.9 


15.7 


16.05 


36.2 




31.5 


" 


16 


61 


22 


81 


31 


44 


37 


57.3 


21 6 


14.0 




36.19 


33 .'4 




26.8 


" 


17 


59 


16 


80 


16 


28 


23 


57.5 


20.5 


12.8 


19.3 


26.24 


34.6 




25.6 




18 


59 


7 


79 


-16 





- 6 


57.8 


15.9 


21 . 1 


6.8 


11.93 


47.5 




18.0 


" 


19 


54 


22 


75 


31 


43 


37 








17.0 


38.04 


39.5 




22.6 


" 


20 


89 


16 


103 


16 


28 


23 


42.9 


19.9 


14 7 


11.9 


16.68 


37.7 


27.0 


32.7 


" 


21 


62 


6 


82 


-17 





- 5 


34.8 


9.9 


24.3 


3.5 


9.86 


54.2 


39.5 


17.7 


(.: 


22 


59 


15 




-53* 






63.2 




21.9 


10.3 


3.42 


71.7 


56.9 


26.6 




23 


175 


54 


81* 


-40* 






93.4 


3,8.1 


32.1 


4.9 


3.0 


80. 


63. 


89.2 




24 


16(5 


43 


84 


15 


' 37 


"28 


58 1 


S5.0 


22.7 


73.9 


24.16 


32.8 


11.7 


65.9 


" 


25 


167 


23 


85 


-11 


6 


2 


57 . 7 


72.6 


18.6 


37.9 


14.52 


37.4 




57.6 


" 


26 


162 


2S 


83 


- 3 


14 


2 


57.9 


73.6 


19 3 


46.5 


17.55 


34.9 


18.6 


59.9 


" 


27 


176 


42 


88 


14 


36 


28 


58.9 


88.6 


19 7|74.4 


23.31 


30.5 


13.5 


70.5 


F 


28 


152 


40 


79 


13 


21 


16 


.... 




... 




20.1 


47.8 































* Temperature of air at entrance and exit of expansion-cylinder. 

t On a basis of 3 lbs. of coal per hour per H.P. of steam-cylinder of com- 
pression-machine and an evaporation of 11.1 lbs. of water per pound of 
combustible from and at 212° F. in the absorption-machine. 

X Per cent of theoretical with no friction. 

§ Loss due to heating during aspiration of gas in the compression-cylinder 
and to radiation and superheating at brine-tank. 

I! Actual, including resistance due to inlet and exit valves. 



PERFORMANCES OF ICE-MAKING MACHINES. 997 

In class A, a German machine, the ice-melting capacity ranges from 46.29 
to 16.14 lbs. of ice per pound of coal, according as the suction pressure 
varies from about 45 to 8 lbs. above the atmosphere, this pressure being the 
condition which mainly controls the economy of compression-machines. 
These results are equivalent to realizing from 72$ to 57'$ of theoretically per- 
fect performances. The higher per cents appear to occur with the higher 
suction-pressures, indicating a greater loss from cylinder-heating (a phe- 
nomenon the reverse of cylinder condensation in steam-engines), as the 
range of the temperature of the gas in the compression-cylinder" is 
greater. 

In E, an American compression-machine, operating on the " dry system, 1 ' 
the percentage of theoretical effect realized ranges from 69.5$ to 62.6$. 
The friction losses are higher for the American machine. The latter's higher 
efficiency may be attributed, therefore, to more perfect displacement. 

The largest " ice-melting capacity " in the American machine is 24.16 lbs. 
This corresponds to the highest suction-pressures used in American practice 
for such refrigeration as is required in beer-storage cellars using the direct- 
expansion system. The conditions most nearly corresponding to American 
brewery practice in the German tests are those in line 5, which give an " ice- 
melting capacity " of 19.07 lbs. 

For the manufacture of artificial ice, the conditions of practice are those 
of lines 3 and 4, and lines 25 and 26. In the former the condensing pressure 
used requires more expense for cooling water than is common in American 
practice. The ice-melting capacity is therefore greater in the German ma- 
chine, being 22.03 and 16.14 lbs. against 17.55 and 14.52 for the American 
apparatus. 

Class B. Sulphur Dioxide or Pictet Machines.— No records are available 
for determination of the "ice-melting capacity" of machines using pure 
sulphur dioxide. This fluid is in use in American machines, but in Europe 
it has given way to the " Pictet fluid," a mixture of about 97$ of sulphur 
dioxide and 3$ of carbonic acid. The presence of the carbonic acid affords 
a temperature about 14 Fahr. degrees lower than, is obtained with pure sul- 
phur dioxide at atmospheric pressure. The latent heat of this mixture has 
never been determined, but is assumed to be equal to that of pure sulphur 
dioxide. 

Tor brewery refrigerating conditions, line 17, we have 26.24 lbs. "ice- 
melting capacity," and for ice-making conditions, line 13, the "ice-melt- 
ing capacity" is 17.47 lbs. These figures are practically as economical 
as those for ammonia, the per cent of theoretical effect realized ranging 
from 65.4 to 57.8. At extremely low temperatures, —15° Fahr., lines 14 and 
18, the per cent realized is as low as 42.5. 

Cylinder-heating.— In compression-machines employing volatile 
vapors the principal cause of the difference between the theoretical and the 
practical result is the heating of the ammonia, by the warm cylinder walls, 
during its entrance into the compressor, thereby expanding it, so that to 
compress a pound of ammonia a greater number of revolutions must be 
made by the compressing-pumps than corresponds to the density of the 
ammonia-gas as it issues from the brine-tank. 

Tests of Ammonia Absorption-machine used in storage- ware- 
houses under approaches to the New York and Brooklyn Bridge. (Eng'g, 
July 22, 1887.) — The circulated fluid consisted of a solution of chloride of cal- 
cium of 1.163 sp. gr. Its specific heat was found to be .827. 

The efficiency of the apparatus for 24 hours was found by taking the 
product of the cubic feet of brine circulating through the pipes by the aver- 
age difference in temperature in the ingoing and outgoing currents, as 
observed at frequent intervals by the specific heat of the brine (827) and its 
weight per cubic foot (73.48). The final product, applying all allowances for 
corrections from various causes, amounted to 6,218,816 heat-units as the 
amount abstracted in 24 hours, equal to the melting of 43,565 lbs. of ice in 
the same time. 

The theoretical heating-power of the coal used in 24 hours was 27,000,000 
heat-units; hence the efficiency of the apparatus was 23$. This is equivalent 
to an ice-melting effect of 16.1 lbs. per lb. of coal having a heating value of 
10,000 B.T.U. per lb. 

A test of a 35-ton absorption -machine in New Haven, Conn., by Prof.. 
Denton (Trans. A. S. M. E., x. 792), gave an ice-melting effect of 20.1 lbs. per 
lb. of coal on a basis of boiler economy equivalent to 3 lbs. of steam per 
I.H.P. in a good non-condensing steam-engine. The ammonia was worked 
between 138 and 23 lbs. pressure above the atmosphere. 



998 ICE-MAKIKG OR REfRIGEEaTIHG MACHINES. 
Performance of a 75-ton Refrigerating-machine. 





'O aj 


73 £ rr 


t?i 


Ti v 




A.O 


3 O <» 


Oj5 




ci- 


oS uX> 


* «$ 


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oS u%> 


03™ £ 




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P-rt 3 


frO — 


a-g 3 






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§£?! 


C V. a $ 




^>>J 




3 c^ 1 


sl!-£ 


5 5* 






3 3 "-J* 


pagj 


x a o3 




.Eoo 
X a o3 


.3 c .3 a 
M a ■- 93 


.3 o.S v 
X a £ u 




esHCQ 


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§ 


r5 


s 


161 lbs. 


Av. high ammonia press, above atmos 


151 lbs. 


152 lbs. 


147 lbs. 


Av. back ammonia press, above atmos 

Av. temperature brine inlet 


28 " 


8.2 " 


13 " 


27.5 " 


36.76° 


6.27° 


14.29° 




Av. temperature brine outlet. 


28.86° 


2.03° 


2.29° 


'28.45° 


Av. range of temperature 


7.9° 


4.24° 


12.00° 


7.91° 


Lbs. of brine circulated per minute 


2281 


2173 


943 


2374 


Av. temp, condensing-water at inlet 


44.65° 


56.65° 


46.9° 


54.00° 


Av. temp, condensing-water at outlet 


83.66° 


85.4° 


85.46° 


82.86° 


Av. range of temperature 


39.01° 


28.75° 


38.56° 


28.80° 


Lbs. water circulated p. min. thro' cond'ser 


442 


315 


257 


601.5 


Lbs. water per min. through jackets 


25 


44 


40 


14 


Range of temp rature in jackets 


24.0° 


16.2° 


16.4° 


29.1° 


Lbs. ammonia circulated per min 


*28.17 


14.68 


16.67 


28.32 


Probable temperature of liquid ammonia, 










entrance to brine-tank 


*71.3° 


*G8° 


*63.7° 


76.7° 


Temp, of amm. corresp. to av. back press. 


+14° 


- 8° 


- 5° 


14° 


Av. temperature of gas leaving brine-tanks 


34.2° 


14.7° 


3 0° 


29.2° 


Temperature of gas entering compressor. . 


*39° 


25° 


10.13° 


34° 


Av. temperature of gas leaving compressor 


213° 


263° 


239° 


221° 


Av. temp, of gas entering condenser 

Temperature due to condensing pressure.. 


200° 


218° 


209° 


168° 


84.5° 


84.0° 


82.5° 


88.0° 


Heat given ammonia: 










By brine, B T.U. per miniute 


14776 


7186 


8824 


14647 


By compressor, B.T.U. per minute 


2786 


2320 


2518 


3020 


By atmosphere, B.T.U. per minute. ... 


140 


147 


167 


141 


Total heat rec. by amm., B.T.U. per min. 


17702 


9653 


11409 


17708 


Heat taken from ammonia: 










By condenser, B.T.U per min 


17242 
608 


9056 

712 


9910 
656 


17359 
406 


By jackets, B.T.U. per min 


By atmosphere, B.T.U. per min 


182 


338 


250 


252 


Total heat rej. by amm., B.T U. per min. . . 


18032 


10106 


10816 


18017 


Dif. of heatrec'd and rej., B.T.U. per min. 


330 


453 


407 


309 


% work of compression removed by jackets. 


22# 


31 % 


26% 


Vi% 


Av. revolutions per min 


58.09 


57.7 


57.88 


58.89 


Mean eff. press, steam-cyl., lbs. per sq. in.. 


32.5 


27.17 


27.83 


32.97 


Mean eff. press, amm.-cyl., lbs. per sq. in . . 


65.9 


53.3 


59.86 


70.54 


Av. H.P. steam-cylinder 


85.00 
65.7 


71.7 
54.7 


73.6 
59.37 


88.63 
71.20 


Av. H.P. ammonia-cylinder 


Friction in per cent of steam HP 


23.0 


24.0 


20.0 


19.67 


Total cooling water, gallons per min. per 


ton per 24 hours 


0.75 


1.185 


0.797 


0.990 


Tons ice-melting capacity per 24 hours 


74.8 


36.43 


44.64 


74.56 


Lbs. ice-refrigerating eff. per lb. coal at 3 










lbs. per H.P. per hour 


24.1 


14.1 


17.27 


23.37 


Cost coal per ton of ice-refrigerating effect 










at $4 per ton 


$0,166 


■,$0,283 


$0,231 


$0,170 


Cost water per ton of ice-refrigerating effect 








at $1 per 1000 cu. ft... 


SO. 128 


$0,200 


$0,136 


$0,169 


Total cost of 1 ton of ice-refrigerating eff... 


$0,294 


$0,483 


$0,467 


$0,339 



Figures marked thus (*) are obtained by calculation; all other figures are 
obtained from experimental data ; temperatures are in Fahrenheit degrees. 



ARTIFICIAL ICE-MANUFACTURE. 



999 



Ammonia Compression-machine. 

Actual Results obtained at the Munich Tests. 
(Prof. Liu.de, Trans. A. S. M. E., xiv. 1419.) 



No. of Test . . , 


' 


2 


3 


4 


5 






Temp, of refrig- (_ Inlet, deg. F 


43.194 


28.344 


13.952 


-0.279 


28.251 


erated brine ) Outlet, t deg. F. . . 


37.054 


22.885 


8.771 


-5.879 


23.072 


Specific heat of brine 


0.861 


0.851 


0.843 


0.837 


0.851 


Quantity of brine circ. per h., cu. ft. 


1,039.38 


908.84 633.89 


414.98 


800.93 


Cold produced, B.T.U. per hour — 


342,909 


2C3.950 172,776 


121,474 


220,284 


Quant, of cooling water per h.. c. ft. 


338.76 


260.83 187.506 


139.99 


97.76 


I.H.P. in steam-engine cylinder (Le). 


15.80 


16.47 


15.28 


14.24 


21.61 


Cold pro- ) Per I.H.P. in comp.-cyl. 


24,813 


18,471 


12,770 


10,140 


11,151 


duced per >Per I H.P. in steam-cyl. 


21.703 


16,026 


11,307 


8,530 


10,194 


h.. B.T.U. ) Per lb. of steam 


1,100.8 


785.6 


564.9 


435.82 


512.12 



Means for Applying the Cold. # (M. C. Bannister, Liverpool 
Eng'g Soc'y, 1890.) — The most useful means for applying the cold to various 
uses is a saturated solution of brine or chloride of magnesium, which 
remains liquid at 5° Fahr. The brine is first cooled by being circulated in 
contact with the refrigerator-tubes, and then distributed through coils of 
pipes, arranged either in the substances requiring a reduction of tempera- 
ture, or in the cold stores or rooms prepared for them; the air coming in 
contact with the cold tubes is immediately chilled, and the moisture in the 
air deposited on the pipes. It tben falls, making room for warmer air, and 
so circulates until the whole room is at the temperature of the brine in the 
pipes. 

In a recent arrangement for refrigerating made by the Linde British Re- 
frigeration Co., the cold brine is circulated through a shallow trough, in 
which revolve a number of shafts, each geared together, and driven by me- 
chanical means. On the shafts are fixed a number of wrought-iron disks, 
partly immersed in the brine, which cool them down to the brine tempera- 
ture as they revolve; over these disks a rapid circulation of air is passed by 
a fan, being cooled by contact with the plates; then it is led into the cham- 
bers requiring refrigeration, from which it is again drawn by the same fan; 
thus all moisture and impurities are removed from the chambers, and de- 
posited in the brine, producing the most perfect antiseptic atmosphere yet 
invented for cold storing; while ihe maximum efficiency of the brine tem- 
perature was always available, the brine being periodically concentrated by 
suitable arrangements. 

Air has also been used as the circulating medium. The ammonia-pipes 
refrigerate the air in a cooling-chamber, and large wooden conduits are used 
to convey it to and return it from the rooms to be cooled. An advantage of 
this system is that by it a room may be refrigerated more quickly than by 
brine-coils. The returning air deposits its moisture in the form of snow oh 
the ammonia-pipes, which is removed by mechanical brushes. 

ARTIFICIAL. ICE-MANUFACTURE. 

Under summer conditions, with condensing water at 70°, artificial ice-ma- 
chines use ammonia at about 190 lbs. above the atmosphere condenser- 
pressure, and 15 lbs. suction-pressure. 

In a compression type of machine the useful circulation of ammonia, 
allowing for the effect of cylinder- heating, is about 13 lbs. per hour per in- 
dicated horse-power of the steam cylinder. This weight of ammonia pro- 
duces about 32 lbs. of ice at 15° from water at 70°. If the ice is made from 
distilled water, as in the "can system," the amount of the latter supplied 
by the boilers is about 33$ greater than the weight of ice obtained. This 
excess represents steam escaping to the atmosphere, from the re-boiler and 
steam-condenser, to purify the distilled water, or free it from air; also, the 
loss through leaks and drips, and loss by melting of the ice in extracting it 
from the cans. The total steam consumed per horse-power is, therefore, 
about 32 X 1.33 = 43,0 lbs. About 7.0 lbs. of this covers the steam -consump- 
tion of the steam-engines driving the brine circulating-pumps, the several 



1000 ICE-MAKING OK REFRIGERATING MACHINES. 

cold-water pumps, and leakage, drips, etc. Consequently, the main steam- 
engine must consume 36 lbs. of steam per hour per I.H.P,, or else live steam 
must be condensed to supply the required amount of distilled water. There 
is, therefore, nothing to be gained by using steam at high rates of expansion 
in the steam-engines, in making artificial ice from distilled water. If the 
cooling water for the ammonia-coils and steam-condenser is not too hard for 
use in the boilers, it may enter the latter at about 1?5° F., by restricting the 
quantity to \y% gallons per minute per ton of ice. With good coal %% lbs. of 
feed-water may then be evaporated, on the average, per lb. of coal. 

The ice made per pound of coal will then be 32 -h -—— — 6.0 lbs. This cor- 
responds with the results of average practice. 

If ice is manufactured by the "plate system," no distilled water is used 
for freezing. Hence the water evaporated by the boilers may be reduced to 
the amount which will drive the steam-motors, and the latter may use steam 
expansively to any extent consistent with the power required to compress 
the ammonia, operate the feed and filter pumps, and the hoisting machinery. 
The latter may require about 15$ of the power needed for compressing the 
ammonia. 

If a compound condensing steam-engine is used for driving the com- 
pressors, the steam per indicated steam horse-power, or per 32 lbs. of net 
ice, may be 14 lbs. per hour, ffhe other motors at 50 lbs. of steam per horse- 
power will use 7.5 lbs. per hour, making the total consumption per steam 
horse-power of the compressor 21.5 lbs. Taking the evaporation at 8 lbs., 
the feed-water temperature being limited to about 110°, the coal per horse- 
power is 2.7 lbs. per hour. The net ice per lb. of coal is then about 32 -s- 2.7 = 
11.8 lbs. The best results with "plate-system 1 ' plants, using a compound 
steam-engine, have thus far afforded about 10^ lbs. of ice per lb. of coal. 

In the " plate system " the ice gradually forms, in from 9 to 14 days, to a 
thickness of about 14 inches, on hollow plates 10 x 14 feet in area, in which 
the cooling fluid circulates. 

In the "can system " the water is frozen in blocks weighing about 300 lbs. 
each, and the freezing is completed in from 50 to 60 hours. The freezing- 
tank area occupied by the "plate system" is, therefore, about four times, 
and the cubic contents about twelve times, as much as is required in the 
" can system." 

The investment for the "plate" is about one third greater than for the 
"can" system. In the latter system ice is being drawn throughout the 24 
hours, and the hoisting is done by hand tackle. In the " plate system " the 
entire daily product is drawn, cut, and stored in a few hours, the hoisting 
being performed by power. The distribution of cost is as follows for the 
two systems, takingthe cost for the "can " or distilled-water system as 100, 
which represents an actual cost of about $1.25 per net ton: 

Can System. Plate System. 

Hoisting and storing ice 14.2 2.8 

Engineers, firemen, and coal-passer 15.0 13.9 

Coal at $3.50 per gross ton 42.2 20.0 

Water pumped directly from a natural source 

at 5 cts. per 1000 cubic feet... 1.3 2.6 

Interest and depreciation at 10$ 24.6 32.7 

Repairs 2.7 3.4 

100.00 75.4 

A compound condensing engine is assumed to be used by the " plate sys- 
tem." 
Test of the New York Hygeia Ice-making Plant.— (By 

Messrs. Hupfel, Griswold, and Mackenzie; Stevens Indicator, Jan. 1894.) 
The final results of the tests were as follows: 

Net ice made per pound of coal, in pounds 7. 12 

Pounds of net ice per hour per horse-power 37.8 

Net ice manufactured per day (12 hours) in tons 97 

Av. pressure of ammonia-gas at condenser, lbs. per sq. in. ab. atmos. 135.2 

Average back pressure of amm.-gas, lbs. per sq. in. above atmos.. . . 15.8 

Average temperature of brine in freezing-tanks, degrees F 19.7 

Total number of cans filled per week 4389 

Ratio of cooling-surface of coils in brine-tank to can-surface 7 to 10 



MARINE ENGIKEERIKG. lOOi 

Ratio of brine in tanks to water in caus 1 to 1 .2 

Ratio of circulating water at condensers to distilled water 26 to 1 

Pounds of water evaporated at boilers per pound of coal 8.085 

Total horse-power developed by compressor-engines 444 

Percentage of ice lost in removing from cans 2.2 

APPROXIMATE DIVISION OF STEAM IN PER CENTS OF TOTAL AMOUNT. 

Compressor-engines 60. 1 

Live steam admitted directly to condensers 19.7 

Steam for pumps, agitator, and elevator engines , 7.6 

Live steam for reboiling distilled water 6.5 

Steam for blowers furnishing draught at boilers 5.6 

Sprinklers for removing ice from cans 0.5 

The precautions taken to insure the purity of the ice are thus described: 
The water which finally leaves the condenser is the accumulation of the 
exhausts from the various pumps and engines, together with an amount of 
live steam injected into it directly from the boilers. This last quantity is 
used to make up any deficit in the amount of water necessary to supply the 
ice-cans. This water on leaving the condensers is violently reboiled, and 
afterwards cooled by running through a coil surface-cooler. It then passes 
through an oil-separator, after which it runs through three charcoal-filters 
and deodorizers, placed in series and containing 28 feet of charcoal. It next 
passes into the supply-tank in which there is an electrical attachment for 
detecting salt. Nitrate-of-silver tests are also made for salt daily. From 
this tank it is fed to the ice-cans, which are carefully covered so that the 
water cannot possibly receive any impurities. 

MAKINE ENGINEERING. 

Rules for Measuring Dimensions and Obtaining- Ton- 
nage of Vessels. (Record of American & Foreign Shipping. American 
Shipmasters' Assn., N. Y. 1890.)— The dimensions to be measured as follows: 

I. Length, L.— From the fore side of stem to the after side of stern-post 
measured at middle line on the upper deck of all vessels, except those hav- 
ing a continuous hurricane-deck extending right fore and aft, in which the 
/ength is to be measured on the range of deck immediately below the hurri- 
cane-deck. 

Vessels having clipper heads, raking forward, or receding stems, or rak- 
ing stern-posts, the length to be the distance of the fore side of stem from 
aft-side of stern-post at the deep-load water-line measured at middle line. 
(The inner or propeller-post to be taken as stern-post in screw-steamers. 

II. Breadth, B.— To be measured over the widest frame at its widest part; 
in other words, the moulded breadth. 

III. Depth, D.— To be measured at the dead-flat frame and at middle line 
of vessel. It shall be the distance from the top of floor-plate to the upper 
side of upper deck-beam in all vessels except those having a continuous 
hurricane-deck, extending right fore and aft, and not intended for the 
American coasting trade, in which the depth is to be the distance from top 
of floor-plate to midway between top of hurricane deck-beam and the top 
of deck-beam of the deck immediately below hurricane-deck. 

In vessels fitted with a continuous hurricane deck, extending right fore 
and aft. and intended for the American coasting trade, the depth is to be 
the distance from top of floor-plate to top of deck-beam of deck immedi- 
atelv below hurricane-deck. 

Rule for Obtaining Tonnage. — Multiply together the length, 
breadth, and depth, and their product by .75; divide the last product by 100; 

T y 7? v T) v 75 
the quotient will be the tonnage. — — : — = tonnage. 

The U. S. Custom-liouse Tonnage Law, May 6, 1864, provides 
that "the register tonnage of a vessel shall be her entire internal cubic 
capacity in tons of 100 cubic feet each.'' 1 This measurement includes all the 
space between upper decks, however many there may be. Explicit direc- 
tions for making the measurements are given in the law. 

The Displacement of a "Vessel (measured in tons of 2240 lbs.) is 
the weight of the volume of water which it displaces. For sea-water it is 
equal to the volume of the vessel beneath the water-line, in cubic feet, 
divided by 35, which figure is the number of cubic feet of sea-water at 60° 



100& MAH1KE EKGitfEERlim. 

F. in a ton of 2240 lbs. For fresh water the divisor is 35.93. The U. S. reg' 
ister tonnage will equal the displacement when the entire internal cubic 
capacity bears to the displacement the ratio of 100 to 35. 

The displacement or gross tonnage is sometimes approximately estimated 
as follows: Let L denote the length in feet of the boat, B its extreme 
breadth in feet, ami D the mean draught in feet; the product of these three 
dimensions will give the volume of a parallelopipedon in cubic feet. Put- 
ting V for this volume, we have V = L X B X D. 

The volume of displacement may then be expressed as a percentage of 
the volume V, known as the " block coefficient.' 1 '' This percentage varies for 
different classes of ships. In racing yachts with very deep keels it varies 
from 22 to 33; in modern merchantmen from 55 to 75; for ordinary small 
boats probably 50 will give a fair estimate. The volume of displacement in 
cubic feet divided by 35 gives the displacement in tons. 

Coefficient of Fineness.— A term used to express the relation be- 
tween the displacement of a ship and the volume of a rectangular prism or 
box whose lineal dimensions are the length, breadth, and draught of the 
ship. 

Coefficient of fineness = f R w ; D being the displacement in tous 

of 35 cubic feet of sea-water to the ton, Lthe length between perpendiculars, 
B the extreme breadth of beam, and W the mean draught of water, all in 
feet. 

Coefficient of Water-lines.— An expression of the relation of the 
displacement to the volume of the prism whose section equals the midship 
section of the ship, and length equal to the length of the ship. 

Coefficient of water-lines = ~-. — — -j T -: r . Seaton 

area of immersed water section X L 
gives the following values: 

Coefficient Coefficient of 

of Fineness. Water-lines. 

Finely-shaped ships 0.55 0.63 

Fairly-shaped ships 0.61 0.67 

Ordinary merchant steamers for speeds of 10 to 

11 knots 0.65 0.72 

Cargo steamers, 9 to 10 knots 0.70 0.76 

Modern cargo steamers of large size 0.78 0.83 

Resistance of Ships.— The resistance of a ship passing through 
water may vary from a number of causes, as speed, form of body, displace- 
ment, midship dimensions, character of wetted surface, fineness of lines, 
etc. The resistance of the water is twofold : 1st. That due to the displace- 
ment of the water at the bow and its replacement at the stern, with the 
consequent formation of waves. 2d. The friction between the wetted sur- 
face of the ship and the water, known as skin resistance. A common ap- 
proximate formula for resistance of vessels is 

Resistance = speed 2 x /(/displacement 2 x a constant, or R = S*D§ x C. 
If D = displacement in pounds, S = speed in feet per minute, R = resist- 
ance in foot-pounds per minute, R = CS^Di. The work done in overcom- 
ing the resistance through a distance equal to S is R X S = CS 3 D%\ and 
if Eis the efficiency of the propeller and machinery combined, the indicated 

horse-power I.H.P. = ~^~ - 

If S = speed in knots, D = displacement in tons, and (7 a constant which 
includes all the constants for form of vessel, efficiency of mechanism, etc., 

I.H.P.= *£. 

The wetted surface varies as the cube root of the square of the displace- 
ment; thus, let L be the length of edge of a cube just immersed, whose dis- 
placement is D and wetted surface W. Then D = L 3 or L = y'D, and 
W = 5xi 2 = 5X( |/2>) a . That is, IF varies as Z>§. 



MARINE ENGINEERING. 



1003 



Another approximate formula is 



I.H.P. = 



area of immersed mid ship section x « 



A" 



The usefulness of these two formulae depends upon the accuracy of the 
so-called "constants " Cand K, which vary with the size and form of the 
ship, and probably also with the speed. Seaton gives the following, which 
may be taken roughly as the values of C and K under the conditions ex- 
pressed : 



General Description of Ship, 



Ships over 400 feet long, finely shaped . . 
300 



Ships over 300 feet long, fairly shaped . . 
Ships over 250 feet long, finely shaped . . . 

Ships over 250 feet long, fairly shaped . . . 

Ships over 200 feet long, finely shaped. . . 

Ships over 200 feet long, fairly shaped . . 
Ships under 200 feet long, finely shaped . 

Ships under 200 feet long, fairly shaped. 



Speed, 


Value 


knots. 


of C. 


15 to 17 


240 


15 " 17 


190 


13 " 15 


240 


11 " 13 


260 


11 " 13 


240 


9 " 11 


260 


13 " 15 


200 


11 " 13 


240 


9 " 11 


260 


11 " 13 


220 


9 " 11 


250 


11 " 12 


220 


9 " 11 


240 


9 " 11 


220 


11 " 12 


200 


10 " 11 


210 


9 " 10 


230 


9 " 10 


200 



Value 
of K. 



620 
500 
650 
700 
650 
700 
580 
660 
700 
620 
680 
600 
640 
620 
550 
580 
620 
600 



Coefficient of Performance of Vessels. -The quotient 



^/(displacement) 2 X (speed in knots) 3 

tons of coal in 24 hours 

gives a quotient of performance which represents the comparative cost of 
propulsion in coal expended. Sixteen vessels with three-stage expansion- 
engines in 1890 gave an average coefficient of 14,810, the range being from 
12,150 to 16,700. 

In 1881 seventeen vessels with two-stage expansion-engines gave an aver- 
age coefficient of 11.710. In 1881 the length of the vessels tested ranged from 
260 to 320, and in 1890 from 295 to 400. The speed in knots divided by the 
square root of the length in feet in 1881 averaged 539; and in 1890, 0.579; 
ranging from 0.520 to 0.641. (Proc. Inst. M E.. July, 1891, p. 329.) 

Defects of the Common Formula for Resistance.— Modern 
experiments throw doubt upon the truth of i he statement that the resistance 
varies as the square of the speed. (See Robt. Mansel's letters in Engineer- 
ing, 1891 ; also his paper on The Mechanical Theory of Steamship Propulsion, 
read before Section G of the Engineering Congress, Chicago, 1893.) 

Seaton says: In small steamers the chief resistance is the skin resistance. 
In very fine steamers at high speeds the amount of power required seems 
excessive when compared with that of ordinary steamers at ordinary speeds. 

In torpedo-launches at certain high speeds the resistance increases at a 
lower rate than the square of the speed. 

In ordinary sea-going and river steamers the reverse seems to be the case. 

Rankine's Formula for total resistance of vessels of the "wave- 
line" type is: 

R = ALBVH1 -f 4 sin 2 + sin* 0), 

in which equation is the mean angle of greatest obliquity of the stream- 
lines, A is a constant multiplier, B the mean wetted girth of the surface ex- 
posed to friction, L the length in feet, and V the speed in knots. The potver 
demanded to impel a ship is thus the product of a constant to be determined 
by experiment, the area of the wetted surface, the cube of the speed, and the 



1004 



MARINE ENGINEERING. 



quantity in the parenthesis, which is known as the "coefficient of augmen- 
tation. 1 ' The last term of the coefficient may be neglected in calculating the 
resistance of ships as too small to be practically important. In applying the 
formula, the mean of the squares of the sines of the angles of maximum 
obliquity of the water-lines is to be taken for sin 2 0, and the rule will then 
read thus: 

To obtain the resistance of a ship of good form, in pounds, multiply the 
length in feet by the mean immersed girth and by the coefficient of augmen- 
tation, and then take the product of this "augmented surface,' 1 as Rankine 
termed it, by the square of the speed in knots, and by the proper constant 
coefficient selected from the following: 

For clean painted vessels, iron hulls A — .01 

For clean coppered vessels A — .009 to .008 

For moderately rough iron vessels A — .011 -f- 

The net, or effective, horse-power demanded will be quite closely obtained 
by multiplying the resistance calculated, as above, by the speed in knots and 
dividing by 326. The gross, or indicated, power is obtained by multiplying 
the last quantity by the reciprocal of the efficiency of the machinery and 
propeller, which usually should be about 0.6. Rankine uses as a divisor in 
this case 200 to 260. 

The form of the vessel, even when designed by skilful and experienced 
naval architects, will often vary to such an extent as to cause the above con- 
stant coefficients to vary somewhat; and the range of variation with good 
forms is found to be from 0.8 to 1.5 the figures given. 

For well-shaped iron vessels, an approximate formula for the horse-power 
SV 3 
required is H.P. = , in which S is the "augmented surface." The ex- 

SV 3 

pression ^5" nas been called by Rankine the coefficient of propulsion. In 

the Hudson River steamer " Mary Powell," according to Thurston, this 
coefficient was as high as 23,500. 

DaV 3 

The expression TT _. has been called the locomotive performance. (See 
M.ir. 
Rankine's Treatise on Shipbuilding, 1864; Thurston's Manual of the Steam- 
engine, part ii. p. 16; also paper by F. T. Bowles, U.S.N., Proc. U. S. Naval 
Institute, 1883.) 

Rankine's method for calculating the resistance is said by Seaton to give 
more accurate and reliable results than those obtained by the older rules, 
but it is criticised as being- difficult and inconvenient of application. 

Dr. Kirk's Method.— This method is generally used on the Clyde. 

The general idea proposed by Dr. Kirk is to reduce "all ships to so definite 
and simple a form that they may be easily compared; and the magnitude of 
certain features of this form shall determine the suitability of the ship for 
speed, etc. 

The form consists of a middle body, which is a rectangular parallelopiped, 
and fore body and after body, prisms having isosceles triangles for bases, 
as shown in Fig. 168. 

D E 




This is called a block model, and is such that its length is equal to that of 
the ship, the depth is equal to the mean draught, the capacity equal to the 
displacement volume, and its area of section equal to the area of im- 



MARINE ENGINEERING. 1005 

mersed midship section. The dimensions of the block model may be obtained 
as follows: 

Let AG - HB = length of fore- or after-body = F\ 
GH = length of middle body = M ; 

KL — mean draught = H ; 

area of immersed midship section 
EK = zz - = B. 

Volume of block = (F + M) x B x H; 

Midship section = B X H; 

Displacement in tons — volume in cubic ft. -v- 35. 

AH = AG + GH = F+M = displacement x 35 -*- (B X H). 

The wetted surface of the block is nearly equal to that of the ship of the 
same length, beam and draught ; usually 2% to 5% greater. In exceedingly 
fine hollow-line ships it may be 8$ greater. 

Area of bottom of block = (F-\- M) X B; 
Area of sides = 2M x H. 



Area of sides of ends = A a/ F 2 + (^) 2 X H"; 
Tangent of half angle of entrance = —=-■ = — . 

From this, by a table of natural tangents, the angle of entrance may be 
obtained: 

Angle of Entrance Fore-body in 
of the Block Model, parts of length. 
Ocean-going steamers, 14 knots and upward. 18° to 15° .3 to .36 

12 to 14 knots 21 to 18 .26 to .3 

cargo steamers, 10 to 12 knots.. 30 to 22 .22 to .26 

E. R. Mumford's Method of Calculating Wetted Surfaces 

is given in a paper by Archibald Denny, Eng'g, Sept. 21, 1894. The following 
is his formula, which gives closely accurate results for medium draughts, 
beams, and finenesses: 

S=(LXDX 1.7) + {LXBXC), 

ill which & = wetted surface in square feet; 

L = length between perpendiculars in feet; 
D = middle draught in feet: 
B — beam in feet; 
C = block coefficient. 

The formula may also be expressed in the form S = L(1.7D -\- BC). 

In the case of twin-screw ships having projecting shaft-casings, or in the 
case of a ship having a deep keel or bilge keels, an addition must be made 
for such projections. The formula gives results which are in general much 
more accurate than those obtained by Kirk's method. It underestimates 
the surface when the beam, draught, or block coefficients are excessive; but 
the error is small except in the case of abnormal forms, such as stern-wheel 
steamers having very excessive beams (nearly one fourth the length), and 
also very full block coefficients. The formula gives a surface about 6$ too 
small for such forms. 

To Find tUe Indicated Horse-power from the Wetted 
Surface. (Seaton.) — In ordinary cases the horse-power per 100 feet of 
wetted surface may be found by assuming that the rate for a speed of 10 
knots is 5, and thar the quantity varies as the cube of the speed. For exam- 
ple: To find the number of I.H.P. necessary to drive a ship at a speed of 15 
knots, having a wetted skin of block model of 16,200 square feet: 

The rate per 100 feet = (15/10)3 x 5 = 16.875. 
Then I.H.P. required = 16.875 X 162 = 2734. 



1006 



MARINE ENGINEERING. 



When the ship is exceptionally well-proportioned, the bottom quite clean, 
and the efficiency of the machinery high, as low a rate as 4 I.H.P. per 100 
feet of wetted skin of block model may be allowed 

The gross indicated horse-power includes the power necessary to over- 
come the friction and other resistance of (he engine itself and the shafting, 
and also the power lost in the propellor. In other words, I.H.P. is no meas- 
ure of the resistance of the ship, and can only be relied on as a means of 
deciding the size of engines for speed, so long as the efficiency of the engine 
and propellor is known definitely, or so long as similar engines and propellers 
are employed in ships to be compared. The former is difficult to obtain, 
and it is nearly impossible in practice to know how much of the power shown 
in the cylinders is employed usefully in overcoming the resistance of the 
ship. The following example is given to show the variation in the efficiency 
of propellers: 

Knots. I.H.P. 

H.M.S. " Amazon," with a 4-bladed screw, gave. 12.064 with 1940 

H.M.S. " Amazon,' 1 with a 2-bladed screw, increased pitch, 

and less revolutions per minute 12.396 " 1663 

H.M.S. "Iris,' 1 with a 4-bladed screw 16.577 " 7503 

H.M.S. " Iris," with 2-bladed screw, increased pitch, less 

revolutions per knot 18.587 " 7556 

Relative Horse-power Required for Different Speeds oi 
Vessels. (Horse-power for 10 knots = 1.) — The horse-power is taken usually 
to vary as the cube of the speed, but in different vessels and at different 
speeds it may vary from the 2.8 power to the 3.5 power, depending upon the 
lines of the vessel and upon the efficiency of the engines, the propeller, etc. 



1*1 

ft5 


4 


6 

.239 


8 

.535 


10 
1. 


12 
1.666 


14 
2.565 


16 
3.729 


18 
5.185 


20 
6.964 


22 
9.095 


24 
11.60 


26 
14.52 


28 


HPCC 


.0769 


17.87 


#2-9 


.0701 


.227 


.524 


1. 


1.697 


2.653 


3.908 


5.499 


7.464 


9.841 


12.67 


15.97 


19.80 


.S3 


.0640 


.216 


.512 


1. 


1 . 72S 


2.744 


4.096 


5.83.2 


8. 


10.65 


13.82 


17.58 


21.95 


, S 3-1 


.0584 


.205 


.501 


1. 


1.760 


2.s>;s 


4.293 


6.185 


8.574 


11.52 


15.09 


19.34 




S3 "2 


.0533 


.195 


.490 


1. 


1.792 


2.935 


4 500 


6.559 


9.189 


12.47 


16.47 


21.28 


26 97 


S3 -3 


.0486 


. 1 S5 


.479 


1. 


1 S.25 


3 036 


4 716 


6 957 


9 849 


13 49 


17 98 


23 41 


29.90 


flS-4 


.0444 


.176 




1. 


1.859 


3 139 


1 943 


7.378 


10.56 


14.60 


19.62 


25.76 


33.14 


S 35 


.0105 


.167 


.458 


1. 


1.893 


3.247 


5.181 


7 824 


11.31 


15.79 


21.42 


28.34 


36.73 



21.67 

24.19 

27. 

30.14 

33.63 

37.54 

41.90 



Example in Use op the Table.— -A certain vessel makes 14 knots speed 
with 587 I.H.P. and 16 knots with 900 I.H.P. What I.H.P. will be required at 
18 knots, the rate of increase of horsepower with increase of speed remain- 
ing constant ? The first step is to find the rate of increase, thus: 14* : 16 x :: 
587 : 900. 

x log 16 - x log 14 = log 900 - log 587; 

a;(0.204120 - 0.146128) = 2.954243 - 2.768638, 

whence x (the exponent of S in formula H.P. cc S x ) = 3 2. 

From the table, for S 3 2 and 16 knots, the I H.P. is 4.5 times the I.H.P. at 
10 knots, .-. H.P. at 10 knots = 900 h- 4.5 = 200. 

From the table, for S 3 " 2 and 18 knots, the I.H.P. is 6.559 times the I.H.P. at 
10 knots: .-. H.P. at 18 knots = 2.10 X 6.559 = 1312 HP. 

Resistance per Horse-power for Different Speeds. (One 
horse-power = 33.000 lbs. resistance overcome through 1 tt. in 1 rain.)— The 
resistances per horse-power for various speeds are as follows: For a speed of 
1 knot, or 6080 feet per hour = 101^ ft. per min., 33,000 -=- 101Jg = 325.658 lbs. 
per horse-power; and for any other speed 325.658 lbs. divided by the speed 
in knots; or for 



1 knot 325.66 lbs. 6 knots 54.28 lbs. 

2 knots 162.83 " 7 " 46.52 " 

3 " 108.55 " 8 " 40.71 " 

4 " 81.41 " 9 " 36.18 " 

5 " 65.13 " 10 " 32.57 " 



11 knots 29.61 lbs. 

12 " 27.14 " 

13 " 25.05 " 

14 " 23.26 " 

15 " 21.71 " 



16 knots 20.35 lbs. 

17 " 19.16 " 

18 " 18.09 " 

19 " 17.14 " 

20 li 16.28 " 



MARINE ENGINEERING. 



1007 



Results of Trials of Steam-vessels of Various Sizes. 

(From Seaton's Marine Engineering.) 



Length, perpendiculars 

Breadth, ext reme 

Mean draught water 

Displacement (tons) 

Area Immersed mid. section. 
Wetted skin 



W3 



Length, fore-body. 

Angle of entrance. 
Displacement X 35 



Length x Imm. mid area"'" 

Speed (knots) 

Indicated horse-power 

I.H.P. per 100 ft. wetted skin .... 

I.H.P. per 100 ft. wetted skin, re 

duced to 10 knots 

I.H.P. ' 

Immersed mid area X S 3 



6 


a 

a 

wo* 




1 


A 


*& 

H 


o 


^ 


08 




90' 0" 
10' 6" 
2' 6" 
29 73 
24? 
903 


171' 9" 
18' 9" 
6' 9^" 
280 
99 
3793 


130' 0" 
31' 0" 
8' 10" 
370 
148 
3754 


286' 0" 
34' 3" 
6' 0" 
800 
200 
8222 


230' 0" 
29' 0" 
13' 6" 
1500 
340 
10,075 


45' 0" 


72' 00" 


42' 6" 


143' 0" 


79' 6" 


12° 40' 


11° 30' 


23° 50' 


13° 21' 


17° 0" 


0.481 


0.576 


0.608 


0.489 


0.671 


22 01 
460 
50.9 


15.3 
798 
21.04 


10.74 
371 
9.88 


17.20 
1490 
18.12 


10.01 
503 
5.00 


4.78 


5.87 


7.97 


3.56 


4.90 


223 


192 


172.8 


293.7 


266 


'556? 


445 


495 


683 


6«J0 



w o 



327' 0" 

35' 0" 
13' 0" 
1900 
336 

15,782 

129' 0" 

11° 26' 

0.605 

17.8 
4751 
30.00 



«?> 


°^V 


co^ 


s 


co'5 






kT 


tog 

o 




270' 0" 


300' 0" 


300' 0" 


370' 0" 


392. 0" 


4.2' 0" 


46' 0" 


46' 0" 


41' 0" 


39 0" 


18' 10" 


18' 2" 


18' 2" 


18' 11" 


21' 4" 


3057 


3290 


3^90 


4635 


5767 


632 


700 


700 


656 


738 


16,008 


18,168 


18,163 


22,633 


26,235 


101' 0' 


135' 6" 


135' 6" 


123' 0" 


118' 0" 


18° 44' 


16° 16' 


16° 16' 


16° 4' 


16° 30' 


0.629 


0.548 


0.548 


0.668 


0.698 


14.966 


18.573 


15.746 


13.80 


12.054 


4015 


7714 


3958 


2500 


1758 


25.08 


42.46 


21.78 


11.04 


6.7 


7.49 


6.634 


5.58 


4.20 


3.83 


175.8 


183.7 


218.2 


292 


320 


527.5 


581.4 


690.5 


689 


735 



tf'm 



Length, perpendiculars.. 

Breadth, extreme 

Mean draught water 

Displacement (tons) 

Area Imm. mid. section 

™ ■ f Wetted skin 

"u a 

5 ■§ i Length, fore-body 

pq ^ [ Angle of entrance 

Displacement X 35 
Lsngth x Imm. mid area 

Speed (knots) 

Indicated horse-power 

I.H.P. per 100 ft, wetted skin . . 

I.H.P. per 100 ft. wetted skin, r 

duced to 10 knots 

Dl X .S* 
I.H.P. ■■"' 

Immersed mid area X S 3 
I.H.P. 



450' 0" 
45' 2" 
23' 7" 
8500 
926 
32,578 

129' 0" 

17° 16' 

0.714 

15.045 
4900 
15.04 



289.3 
642.5 



ioos 



MARINE EJSTGIKEERIHG. 



Results of Progressive Speed Trials in Typical Vessels. 

(Eng'g, April 15, 1892, p. 463.) 





eg 
O 

6 


CO 

A - ■ * 




In 


„ U 


£0 


'5d: 




o 
E- 








H-3 


■So 


~ m =« 


Length (in feet) 


135 


230 


265 


300 


360 


375 


525 


Breadth " 4V 
Draught (mean 




14 

5' 1" 


27 
8' 3" 


41 
16' 6" 


43 
16'2" 


60 
23' 9" 


65 
25' 9" 


63 


) on trial 


21' 3" 




103 


735 


2800 


3330 


7390 


9100 


11550 


I.H.P.— 10 knot 
14 " 
18 " 
20 " 




110 
260 
870 
1130 


450 
1100 
2500 
3500 


700 
2100 
6400 
10000 


800 
2400 
6000 
9000 


1000 
3000 
7500 
11000 


1500 
4000 
9000 
12500 


2000 




4600 




10000 




14500 






Speed 


Ratio of 
speed 3 


















10 


1. 


Ratio of H.P.= 


1 


1 


1 


1 


1 


1 


1 


14 


2.744 


" " = 


2 36 


2.44 


3 


3 


3 


2.67 


2.3 


18 


5.832 


44 44 _ 


7.91 


5.56 


9.14 


7.5 


7.5 


6. 


5 


20 


8. 


44 44 _ 


10.27 


7.78 


14.14 


11.25 


11 


8.42 


7.25 


Admiralty coeff . f 10 knots. 


200 


181 


284 


279 


380 


290 


255 


C-f* S3 i! 4 8 << 


232 


203 


259 


255 


347 


298 


304 


147 


190 


181 


217 


295 


282 


297 


I.H.P. ^ u 


156 


186 


159 


198 


276 


278 


281 



The figures for I.H.P. are " round." The " Medusa's " figures for 20 knots 
are from trial on Stokes Bay. and show the retarding effect of shallow water. 
The figures for the other ships for 20 knots are estimated for deep water. 

More accurate methods than those above given for estimating the 
horse-power required for any proposed ship are: 1. Estimations calculated 
from the results of trials of " similar" vessels driven at " corresponding " 
speeds; " similar " vessels being those that have the same ratio of length to 
breadth and to draught, and the same coefficient of fineness, and " corre- 
sponding" speeds those which are proportional to the square roots of 
the lengths of the respective vessels. Froude found that the resistances of 
such vessels varied almost exactly as wetted surface x (speed) 2 . 

2. The method employed by the British Admiralty and by some Clyde 
shipbuilders, viz., ascertaining the resistance of a model of the vessel, 12 to 
20 ft. long, in a tank, and calculating the power from the results obtained. 

Speed. On Canals.— A great loss of speed occurs when a steam-vessel 
passes from open water into a more or less restricted channel. The average 
speed of vessels in the Suez Canal in 1882 was only b% statute miles per hour. 
{Enq'g. Feb. 15, 1884, p. 139.) 

Estimated Displacement, Horse-power, etc. -The table on 
the next page, calculated by the author, will be found convenient for mak- 
ing approximate estimates. 2 

The figures in 7th column are calculated by the formula H.P. = S 3 Ds -*- c, 
in which c — 200 for vessels under 200 ft. long when C = .65, and 210 
when C = .55; c = 200 for vessels 200 to 400 ft. long when C = .75, 220 when 
C = .65, 240 when C — .55; c = 230 for vessels over 400 ft. long when C = .75, 
250 when C = .65, 260 when C = .55. 

The figures in the 8th column are based on 5 H.P. per 100 sq. ft. of wetted 
surface. 

The diame ters of screw in the 9th column are from for mula D — 
3.31 |/I.H.P., and in the 10th column from formula D = 2.71 |/l7H.P. 

To find the diameter of screw for any other speed than 10 knots, revolu- 
tions being 100 per minute, multiply the diameter given in the table by the 
5th root of the cube of the given speed h- 10. For any other revolutions per 
minute than 100, divide by the revolutions and multiply by 100. 

To find the approximate horse-power for any other speed than 10 knots, 
multiply the horse-power given in the table by the cube of the ratio of the 
given speed to 10, or by the relative figure from table on p. 1006. 



MARINE ENGINEERING. 



1009 



Estimated Displacement, Horse-power, etc., of Steam- 
vessels of Various Sizes. 









■g 


Displace- 




Estimat 


(1 Horse- 


Diam. of 


Screw for 10 


lis 


g«] 


5 "S 


•3 =° 


LBDx C 


Wetted Surface 

Ul.lD + BC) 

Sq. ft. 


power at 


10 knots. 


knots sp 
revs, pe 


eed and 100 


Calc. 
from Dis- 


Calc. irom 
Wetted 




►j 51 " 


35 


If Fitch = 


If Pitch = 




3 


1.5 


.55 


tons. 




placem't. 


Surface. 


Diam. 


1.4 Diam. 


12 


.85 


48 


4.3 


2.4 


4.4 


3.6 


»■{ 


3 


1.5 


.55 


1.13 


64 


5.2 


3.2 


4.6 


3.8 


4 


2 


.65 


2.38 


96 


8.9 


4.8 


5.1 


4.2 


20 j 


3 


1.5 


.55 


1.41 


80 


6.0 


4.0 


4.7 


3.9 


4 


2 


.65 


2.97 


120 


10.3 


6.0 


5.3 


4.3 


24 -j 


3.5 


1.5 


.55 


1.98 


104 


7.5 


5.2 


5 


4.1 


4.5 


2 


.65 


4.01 


152 


12.6 


7.6 


5.5 


4.5 


30 j 


4 


2 


.55 


3.77 


168 


11.5 


8.4 


5.4 


4.4 


5 


2.5 


.65 


6.96 


224 


18.2 


11.2 


5.9 


4.8 


40 ] 


4.5 


2 


.55 


5.66 


235 


15.1 


11.8 


5.7 


4.7 


6 


2.5 


.65 


11.1 


326 


24.9 


16.3 


6.3 


5.2 


50 -j 


6 


3 


.55 


14.1 


420 


27.8 


21.0 


6.4 


5.4 


8 


3.5 


.65 


26 


558 


43.9 


27.9 


7.1 


5.8 


60 | 


8 


3.5 


.55 


26.4 


621 


42.2 


81.1 


7.0 


5.7 


10 


4 


.65 


44.6 


798 


62.9 


39 9 


7.6 


6.2 


«.{ 


10 


4 


.55 


44 


861 


59.4 


43.1 


7.5 


6.1 


12 


4.5 


.65 


70.2 


1082 


85.1 


54.1 


8.1 


6.6 


80 -j 


13 


4.5 


.55 


67.9 


1140 


79.2 


57.0 


7.9 


6.5 


14 


5 


.65 


104.0 


1408 


111 


70 4 


8.5 


7.0 


90 1 


13 


5 


.55 


91.9 


1408 


97 


70.4 


8.3 


6.8 


16 


6 


.65 


160 


1854 


147 


92.7 


9 


7.3 


I 


13 


5 


.55 


102 


1565 


104 


78.3 


8.4 


6.9 


100^ 


15 


5.5 


.65 


153 


1910 


143 


95.5 


8.9 


7.3 


1 


17 


6 


.75 


219 


2295 


202 


115 


9.6 


7.8 


( 


14 


5.5 


.55 


145 


2046 


131 


102 


8.8 


7.2 


120^ 


16 


6 


.65 


214 


2472 


179 


124 


9.4 


r i'.6 


1 


18 


6.5 


.75 


301 


2946 


250 


147 


10 


8.2 


I 


16 


6 


.55 


211 


2660 


169 


133 


9.2 


7.4 


140 -{ 


IS 


6.5 


.65 


306 


3185 


227 


159 


9.8 


8.0 


( 


•JO 


7 


.75 


420 


3766 


312 


188 


10.5 


8.5 


I 


17 


6.5 


.55 


278 


3264 


203 


163 


9.6 


7.8 


160^ 


19 


7 


.65 


395 


3880 


269 


194 


10.1 


8.3 


1 


2i 


7.5 


.75 


540 


4560 


368 


228 


10.8 


8.8 


( 


•20 


7 


.55 


396 


4122 


257 


206 


10.1 


8.2 


180^ 


2-2 


7.5 


.65 


552 


4869 


337 


243 


10.6 


8.7 


1 


24 


8 


.75 


741 


5688 


455 


284 


11.3 


9.2 




22 


7 


.55 


484 


4800 


257 


240 


10.1 


8.2 


200^ 


•25 


8 


.65 


743 


5970 


373 


299 


10.8 


8.8 


| 


•28 


9 


.75 


1080 


7260 


526 


363 


11.6 


9.5 


I 


28 


8 


.55 


880 


7250 


383 


363 


10.9 


8.9 


250^ 


32 


10 


.65 


1486 


9450 


592 


473 


11.9 


9.7 


( 


36 


12 


.75 


2314 


11850 


875 


593 


12.8 


10.5 


( 


32 


10 


.55 


1509 


10380 


548 


519 


11.7 


9.6 


3(XK 


36 


12 


.65 


2407 


13140 


806 


657 


12.6 


10.4 


j 


40 


14 


75 


3600 


17140 


1175 


857 


13.6 


11.1 


( 


38 


12 


'.55 


2508 


14455 


769 


723 


1-2.5 


10.2 


350 i 


42 


14 


.65 


3822 


17885 


1111 


894 


13.5 


11.0 


I 


46 


16 


.75 


5520 


21595 


1562 


1080 


14.4 


11.8 




44 


14 


.55 


3872 


19200 


1028 


960 


13.3 


10.8 


400^ 


48 


16 


.65 


• 5705 


23360 


1451 


1168 


14.2 


11.6 


1 


52 


18 


.75 


8023 


27840 


2006 


1392 


15.2 


12.4 


I 


50 


16 


.55 


5657 


24515 


1221 


1226 


13.7 


11.2 


450^ 


54 


18 


.65 


8123 


29565 


1616 


1478 


14.5 


11.9 


( 


5S 


20 


.75 


11157 


31875 


2171 


1744 


15.4 


12.6 




52 


18 


.55 


7354 


29600 


1454 


1480 


14.2 


11.6 


500 -j 


56 


•20 


.65 


10400 


35200 


1966 


1760 


15.1 


12.4 


60 


•22 


.75 


14143 


41200 


2543 


2060 


15.9 


13.0 


( 


56 


20 


.55 


9680 


36245 


1747 


1812 


14.7 


12 


550-^ 


60 


•jo 


.65 


13483 


42735 


2266 


2137 


15 5 


12.7 


i 


64 


34 


.75 


18103 


49665 


2998 


24S3 . 


16.4 


13.4 


( 


60 


22 


.55 


12446 


42900 


2065 


2145 


15.2 


12.5 


600^ 


64 


•24 


.65 


17115 


50220 


2656 


2511 


15.4 


13.1 


1 


68 


•26 


.75 


22731 


58020 


3489 


2901 


16.9 


13.8 



1010 MARINE ENGINEERING. 

THE SCREW-PROPELIEB. 

The "pitch" of a propeller is the distance which any point in a blade, 
describing a helix, will travel in the direction of the axis during one revolu- 
tion, the point being assumed to move around the axis. The pitch of a 
propeller with a uniform pitch is equal to the distance a propeller will 
advance during one revolution, provided there is no slip. In a case of this 
kind, the term "pitch' 1 is analogous to the term "pitch of the thread" of 
an ordinary single-threaded screw. 

Let P = pitch of screw in feet, R = number of revolutions per second, 
V = velocity of stream from the propeller = P x R, v = velocity of the ship 
in feet per second, V — v = slip, A = area in square feet of section of stream 
from the screw, approximately the area of a circle of the same diameter, 
A X V = volume of water projected astern from the ship in cubic feet per 
second. Taking the weight of a cubic foot of sea-water at 64 lbs., and the 
force of gravity at 32, we have from the common formula for force of accel- 

V\ W v-i JV 

eration, viz.: F = M-f = — -+, or F = — v x , when t = 1 second, v x being 

t g t 9 

the acceleration. 

64 AV 
Thrust of screw in pounds = — — — (V - v) = 2AV(V — v). 

Rankine (Rules, Tables, and Data, p. 275) gives the following: To calculate 
the thrust of a propelling instrument (jet, paddle, or screw) in pounds, 
multiply together the transverse sectional area, in square feet, of the stream 
driven astern by the propeller; the speed of the stream relatively to the snip 
in knots; the real slip, or part of that speed which is impressed on that 
stream by the propeller, also in knots; and the constant 5.66 for sea- water, 
or 5.5 for fresh water. If S = speed of the screw in knots, s = speed of ship 
in knots, A — area of the stream in square feet (of sea-water), 

Thrust in pounds = A X S(S - s) X 5.66. 

The real slip is the velocity (relative to water at rest) of the water pro- 
jected sternward; the apparent slip is the difference between the speed of 
the ship and the speed of the screw; i.e., the product of the pitch of the 
screw by the number of revolutions. 

This apparent slip is sometimes negative, due to the working of the screw 
in disturbed water which has a forward velocity, following the ship. Nega- 
tive apparent slip is an indication that the propeller is not suited to the 
ship. 

The apparent slip should generally be about 8% to 10$ at full speed in well- 
formed vessels with moderately fine lines; in bluff cargo boats it rarely 
exceeds 5$. 

The effective area of a screw is the sectional area of the stream of water 
laid hold of by the propeller, and is generally, if not always, greater than 
the actual area, in a ratio which in good ordinary examples is 1.2 or there- 
abouts, and is sometimes as high as 1.4; a fact probably due to the stiffness 
of the water, which communicates motion laterally amongst its particles. 
(Rankine's Shipbuilding, p. 89.) 

Prof. D. S. Jacobus, Trans. A. S. M. E., xi. 1028, found the ratio of the ef- 
fective to the actual disk area of the screws of different vessels to be as 
follows : 

Tug-boat, with ordinary true-pitch screw 1 .42 

" " screw having blades projecting backward 57 

Ferryboat" Bergen," with or-) at speed of 12.09 stat. miles per hour. 1.53 

diuary true-pitch screw ) " " 13.4 " . " " " 1.48 

Steamer " Homer Ramsdell," with ordinary true-pitch screw 1.20 

Size of Screw.— Seaton says: The size of a screw depends on so many 
things that it is very difficult to lay down any rule for guidance, and much 
must always be left to the experience of the designer, to allow for all the 
circumstances of each particular case. The following rules are given for 
ordinary cases. (Seaton and Rounthwaite's Pocket-book): 

101339 
P — pitch of propeller in feet = _ — -, in which S = speed in knots, 

R — revolutions per minute, and x = percentage of apparent slip 
112.6S 



For a slip of 10$, pitch = - 



R 



THE SCREW PROPELLER. 



1011 



D — diameter of 



propeller = K / , ' " . 3 , K beii 



being a coefficient given 



in the table below. If K 



/ I.H.P. 
= 20, D - 20000 JU -p- 



Total developed area of blades = CJU ' ' ', in 



which C is a coefficient 



to be taken from the table. 
Another formula for pitch, given in Seaton's Marine Engineering, is 

P = — a/ ' D2 ' , in which C = 737 for ordinary vessels, and 660 for slow- 
speed cargo vessels with full lines. 



Thickness of blade at root 



•V- 



X k, in which d = diameter of tail- 



shaft in inches, n = number of blades, b = breadth of blade in inches where 
it joins the boss, measured parallel to the shaft axis; k = 4 for cast iron, 1.5 
for cast steel, 2 for gun-metal. 1.5 for high-class bronze. 

Thickness of blade at tip: Cast. iron MD 4- A in.; cast steel .03D -f- .4 in.: 
gun-metal .03 D + .2 in. ; high-class bronze .02Z) +. 3 in., where D = diameter 
of propeller in feet. 

Propeller Coefficients. 





83 




hi 


M 


^5 


i 




P ajs 

P. .5 

< 




d 


o 


goj 


Description of Vessel. 


3 02 


= CQ u 


CO 

> 


0) 

3 

> 


"3.S.3 

P 


Bluff cargo boats 


8-10 


One 


4 


17 -17 5 


19 -17.5 


Cast iron 


Cargo, moderate lines. . . 


10-13 


" 


4 


18 -19 


17 -15.5 


" " 


Pass, and mail, fine lines. 


13-17 


" 


4 


19.5-20.5 


15 -13 


C. I. or S. 


" " " " " 


13-17 


Twin 


4 


20.5-21-5 


14.5-1-2.5 


" " " 


" " " very fine. 


17-22 


One 


4 


21 -22 


12.5-11 


G. M. or B 


" " " " " 


17-22 


Twin 


3 


22 -23 


10.5- 9 


n " " 


Naval vessels, " " 


16-22 




4 


21 -22.5 


11.5-10.5 


" " " 


" " " " 


16-22 


" 


3 


22 -23.5 


8.5- 7 


" " " 


Torpedo-boats, " ' ; 


20-26 


One 


3 


25 


7- 6 


B. or F S. 



C.I., cast iron; G. M., gun-metal; B., bronze 



From the formulee D =i 20000 



/ I.K.P. 

V (P X P) 3 

^400 X I.H.P. 



, if P = D 



S., steel; F. S., fo rged steel. 
, • 737 Vl.H.P. 

and R = 100, we obtain D = -j/400 X I. H.P. = 3 31^1 -H.P. 

If P = 1.4D and E = 100, then D = ^145.8 X I.H.P. = 2.71 fl.H.P. 

From these two formulae the figures for diameter of screw in the table on 
page 1009 have been calculated. They may be used as rough approximations 
to the correct diameter of screw for any' given horse-power, for a speed of 
10 knots and 100 revolutions per minute. 

For any other number of revolutions per minute multiply the figures in 
the table by 100 and divide by the given number of revolutions.. For any 
other speed than 10 knots, since the I.H.P. varies approximately as the cube 
of the speed, and the diameter of the screw as the 5th root of the I.H.P., 
multiply the diameter given for 10 knots by the 5th root of the cube of one 
tenth of the given speed. Or, multiply by the following factors: 

For speed of knots: 

4 5 6 7 8 9 11 12 13 14 15 16 

\/(S ~r- 10)3 

= .577 .660 



.807 .875 .939 1.059 1.116 



.170 l.: 



. 1.275 1.327 



1012 



MARINE ENGINEERING. 



18 19 20 21 



22 23 2 
605 1.648 1.1 



26 27 28 
! 1.774 1.815 1.855 



ms +- 10)3 

= 1.375 1.423 1.470 1.515 1.561 1.605 1.648 1.691 1.' 

For more accurate determinations of diameter and pitch of screw, the 
formulae and coefficients given by Seaton, quoted above, should be used. 

Efficiency of tlie Propeller.— According to Rankine, if the slip of 
the water be s, its weight W, the resistance R, and the speed of the ship v, 



R ■■ 



Ws 



Rv -■ 



Wsv 



This impelling action must, to secure maximum efficiency of propeller, be 
effected by an instrument which takes hold of the fluid without shock or 
disturbance of the surrounding mass, and, by a steady acceleration, gives it 
the required final velocity of discharge. The velocity of the propeller over- 
coming the resistance R would then be 

v + (v + s) _ v s, 

and the work performed would be 

_,/ , s\ Wvs . Ws* 

the first of the last two terms being useful, the second the minimum lost 
work; the latter being the wasted energy of the water thrown backward. 
The efficiency is 

*=* + ■(«+!); 

and this is the limit attainable with a perfect propelling instrument, which 
iimit is approached the more nearly as the conditions above prescribed are 
the more nearly fulfilled. The efficiency of the propelling instrument is 
probably rarely much above 0.60, and never above 0.80. 

In designing the screw-propeller, as was shown by Dr. Froude, the best 
angle for the surface is that of 45° with the plane of the disk; but as all 
parts of the blade cannot be given the same angle, it should, where practi- 
cable, be so proportioned that the " pitch-angle at the centre of effort" 
should be made 45°. The maximum possible efficiency is then, according 
to Froude, 77$. 

In order that the water should be taken on without shock and discharged 
with maximum backward velocity, the screw must have an axially increas- 
ing pitch. 

The true screw is by far the more usual form of propeller, in all steamers, 
both merchant and naval. (Thurston, Manual of the Steam-engine, part ii., 
p. 176.) 

The combined efficiency of screw, shaft, engine, etc., is generelly taken 
at 50$. In some cases it may reach 60% or 65$. Rankine takes the effective 
H.P. to equal the I.H.P. -s- 1.63. 



Pitch-ratio and Slip for Screws of Standard Form, 


Pitch-ratio. 


Real Slip of 
Screw. 


Pitch- ratio. 


Real Slip of 
Screw. 


.8- 


15.55 


1.7 


21.3 


.9 


16.22 


1.8 


21.8 


1.0 


16.88 


1.9 


22.4 


1.1 


17.55 


2.0 


22.9 


1.2 


18.2 


2.1 


23.5 


1.3 


18.8 


2.2 


24.0 


1.4 


19.5 


2.3 


24.5 


1.5 


20.1 


2.4 


25.0 


1.6 


20.7 


2.5 


25.4 



THE PADDLE-WHEEL. 1013 

Results of Recent Researches on the efficiency of screw-propel- 
lers are summarized by S. W. Baruaby, in a paper read before section G of 
the Engineering Congress. Chicago, 1893. He states that the following gen- 
eral principles have been established: 

(a) There is a definite amount of real slip at which, and at which only, 
maximum efficiency can be obtained with a screw of any given type, and 
this amount varies with the pitch-ratio. The slip-ratio proper to a given 
ratio of pitch to diameter has been discovered and tabulated for a screw 
of a standard type, as below (see table on page 1012): 

(b) Screws of large pitch-ratio, besides being less efficient in themselves, 
add to the resistance of the hull by an amount bearing some proportion to 
their distance from it, and to the amount of rotation left in the race. 

(c) The best pitch-ratio lies probably between 1.1 and 1.5. 

(d) The fuller the lines of the vessel, the less the pitch-ratio should be. 

(e) Coarse-pitched screws should be placed further from the stern than 
fine-pitched ones. 

(/) Apparent negative slip is a natural result of abnormal proportions of 
propellers. 

(g) Three blades are to be preferred for high-speed vessels, but when the 
diameter is unduly restricted, four or even more may be advantageously 
employed. 

(h) An efficient form of blade is an ellipse having a minor axis equal to 
four tenths the major axis. 

(i) The pitch of wide-bladed screws should increase from forward to aft, 
but a uniform pitch gives satisfactory results when the blades are narrow, 
and the amount of the pitch variation should be a function of the "width of 
the blade. 

(j) A considerable inclination of screw shaft produces vibration, and with 
right-handed twin-screws turning outwards, if the shafts are inclined at 
all, it should be upwards and outwards from the propellers. 

For results of experiments with screw-propellers, see F. C. Marshall, Proc. 
Inst. M. E. 1881; R. E. Froude, Trans. Institution of Naval Architects, 1886; 
G. A. Calvert, Trans. Institution of Naval Architects 1887; and S. W. Bar- 
uaby, Proc. Inst. Civil Eng'rs 1890, vol. cii. 

One of the most important results deduced from experiments on model 
screws is that they appear to have practically equal efficiencies throughout 
a wide range both in pitch-ratio and in surface-ratio; so that great latitude 
is left to the designer in regard to the form of the propeller. Another im- 
portant feature is that, although these experiments are not a direct guide to 
the selection of the most efficient propeller for a particular ship, they sup- 
ply the means of analyzing the performances of screws fitted to vessels, and 
of thus indirectly determining what are likely to be the best dimensions of 
screw for a vessel of a class whose results are known. Thus a great ad- 
vance has been made on the old method of trial upon the ship itself, which 
was the origin of almost every conceivable erroneous view respecting the 
screw-propeller. (Proc. Inst. M. E., July, 1891.) 

THE PADDLE-WHEEL. 

Paddle-wheels with Radial Floats. (Seaton's Marine En- 
gineering.) — The effective diameter of a radial wheel is usually taken from 
the centres of opposite floats; but it is difficult to say what is absolutely 
that diameter, as much depends on the form of float, the amount of dip, 
and the waves set in motion by the wheel. The slip of a radial wheel is 
from 15 to 30 per cent, depending on the size of float. 

Area of one float = ..'.- ' / X C. 

D is the effective diameter in feet, and C is a multiplier, varying from 
0.25 in tugs to 0.175 in fast-running light steamers. 

The breadth of the float is usually about *4 its length, and its thickness 
about % its breadth. The number of floats varies directly with the diam- 
eter, and there should be one float for every foot of diameter. 

(For a discussion of the action of the radial wheel, see Thurston, Manual 
of the Steam-engine, part ii., p, 182.) 

Feathering Paddle-wheels. (Seaton.) — The diameter of a 
feathering-wheel is found as follows : The amount of slip varies from 12 to 
20 per cent, although when the floats are small or the resistance great it 



1014 MARINE ENGINEERING. 

is as high as 25 per cent; a well-designed wheel on a well-formed ship should 
not exceed 15 per cent under ordinary circumstances. 

If K is the speed of the ship in knots, S the percentage of slip, and B the 
revolutions per minute, 

Diameter of wheel at centres = n ^ n . 
3.14 X B 

The diameter, however, must be such as will suit the structure of the 
ship, so that a modification may be necessary on this account, and the 
revolutions altered to suit it. 

The diameter will also depend on the amount of " dip " or immersion of 
float. 

When a ship is working always in smootli water the immersion of the top 
edge should not exceed V% the breadth of the float; and for general service 
at sea an immension of ^ the breadth of the float is sufficient. If the ship 
is intended to carry cargo, the immersion when light need not be more than 
2 or 3 inches, and should not be more than the breadth of float when at the 
deepest draught; indeed, the efficiency of the wheel falls off rapidly with 
the immersion of the wheel. 

I.H P 
Area of one float = — — — X O. 

C is a multiplier, varying from 0.3 to 0.35; D is the diameter of the wheel 
to the float centres, in feet. 

The number of floats = ^(Z) + 2). 

The breadth of the float = 0.35 X the length. 

The thickness of floats — 1/12 the breadth. 

Diameter of gudgeons = thickness of float. 
Seaton and Rounthwaite's Pocket-book gives: 

Number of floats = , 

Vb 

where B is number of revolutions per minute. 

a v, . . \. I.H.P. X 33000 X K 

Area of one float (in square feet) = — — — — , 

JS X \J-) X B) 

where N = number of floats in one wheel. 

For vessels plying always in smooth water K = 1200. For sea-going 
steamers K — 1400. For tugs and such craft as require to stop and start 
frequently in a tide-way K — 1600. 

It will be quite accurate enough if the last four figures of the cube 
(D X BY be taken as ciphers. 

For illustrated description of the feathering paddle-wheel see Seaton 's 
Marine Engineering, or Seaton and Rounthwaite's Pocket-book. The diam- 
eter of a feathering -wheel is about one half that of a radial wheel for equal 
efficiency. (Thurston.) 

Efficiency of Paddle-wheels.— Computations by Prof. Thurston 
of the efficiency of propulsion by paddle-wheels give for light river steamers 
with ratio of velocity of the vessel, v, to velocity of the paddle -float at 

centre of pressure, V, or — , = -, with a dip = 3/20 radius of the wheel, and 

a slip of 25 per cent, an efficiency of .714; and for ocean steamers with 

= % radius, an efficiency of .685. 

JET-PROPULSION. 

Numerous experiments have been made in driving a vessel by the 
reaction of a jet of water pumped through an orifice in the stern, but 
they have all resulted in commercial failure. Two jet-propulsion steamers, 
the' " Waterwitch," 1100 tons, and the 'Squirt," a small torpedo-boat, 
were built by the British Government. The former was tried in 1867, and 
gave an efficiency of apparatus of only 18 per cent. .The latter gave a speed 
of 12 knots, as against 17 knots attained by a sister-ship having a screw and 
equal steam-power. The mathematical theory of the efficiency of the jet 
was discussed by Rankine in Tlie Engineer, Jan. 11, 1867, and he showed that 
the greater the quantity of water operated on by a jet-propeller, the greater 



RECEXT PRACTICE IK MARtHE ENGtKES. 1015 

is the efficiency. In defiance both of the theory and of the results of earlier 
experiments, and also of the opinions of many naval engineers, more than 
$■200,000 were spent in 1888-90 in New York upon two experimental boats, the 
'• Prima Vista ' 1 and the " Evolution," in which the jet was made of very small 
size, in the latter case only %-incli diameter, and with a pressure of 2500 
lbs. per square inch. As had been predicted, the vessel was a total failure. 
(See article by the author in Mechanics, March, 1891.) 

The theory of the jet-propeller is similar to that of the screw-propeller. 
If A = the area of the jet in square feet, Fits velocity with reference to the 
orifice, in feet per second, v — the velocity of the ship in reference to the 
earth, then the thrust of the jet (see Screw-propeller, ante) k -lAV (V - v). 
The work done on the vessel is 2AV(V— v)v, and the work wasted on the 
rearward projection of the jet is ^ X 2AV(V - v) 2 . The efficiency is 

n JTy/T ^ r 1 — ; — TT^-rp rs = r= • This expression equals unity when 

2AV(V — v)v -\- Al (V- v) 2 \ + v 

V = v, that is, when the velocity of the jet with reference to the earth, or 

V — v, = 0; but then the thrust of the propeller is also 0. The greater the 
lvalue of Fas compared with v. the less the efficiency. For V = 20v, as was 
proposed in the " Evolution, " the efficiency of the jet would be less than 10 
per cent, and this would be further reduced by the friction of the pumping 
mechanism and of the water in pipes. 

The whole theory of propulsion may be summed up in Rankine's words: 
"That propeller is the best, other things being equal, which drives astern 
the largest body of water at the lowest velocity." 

It is practically impossible to devise any system of hydraulic or jet propul- 
sion which can compare favorably, under these conditions, with the screw 
or the paddle-wheel. 

Reaction of a Jet.— If a jet of water issues horizontally from a ves- 
sel, the reaction on the side of the vessel opposite the orifice is equal to the 
weight of a column of water the section of which is the area of the orifice, 
and the height is twice the head. 

The propelling force in jet-propulsion is the reaction of the stream issuing 
from the orifice, and it is the same whether the jet is discharged under 
water, in the open air, or against a solid wall. For proof, see account of 
trials by C. J. Everett, Jr., given by Prof. J. Burkitt Webb, Trans. A. S. M. 
E., xii. 904. 

RECENT PRACTICE IN MARINE ENGINES. 

(From a paper by A. Blechynden on Marine Engineering during the past 
Decade, Proc. Inst. M. E., July, 1891.) 

Since 1881 the three-stage-expansion eneine has become the rule, and the 
boiler-pressure has been increased to 160 lbs. and even as high as 200 lbs. per 
square inch. Four-stage-expansion engines of various forms have also been 
adopted. 

Forced. Draught has become the rule in all vessels for naval service, 
and is comparatively common in both passenger and cargo vessels. By this 
means it is possible considerably to augment the power obtained from a 
given boiler: and so long as it is kept within certain limits it need result in 
no injury to the boiler, but when pushed too far the increase is sometimes 
purchased at considerable cost. 

In regard to the economy of forced draught, an examination of the ap- 
pended table (page 1018) will show that while the mean consumption of coal 
in those steamers working under natural draught is 1.573 lbs. per indicated 
horse-power per hour, it is only 1.336 lbs. in those fitted with forced draught. 
This is equivalent to an economy of 15%. Part of this economy, however, 
may be due to the other heat-saving appliances with which the latter 
steamers are fitted. 

Boilers.— As a material for boilers, iron is now a thing of the past, 
though it seems probable that it will continue yet awhile to be the material 
for tubes. Steel plates can be procured at 132 square feet superficial area 
and iy^ inches thick. For purely boiler work a punching-machine has be- 
come obsolete in marine-engine work. 

The increased pressures of steam have also caused attention to be directed 
to the furnace, and have led to the adoption of various artifices in the shape 
of corrugated, ribbed, and spiral flues, with the object of giving increased 
strength against collapse without abnormally increasing the thickness of 
the plate. A thick furnace-plate is viewed by many engineers with great 



1016 MARINE ENGINEERING. 

suspicion; and the advisers of the Board of Trade have fixed the limit of 
thickness for furnace-plates at % inch ; but whether this limitation will 
stand in the light of prolonged experience remains to be seen. It is a fact 
generally accepted that the conditions of the surfaces of a plate are far 
greater factors in its resistance to the transmission of heat than either the 
material or the thickness. With a plate free from lamination, thickness 
being a mere secondary element, it would appear that a furnace-plate might 
be increased from % hich to % irch thickness without increasing its resist- 
ance more than 1J4#. So convinced have some engineers become of the 
soundness of this view that they have adopted flues % i"ch thick. 

Piston-valves.— Since higher steam -pressures have become common, 
piston-valves have become the rule for the high-pressure cjdiuder, and are 
not unusual for the intermediate. When well designed they have the great 
advantage of being almost free from friction, so far as the valve itself is 
concerned. In the earlier piston-valves it was customary to fit spring 
rings, which were a frequent source of trouble and absorbed a large amount 
of power in friction; but in recent practice it has become usual to fit spring- 
less adjustable sleeves. 

For low-pressure cylinders piston-valves are not in favor; if fitted with 
spring rings their friction is about as great as and occasionally greater than 
that of a well-balanced slide-valve; while if fitted with springless rings there 
is always some leakage, which is irrecoverable. But the large port-clear- 
ances inseparable froin the use of piston -valves are most objectionable; 
and with triple engines this is especially so, because with the customary 
late cut-off it becomes difficult to compress sufficiently for iusuring econo- 
my and smoothness of working when in " full gear, 1 ' without some special 
device. 

Steam-pipes.— The failures of copper steam-pipes on large vessels 
have drawn serious attention both to the material and the modes of con- 
struction of the pipes. As the brazed joint is liable to be imperfect, it is 
proposed to substitute solid drawn tubes, but as these are not made of large 
sizes two or more tubes may be needed to take the place of one brazed tube. 
Reinforcing the. ordinary brazed tubes by serving them with steel or copper 
wire, or by hooping them at intervals with steel or iron bands, has been 
tried aud found to answer perfectly. 

Auxiliary Supply of Fresh Water— Evaporators.— To make 
up the losses of water due to escape of steam from safety-valves, leakage at 
glands, joints, etc., either a reserve supply of fresh water is carried in tanks, 
or the supplementary feed is distilled from sea-water by special apparatus 
provided for the purpose. In practice the distillation is effected by passing 
steam, say from the first receiver, through a nest of tubes inside a still or 
evaporator, of which the steam-space is connected either with the second 
receiver or with the condenser. The temperature of the steam inside the 
tubes being higher than that of the steam either in the second receiver or in 
the condenser, the result is that the water inside the still is evaporated, and 
passes with the rest of the steam into the condenser, where it is condensed 
and serves to make up the loss. This plan localizes the trouble of the de- 
posit, and frees it from its dangerous character, because an evaporator can- 
not become overheated like a boiler, even though it be neglected until it 
salts up solid; and if the same precautions are taken in working the evapo- 
rator which used to be adopted with low-pressure boilers when they were 
fed with salt water, no serious trouble should result. 

Weir's Feed-water Heater.— The principle of a method of heating 
feed-water introduced by Mr. James Weir and widely adopted in the 
marine service is founded on the fact that, if the feed-water as it is drawn 
from the hot-well be raised in temperature by the heat of a portion of steam 
introduced into it from one of the steam-receivers, the decrease of the coal 
necessary to generate steam from the water of the higher temperature bears 
a greater ratio to the coal required without feed-heating than the power 
which would be developed in the cylinder by that portion of steam would 
bear to the whole power developed when passing all the steam through all 
the cylinders. Suppose a triple-expansion engine were working under the 
following conditions without feed-heating: boiler-pressure 150 lbs.; I.H.P. in 
high-pressure cylinder 398, in intermediate and low-pressure cylinders to- 
gether 790, total 1188. The temperature of hot-well 100° F. Then with feed- 
heating the same engine might work as follows: the feed might be heated to 
220° F., and the percentage of steam from the first receiver required to heat 
it would be 10.9$; the I.H.P. in the h.p. cylinder would be as before 398, and 
in the three cylinders it would be 1103, or 93$ of the power developed without 



RECENT PRACTICE IN MARINE ENGINES. 



1017 



feed-heating. Meanwhile the heat to be added to each pound of the feed- water 
at 220° F. for converting it into steam would be 1005 units against 1125 units 
with feed at 100° F., equivalent to an expenditure of only 89.4$ of the heat 
required without, feed-heating. Hence the expenditure of heat in relation 
to power would be 89.4 -:- 93.0 = 96.4$, equivalent to a heat economy of 3.6$. 
If the steam for heating can be taken from the low-pressure receiver, the 
economy is about doubled. 

Passenger Steamers fitted with Twin Screws. 



Vessels. 


£ £ <*> 


S3 

n 


Cylinders, two sets 
in all. 


I** 


1 

0> A 




Diameters. 


Stro. 




City of New York \ 

" " Paris i 

Majestic (_ 

Teutonic ( 


Feet 
525 

565 
500 
463^ 

440 

415 
460 


Feet 
63M 

58 

55^ 

51 

48 
54J^ 


Inches 
45, 71, 113 

43, 68, 110 

40, 67, 106 

41, 66, 101 

32, 51, 82 

34, 54, 85 
34^, 57^, 92 


In. 
60 

60 
66 
66 

54 

51 
60 


Lbs. 
150 

180 
160 
160 

160 

160 
170 


I.H.P. 
20,000 

18,000 
11,500 
12,500 

10,125 

10,000 
11,656 


Columbia 

Empress of In di a ) 

" " Japau V 

" " China ) 
Orel 


Scot 



Comparative Results of Working of Marine Engines, 
1872, 1881, and 1891. 



Boilers, Engines, and Coal. 


1872. 


1881. 


1891. 




52.4 
4.410 
55.67 
376 
2.110 


77.4 
3.917 
59.76 
467 
1.828 


158.5 


Heating-surface per horse-power, sq. ft 

Revolutions per minute, revs 


3.275 
63.75 
529 


Coal per horse-power per hour, lbs 


1 522 



Weight of Three - stage - expansion Engines in Nine 
Steamers in Relation to Indicated Horse-power and 
to Cylinder-capacity. 



* 


Weight of 
Machinery. 


Relative Weight of 


Machinery. 












Per Indicated Horse - 


Engine-room 
per cu. ft. 

of Cylinder- 
capacity. 


S£ be 


Type of 


W 


c 2 


53 o 

o p 


I 
o 




power. 




Machinery. 


6 


Engine- 
room. 


Boiler- 
room. 


Total 






tons. 


tons. 


tons. 


lbs. 


lbs. 


lbs. 


tons. 


tons. 




1 


681 


662 


1313 


226 


220 


446 


1.30 


3.75 


Mercantile 


2 


638 


619 


1257 


259 


251 


510 


1.46 


4.10 


" 


3 


134 


128 


262 


207 


198 


405 


1.23 


3.23 


" 


4 


38.8 


46.2 


85 


170 


203 


373 


1.29 


3.30 


" 


5 


719 


695 


1414 


167 


162 


329 


1.41 


3.44 


" 


6 


75.2 


107.8 


183 


141 


202 


343 


1.37 


3.37 


" 


7 


44 


61 


105 


77 


108 


185 


1.21 


2.72J 


Naval 
horizontal 


8 


73.5 


109 


182.5 


78 


116 


194 


1.11 


2.78 


do. 


9 


202 


429 


691 


62.5 


102 


165 


0.82 


2.70 | 


Naval 
vertical 



1018 



MARINE ENGINEERING. 



! " 



WWKW W pdK KK 






•d'H'I J8CI 



- ~ . • "■..--•;. 



MTIOTJ 
.I9d 9)V.lg 

jo -y bs .iad 
!>ii.inq l^oo 



•aj-ejS jo -^j 
bs aad VTH'I 















'd'H'I I^SoSoiod 
•13d I O^hUmotw 



suormiOAan 



•3 rassa.Kl k *S '5 "5 "* S 5? .^ ^S 0000 

-tuua^s 






•aoBj.ms I*- III! 



-JUS SlIJIOOO i ^jHTjrt^HOfeo'rHr-'oOHfNINNsrri; 



'OOOHOOHICC 



^ ^ 






CONSTRUCTION OF BUILDINGS. 



1019 



Dimensions, Indicated Horse - power, and Cylinder - 
capacity of Three - stage - expansion .Engines in Nine 
Steamers. 



o ^ 
53 


Single or 
Twin Screws. 


Cylinders. 


<b 53 


a .5 

— — 33 




53& 
11 


Heating-sur- 
face. 




Diameters. 


Stroke 


Total. 


Per 
I.H.P. 






ins. 


ins. 


revs. 


lbs. 


I.H.P. 


cu.ft. 


sq. ft. 


sq. ft. 


j 


Single 


40 66 100 


72 


64.5 


160 


6751 


522 


17,640 


2.62 


2 


" 


39 61 97 


66 


67.8 


160 


5525 


436 


15,107 


2.73 


3 


" 


23 38 61 


42 


83 


160 


1450 


109 


3,973 


2.73 


4 


" 


17 26^ 42 


24 


90 


150 


510 


30 


1,403 


2.75 


5 


Twin 


32 54 82 


54 


88 


160 


9625 


508 


20,193 


2.10 


6 




15 24 38 


27 


113 


150 


1194 


55 


3,200 


2.68 


7 


Single 


20 30 45 


24 


191 


145 


1265 


36.3 


2,227 


1.76 


8 


Twin 


18^ 29 43 


24 


182.5 


140 


2105 


66.2 


3,928 


1.87 


9 


" 


33J^ 49 74 


39 


145 


150 


9400 


319 


15,882 


1.62 



CONSTRUCTION OF BUILDINGS.* 

(Extract from the Building Laws of the City of New "York, 1893.) 
"Walls of Warehouses, Stores, Factories, and Stables.— 

25 feet or less in width between walls, not less than 12 in. to height of 40 ft.; 
If 40 to 60 ft. in height, not less than 16 in. to 40 ft., and 12 in. thence to top; 
60 to 80 " " " " " 20 " 25 " 16 

75 to 85 " " " " " 24 " 20 ft.; 20 in. to 60 ft., and 16 in. 

to top ; 
85 to 100 ft. in height, not less than 28 in. to 25 ft. ; 24 in. to 50 ft.; 20 in" 

to 75 ft., and 16 in. to top; 
Over 100 ft. in height, each additional 25 ft. in height, or part thereof , next 
above the curb, shall be increased 4 inches in thickness, the upper 100 
feet remaining the same as specified for a wall of that weight. 
If walls are over 25 feet apart, the bearing-walls shall be 4 inches thicker 
than above specified for every 12^ feet or fraction thereof that said walls 
are more than 25 feet apart 

Strength of Floors, Roofs, and Supports. 

Floors calculated to bear 
safely per sq. ft., in addition 
to their own weight. 
Floors of dwelling, tenement, apartment-house or hotel, not 

less than 70 lbs. 

Floors of office-building, not less than 100 " 

public-assembly ouilding, not less than 120 " 

" store, factory, warehouse, etc., not less than 150 " 

Roofs of all buildings, not less than 50 " 

Every floor shall be of sufficient strength to bear safely the weight to be 
imposed thereon, in addition to the weight of the materials of which the 
floor is composed. 

Columns and Posts.— The strength of all columns and posts shall 
be computed according to Gordon's formulae, and the crushing weights in 
pounds, to the square inch of section, for the following-named materials, 
shall be taken as the coefficients in said formulae, namely: Cast iron, 80.000; 
*The limitations of space forbid any extended treatment of this subject. 
Much valuable information upon it will be found in Trautwine's Civil Engi- 
neer's Pocket-book, and in Kidder's Architect's and Builder's Pocket-book. 
The latter in its preface mentions the following works of reference: " Notes 
on Building Construction," 3 vols., Rivingtons, publishers, Boston ; "Building 
Superintendence," by T. M.Clark (J. R. Osgood & Co., Boston.); "The 
American House Carpenter,''' by R. G. Hatfield; "Graphical Analysis of 
Roof-trusses," by Prof. C. E. Greene; "The Fire Protection of Mills," by C. 
J. H. Woodbury; "House Drainage and Water Service," by James C. 
Bayles; "The Builder's Guide and Estimator's Price- book," and " Plaster- 
ing Mortars and Cements," by Fred. T. Hodgson; "Foundations and Con- 
crete Works," and "Art of Building," by E. Dobson, Weale's Series, London. 



1020 COKSTBUoTlOff OF BUILMHGS. 

wrought or rolled iron, 40,000; rolled steel, 48,000; white pine and spruce, 
3500; pitch or Georgia pine, 5000; American oak, C000. The breaking strength 
of wooden beams and girders shall be computed according to the formulae 
in which the constants for transverse strains for central load shall be as 
follows, namely: Hemlock, 400; white pine, 450; spruce, 450; pitch or Georgia 
pine, 550; American oak, 550; and for wooden beams and girders carrying a 
uniformly distributed load the constants will be doubled. The factors of 
safety shall be as one to four for all beams, girders, and other pieces subject 
to a transverse strain; as one to four for all posts, columns, and other 
vertical supports when of wrought iron or rolled steel; as one to five for 
other materials, subject to a compressive strain; as one to six for tie- 
rods, tie-beams, and other pieces subject to a tensile strain. Good, solid, 
natural earth shall be deemed to safely sustain a load of four tons to the 
superficial foot, or as otherwise determined by the superintendent of build- 
ings, and the width of footing-courses shall be at least sufficient to meet this 
requirement. In computing the width of walls, a cubic foot of brickwork 
shall be deemed to weigh 115 lbs. Sandstone, white marble, granite, and 
other kinds of building-stone shall deemed to weigh 160 lbs. per cubic foot. 
The safe-bearing load to apply to good brickwork shall be taken at 8 tons 
per superficial foot when good lime mortar is used, ll^j tons per superficial 
root when good lime and cement mortar mixed is used, and 15 Ions per sup- 
erficial foot when good cement mortar is used. 

Fire-proof Buildings— Iron and Steel Columns.— All cast- 
iron, wrought-iron, or rolled-steel columns shall be made true and smooth 
at both ends, and shall rest on iron or steel becKplates, and have iron or 
steel cap-plates, which shall also be made true. All iron or steel trimmer- 
beams, headers, and tail-beams shall be suitably framed and connected to- 
gether, and the iron girders, columns, beams, trusses, and all other ironwork 
of all floors and roofs shall be strapped, bolted, anchored, and connected to- 
gether, and to the walls, in a strong and substantial manner. Where beams 
are framed into headers, the angle-irons, which are bolted to the tail-beams, 
shall have at least two bolts for all beams over 7 inches in depth, and three 
bolts for all beams 12 inches and over in depth, and these bolts shall not be 
less than %, inch in diameter. Each one of such angles or knees, when bolted 
to girders, shall have the same number of bolts as stated for the other leg. 
The angle-iron in no case shall be less in thickness than the header or trim- 
mer to which it is bolted, and the width of angle in no case shall be less than 
one third the depth of beam, excepting that no angle-knee shall be less than 
2}4 inches wide, nor required to be more than 6 inches wide. All wrought- 
iron or rolled-steel beams 8 inches deep and under shall have bearings equal 
to their depth, if resting on a wall; 9 to 12 inch beams shall have a bearing 
of 10 inches, and all beams more than 12 inches in depth shall have bearings 
of not less than 12 inches if resting on a wall. Where beams rest on iron 
supports, and are properly tied to the same, no greater bearings shall be re- 
quired than one third of the depth of the beams. Iron or steel floor-beams 
shall be so arranged as to spacing and length of beams that the load to be 
supported by them, together with the weights of the materials used in the 
construction of the said floors, shall not cause a deflection of the said beams 
of more than 1/30 of an inch per linear foot of span; and they shall be tied 
together at intervals of not more than eight times the depth of the beam. 

Under the ends of all iron or steel beams, where they rest on the walls, a 
stone or cast-iron template shall be built into the walls. Said template shall 
be 8 inches wide in 12 -inch walls, and in all walls of greater thickness said 
template shall be 12 inches wide; and such templates, if of stone, shall not be 
in any case less than 2^ inches in thickness, and no template shall be less 
than 12 inches long. 

No cast-iron post or column shall be used in any building of a less average 
thickness of shaft than three quarters of an inch, nor shall it have an un- 
supported length of more than twenty times its least lateral dimensions or 
diameter. No wrought-iron or rolled-steel column shall have an unsupported 
length of more than thirty times its least lateral dimension or diameter, nor 
shall its metal be less than one fourth of an inch in thickness. 

Lintels, Bearings and Supports.— All iron or steel lintels shall 
have bearings proportionate to the weight to be imposed thereon, but no 
lintel used to span any opening more than 10 feet in width shall have a bear- 
ing less than 12 inches at each end, if resting on a wall; but if resting on an 
iron post, such lintel shall have a bearing of at least 6 inches at each end, 
by the thickness of the wall to be supported 

Strains on Girders and Bivets.— Rolled iron or steel beam gir- 



STRENGTH OF FLOORS. 1021 

ders, or riveted iron or steel plate gilders used as lintels or as girders, 
carrying a wall or door or both, shall be so proportioned that the loads 
which may come upon them shall not produce strains in tension or com- 
pression upon the flanges of more than 12,000 lbs. for iron, nor more than 
15.000 lbs. for steel per square inch of the gross section of each of such 
flanges, nor a shearing strain upon the web-plate of more than 6000 lbs. per 
square inch of section of such web-plate, if of iron, nor more than 7000 
pounds if of steel; but no web-plate shall be less than J4 inch, in 
thickness. Rivets in plate girders shall not be less than % inch in diameter, 
and shall not be spaced more than 6 inches apart in any case. They shall be 
so spaced that their shearing strains shall not exceed 9000 lbs. per square 
inch, on their diameter, multiplied by the thickness of the plates through 
which they pass. The riveted plate girders shall be proportioned upon the 
supposition that the bending or chord strains are resisted entirely by the 
upper and lower flanges, and that the shearing strains are resisted entirely 
by the web-plate. No part of the web shall be estimated as flange area, nor 
more than one half of that portion of the angle-iron which lies against the 
web. The distance between the centres of gravity of the flange areas will 
be considered as the effective depth of the girder. 

The building laws of the City of New York contain a great amount of de- 
tail in addition to the extracts above, and penalties are provided for viola- 
tion. See An Act creating a Department of Buildings, etc., Chapter 275, 
Laws of 1892. Pamphlet copy published by Baker, Voorhies & Co., New 
York. 

MAXIMUM IiOAD ON FLOORS. 

(Eng'g, Nov. 18, 1892. p. 644.)— Maximum load per square foot of floor 

surface due to the weight of a dense crowd. Considerable variation is 

apparent in the figures given by many authorities, as the following table 

shows: 

Authorities Weight of Crowd, 

Authorities. lbs _ per gq ft 

French practice, quoted by Trautwine and Stoney 41 

Hatfield (" Transverse Strains," p. 80) 70 

Mr. Page, London, quoted by Trautwine 84 

Maximum load on American highway bridges according to 

Waddell's general specifications 100 

Mr. Nash, architect of Buckingham Palace 120 

Experiments by Prof. W. N. Kernot, at Melbourne \ ut 1 

Experiments by Mr. B. B. Stoney (" On Stresses," p. 617) . . . 147.4 

The highest results were obtained by crowding a number of persons pre- 
viously weighed into a small room, the men being tightly packed so as to 
resemble such a crowd as frequently occurs on the stairways and platforms 
of a theatre or other public building. 

STRENGTH OF FLOORS. 
(From circular of the Boston Manufacturers 1 Mutual Insurance Co.) 

The following tables were prepared by C. J. H. Woodbury, for determining 
safe loads on floors. Care should be observed to select the figure giving the 
greatest possible amount and concentration of load as the one which may 
be put upon any beam or set of floor-beams; and in no case should beams be 
subjected to greater loads than those specified, unless a lower factor of 
safety is warranted under the advice of a competent engineer. 

Wiienever and wherever solid beams or heavy timbers are made use of in 
the construction of a factory or warehouse, they should not be painted, var- 
nished or oiled, filled or encased in impervious concrete, air-proof plastering, 
or metal for at least three years, lest fermentation should destroy them by 
what is called " dry rot." 

It is, on the whole, safer to make floor-beams in two parts, with a small 
open space between, so that proper ventilation may be secured, even if the 
outside should be inadvertently painted or filled. 

These tables apply to distributed loads, but the first can be used in respect 
to floors which may carry concentrated loads by using half the figure given 
in the table, since a beam will bear twice as much load when evenly distrib- 
uted over its length as it would if the load was concentrated in the centre 
of the span. 

The weight of the floor should be deducted from the figure given in the 
table, in order to ascertain the net load which may be placed upon any floor. 
The weight of spruee may be taken at 36 lbs. per cubic foot, and that of 
Southern pine at 48 lbs. per cubic foot. 



1022 CO^TRtTCTtOH OF BUILDINGS. 

Table I was computed upon a working modulus of rupture of Southern 
pine at 2160 lbs., using a factor of safety of six. It can also be applied to 
ascertaining the strength of spruce beams if the figures given in the table 
are multiplied by 0.78; or in designing a floor to be sustained by spruce 
beams, multiply the required load by 1.28, and use the dimensions as given 
by the table. 

Theses tables are computed for beams one inch in width, because the 
strength of beams increases directly as the width when the beams are broad 
enough not to cripple. 

Example.— Required the safe load per square foot of floor, which may be 
safely sustained by a floor on Southern pine 10 x 14 inch beams, 8 feet on 
centres, and 20 feet span. In Table I a 1 X 14 inch beam, 20 feet span, will 
sustain 118 lbs. per foot of span; and for a beam 10 inches wide the load 
would be 1180 lbs. per foot of span, or 147^ lbs. per square foot of floor fur 
Southern-pine beams. From this should be deducted the weight of the floor, 
which would amount to 17^2 lbs. per square foot, leaving 130 lbs. per square 
foot as a safe load to be carried upon such a floor. If the beams are of 
spruce, the result of 147^ lbs. would be multiplied by 0.78, reducing the load 
to 115 lbs. The weight of the floor, in this instance amounting to 16 lbs., 
would leave the safe net load as 90 lbs. per square foot for spruce beams. 

Table II applies to the design of floors whose strength must be in excess 
of that necessary to sustain the weight, in order to meet the conditions of 
delicate or rapidly moving machinery, to the end that the vibration or dis- 
tortion of the floor may be reduced to the least practicable limit. 

In the table the limit is that of load which would cause a bending of the 
beams to a curve of which the average radius would be 1250 feet. 

This table is based upon a modulus <»f elasticity obtained from observa- 
tions upon the deflection of loaded storehouse floors, and is taken at 2,000,000 
lbs. for Southern pine; the same table can be applied to spruce, whose 
modulus of elasticity is taken as 1,200,000 lbs., if six tenths of the load for 
Southern pine is taken as the proper load for spruce; or, in ihe matter of 
designing, the load should be increased one and two thirds times, and the 
dimension of timbers for this iucreased load as found in the table should be 
used for spruce. 

It can also be applied to beams and floor-timbers which are supported at 
each end and in the middle, remembering that the deflection of a beam 
supported in that manner is only four tenths that of a beam of equal span 
which rests at each end; that is to say, the floor-planks are two and one 
half times as stiff, cut two bays in length, as they would be if cut only one 
bay in length. When a floor-plank two bays in length is evenly loaded, 
three sixteenths of the load on the plank is su-tained by the beani at each 
end of the plank, and ten sixteenths by the beam under the middle of the 
plank; so that for a completed floor three eighths of the load would be sus- 
tained by the beams under the joints of the plank, and five eighths of the load 
by the beams under the middle of the plank: this is the reason of the impor- 
tance of breaking joints in a floor-plank every three feet in order that each 
beam shall receive an identical load. If it were not so, three eighths of the 
whole load upon the floor would be sustained by every other beam, and five 
eighths of the load by the corresponding alternate beams. 

Repeating the former example for the load on a mill floor on Southern- 
pine beams 10 X 14 inches, and 20 feet span, laid 8 feet on centres: In Table 
II a 1 X 14 inch beam should receive 61 lbs. per foot of span, or 75 lbs. per 
sq. ft. of floor, for Southern-pine beams. Deducting the weight of the floor, 
17^2 lbs. per sq. ft., leaves 57 lbs. per sq. ft. as the advisable load. 

If the beams are of spruce, the result of 75 lbs. should be multiplied by 0.6, 
reducing the load to 45 lbs. The weight of the floor, in this instance amount- 
ing to 16 lbs., would leave the net load as 29 lbs. for spruce beams. 

If the beams were two spans in length, they could, under these conditions, 
support two and a half times as much load with an equal amount of deflec- 
tion, unless such load should exceed the limit of safe load as found by Table 
I, as would be the case under the conditions of this problem. 

Mill Columns.— Timber posts offer more resistance to fire than iron 
pillars, anct have generally displaced them in millwork. Experiments 
made on the testing-machine at the U. S. Arsenal at Watertown, Mass., 
show that sound timber posts of the proportions customarily used in mill- 
work yield by direct crushing, the strength being directly as the area at the 
smallest part. The columns yielded at about 4500 lbs. per square inch, con- 
firming the general practice of allowing 600 lbs. per square inch, as a safe 
load. Square columns are one fourth stronger than round ones of the same 
diameter. 



STRENGTH OF FLOORS. 



1023 



I. Safe Distributed Loads upon Southern-pine Beams 
One Inch in Width. 

(C. J. H. Woodbury.) 
(If the load is concentrated at the centre of the span, the beams will sus- 
tain half the amount as given in the table.) 



Oi 


Depth of Beam in inches. 


s 


2 | 3 


,« 


5 | C 


I* 


|8| 9 


| 10 | 11 | 12 


I 13 I U 


1 » 


N 


ft 

Xfl 






Load in 


pounds per foot of Span. 






5 


38 


86 


154 


240 


346 


470 


614 


778 


9(50 














6 


27 


R0 


107 


T67 


24(1 


327 


427 


54 C 


lit;? 


807 












7 


20 


44 


7S 


122 


176 


240 


814 


397 


490 


59:- 


705 


828 








8 


15 


34 


60 


94 


185 


1S4 


210 


304 


875 


454 


540 


684 


785 






9 




27 


47 


74 


107 


145 


190 


240 


296 




427 


501 


581 


667 


759 


10 




22 


38 


m 


86 


118 


154 


194 


240 




346 


406 


470 


540 


614 


11 






32 


m 


71 


97 


127 


161 


198 


240 


286 






446 


508 


12 






27 


42 


fin 


82 


107 


135 


167 


202 


240 


2S2 


827 


375 


474 


18 








36 


51 


70 


90 


115 


142 


172 


205 


240 


278 


320 


364 


14 








31 


44 


60 


78 


99 


123 


148 


176 


207 


240 


276 


314 


15 








27 


38 


52 


68 


86 


107 


129 


154 


180 


209 


240 


273 


16 










34 
30 


46 
41 


60 

58 


76 
67 


94 

88 


113 

101 


135 

120 


158 
140 


184 
168 


211 

187 


240 


17 










217 


18 












36 


47 


60 


74 


90 


107 


125 


145 


167 


190 


19 














48 

38 


54 
49 
44 


66 
60 
54 


80 
73 
66 


96 

86 
78 


112 

101 
92 


130 
118 
107 


150 
135 
122 


170 


20 














154 


21 














139 


22 


















50 


60 


71 


84 


97 


112 


127 


23 


















45 


55 


65 




89 


102 


116 


24 




















50 


60 


70 


82 


94 


107 


25 




















46 


55 


65 


75 


86 


98 



Distributed Loads upon Southern-pine Beams suffi- 
cient to produce Standard Limit of Deflection. 

(C. J. H. Woodbury.) 



t" 








Depth of Beam in 


inches. 






a' 
o ^ 


tl ~l 


2| 3 I 4 


5 


6 


r 1 si 9 1 10 


| 11 | 12 


1 13 I 14 1 15 


I 16 




















ft 

CO 


Load in pounds per foot of Span. 







3 


10 


23 


44 


77 


122 


182 


















.0300 


6 


2 


7 


16 


31 


53 


85 


126 


180 


247 














.0432 


7 




5 


12 


23 


39 


62 


93 


132 


181 


241 












.0588 


8 




4 


9 


17 


30 


48 


71 


101 


139 


185 


241 


305 








.0768 


9 






7 


14 


24 


38 


56 


80 


110 


146 


190 


241 


301 






.0972 


10 






6 


11 


19 


30 


46 


65 


89 


118 


154 


195 


244 


800 




.1200 


11 








9 


16 
13 


25 
21 


38 
32 


54 
45 


73 
62 


98 
82 


127 

107 


161 
136 


202 
- 169 


218 
208 


301 
258 


.1452 


12 








.1728 


13 










11 


18 


27 


38 


53 


70 


91 


116 


144 


178 


215 


.2028 


14 












16 
14 


23 

20 


33 

29 


45 
40 


60 
53 


78 
68 


100 

87 


124 
108 


153 
133 


186 
162 


.2352 


15 












.2700 


16 














18 


25 


35 


46 


60 


76 


95 


117 


147 


.3072 


17 














16 


22 
20 


31 

27 


41 

37 


53 

47 


68 
60 


84 


104 
93 


126 
112 


.3468 


18 










.3888 


19 
















18 


25 


33 


43 


54 


68 


83 


101 


.4332 


90 


















22 
20 


30 
27 
24 


38 
35 
32 


49 
44 
40 


61 
55 
.50 


75 
68 
62 


91 

83 
75 


.4800 


21 








.5292 


22 


















.5808 


23 




















22 


29 


37 


46 


57 


69 


.6348 


24 






















27' 


34 


42 


52 


63 


.6912 


25 






















25 


31 


39 


48 


58 


.7500 



1024 ELECTKICAL ENGINEERING. 



ELECTRICAL ENGINEERING. 

STANDARDS OF MEASUREMENT. 

O.G.S. (Centimetre 9 Gramme, Second) or "Absolute" 
System of Physical Measurements s 

Unit of space or distance = 1 centimetre, cm.; 

Unit of mass = 1 gramme, gm. ; 

Unit of time = 1 second, s.; 

Unit of velocity = space -4- time — 1 centimetre in 1 second; 

Unit of acceleration = change of 1 unit of velocity in 1 second; 

Acceleration due to gravity, at Paris, = 981 centimetres in 1 second; 

Unit of force = 1 dyne = ~ gramme = •°° 22046 lb. = .000002247 lb. 

A dyne is that force which, acting on a mass of one gramme during one 
second, will give it a velocity of one centimetre per second. The weight of 
one gramme in latitude 40° to 45° is about 980 dynes, at the equator 973 dynes, 
and at the poles nearly 984 dynes. Taking the value of g, the acceleration 
due to gravity, in British measures at 32.185 feet per second at Paris, and the 
metre = 39.37 inches, we have 

1 gramme = 32.185 x 12 -f- .3937 = 981.00 dynes. 

Unit of work = 1 erg = 1 dyne-centimetre = .00000007373 foot-pound ; 
Unit of power = 1 watt = 10 million ergs per second, 

= .7373 foot-pound per second, 
7373 1 

= ^^- = — of 1 horse-power = .00134- H.P. 

C.G.S. Unit of magnetism = the quantity which attracts or repels an 
equal quantity at a centimetre's distance with the force of 1 dyne. 

C.G.S. Unit of electrical current = the current which, flowing through a 
length of 1 centimetre of wire, acts with a force of 1 dyne upon a unit of 
magnetism distant 1 centimetre from every point of the wire. The ampere, 
the commercial unit of current, is one tenth of the C.G.S. unit. 

The Practical Units used in Electrical Calculations are: 

Ampere, the unit of current strength, or rate of flow, represented by C. 

Volt, the unit of electro-motive force, electrical pressure, or difference of 
potential, represented by E. 

Ohm, the unit of resistance, represented by R. 

Coulomb (or ampere-second), the unit of quantity, Q. 

Ampere-hour = 3600 coulombs, Q'. 

Watt (ampere-volt, or volt -ampere), the unit of power, P. 

Joule (volt-coulomb), the unit of energy or work, W. 

Farad, the unit of capacity, represented by K. 

Henry, the unit of induction. 

Using letters to represent the units, the relations between them may be 
expressed by the following formulae, in which t represents one second and 
Tone hour: 

C = | , Q=Ct, Q> = CT, K=Q, W=QE, P = CE. 

As these relations contain no coefficient other than unity, the letters may 
represent any quantities given in terms of those units. For example, if E 
represents the number of volts electro-motive force, and R the number of 
ohms resistance in a circuit, then their ratio _£"-=- R will give the number of 
amperes current strength in that circuit. 

The above six formulae can be combined by substitution or elimination, 
so as to give the relations between any of the quantities. The most impor- 
tant of these are the following : 

Q = ^t, K = j^t, W=CEt = ^t= C*Rt = Pt, 
_ E* ^ 9r> W QE 



STANDARDS OF MEASUREMENT. 1025 

The definitions of these units as adopted at the International Electrical 
Congress at Chicago in 1893, and as established by Act of Congress of the 
United States, July 12, 1894, are as follows: 

The ohm is substantially equal to 10 9 (or 1 ,000,000,000) units or resistance 
of the C.G.S. system, and is represented by the resistance offered to an un- 
varying electric current by a column of mercury at 32° F., 14.4521 grammes 
in mass, of a constant cross-sectional area, and of the length of 100.3 centi- 
metres. 

The ampere is 1/10 of the unit of current of the C.G.S. system, and is the 
practical equivalent of the unvarying current which when passed through 
a solution of nitrate of silver in water in accordance with standard speci- 
fications deposits silver at the rate of .001118 gramme per second. 

The volt is the electro-motive force that, steadily applied to a conductor 
whose resistance is one ohm, will produce a current of one ampere, and is 
practically equivalent to 1000/1434 (or .6974) of the electro-motive force be- 
tween the poles or electrodes of a Clark's cell at a temperature of 15° C, 
and prepared in the manner described in the standard specifications. 

The coulomb is the quantity of electricity transferred by a current of one 
ampere in one second. 

The farad is the capacity of a condenser charged to a potential of one 
volt by one coulomb of electricity. 

The joule is equal to 10,000,000 units of work in the C.G.S. system, and is 
practically equivalent to the energy expended in one second by an ampere 
in an ohm. 

The watt is equal to 10,000,000 units of power in the C.G.S. system, and is 
practically equivalent to the work done at the rate of one joule per second. 

The henry is the induction in a circuit when the electro-motive force in- 
duced in this circuit is one volt, while the inducing current varies at the rate 
of one ampere per second. 

The ohm, volt, etc., as above defined, are called the "international !: ohm, 
volt, etc., to distinguish them from the " legal " ohm, B.A. unit, etc. 

The value of the ohm, determined by a committee of the British Associa- 
tion in 1863, called the B.A. unit, was the resistance of a certain piece of 
copper wire preserved in London. The so-called " legal " ohm, as adopted 
at the International Congress of Electricians in Paris in 1884, was a correc- 
tion of the B.A. unit, and was defined as the resistance of a column of 
mercury 1 square millimetre in section and 106 centimetres long, at a tem- 
perature of 32° F. 

1 legal ohm = 1.0112 B.A. units, 1 B.A. unit = 0.9889 legal ohm; 

1 international ohm = 1.0136 " " 1 " " - 0.9866 int. ohm; 

1 " " = 1.0023 legal ohm, 1 legal ohm = 0.9977 ■« " 

Derived Units. 
1 megohm = 1 million ohms; 
1 microhm = 1 millionth of an ohm; 
1 milliampere = 1/1000 of an ampere; 
1 micro-farad — 1 millionth of a farad. 
Relations of Various Units. 

1 ampere =1 coulomb per second ; 

1 volt-ampere =1 watt = 1 volt-coulomb per second; 

( = .7373 foot-pound per second, 

1 watt < — .0009477 heat-units per second (Fahr.), 

/ = 1/746 of one horse-power; 
1 = .7373 foot-pound, 

1 joule •< = work done by one watt in one second, 

( = .0009477 heat-umt; 

1 British thermal unit = 1055.2 joules; 

l = 737.3 foot-pound per second, 

1 kilowatt, or 1000 watts ■< = .9477 heat-units per second, 

{ = 1000/746 or 1.3405 horse-powers; 
1 Kilowatt-hours, I = 1.3405 horse-power hours, 

1000 volt-ampere hours, •< = 2,654,200 foot-pounds, 

1 British Board of Trade unit, ( = 3416 heat-units; 

* , „,„ „^„ r ^„ i = 746 watts = 146 volt -amperes, 

1 horse-power -> = 83 ,000 foot-pounds per minute. 

The ohm, ampere, and volt are defined in terms of one another as follows: 
Ohm, the resistance of a conductor through which a current of one ampere 
will pass when the electro-motive force is one volt. Ampere, the quantity 



1026 



ELECTRICAL ENGINEERING. 



00 

s 

6 

1) 

> 

H 


1,055 watt seconds. 
778 ft.-lbs. 

107.6 kilogram metres. 
.000293 K. W. hour. 
.000393 H.P. hour. 
.0000688 lbs. carbon oxi- 
dized. 
.001036 lbs. water evap. 
from and at 212° F. 


$?& 

53 &v 

?>^ 2? 
^oo 


7.233 ft.-lbs. 
.00000365 H.P. hour. 
.00000272 K. W. hour. 
.0093 heat-units. 


14,544 heat-units. 

1.11 lb. Anth'cite coal ox. 
2.5 lbs. dry wood oxidized. 
21 cu. ft. illuminating-gas. 
4.26 K. W. hours. 
5.71 H.P. hours. 
11,315,000 ft.-lbs. 

15 lbs. of water evap. from 
and at 212° F. 


.283 K. W. hour. 
379 H.P. hour. 
965.7 heat-units. 
103,900 k. g. m. 
1,019.000 joules. 
751,300 ft.-lbs. 

.0664 lb. of carbon oxi- 
dized. 


5 


3 3 


? • si 

gcc E 

W 53 53 

h aa 


§ II 
big 


lib. 
Carbon 
Oxidized 
with per- 
fect Effi- 
ciency = 


Su-I ii 


5 
* 

o 

a 

0j 

a 

O) 


746 watts. 
.746 K. W. 
33,000 ft.-lbs. per minute. 
550 ft.-lbs. per second. 
2,545 heat-units per hour. 
42.4 heat-units per minute. 
.707 heat-units per second. 
.175 lbs. carbon oxidized per 
hour. 
2.64 lbs. water evap. per hour 
from and at 212° F. 


1 watt second. 
.000000278 K. W. hour. 
.102 k. g. m. 
.0009477 heat-units. 
.7373 ft -lb. 


1.356 joules. 
.1383 k.g. m. 
.000000377 K. W. hours. 
.001285 heat-units. 
.0000005 H.P. hour. 


1 joule per second. 

.00134 H.P. 
3.412 heat-units per hour. 

.7373 ft.-lbs. per second. 

.0035 lbs. water evap. per hr. 
44.24 ft.-lbs. per minute. 


8.19 heat-units per sq. ft. per 
minute. 
6371 ft.-lbs. per sq. ft. per min- 
ute. 
.193 H.P. per sq. ft. 


S 


II 


II 

o 
•-a 


~% II 


II 
"la 


> 53 o 


5 
a> 

6 

s 
a> 

> 
a 

"3 

> 
'3 
cr 
W 


1,000 watt hours. 

1.34 horse-power hours. 
2,654,200 ft.-lbs. 
3,600,000 joules. 
3,412 heat-units. 
367,000 kilogram metres. 

.235 lb. carbon oxidized 
with perfect efficiency. 
3.53 lbs. water evap. from 
and at 212° F. 
22.15 lbs. of water raised 
from 62° to 212° F. 


.746 K. W. hours. 
1,980,000 ft.-lbs. 
2,545 heat-units. 
273,740 k.g. m. 

.175 lb. carbon oxidized 

with perfect efficiency. 
2.64 lbs. water evaporated 

from aud at 212° F. 
17.0 lbs. water raised from 
62° F. to 212° F. 


1,000 watts. 

1.34 horse-power. 
2,654,200 ft.-lbs. per hour. 
44,240 ft.-lbs. per minute. 
737.3 ft.-lbs. per second. 
3,412 heat-units per hour. 
56.0 heat-units per minute. 
.948 heat-unit per second. 
.22751b. carbon oxidized 
per hour. 
3.53 lbs. water evap. per 
hour from and at 212° F. 


I 










p 


II 

=1 




6 

"3 


II 
ia 



FLOW OF WATER AttD ELECTRICTY. 102? 

Of current which will flow through a resistance of one ohm when the electro- 
motive force is one volt. Volt, the electro-motive force required to cause a 
current of one ampere to flow through a resistance of one ohm. 
Units of the Magnetic Circuit.— (See Electro-magnets, page 1058.) 
For Methods of making Electrical Measurements, Test- 
ing, etc., see Munroe & Jamieson's Pocket-Book of Electrical Rules, 
Tables, and Data; S. P. Thompson's Dynamo-Electric Machinery; and works 
I on Electrical Engineering. 

Equivalent Electrical and Mechanical Units.— H. Ward 
Leonard published in The Electrical Engineer. Feb. 25, 1895, a table of use- 
ful equivalents of electrical and mechanical units, from which the table on 
page 1026 is taken, with some modifications. 

ANALOGIES BETWEEN THE FLOW OF WATER AND 
ELECTRICITY. 

Water. Electricity. 

Head, difference of level, in feet. (Volts; electro-motive force; differ- 

Difference of pressure per sq. in., in •< ence of potential or of pressure; E. 
lbs. ( or E.M.F. 

lS rt^SS&fi5^ ^?h I °Se^s S d1ISl? asKSTol 
arer^^^^s^dr] S^^^SS^^ 

Rate of flow, as cubic ft. per second, f A 5XnlftVoTS?renTrl?e offlol'-'i 

gallons per minute, etc., or volume \ ~_lSmbM second' 
divided by the time. In the mining { ampeie - l coulomb pei second, 
regions sometimes expressed in | volts E 

" miners' inches." I Amperes = ^^ ; C = — ; E = CR. 

Quantity, usually measured in cubic 1 
feet or gallons, but is also equiva- | Coulomb, unit of quantity, Q, = rate 
lent to rate of flow X time, as J- of flow X time, as ampere-seconds, 
cubic feet per second for so many j 1 ampere-hour = 3600 coulombs, 
hours. J 

'Joule, volt-coulomb, W, the unit of 



Work, or energy, measured in foot- 
pounds; product of weight of fall- 
ing water into height of fall: ''" 



work, — product of quantity by the 
electro-motive force — volt-ampere- 
second. 1 joule= .7373 foot-pound. 



pumping, product of quantity in -J If C (amperes) = rate of flow, and 



cubic feet into the pressure in 
per square foot against which the 
water is pumped. 



E (volts) = difference of pressure 
between two points in a circuit, 
energy expended = CEt, = C' 2 Et, 
since E = CR. 



Power, rate of work. Horse-power,ft.- , 

lbs. of work done in 1 min.-s- 33,000. I Watt, unit of power, P, = volts X 
In falling water, pounds falling in | amperes, = current or rate of flow 

one second -+- 550. In water flowing \~ X difference of potential. 

in pipes, rate of flow in cubic feet I 1 watt = .7373 foot-pound per second 

per second X pressure resisting the | = 1/746 of a horse-power. 

flow in lbs. per sq. ft. -r- 550. J 

Analogy between the Ampere and the Miner's Inch. 
(T. O'Connor Sloane.) — The miner's inch is defined as the quantity of water 
which will flow through an aperture an inch square in a board two inches 
thick, under a head of water of six inches. Here, as in the case of the am- 
pere, we have no reference to any abstract quantity, such as gallons or 
pounds. There is no reference to time. It is simply a rate of flow. We 
may consider the head of water, six inches, as the representative of electri- 
cal pressure; i.e., one volt. The aperture restricting the flow of water may 
be assumed to represent the resistance of one ohm; the flow through a re- 
sistance of one ohm under the pressure of one volt is one ampeie; the flow 
through the resistance of a one-inch hole two inches long under the pressure 
of six inches to the upper edge of the opening is one miner's inch. 

The miner's inch-second is the correct analogue of the ampere-second; the 
one denotes a specific quantity of water, 0.194 gallon; the other a specific 
quantity of electricity, a coulomb. 



1028 



ELECTRICAL ENGINEERING. 



ELECTRICAL. RESISTANCE. 



Laws of Electrical Resistance.— The resistance, R, of any con- 
ductor varies directly as its length, /, and inversely as its sectional area, s, 

or R cc — . 
s 
Example.— If one foot of copper wire .01 in. diameter has a resistance of 
.10323 ohm, what will be the resistance of a mile of wire .3 in. diam. at the 
same temperature ? The sectional areas being proportional to the squares 
of the diameters, the ratio of the areas is .3 2 : .01 2 = 900 to 1. The lengths 
are as 5280 to 1. The resistances being directly as the lengths and inversely 
as the sectional areas, the resistance of the second wire is .10323 x 5280 -r- 
900 = .6056 ohm. 



Conductance, c, is the inverse of resistance. R = 



I 



sR 



If c and c 2 



represent the conductances, and R and i? 2 the respective resistance of two 
substances of the same length and section, then c : e% : : R 2 : R. 

Equivalent Conductors.— With two conductors of length I, Z x , of 
conductances c, c 1? and sectional areas .s, s lf we have the same resistance, 

and one may be substituted for the other when — = — — . 

CS c'lSi 

The specific resistance, also called resistivity, a, of a material of unit 
length and section is its resistance as compared with the resistance of a 
standard conductor, such as pure copper. Conductivity, or specific con- 
ductance, is the reciprocal of resistivity. 



I 



al 



If two wires have lengths I, l x , areas s, s l7 and specific resistances a, a x , their 



actual resistances are R= — , R t = , and — 

s s x R x 



als x 
dills' 



Electrical Conductivity of Different Metals and Alloys. 

— Lazare Weiler presented to the Societe Internationale des Electriciens the 
results of his experiments upon the relative electrical conductivity of certain 
metals and alloys, as here appended : 



1. Pure silver 100 

2. Pure copper 100 

3. Refined and crystallized 

copper 99.9 

4. Telegraphic silicious bronze 98 

5. Alloy of copper and silver 

(50$) 86.65 

6. Pure gold 78 

7. Silicide of copper, 4$ Si 75 

8. Silicide of copper, 12$ Si. . . 54.7 

9. Pure aluminum 54.2 

10. Tin with 12% of sodium... 46.9 

11. Telephonic silicious bronze 35 

12. Copper with 10% of lead .... 30 

13. Pure zinc .. 29.9 

14. Telephonic phosphor - 

bronze 29 

15. Silicious brass, 25$ zinc 26.49 

16. Brass with 35$ of zinc 21.5 



17. Phosphor tin 17.7 

18. Alloy of gold and silver 

(50$) 16.12 

19. Swedish iron 16 

20. Pure Banca tin 15.45 

21. Antimonial copper 12.7 

22. Aluminum bronze (10$) .... 12.6 

23. Siemens steel 12 

24. Pure platinum 10.6 

25. Copper with 10$ of nickel.. 10.6 

26. Cadmium amalgam (15%). 10.2 

27. Dronier mercurial bronze.. 10.14 

28. Arsenical copper (10$) 9.1 

29. Pure lead .. 8.88 

30. Bronze with 20$ of tin 8.4 

31. Pure nickel 7.89 

32. Phosphor-bronze, 10$ tin .. 6.5 

33. Phosphor-copper, 9$ phos.. 4.9 

34. Antimony 3.88 



The above comparative resistances may be reduced to ohms on the basis 
that a wire of soft copper one milimetre in diameter at a temperature of 
0° C. has a resistance of .02029 international ohms per metre; or a wire .001 
inch diam. has a resistance of 9.59 international ohms per foot. 



ELECTRICAL RESISTANCE. 



1029 



Relative Conductivities of Different Metals at 0° and 
100° C. (Mattiiiessen.) 





Conductivities. 


c. 

F. 


Metals. 


Conductivities. 


Metals. 


At 0° C. 
" 32° F. 


At 100° 
" 212° 


At 0° C. 
" 32° F. 


At 100° C. 
" 212° F. 




100 
99.95 
77. 9G 
29.02 
23.72 
18.00 
16.80 


71.56 
70.27 
55.90 

20.67 
16.77 


Tin 


12.36 
8.32 
4.76 
4.62 
1.60 
1.245 






8.67 


Copper, hard 

Gold, hard 

Zinc, pressed 




5 86 




3 33 


Antimony. . 
Mercury, pu 
Bismuth 


■e. . 


3.26 


Platinum, soft. .. 
Iron, soft 
















Conductors and Insulators in Order of their Value. 


Conductc 
All metals 
Well- burned ehai 
Plumbago 
Acid solutions 
Saline solutions 
Metallic ores 
Animal fluids 
Living vegetable 
Moist earth 
Water 


>rs. 
coal 

substance 


3 




Insulators (Non-conductors). 
Dry Air Ebonite 
Shellac Gutta-percha 
Paraffin India-rubber 
Amber Silk 
Resins Dry Paper 
Sulphur Parchment 
Wax Dry Leather 
Jet Porcelain 
Glass Oils 
Mica 



According to Culley, the resistance of distilled water is 6754 million times 
as great as that of copper. 

Resistance Varies witli Temperature.— For every degree Cen- 
tigrade the resistance of copper increases about 0.4#, or for every degree F. 
1%. Thus a piece of copper wire having a resistance of 10 ohms at 32° 
would have a resistance of 11.11 ohms at 82° F. 

The following table shows the amount of resistance of a few substances 
used for various electrical purposes by which 1 ohm is increased by a rise 
of temperature 1° F., or 1° C. 

Rise of R. of 1 Ohm when Heated- 



Material. 



1°F. 



Platinoid 00013 

Platinum-silver 00018 

German silver (see below) .00024 

Gold, silver 00036 

Cast iron 00044 

Copper 00222 



1° C. 
.00021 
.00031 
.00044 
.00065 
.00080 
.00400 



Annealing.— The degree of hardness or softness of a metal or alloy 
affects its resistance. Resistance is lessened by annealing. Matthiessen 
gives the following relative conductivities for copper and silver, the com- 
parison being made with pure silver at 100° C. : 



Metals. Temp. C. 

Copper 11° 

Silver 14.6° 



Hard. 
95.31 
95.36 



Annealed. 



Dr. Siemens compared the conductivities of copper, silver, and brass with 
pure mercury at 0° C, with the following results: 

Metal. Hard. Annealed. 

Copper 52.207 55.253 

Silver 56.252 64.380 

Brass 11.439 13.502 

Edward Weston (Proc. Electrical Congress 1893, p. 179) says that the re- 
sistance of German silver depends on its composition. Mathiessen gives it as 
nearly 13 times that of copper, with a temperature coefficient of .0004433 per 
degree C. Weston, however, has found copper-nickel-zinc alloys (German 



1030 ELECTRICAL ENGINEERING. 

silver) which had a resistance of nearly 28 times that of copper, and a tem- 
perature coefficient of about one half that given by Matthiessen. Kennelly 
and Fessenden (Proc. Elec. Cong., p. 186) find that copper has a uniform 
temperature coefficient of 0.40t$ per degree C, between the limits of 20° and 
250° C. 

Standard of Resistance of Copper Wire. (Trans. A. I. E. E., 
Sept. and Nov. 1890.)— Matthiessen's standard is: A hard-drawn copper wire, 
1 metre long, weighing 1 gramme has a resistance of 0.1469 B.A. unit at 
0° C. (1 B.A. unit = 0.9889 legal ohm = 0.9866 international ohm.) Resist- 
ance of hard copper = 1.0226 times that of soft copper. Relative conducting 
power (Matthiessen): silver, 100; hard or unannealed copper, 99.95; soft or 
annealed copper, 102.21. Conductivity of copper at other temperatures than 
0°C, 

Ct = C' (l - .00387* -f .000009009* 2 ). 

The resistance is the reciprocal of the conductivity, and is 

Rt = R (l -f- .00387* + .00000597* 2 ). 

A committee of the Am. Inst. Electrical Engineers recommend the follow- 
ing as the most correct form of the Matthiessen standard, taking 8.89 as the 
sp. gr. of pure copper : 

A soft copper wire 1 metre long and 1 mm. diam. has an electrical resist- 
ance of .02057 B.A. unit at 0° C. From this the resistance of a soft copper 
wire 1 foot long and .001 in. diam. (mil-foot) is found to be 9.720 B.A. units 
at 0° C. 

Standard Resistance at 0° C. B.A. Units. LegalOhms. ^Ss^' 

Metre-millimetre, soft copper 02057 .02034 .02029 

Cubic centimetre " " 000001616 .000001598 .000001593 

Mil-foot " " 9.720 9.612 9.590 

1 mil-foot, of soft copper at 10°. 22 C. or 50°. 4 F. . . 10 9.977 

" " " " " 15°. 5 " 59°. 9 F... 10.20 10.175 

" " " " " 23°.9 " 75° F... 10.53 10.505 

For tables of the resistance of copper wire, see pages 218 to 220, also 
pp. 1034, 1035. 

Taking Matthiessen's standard of pure copper as 100$. some refined metal 
has exhibited an electrical conductivity equivalent to 103$. 

Matthiessen found that impurities in copper sufficient to decrease its 
density from 8.94 to 8.90 produced a marked increase of electrical resistance. 

ELECTRIC CURRENTS. 

Olim's Law.- This law expresses the relation between the three fun- 
damental units of resistance, electrical pressure, and current. It is : 

„ electrical pressure _, E ; E 

Current = r— (?=-=•; whence E = CR, and R = —.. 

resistance R C 

In terms of the units of the three quantities, 

volts ... . , volts 

Amperes = ■ ■ ■ ; vOlts = amperes X ohms; ohms = . 

ohms amperes 

Examples: Simple Circuits.— 1. If the source has an effective electrical 
pressure of 100 volts, and the resistance is two ohms, what is the current ? 

^ E 100 * n 

C = -^r = -Q- = 50 amperes. 

2. What pressure will give a current of 50 amperes through a resistance of 

2 ohms ? E = CR = 50 X 2 = 100 volts. 

3. What resistance is required to obtain a current of 50 amperes when the 

pressure is 100 volts ? R — -— — — - = 2 ohms. 

The following examples are from R. E. Day's " Electric Light Arithmetic:" 
1. The internal resistance of a certain Brush dynamo-machine is 10.9 ohms, 
and the external resistance is 73 ohms; the electro-motive force of the ma- 
chine being 839 volts. Find the strength of the current flowing in the circuit. 
E = 839; R = 73 + 10.9 = 83.9 ohms; 
C = E -v- R = 839 -*- 83.9 = 10 amperes. 



ELECTRIC CURRENTS. 1031 

2. Three arc lamps in series have a resistance of 9.36 ohms, while the re- 
sistance of the leading wires is 1.1 ohm, and that of the dynamo is 2.8 ohms. 
Find what must be the electro-motive force of the machine when the strength 
of the current produced is 14.8 amperes. 

ij = 2.8 -f- 9.36 + 1.1 = 13.26 ohms; C = 14.8 amperes; 
E = C X B = 13.26 X 14.8 = 196.3 volts. 

3. Calculate from the following data the average resistance of each of 
three arc lamps arranged in series. The electro-motive force of the machine 
is 244 volts and its resistance is 3.7 ohms, while that of the leading wires is 2 
ohms, and the. strength of current through each lamp is 21 amperes. 

If x represent the average resistance in ohms of each lamp, then the total 
resistance of the circuit is B = Sx + 2 -f 3.7. 

But by Ohm's law B - E -s- C, .'. 3a -f- 5.7 = 244/21 = 11.61 ohms, whence 
x = 1.97 ohms, nearly. 

4. Three Maxim incandescent lamps were placed in series. The average 
resistance, when hot, of each lamp was 39.3 ohms, and that of the dynamo 
and leading wires 11.2 ohms. What electro-motive force was required to 
maintain a current of 1.2 amperes through this circuit ? 

In this case we have 

B = 3 X 39.3 + 11.2 = 129.1 ohms, and 
C = 1.2 ampere; 
and therefore, by Ohm's law, 

E = C X B = 1.2 X 129.1 = 154.9 volts. 

_. The resistance of the arc of a certain Brush lamp was 3.8 ohms when a 
current of 10 amperes was flowing through it. What was the electro-motive 
force between the two terminals ? 

E - C X B = 10 X 3.8 = 38 volts. 

6. Twenty-five exactly similar galvanic cells, each of which had an aver- 
age internal resistance of 15 ohms, were joined up in series to one incandes- 
cent lamp of 70 ohms resistance, and produced a current of 0.112 amperes. 
What would be the strength of current produced by a series of 30 such cells 
through 2 lamps, each of 30 ohms resistance ? 

The data of the first part of the problem enable us to determine the 
average electro-motive force of each cell of the battery. Let this be repre- 
sented by E; then we have 

25.EJ = C X B = .112 X (25 X 15 + 70) = .112 X 445; 



Then from the data in the second part of the problem, we have, by Ohm's 
law, 

Divided. Circuits.— If the circuit has two paths, the total current in 
both divides itself inversely as the resistances. 

If B and B, are the resistances of the two branches, and C and C x the cur- 
C B 
rents, C X B = Cj X i?i, and — = -£-', whence 

°~ b ' Ci -rT' r -~c~' ^".cr 

In the case of the double circuit, one circuit is said to be in shunt to the 
other, or the circuits are in multiple arc or in parallel. 

Conductors in Series.— If conductors are arranged one after the 
other they are said to be in series, and the total resistance is the sum of their 
several resistances. B = B^ + B% -\- B a . 

Internal Resistance.— In a simple circuit we have two resistances, 
fhat of the circuit B and that of the internal parts of the source, called in^ 



1032 ELECTRICAL ENGINEERING. 

ternal resistance, r. The formula of Ohm's law when the internal resistance j tt 

is considered is C = -j—-. — . 
K -f- r 
Total or Joint Resistance of Two Branches.— Let C be the I 

total current, and C u C 2 the currents in branches whose resistances respect- 
ively are R u i? 2 . Then C* = C x + C 2 ; C = ^ ; C x = — -; C\ = ~\ or, \i E = 

1, *C = ^r = s- +^5-, whence K = D *, ' , which is the joint resistance of 

K K 1 xr, 2 Kx -f- K a 

J? x and i?„. 
Similarly, the joint resistances of three branches have resistances respect- 

ively of R^R,, R t , is R = BiBt *££\ BaRt - 
When the branch resistances are equal, the formula becomes 



#i n ~ * X n n 

where R x = the resistance of one branch, and n — the number of branches. 

Kirchhoff's I<aws.— 1. The sum of the currents in all the wires which 
meet in a point is nothing. 

2. The sum of all the products of the currents and resistances in all the 
branches forming a closed circuit is equal to the sum of all the electrical 
pressures in the same circuit. 

When E = E r -f E? + E s , etc., and C = C x + C 2 + C 3 , etc., and R is the 
total resistance of RxR^R^ etc., then 

#! + # 2 + # 3 , etc. = Ciffx + C 2 i? a + C3.K3, etc. 

Power of the Circuit.— The power, or rate of work, in watts = 
current in amperes X resistance in ohms = C X E. Since C = E -+- R, 

watts — —- = electro-motive force 2 h- resistance. 

Example.— What H.P. is required to supply 100 lamps of 40 ohms resist- 
ance each, requiring an electro-motive force of 60 volts ? 

E" 2 60 2 
The number of volt-amperes for each lamp is — - = -- , 1 volt-ampere = 

60 s 
.00134 H.P.; therefore — X 100 X .00134 = 12 H.P. (electrical) very nearly. 

If the loss in the dynamo is 20 per cent, then 12 H.P. is 80 per cent of the 
12 
actual H.P. required; which therefore is — = 15 H.P. 

Heat Generated by a Current.— Joule's law shows that the heat 
developed in a conductor is directly proportional, 1st, to its resistance; 2d, 
to the square of the current strength; and 3d, to the time during which the 
current flows, or H = C*Rt. Since C = E -*- R, 

n E EH 

,*» = ,_, = _.. 

Or, heat = current 2 X resistance x time 

— electro -motive force x current X time 
= electro-motive force 2 X time -f- resistance. 

Q = quantity of electricity flowing = Ct — — i. 

H — EQ; or heat = electro-motive force x quantity. 

The electro-motive force here is that causing the flow, or the difference in 
potential between the ends of the conductor. 

The electrical unit of heat, or "joule " = 10 T ergs = neat generated in one 
second by a current of 1 ampere flowing through a resistance of one ohm = 
.239 gramme Of water raised 1° C. H — C' 2 Rt x .239 gramme calories = 
C*Rt X .0009478 British thermal units. 

In electric lighting the energy of the current is converted into heat in the 
lamps. The resistance of the lamp is made great so that the required 
quantity of heat may be developed, while in the wire leading to and froni 



ELECTRIC CURRENTS. 



1033 



the lantp the resistance is made as small as is commercially practicable, so 
that as little energy as possible may be wasted in heating the wire. The 
transformations of energy from the fuel burned in tbe boiler to the electric 
light are the following: 

Heat energy is transformed into mechanical energy by means of the boiler 
and engine. 
Mechanical energy is transformed into electrical energy in the dynamo. 
Electrical euergy is transformed into heat 411 the electric light. 
The heat generated in a conductor is the equivalent of tbe energy causing 
the flow. Thus, rate of expenditure of energy in watts = electro-motive 
force in volts X current in amperes = EC, and the energy in joules = watts 
X time in seconds = ECt. Heat = C 2 Rt = ECt. 
f Heating of Conductors. (From Kapp's Electrical Transmission 
1 of Energy.)— It becomes a matter of great importance to determine before- 
I hand what rise in temperature is to be expected in each given case, and if 
that rise should be found to be greater than appears safe, provision must be 
made to increase the rate at which h*at is carried off. This can generally 
be done by increasing the superficial area of the conductor. Say we have 
one circular conductor of 1 square inch area, and find that with 1000 amperes 
flowing it would become too hot. Now by splitting up this conductor into 
10 separate wires each one tenth of a square inch cross-seclional area, we 
have not altered the total amount of energy transformed into heat, but we 
have increased the surface exposed to the cooling action of the surrounding 
air in the ratio of 1 : ^10, and therefore the ten thin wires can dissipate more 
than three times the heat, as compared with the single thick wire. 
Heating of Wires of Subaqueous and Aerial Cables (in- 
sulated with Gutta-percha). (Prof. Forbes.) 
Diameter of cable -4- Diameter of conductor = 4. 
Temperature of air = 20° C. = 68* F. 
t = excess of temperature of conductor over air. . 



Diameter in centi- 
metres and mils. 




Current in amperes. 




Cm. 


Mils. 


t = 1° C 


t = 9° C. 


t = 25° C. 


t = 49° C. 


* = 81° C. 


= 1.8° F. 


= 16.2° F. 


= 45° F. 


= 92.2° F. 


= 145.8° F. 


.1 


40 


3.7 


11.0 


17.8 


24.0 


29.5 


.2 


80 


9.1 


27.0 


43.8 


59.0 


72.5 


.3 


120 


.15.0 


44.4 


72.1 


97.3 


119 


.4 


160 


21.2 


62.5 


102 


137 


168 


.5 


200 


27.4 


81.0 


131 


177 


218 


.6 


240 


33.7 


100 


164 


219 


268 


7 


280 


40.1 


119 


192 


259 


319 


.8 


310 


46.4 


137 


223 


301 


369 


.9 


350 


52.9 


157 


253 


342 


420 


1.0 


390 


59.3 


175 


285 


384 


472 


2.0 


780 


124 


367 


595 


803 


988 


3.0 


1180 


189 


559 


908 


1225 


1503 


4.0 


1570 


254 


753 


1221 


1646 


2021 


5.0 


1970 


319 


945 


1534 


2068 


2523 


6.0 


2360 


385 


1138 


1846 


2491 


3058 


7.0 


2760 


450 


1330 


2158 


2846 


3575 


8.0 


3150 


514 


1525 


2472 


3335 


4094 


9.0 


3540 


580 


1716 


2785 


3755 


4611 


10.0 


3940 


645 


1909 


3097 


4178 


5130 



Prof. Forbes states that an insulated wire carries a greater current without 
overheating than a bare wire if the diameter be not too great. Assuming 
the diameter of the cable to be twice the diam. of the conductor, a greater 
cnrrent can be carried in insulated wires than in bare wires up to 1.9 inch 
diam. of conductor. If diam. of cable = 4 times diam. of conductor, this is 
the case up to 1.1 inch diam. of conductor. 

Copper-wire Table.— The table on pages 1034 and 1035 is abridged 
from one computed by the Committee on Units and Standards of the Ameri- 
can Institute of Electrical Engineers (Trans. Oct. 1893). 



1034 



ELECTRICAL EtfGttfEEKlXG. 



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000000090000000' odd 



ELECTRIC CURRENTS. 



1035 



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i r b z z z z z 'z z z~ 



O -•: :: C — -- - "" X '. • I - ~ 

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o*ffl*M?. ;- 



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JOHN'JOHCIN' ■ ' ■■•: — ' - — - — — . /■:-. — r ~ r. 7 7 / --h(,i--i.-;hhc.oOO 

HBIXCl'I-.S'.Hi-II^IOC--.' ■ - r - - ' - -h -< o © © o - . 

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OOO' ooo o 00 = = 00=: ©©©©©©©©©©©o©©©©© 


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m 






1036 ELECTRICAL ENGINEERING. 






The data from which the foregoing table has been computed are as follows: 
Matthiessen's standard resistivity, Matthiessen's temperature coefficients, 
specific gravity of copper = 8.89. Resistance in terms of the international 
ohm. 

Matthiessen's standard 1 metre-gramme of hard-drawn copper = 0.1469 
B. A. U. @, 0° C. Ratio of resistivity hard to soft copper 1 .0226. 

Matthiessen's standard 1 metre-gramme of soft-drawn copper = 0.14365 
B. A. U. @, 0° C. One B. A. U. = 0.9866 international ohm. 

Matthiessen's standard 1 metre-gramme of soft-drawn copper = 0.141729 
international ohm @ 0° C. 

Temperature coefficients of resistance for 20° C, 50° C, and 80° C , 1 07968 
1.20625, and 1.33681 respectively. 1 foot = 0.3048028 metre, 1 pound =3 
453.59256 grammes. 

Heating of Coils.— To calculate the heating of a coil, given the cool- 
ing surface and its resistance. (Forbes.) 
Let p = the resistance of a coil in ohms at the permissible temperature 
(the resistance (cold) must be increased by 1/5 of its value to give p) ; 
S = the surface exposed to the air measured in square centimetres 

(1 square cm. = .155 square inch; 1 sq. in. = 6.45 square cm.); 
t = the rise in temperature, centigrade scale; 
C = the current in amperes. 

.24C 2 p = heat generated = etS. 
where e is McFarlane's constant, varying from .0002 to .0003. The latter 
value may be taken. If 50° C. be the permissible rise in temperature, 



j / .0003 X 50 X S _ .. . /S 

'-y ^xp - - 25 v p 



Example.— The resistance of the field-magnets of a dynamo is 1.5 ohms 
cold, and the surface exposed to the air is 1 square metre; find the current 
to heat it not more than 50° C. 

Here 8 = 10,000; p = 1.8 ohms; and C = .25i/^y^ = 33.5 amperes. 

For the heating of coils of field-magnets Mr. C. Hering gives 1 watt of 
energy dissipated for every 223 square inches of cooling-surface for each 
degree F. of difference between the temperature of the coil and the sur- 
rounding air. 

W = CE - 1/223TS = 0.004476TS, in which W = watts lost in coil, T = 
degrees Fahr., and £>' = squai*e iuches. 

C = 2^r= is the greatest curreut which can be used in the magnet coils of 
a shunt machine having a certain pressure in order that they do not heat 
above a certain temperature. Thus for a rise of temperature of 50° F. above 
the surrounding air, 

C = 2^=, - .224 — . Substituting for E its equivalent CR, we get 



°=\/'- m i- 



If 80° F. is the maximum difference of temperature, 

C -223E = - 36 E = - 6( y B 

The formula can be used for series machines when C is known, for writing 

C*R - 1/2232-5, we get R = ^L. 

With a permissible rise of 50° F. or 80° F., we have respectively, 
.2245 . „ fl 

The surface area of the coil in square inches may be found from 
22SW _ 22SCE _ 223C 2 i? 
~ T '" T ~ T ' 



ELECTRIC CURRENTS. 



1037 



For a rise of temperature of 50° F. or 80° F., respectively, the surface will 
be 



223 W 223 W 

S = ^ r =iAtW; and .9 = ^ 



: 2.8W. 



Fusion of Wires.— W. H. Preece gives a formula for the current re- 
quired to fuse wires of different metals, viz.: C = adh in which d is the 
diameter in inches and a a coefficient whose value for different metals is as 
follows: Copper 10214; aluminum 7585; platinum 5172; German silver 5230; 
platinoid 4750; iron 3148; tin, 1642: lead, 1379; alloy of 2 lead and 1 tin, 1318. 

.Diameters of Various Wires which will be Fused by a 
given Current. 



Formula, d =(- ) 3 ; a = 
3148 for iron. 



1642 for tin = 1379 for lead = 10244 for copper = 





Tin Wire. 


Lead Wire. 


Copper Wire. 


Iron 


Wire. 


Current, 


















in 
amperes. 


Diam. 
inches. 


£T£- 


Diam. 
inches. 


ISR* 


Diam. 
inches. 


Approx. 

S.W. G. 


Diam. 
inches. 


Approx. 
S.W. G. 


1 


.0072 


36 


.0081 


35 


.0021 


47 


.0047 


40 


2 


.0113 


31 


.0128 


30 


.0034 


43 


.0074 


36 


3 


.0149 


28 


.0168 


27 


.0044 


41 


.0097 


33 


4 


.0181 


26 


.0203 


25 


.0053 


39 


.0117 


31 


5 


.0210 


25 


.0236 


23 


.0062 


38 


.0136 


29 


10 


.0334 


21 


.0375 


20 


.0098 


33 


.0216 


24 


15 


.0437 


19 


.0491 


18 


.0129 


30 


.0283 


22 


20 


.0529 


17 


.0595 


17 


.0156 


28 


.0343 


20.5 


25 


.0614 


16 


.0690 


15 


.0181 


26 


.0398 


19 


30 


.0694 


15 


.0779 


14 


.0205 


25 


.0450 


18.5 


35 


.0769 


14.5 


.0864 


13.5 


.0227 


21 


.0498 


18 


40 


.0840 


13.5 


.0944 


13 


.0248 


23 


.0545 


17 


45 


•0909 


13 


.1021 


12 


.0268 


22 


.0589 


16.5 


50 


.0975 


12.5 


.1095 


11.5 


.0288 


22 


.0632 


16 


60 


.1101 


11 


.1237 


10 


.0325 


21 


.0714 


15 


70 


. 1220 


10 


.1371 


9.5 


.0360 


20 


.0791 


14 


80 


.1334 


9.5 


.1499 


8.5 


.0394 


19 


.0864 


13.5 


90 


.1443 


9 


.1621 


8 


.0426 


18.5 


.0935 


13 


100 


.1548 


8.5 


.1739 


7 


.0457 


18 


.1003 


12 


120 


.1748 


7 


.1964 


6 


.0516 


17.5 


.1133 


11 


140 


.1937 


6 


.2176 


5 


.0572 


17 


.1255 


10 


160 


.2118 


5 


.2379 


4 


.0625 


16 


.1372 


9.5 


180 


.2291 


4 


.2573 


3 


.0676 


16 


.1484 


9 


200 


.2457 


3.5 


.2760 


2 


.0725 


15 


.1592 


8 


250 


.2851 


1.5 


.3203 





.0841 


13.5 


.1848 


6.5 


300 


.3220 





.3617 


00.5 


.0950 


12.5 


.2086 


5 



Current in Amperes Required to Fuse Wires According 
to the Formula C — ad$- 



No. 


Diameter, 


■d§- 


Tin. 


Lead 


Copper 


Iron. 


S.W. G. 


inches. 


a = 1642. 


a = 1379. 


a = 10244 


a = 3148. 


14 


.080 


.022627 


37.15 


31.20 


231.8 


71.22 


16 


.064 


.016191 


26.58 


22.32 


165.8 


50.96 


18 


.048 


.010516 


17.27 


14.50 


107.7 


33.10 


20 


.036 


.006831 


11.22 


9.419 


69.97 


21.50 


22 


.028 


.004685 


7.692 


6.461 


48.00 


14.75 


24 


.022 


.003263 


5.357 


4.499 


33.43 


10.27 


26 


.018 


.002415 


3.965 


3.330 


24.74 


7.602 


28 


.0148 


.001801 


2.956 


2.483 


18.44 


5.667 


30 


.0124 


.001381 


2.267 


1 904 


14.15 


4.347 


32 


.0108 


.001122 


1.843 


1.548 


11.50 


3.533 



1038 ELECTRICAL ENGINEERING. 

ELECTRIC TRANSMISSION. 

Cross-section of Wire Required for a Given Current.— 

Constant Current (Series) System. —The cross-sectional area of copper 
necessary in any circuit for a given constant current depends on the differ- 
ence between the pressure at the generating station and the maximum 
pressure required by all the apparatus on the circuit, and on the total length 
of the circuit. The following formulae are given in "Practical Electrical 
Engineering: 1 ' 

If V = pressure in volts at generators; 

v = sum of all the pressures (in volts) required by apparatus supplied 

in the circuit; 
n = total length (going and return) of circuit in miles; 
C = current in amperes; 
r = resistance of 1 mile of copper-conductor of 1 square inch sectional 

area in ohms; 
a = required cross sectional area of copper in square inches,— 

nrC 

a = — . 

V — v 

If we take the temperature of the conductor when the current has been 
flowing for some time through it, as 80° F., 

_.._. . , 0.0455nC 
r — 0.04oo ohm, and a = — ^ . 



It generally happens, however, that we are not tied down to a particular 
value of V, as the pressure at the generators can be varied by a few volts to 
suit requirements. In this case it is usual to fix upon a current density and 
determine the cross-sectional area of copper in accordance with it. 

If D = current density in amperes per square inch determined upon, 

C 

The current density is frequently taken at 100C amperes to the square inch, 
but should in general be determined by economical considerations for 
every case in question. 

Allowable Current Density in Insulated Cables. — Experiments on 
insulated cables in casing gave the results shown below, but they need con- 
firmation or correction of the current densities permissible in different sizes 
of insulated cables run underground. C and D are the current in amperes 
and the current density in amperes per square inch, respectively, which will 
raise the temperature of the conductor by the number of degrees Fahr. 
indicated by the suffix. 

No. S.W.G.* of 
Strands, each Wire. 



Area of 










Strand in 


c J8 


D 18 


C50 


D 50 


square inches. 










0.0072 


18 


2,500 


28 


3.900 


0.0357 


59 


1,400 


95 


2,700 


0.0975 


126 


1,300 


205 


2,100 



Constant Pressure (Parallel System).— To determine the loss in 
pressure in a feeder of given size in the case of two-wire parallel distribution. 

Let a — cross-sectional area of copper of one conductor of the feeder in 
square inches; 
n = length of feeder (going and return) in miles; 
C = current in amperes; 
V — v = loss of pressure in feeder in volts; 

r — resistance of 1 mile of copper conductor of 1 square inch sec- 
tional area in ohms. 



* Standard (British) Wire-gauge. 



ELECTRIC TRANSMISSION. 1039 

If the temperature of the conductor with this current flowing in it is 
assumed to be 80° F., 

0455n(7 

r — 0.0455 ohm, and V - v = — . 

a 

Three-wire Feeder.— In the case of a three-wire feeder, let p x q x and 
p, 2 q 2 represent the two outer conductors, and let p'q' represent the middle 
conductor, p x , p', p 2 being at the feeding-point and q x , q\ g 2 at the generat- 
ing station, and let 

a = cross-sectional area of each of the outer conductors in square inches; 
a' — cross-sectional area of middle conductor; 
n = length in miles of each conductor of feeder ; 
V x = pressure between p x and p' in volts at generating station; 
V 2 — pressure between p' and p 2 in volts at generating station; 
v x — pressure between q x and q' in volts at feeding-point; 
r 2 = pressure between q' and g 2 in volts at feeding-point; 
C x = current in p x q x in amperes; 
C 2 = current in ptfi in amperes; 

r --■ resistance of 1 mile of copper conductor of 1 square inch sectional 
area in ohms. 
Then 

Oi.C.-Cj, TT i C 2 C x - c 2 , 



V x - v x - nr ) — H ; — ; — - !- ; F 2 — v 2 = rar J 



a' j ' ' ■* (a a' J 

It will be noticed that if t^ = v 2 , and if CJis greater than C 2 , Vi is greater 
than V 2 by twice the loss of pressure in the middle wire; this result shows 
that the regulators must be in circuit with the two outer conductors. 

It is usual to make a' half a; then, if the greatest want of balance between 
t lie loads of the two sections of the three-wire system is m* per cent of the 
maximum load of the more heavily loaded section, and if C x is the maximum 
current in either of the outer conductors of the feeder under consideration, 

C 2 will not be less than G x \\ - tkr), and consequently C x — C 2 will not be 

mC 
greater than - x 



100 
We have then 



... -f m TT nrC x 200 - m 

-_-; F 2 -t, 2= — X-^-; 



so that if v x and v 2 are each equal to V— the pressure required to be main- 
tained constant at the feeding-point — we can calculate V x and F" 2 for given 
values of n, a, and C x , employing the value of m, which we estimate should 
be the maximum it can have. 

These last expressions show that the difference in the pressures required 
at the station across the two sections of a three-wire feeder increases with 
the current carried by the feeder ; hence the regulators on each of the outer 
conductors should be equivalent to a variable resistance having at least 
nrm , 

-r^- ohms as a maximum. 
100a 

It is usual to make the area of the middle conductor one half of that of 
each of the outer conductors, but this is not invariably the case. 

Sliort-circuiting.— From the law C= — it is seen that with any pres- 
sure E the current Cwill become very great if R is made very small. In 
short-circuiting the resistance becomes-small and the current therefore great. 
Hence the dangers of short-circuiting a current. 

Economy of Electric Transmission. (R. G. Blaine, Eng'g, June 
5, 1891.)— Sir W. Thomson's rule for the most economical section of conductor 

* The value to be assigned to m may vary from 10 to 25, according to the 
case exercised in connecting customers to one section or the other, or both, 
and according to the local conditions. At a certain station supplying current 
on the three-wire low-pressure system to about 25,000 8-c.p. lamps, we were 
informed that in had never exceeded 7 or 8. 



1040 



ELECTRICAL ENGINEERING. 



is that for which the "annual interest on capital outlay is equal to the 
annual cost of energy wasted," and its practical outcome is that the area of 

17 
the copper conductor should be such that its resistance per mile = — 

(C being the current in amperes). 

Tables have been compiled by Professor Forbes and others in accordance 
with modifications of Sir W. Thomson's rule. For a given entering horse- 
power the question is merely one as to what current density, or how many 
amperes per square inch of conductor, should be employed. Sir W. Thom- 
son's rule gives about 393 amperes per square inch, and Professor Forbes's 
tables— for a medium cost of one electrical horse-power per hour— give a 
current density of about 380 amperes per square inch as most economical. 

When a given horse-power is to be delivered at a given distance, the case 
is somewhat different, and Professors Ay rton and Perry (Electrician, March, 
1886) have shown that in that case both the current and resistance are 
variables, and that their most economical values may be found from the foK 
lowing formulas: 



and r = 



J>2 



sin 4> 



nw (1 + sin <£) a ' 



in which C = the proper current in amperes; r = resistance in ohms per 
mile which should be given to the conductor; P = pressure at entrance in 
volts; n = number of miles of conductor; w = power delivered in watts; 
(J> = such an angle that tan <f> = nt-i-P, t being a constant depending on 
the price of copper, the cost of one electrical horse-power, interest, etc.: it 
may be taken as about 17. 

In this case the current density should not remain constant, but should 
diminish as the length increases, being in all cases less than that calculated 
by Sir W. Thomson's rule. 

Example.— If the current for an electric railway is sent in at 200 volts, 100 
horse-power being delivered, find the waste of power in heating the con- 
ductor, the distance being 5 miles and there being a return conductor. 

Here n = 10, t = 17, P = 200; tan 4> = 170 -=- 200 = .85, <f> = 40° 22', sin <f> - 
.6477. 

Hence most economical resistance 



200 2 



10 X 74600 
or .1279 ohm in its total length. 
The most economical current, C 
C*R 



.6477 
1.6477 2 ' 



.01279 ohm per mile, 



_ 74600 
200 
614.58 2 X .1271 



1.6477 = 614.58 amperes, and W, 



= 64.75 horse-power. 
The following tables show the power wasted as heat in the conductor. 



Horse-power Wasted in Transmitting Power Electrically to a Given 
Distance, the Entering Power being Fixed. Pressure at Entrance, 
200 Volts. Current Density, 380 Amperes per Square Inch. 





Horse-power Wasted, the 










Horse-power 
sent in.* 


Power is Transmitted being 

one Mile (there being a 

Return Conductor). 


Horse-power Wasted. 
Distance Five Miles. 


10 


1.663 


8.318 


20 


3.327 


16 636 


40 


6.654 


33.27 


50 


8.318 


41.59 


80 


13.308 


66.54 


100 


16.636 


83.18 


200 


33.272 


166.36 



* That is, horse-power at the generator terminals. 



ELECTRIC TRANSMISSION. 



1041 





Pressure at 


Entrance, 2000 Volts. 




Horse- 
power 
sent in. 


Horse-power 

Wasted. Distance 

One Mile (there 

being a Return 

Conductor). 


Horse- 
power 
Wasted. Dis- 
tance Five 
Miles. 


Horse- 
power 
Wasted. 
Distance Ten 
Miles. 


Horse-power 
Wasted. 
Distance 

Twenty Miles. 


100 
200 
400 
500 
800 
1000 
2000 


1.663 
3.327 
6.654 

8.318 
13.308 
16.636 
33.272 


8.318 
16.636 
33.272 
41.59 
66.54 
83.18 
166.36 


16.636 
33.272 
66.54 
83.18 
133.08 
166.36 
332.72 


33.27 
66.54 
133.08 
166.36 
266.17 
332.72 
665.44 



It will be seen from these numbers that when the current density is fixed 
the power wasted is proportional to the entering horse-power and the length 
of the conductor, and is inversely proportional to the potential. For a 
copper conductor the rule may be simply stated as 



W = 16.6! 



o# 



XI, 



E being the horse-power and P the pressure at entrance, and I the length of 
the conductor in miles. 

Horse-power Wasted in Electric Transmission to a Given Distance, 
the Power to be Delivered at the Distant End being Fixed. Pres- 
sure at Entrance, 200 Volts. Current and Resistance Calculated 
by Ayrton and Perry's Rules. 





Horse-power Wasted, 








the Distance to which 


Horse-power 


Horse-power 


Horse-power 


the Power is Transmitted 


Wasted. 


Wasted. 


Delivered, 


being One Mile (there 


Distance Five 


Distance Ten 




being a Return 


Miles. 


Miles. 




Conductor). 






10 


1.676 


6.476 


8.620 


20 


3.352 


12.952 


17.24 


40 


6.704 


25.904 


34.48 


50 


8.38 


32.38 


43.10 


80 


13.408 


51.808 


68.96 


100 


16.76 


64.86 


86.20 


200 


33.52 


129.52 


172.4 



Pressure at Entrance, 2000 Volts. 



Horse-power 
Delivered. 


Horse-power 


Horse-power 


Horse-power 


Wasted. Distance 


Wasted. Distance 


Wasted. Distance 


One Mile. 


Five Miles. 


Ten Miles. 


100 


1.716 


8.484 


16.763 


200 


3.432 


16.968 


33.526 


400 


6.864 


33.938 


67.052 


500 


8.58 


42.42 


83.815 


800 


13.728 


67.87 


134.104 


1000 


17.16 


84.84 


167 63 


2000 


34.32 


169.68 


335.26 



If H = horse-power sent in, w = power delivered in watts, C = current in 
amperes, r = resistance in ohms per mile, P = pressure at entrance in 
volts, and n = number of miles of conductor, 

(w + CV) -f- 746 = H; iv = 746H - <Pr\ 



1042 



ELECTRICAL ENGINEERING. 



and the formulae for best current and resistance become 

P 2 sin 4 



Energy wasted as heat in watts per mile = Ch- 



Horse-power wasted per mil'e= W x = 



n(7462f - CV l + sin<f>' 
4(jHsin4> 



Hsin 



n + sin </>' 

(4> = angle whose tangent = nt-i- P, and the value of t corresponding to a 
current density of 380 amperes per sq. in. is 16.636.) 

TABLE OF EL.ECTRICAL. HORSE-POWERS. 



Volts X Amperes 
746 



= H.P., or 1 volt-ampere = .0013405 H.P. 



Read amperes at top and volts at side, or vice versa. 



%i 












Volts or Amperes 












£c 






r> 


























B u 


1 
00134 


10 


20 


30 


40 

.0536 


50 
.0570 


60 


70 


80 


90 


100 


110 


120 


1 


.0134 


.0268 


.0402 


.0804 


.0938 


.1072 


.1206 


.1341 


.1475 


.1609 


2 


00268 


.0268 


.0536 


.0804 


.1072 


.1341 


.1609 


.1877 


.2145 


.2413 


.2681 


.2949 


.3217 


3 


(10402 


.0402 


.0804 


.1206 


.1609 


.2011 


.2413 


.2815 


.3217 


.3619 


.4022 


.4424 


.4826 


4 


00530 


.0536 


.1072 


.1609 


.21451 


.2681 


.3217 


,375:', 


.4290 


.4826 


.5362 


.5898 


.6434 


6 


00670 


.0670 


.1341 


.2011 


.2681 


.3351 


.4022 


.4692 


.5362 


.6032 


.6703 


.7373 


.8043 


6 


00804 


.0804 


' .1609 


.2413 


.3217 


.4022 


.4826 


.5630 


.6434 


.7239 


.8043 


.8847 


.9652 


7 


■ 


.0938 


.1877 


.2815 


.3753 


,1602 


.5630 


.6568 


.7507 


.8445 


.9384 


1.032 


1.126 


8 


oio;- 


.1072 


.2145 


.3217 


1290 


5362 


.6434 


.7507 


.8579 


.9652 


1.072 


1 180 


1.287 


9 


oi-106 


,1206 


.2413 


.3619 


.4826 


.6033 


.7239 


.8445 


.9652 


1.086 


1.206 


1.327 


1.448 


10 


01341 


.1341 


.2681 


.4022 


.5362 


.6703 


.8043 


.9383 


1.072 


1.206 


1.341 


1.475 


1.609 


11 


.01475 


.1475 


.2949 


.4424 


.5898 


.7373 


.8847 


1.032 


1.180 


1.327 


1.475 


1.622 


1.709 


12 




.1609 


.3217 


■1826 


.6434 


.8043 


.9652 


1.126 


1.287 


1.448 


1.609 


1.769 


1.930 


13 




.1743 


.34S5 


.5228 


.6970 


.8713 


1.046 


1.220 


1.394 


1.568 


1.743 


1.917 


2.091 


14 


.01877 


.1877 


.3753 


.5630 


.7507 


.9384 


1.126 


1.314 


1.501 


1.689 


1.877 


2.064 


2.252 


15 


.02011 


.2011 


.4022 


.6032 


.8043 


1.005 


1.206 


1.408 


1.609 


1.810 


2.011 


2.212 


2.413 


16 


.02145 


.2145 


.4290 


.6434 


.8579 


1.072 


1.287 


1.501 


1.716 


1.930 


2.145 


2.359 


2.574 


17 




.2279 


.4558 


.6837 


.9115 1.139 


1.367 


1.595 


1.823 


2.051 


2.279 


2.507 


2-735 


18 


.02413 


.2413 


.4826 


.7239 


.9652 1.206 


1.448 


1.689 


1.930 


2.172 


2.413 


2.654 


2.895 


19 


.02547 


.2547 


.5094 


.7641 


1.019 


1.273 


1.528 


1.783 


2.037 


2.292 


2.547 


2.801 


3.056 


20 


.02681 


.2681 


.5362 


.8043 


1.072 


1.340 


1.609 


1.877 


2.145 


2.413 


2.681 


2.949 


3.217 


21 


.02815 


.2815 


.5630 


.8445 


1.126 


1.408 


1 689 


1.971 


2.252 


2.533 


2.815 


3.097 


3.378 


22 


.02949 


.2949 


.5898 


.8847 


1.180 


1.475 


1.769 


2.064 


2.359 


2.654 


2.949 


3.244 


3.539 


2.3 


.0308.'! 


.3083 


.6166 


.9249 


1.233 


1.542 


1.850 


2.158 


2.467 


2.775 


3.083 


3.391 


3.700 


24 


.03217 


.3217 


.6434 


.9652 


1.287 


1.609 


1.93,0 


2.252 


2.574 


2.895 


3.217 


3 539 


3.861 


25 


.03351 


.3351 


.6703 


1.005 


1.341 


1.676 


2.011 


2.346 


2.681 


3.016 


3.351 


3.686 


4.022 


26 


.03485 


.3485 


.6971 


1.046 


1.394 


1.743 


2.091 


2.440 


2.788 


3.137 


3.485 


3.834 


4.182 


27 


.03619 


.:;0l! 


.7239 


1.086 


1.448 


1.810 


2.172 


3.534 


2.895 


3.257 


3.619 


3.981 


4.343 


28 


.0375; 


.3,753 


.7507 


1.126 


1.501 


1.877 


2.252 


2.6,27 


3.003 


3.378 


3.753 


4.129 


4.504 


29 


.03887 


.3887 


.7775 


1.166 


1.555 


1.944 


2.332 


2.721 


3.110 


3.499 


3.887 


4.276 


4.665 


30 


.01022 


.4022 


.8043 


1.206 


1.609 


2.011 


2.413 


2.815 


3.217 


3.619 


4.022 


4.424 


4.826 


31 


.04156 


.4156 


.8311 


1.247 


1.662 


2.078 


2.493 


2.909 


3.324 


3.740 


4.156 


4.571 


4.987 


32 


.04290 




.8579 


1.287 


1.716 


2.145 


2.574 


3.003 


3 432 


3.861 


4.290 


4.719 


5.148 


33 


■ 


.4424 


.8847 


1.327 


1.769 


2.212 


2.654 


3.097 


3.539 


3.986 


4.424 


4.866 


5.308 


34 


.0455b 


.4558 


.911E 


1.367 


1.823 


3.3,0 


2.735 


3.190 


3.646 


4.102 


4.558 


5.013 


5.469 


35 


.04692 


.4692 


.9384 


1.408 


1.877 


2.346 


2.815 


3.284 


3.753 


4.223 


4.692 


5.161 


5.630 


36 


.0482t 


.4826 


.9655 


1.448 


1.930 


2.413 


2.895 


3.378 


3.861 


4.343 


4.826 


5.308 


5.791 


37 


0!96l 


.4960 


.'J92( 


1.488 


1.984 


2.480 


2.976 


3.472 


3.968 


4.464 


4.960 


5.456 


5.952 


38 


.05094 


50!) 


1.019 


1.528 


2.038 


2.547 


3.056 


3.566 


4.075 


4.585 


5.094 


5.603 


6.113 


3S 


.0522b 


.522b 


1.046 


1.568 


2.091 


2.614 


3.137 


3.660 


4.182 


4.705 


5.228 


5.751 


6.274 


« 


.05365 


.5365 


1.072 


1.609 


2.145 


2 681 


3.217 


3.753 


4.290 


4.826 


5.362 


5.898 


6.434 


41 


.05491 


.5496 


1.099 


1.649 


2.198 


2.748 


3.298 


3.847 


4.397 


4.946 


5.496 


6.046 


6.595 


45 


.0503< 


.5631 


1.126 




2.252 


2.815 


3.378 


3.941 


4.504 


5.067 


5.630 


6.193 


6.756 


±: 


.0576 J 


.5761 


1.153 


1.729 


2.300 


2.882 


3.458 


4.035 


4.611 


5.187 


5.764 


6.341 


6.917 


41 


.0589. 


,589b 


1.180 


1.769 


2.359 


2.949 


3.539 


4.129 


4.719 


5.308 


5.898 


6.488 


7.078 


n 


.0603. 


.603; 


1.206 


1.810 


2.413 


3.016 


3.619 


4.223 


4.826 


5.439 


6.032 


6.635 


7.239 


u 


.0616 


.616C 


1.233 


1.850 


2.467 


3.083 


3.700 


4.316 


4.933 


5.550 


6.166 


6.783 


7.400 


4 r 


.0630 


.630( 


1.260 


1.890 


2.520 


3.150 


3.780 


4.410 


5.040 


5.070 


6.300 


6.930 


7.560 


a 


.0643 


.013- 


1.287 


1.930 


2.574 


3.217 


3.861 


4.504 


5.148 


5.791 


6.434 


7.078 


7.721 


4 


.0656 


.656 


1.314 


1.970 


2.627 


3.284 


3.941 


4.598 


5.255 


5.912 


6.568 


7.225 


7.882 


5( 


.0670 


.670 


1.341 


2.011 


2.681 


3.351 


4.022 


4.692 


5.362 


6.032 


6.703 


7.373 


8.043 



TABLE OF ELECTRICAL HORSE-POWERS. 



1043 



TABLE: OF ELECTRICAL HORSE-POAVERS- (Continued.) 



So 










Volts o 


• Amperes. 












< s 
























1 10 


20 


J 30 


40 


50 1 60 


70 


80 


90 


100 


110 


120 


55 


.0737?! .7373 


1.475 


2.212 


2.949 


3.686 4.424 


5.161 


5.89S 


6.635 


7.373 


8.110 


8.847 


60 


.0804J, -8043 
Ml\ i } .8713 


1.6(1! 


2.413 


3.217 


4.022 


4.826 


5.630 


6.434 


7.289 


8.043 


8.847 


9.652 


65 


1.743 


2.614 


3.485 


4.357 


5.228 


6.099 


6.970 


7.842 


8.713 


9.584 


10.46 


70 


.0938* .9384 


1.87; 


2.815 


3.753 


4.61C 


5.630 


6.568 


7.507 


8.445 


9.884 


10.32 


11.26 


75 


.1005* 1.005 


2.011 


3.016 


4.021 


5.027 


6.032 


7.037 


8.043 


9.048 


10.05 


11.06 


12.06 


80 


.10724 1.072 


2.145 


3.217 


4.290 


5.362 6.434 


7.507 


8.579 


9.652 


10.72 


11.80 


12.87 


85 


.11391 1.139 


•.'.270 


3.418 


4.558 


5.6971 6.836 


7.976 


9.115 


10.26 


11.39 


12.53 


13.67 


90 


.12065, 1.206 


2.413 


3.619 


4.826 


6.032 7.239 


8.445 


9.652 


10.86 


12.06 


13.27 


14.48 


95 


.12735 1.273 


2.547 


3.820 


5.001 


6.367 1 7.641 


8.914 


10.18 


11.46 


12.73 


14.01 


15.28 


100 


.13405 1.341 


2.681 


4.022 


5.362 


6.703J 8.043 


9.384 


10.72 


12.06 


13.41 


14.75 


16.09 


200 


.26810 2.681 


5.362 


8.043 


10.72 


13.41 


16.09 


18.77 


21.45 


24.13 


26.81 


29.49 


32.17 


300 


.402151 4.022 


8.013 


12.06 


16.09 


20.11 


24.13 


28.15 


32.17 


36.19 


40.22 


44.24 


48.26 


400 


.53620: 5.362 


10.72 


16.09 


21.45 


26.81 


32.17 


37.53 


42.90 


48.26 


53.62 58.98 


64.34 


500 


.67025! 6.703 


13.41 


20.11 


26.81 


33.51 


40.22 


46.92 


53.62 


60.32 


67.03. 73.73 


80.43 


600 


.80430 


8.043 


16.09 


24.13 


32.17 


40.22 


48.26 


56.30 


64.34 


72.39 


80.43 


88.47 


96.52 


700 


.93835 


9.384 


18.77 


28.15 


37.53 


46.92 


56.30 


65.68 


75.07 


84.45 


93.84 


103.2 


112.6 


800 


1.0724 


10.72 


21.45 


32.17 


42.90 


53.62 


64.34 


75.07 


85.79 


96.52 


107.2 


118.0 


128.7 


900 


1.2065 


12.06 


24.13 


36.19 


48.26 


60.32 


72.39 


84.45 


96.52 


108.6 


120.6 


132.7 


144.8 


1 ,001) 


1.3405 


13.41 


26.81 


40 22 


53.62 


67.03 


80.43 


9.3.84 


107.2 


120.6 


134.1 


147.5 


160.9 


2,000 


2.6810 


26.81 


53.62 


80.43 


107.2 


134.1 160.9 


187.7 


214.5 


241.3 


268.1 


294.9 


321.7 


3,000 


4.0215 


40.22 


80.43 


120.6 


160.9 


201.1 241.3 


281.5 


321.7 


361.9 


402.2 


442.4 


482.6 


4,000 


5.3620 


53.62 


107.2 160.9 


214.5 


268.1 321.7 


375.3 


429.0 


482 6 


536.2 


589.8 


643.4 


5, 00(1 


6.7025 


67.03 


134.1 201.1 


268.1 


335.1 1402.2 


469.2 


536.2 


608.2 


670.3 


737.3 


804.3 


6.000 


8.0130 


80.43 


160.9 241.3 


321.7 


402.2 1482.6 


563.(1 


643.4 


723.9 


804.3 


8S4.7 


965.2 


7,000 


9.3835 


93.84 


187.7 ,381.5 


375.3 


469.2 563.0 


656.8 


750.7 


S44!5 


938.4 


1032 


1126 


8,000 


10.724 


107.2 


214.5 321.7 


429.0 


536.2 1643.4 


750.7 


857.9 


965.2 


1072 


1180 


1287 


9.000 


12.06:. 


120.6 


241.3 361.9 


482. 6 


603.2 1723.9 


844.5 


965.2 


1086 


1206 


1327 


1448 


10,000 


13.405 


134.1 


268.1 402.2 


536.2 


670.3 804.3 


938.3 


1072 


1206 


1341 


1475 


1009 



"Wire Table.— The wire table on the following page (from a circular of 
the Westinghouse El. & Mfg. Co.) shows at a glance the size of wire neces- 
sary for the transmission of any given current over a known distance with 
a given amount of drop, for 100-volt and 500-volt circuits, with varying 
losses. The formula by which this table has been calculated is 



D x 1000_ 
CX2L~ 



B, 



in which D equals the volts drop in electro-motive force, Cthe current, L the 
distance from the dynamo to the point of distribution, and R the line resist- 
ance in ohms per thousand feet. 

Example 1.— Required the size of wire necessary to carry a current of GO 
amperes a distance of 050 feet with a loss of b% at 100 volts. 

Referring to the table, under 60 amperes, we find the given distance, 050 
feet. In the same horizontal line and under 5% drop at 100 volts, we find No. 
000 wire, which is the size required. 

Example 2.— What size will be required for 10 amperes 2000 feet, with a 
drop of 10% at 500 volts. 

Under 10 amperes find 1930— the nearest figure to 2000— and in the same 
horizontal line under 10$ at 500 volts find No. 11, the size required. 

Wiring Formulae for Incandescent Lighting. (W. D. 
Weaver, Elec. World, Oct. 15, 1892.) — A formula for calculating wiring 
tables is 

. 2150LC 
A — - 



aE* 



aE 



where A = section in circular mils; W = watt rating of lamps; E— volt- 
age; L — distance to centre of distribution, in feet; N — number of lamps; 
a — percentage of drop; C = current in amperes. 

Example.— Volts, 50; amperes, 100; feet to centre of distribution, 100; 
drop, 2%. 

2150 X 100 x 100 01Rn __ . , 

— 215,000 circular mils, 



or about 0000 B. 



2 X 50 
X S. gauge. 



1044 



ELECTRICAL ENGINEERING. 





f- 




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o 





o 

> 


§ 


m 


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cd 
























O 

















e 




















fc 







IO KS •* 50 t- t-HMOOO 

■*oio5!ov £- to -* co eo 


s 

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iisia * gs ^ s 


^TdSiOWt-OO^MMeQNWHH 


nisi i- ss - 




lO CO 0-5 C- -H .-I CO P 5 3i 


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Stm oioiNot. S-*eo 



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ELECTRIC TRANSMISSION. 



1045 



The horse-power and efficiency of a motor being given, the size of the con- 
ducting wire in circular mils can be found from the following formula: 

_ 160,400,000 X H.P. X L 
~ a£ 2 X efficiency 

Example.— Horse-power, 10; volts, 500; drop, %\ feed to distributing 
point, 600: efficiency of motor, 75$. 

160,400,000X10X600 •„. . . .. . . __ D _ ■ _ 

A = ■ ' ' — enn ^ = 17,109 circular mils, or about No. 8 B. & S. 
3 X 500 X 500 X 75 

Cost of Copper for Long-distance Transmission. 

(Westinghouse El. & Mfg. Co.) 

Cost of Copper required for the Delivery of One Mechanical Horse- 
power at Motor Shaft with 1000, 2000, 3000, 4000, 5000, and 10,000 Volts 
at Motor Terminals, or at Terminals of Lowering Transformers. 

Loss of energy in conductors (drop), equals 20$. 
Distances equal one to twenty miles. 
Motor efficiency equals 90$. 

Length of conductor per mile of single distance, 11,000 feet, to allow for 
sag. 
Cost of copper equals 16 cents per pound. 



Miles. 


1000 v. 


2000 v. 


3000 v. 


4000 v. 


5000 v. 


10,000 v. 


1 


$2.08 


$0.52 


$0.23 


$0.13 


$0.08 


$0.02 


3 


8.33 


2.08 


0.93 


0.52 


0.33 


0.08 


3 


18.70 


4.68 


2.08 


1.17 


0.75 


0.19 


4 


33.30 


8.32 


3.70 


2.08 


1.33 


0.33 


5 


52.05 


13.00 


5.78 


3.25 


2.08 


0.52 


6 


74.90 


18.70 


8.32 


4.68 


3.00 


0.75 


7 


102.00 


25.50 


11.30 


6.37 


4.08 


1.02 


8 


133.25 


33.30 


14.80 


8.32 


5.33 


1.33 


9 


168.60 


42.20 


18.70 


10.50 


6.74 


1.69 


10 


208.19 


52.05 


23.14 


13.01 


8.33 


2.08 


11 


251.90 


63.00 


28 00 


15.75 


10.08 


2.52 


12 


299.80 


75.00 


33.30 


18.70 


12.00 


3.00 


13 


352.00 


88.00 


39.00 


22.00 


14.08 


3.52 


* 14 


408 00 


102.00 


45.30 


25.50 


16.32 


4.08 


15 


468.00 


117.00 


52.00 


29.25 


' 18.72 


4.68 


16 


533.00 


133.00 


59.00 


33.30 


21.32 


5.33 


17 


600.00 


150.00 


67.00 


37.60 


24.00 


6.00 


18 


675.00 


169.00 


75.00 


42.20 


27.00 


6.75 


19 


750.00 


188.00 


83.50 


47.00 


30.00 


7.50 


20 


833.00 


208.00 


92.60 


52.00 


33.32 


8.33 



A Graphical Method of calculating leads for wiring for electric 
lighting is described by Carl Hering in Trans. A. I. E. E., 1891. He furnishes 
a chart containing three sets of diagonal straight -line diagrams so con- 
nected that the examples under the general formula for wiring may be 
solved without calculation by simply locating three points in succession on 
the chart. 

The general principle upon which the chart is based is that for any 
formula containing three variable quantities, one of which is the product 
or the quotient of the other two, the "curves' 1 '' representing their relative 
values may always be represented by a series of straight diagonal lines 
drawn through the centre or zero-point. Such a set of lines will therefore 
enable one to make any calculations graphically for that formula. For 
instance, horse-power = volts X amperes; the constant 746 does not con- 
cern us at present. A series of diagonal lines properly spaced will there- 
fore give directly either the horse-power, the volts, or the amperes, when 
the other two are given. 

One scale is vertical, the other horizontal, and the diagonal lines (or the 
hyperbolas) each represent one unit (or a number of units) of the third 
scale. To make the "curves" straight lines the diagonals must be made 



1046 



ELECTRICAL ENGINEERING. 



Cost of Copper required to deliver One Mechanical Horse-power at 
Motor-shaft with Varying Percentages of Loss in Conductors, upon 
the Assumption that the Potential at Motor Terminals is in Each 
Case 3000 Volts. 

Distances equal one to twenty miles. 
Motor efficiency equals 90$. 

Length of conductor per mile of single distance, 11,000 feet, to allow for 
sag. 
Cost of copper equals 16 cents per pound. 



Miles. 


10$ 


15$ 


20$ 


25$ 


30$ 


1 


$0.52 


$0.33 


$0.23 


$0.17 


$0.13 


2 


2.08 


1.31 


0.93 


0.69 


0.54 


3 


4.68 


2.95 


2.08 


1.55 


1.21 


4 


8.32 


5.25 


3.70 


2.77 


2.15 


5 


13.00 


8.20 


5.78 


4.33 


3.37 


6 


18.70 


11.75 


8.32 


6.23 


4.85 


7 


25.50 


16.00 


11.30 


8.45 


6.60 


8 


33.30 


21.00 


14.80 


11.00 


8.60 


9 


42.20 


26.60 


18.75 


14.00 


10.90 


10 


52.05 


32.78 


23.14 


17.31 


13.50 


11 


63.00 


39.75 


28.00 


21.00 


16.30 


12 


75.00 


47.20 


33.30 


24.90 


19.40 


13 


88.00 


55.30 


39.00 


29.20 


22.80 


14 


102.00 


64.20 


45.30 


33.90 


26.40 


15 


117.00 


73.75 


52.00 


38.90 


30.30 


16 


133.00 


83.80 


59.00 


44.30 


34.50 


17 


150.00 


94.75 


67.00 


50.00 


39.00 


18 


169.00 


106.00 


75.00 


56.20 


43.80 


19 


188.00 


118.00 


83.50 


62.50 


48.70 


20 


208.00 


131.00 


92.60 


69.25 


54.00 



to represent one of the two quantities which is equal to the quotient of the 
other two, and not the one which is equal to the product of the other two, 
because the curves would then be hyperbolas. In the example given the 
diagonals must represent volts or amperes, but not horse-powers. The con- 
stants in such formula? affect only the positions of the diagonals; although 
they increase considerably the work of arithmetically calculating the results, 
they do not affect in the least the graphical calculations after the diagrams 
are once drawn. 
The general formula for wiring is : 



Cross-section = 



current for one lamp x No. of lamps x distance X constant 
loss in volts 

containing six quantities only, one of which is always constant, being equal 
to twice the mil-foot resistance of copper, if the cross-section is in circular 
mils. Calculations involving three of these five quantities may readily be 
made graphically by means of a single set of diagonal lines. 

In Mr. Hering's method the formula is split up into three smaller ones, 
each of which contains no more than three variable quantities. Each 
formula can then be calculated separately by a simple diagram, as de- 
scribed, thus permitting the whole formula to be calculated graphically. 

To do this, let the first diagram perform the calculation, 

_ current for one lamp 
loss in volts ' 



in which a; is a mere auxiliary quantity, 
perform the next calculation, 

y = x X number of lamps; 

and a third diagram the final calculation, 

cross-section = y X distance. 



Let a second similar diagram 



ELECTRIC TRANSMISSION. 



1047 



The constant may be combined with any one of these, it is immaterial 
which one. This triple calculation may at first seem to complicate matters 
on account of the new quantities, x and y. These, however, are easily 
elimiuated by the simple device of placing the three diagrams together, side 
by side, in such a position that the two x scales coincide, and similarly the 
two y scales. By doing this one has merely to pass directly from one set of 
diagonals to the next to perform the successive steps of the calculation, 
without being concerned about the intermediate auxiliary quantities. These 
intermediate quantities correspond, and are equal to the successive products 
or quotients which are obtained in the successive arithmetical multiplications 
and divisions of these five quantities in the formula, which cannot, of 
course, be eliminated in making the calculations arithmetically. 

Weight of Copper required for Long- distance Trans- 
mission.— W. F. C. Hasson (Trans. Tech. Socy. of the Pacific Coast, vol. 



x. No. 4) gives the following formula: 

D 2 
W=~H..F. 



(100 - £), n 



266.5, 



where IF is the weight of copper wire in pounds; D, the distance in miles; 
E. the E.M.F. at the motor in hundreds of volts; H.P., the horse-power 
delivered to the motor; L, the per cent of line loss. 

Thus, to transmit 200 horse-power ten miles with 10 per cent loss, and 
have 3000 volts at the motor, we have 



Vtt . 10 X 10 M (100 - 
W = 80*80 X2 °° X So" 



10) 



X 266.5 = 53,300 lbs. 



Efficiency of Long-distance Transmission. (F. R. Hart, 
Potver, Feb. 1892.)— The mechanical efficiency of a system is the ratio of the 
power delivered to the dynamo-electric machines at one end of the line to 
the power delivered by the electric motors at the distant end. The com- 
mercial efficiency of a dynamo or motor varies with its load. The .maximum 
efficiency of good machines should not be under 90$ and is seldom above 
92$. Under the most favorable conditions, then, we must expect a loss of 
say 9$ in the dynamo and 9$ in the motor. The loss in transmission, due to 
fall in electrical pressure or " drop " in the line, is governed by the size of 
the wires, the other conditions remaining the same. For a long-distance 
transmission plant this will vary from 5$ upwards. With a loss of 5$ in the 
line, the total efficiency of transmission will be slightly under 79$. With a 
loss of 10$ in the line, the efficiency would be slightly 'under 75$. We may 
call 80$ the practical limit of the efficiency with the apparatus of to-day. 
The methods for long-distance power transmission by electricity may be 
divided into three general classes: (1) Those using continuous current; (2) 
those using alternating current; and (3) regenerating or " motor-dynamo" 
systems. The subdivisions of each of these general classes are tabulated as 
follows: 



Continuous 
current 



Alternating 
current 



Regenerating 
systems 



One machine. 

Machines in parallel. 

One machine. 

Machines in parallel. 

Machines in series. 

2 machines in series. 

Machines in multiple series. 

Machines in series. 

Without conversions. 

With conversions. 

Without conversions. 

With conversions. 
f Alternating continuous. 

I Alternating converter; line converter; alternating con- 
■{ tinuous. 
I Continuous-continuous. 
I. Partial reconversion of any system. 

The relative advantages of these systems vary with each particular trans- 
mission problem, but in a general way may be tabulated as below. 



f Low 

voltage 

| High 

[ voltage 



[ Multiple-wire 

J Alternating single phase 

[ Alternating multiphase 



1048 



ELECTRICAL ENGINEERING. 





System. 


Advantages. 


Disadvantages. 




( Low voltage. 


Safety, simplicity. 


Expense for copper. 





( High voltage. 


Economy, simplicity. 


Danger, difficulty of 
building machines. 


p 

p 

o 


3-wire. 


Low voltage on machines 
and saving in copper. 


Not saving enough in 


U 


Multiple-wire. 


Low voltage at machines 
and saving in copper. 


tances. Necessity for 
" balanced " system. 




Single phase. 


Economy of copper. 


Cannot start under load. 
Low efficiency. 


_p 

p 


Multiphase. 


Economy of copper, syn- 
chronous speed unnec- 
essary; applicable to 
very long distances. 


Complexity. Lower ef- 
ficiency of terminal 
apparatus. Not as yet 
"standard.' 1 


< 


Motor-dynamo. 


High-voltage transmis- 
sion. Low-voltage de- 
livery. 


Expensive. 
Low efficiency. 



There are many factors which govern the selection of a system. For each 
problem considered there will be found certain fixed and certain unfixed 
conditions. In general the fixed factors are: (1) capacity of source of 
power; (2) cost of power at source; (3) cost of power by other means at point 
of delivery; (4) danger considerations at motors; (5) operation conditions; 
(6) construction conditions (length of line, character of country, etc.). The 
partly fixed conditions are: (7) power which must be delivered, i.e., the effi- 
ciency of the system; (8) size and number of delivery units. The variable 
conditions are: (9) initial voltage; (10) pounds of copper on line; (11) origi- 
nal cost of all apparatus and construction: (12) expenses, operating (fixed 
charges, interest, depreciation, taxes, insurance, etc.); (13) liability of trouble 
and stoppages; (14) danger at station and on line; (15) convenience in oper- 
ating, making changes, extensions, etc. Assuming that the cost of dyna- 
mos, motors, etc., will be approximately the same whatever the initial 
pressure, the great variation in the cost of wire at different pressures is 
shown by Mr. Hart in the following figures, giving the weights of copper 
required for transmitting 100 horse-power 5 miles : 

"Voltage. Drop 10 per cent. Drop 20 per cent. 

2,000 16,800 lbs. 8,400 lbs. 

3,000 7,400 " 3,700 " 

10,000 620 " 310 " 

Efficiency of a. Combined Engine and Dynamo. — A com- 
pound double - crank Willans engine mounted on a single base with a 
dynamo of the Edison-Hopkinson type was tested in 1890, with results as 
follows : The low-pressure cylinder is 14 in. diam., 13 in. stroke; steam- 
pressure 120 lbs. It is coupled to a dynamo constructed for an output of 475 
amperes at 110 volts when driven at 430 revolutions per minute. The arma- 
ture is of the bar construction, is plain shunt-wound, and is fitted with a 
commutator of hard-drawn copper with mica insulation. Four brushes are 
carried on each rocker-arm. 

Resistance of magnets 16. ohms 

Resistance of armature ... 0.0055 " 

I.H.P 83.3 

E.H.P 72.2 

Total efficiency 86.7 per cent 

Consumption of water per I.H.P. hour 21.6 pounds 

Consumption of water per E.H.P. hour 25 " 

The engine and dynamo were worked above their full normal output, 
which fact would tend to slightly increase the efficiency. 

The electrical losses were : Loss in magnet coils, 756 watts, equal to 1.4$; 
loss in armature coil, 1386 watts, equal to 2.6$; so that the electrical efficiency 



ELECTRIC TRANSMISSION. 



1049 



of the machine due to ohmic resistance alone was 96$. The remainder of 
the losses, a little over 8 horse-power, is clue to friction of engine and 
dynamo, hysteresis, and the like. 

Electrical Efficiency of a Generator and Motor.— A twelve- 
mile transmission of power at Bodie, Cal., is described by T. H. Leggett 
(Trans. A. I. M E. 1894). A single-phase alternating current is used. The 
generator is a Westinghouse 120 K. W. constant-potential 12-pole machine, 
speed 860 to 870 revs, per min. The motor is a synchronous constant-po- 
tential machine of 120 horse-power. It is brought up to speed by a 10-H.P. 
Tesla starting motor. Tests of the electrical efficiency of the generator and 
motor gave the following results : 

Test on Generator. 





Amperes 


Volts. 


Watts. 




15.8 
18.2 


60 

78 


948 




1419 6 


Resistance of armature, 1.6618 ohms. 


664.72 








3032.32 


Load 


20 


3414 


68280 



Apparent electrical efficiency of generator, 95.559$. 
Test on Motor. 





Amperes 


Volts. 


Watts. 


Self-excited field 


52 


62.4 


3244.8 


Resistance of armature, 1.4 ohms. 


560 








3804.08 




20 


3110 


62200 







Apparent electrical efficiency of motor, 93.883$. 

Efficiency of an Electrical Pumping-plant. (Eng. & M. 
Jour., Feb. 7, 1891.)— A pumping-plant at a mine at Normanton, England, 
was tested, with results given below: 

Above ground there is a pair of 20^ x 48-in. engines running at 20 revs, per 
min., driving two series dynamos giving 690 volts and 59 amperes. The cur- 
rent from each dynamo is carried into the mine by an insulated cable about 
3000 feet long. There they are connected to two 50-h.p. motors which oper- 
ate a pair of differential ram-pumps, with rams 6 in. and 4^ in. diam. and 
24 in. stroke. The total head against which the pumps operate is 890 feet. 
Connected to the same dynamos there is also a set of gearing for driving a 
hauling plant on a continuous-rope system, and a set of three-throw ram- 
pumps with 6-inch rams and 12-inch stroke can also be thrown into gear. 
The connections are so made that either motor can operate any or all three 
of the sets of machinery just described. Indicator-diagrams gave the fol- 
lowing results: 

Friction of engine 6.9 H.P. 9.4$ 

Belt and dynamo friction 4.8 " 6.5$ 

Leads and motor 6.7 " 9.4$ 

Motor belt, gearing and pumps empty 10.2 " 14.0$ 

Load of 117 gallons through 890 feet 31.5 " 43.1$ 

Water friction in pumps and rising main 12.9 " 17.6$ 

73.0 H.P. 100.0$ 

At the time when these data were obtained the total efficiency of the plant 
was 43. 1$, but in a later test it rose to 47$. 

References on Power Distribution.— Kapp, Electric Transmis- 
sion of Energy; Badt, Electric Transmission Handbook; Martin and Wetzler, 
The Electric Motor and its Applications; Hospitalier, Poly phased Electric 
Currents, 



1050 ELECTRICAL ENGINEERING. 

EL.ECTRIC RAILWAYS. 

Space will not admit of a proper treatment of this subject in this work. 
Consult Crosby and Bell, The Electric Railway in Theory and Practice, 
price $250; FairchiM, Street Railways, price $4.00; Merrill, Reference 
Book of Tables and Formulae for Street Railway Engineers, price $1.00. 

Test of a Street Railway Plant.— A test of a small electric-rail- 
way plant is reported by Jesse M. Smith in Trans. A. S. M. E., vol. xv. The 
following are some of the results obtained: 

Friction of engine, air-pump, and boiler feed-pump; main belt off 9.22 I.H.P. 
Friction of engine, air and feed pumps, and dynamo, brushes off. 11.34 I.H.P. 

Friction of dynamo and belt 2.12 I.H.P. 

Power consumed by engine, air and feed pumps and dynamo, 

with brushes on and main circuit open 14.34 I.H.P. 

Power required to charge fields of dynamo 3.00 I.H.P. 

Rated capacity of engine and dynamo each 150 I.H.P. 

Power developed by engine min. 21.27; max. 141.4; mean, 70.1 I.H.P. 

Volts developed by dynamo — range, 480 to 520; average, 501 volts 

Amperes developed by dynamo max. 200; min. 4.7; average, 67 amperes 

Average watts delivered by dynamo 33,567 Watts 

Average electrical horse-power delivered by dynamo 45 E.H.P. 

Average I.H.P. deFd to pulley of dynamo, estimating friction of 

armature shaft to be the same as friction of belt 59.8 I.H.P. 

Average commercial efficiency of dynamo 45 -*- 59.8 = 75.25$ 

Average number of cars in use during test 2.89 cars. 

Number of single trips of cars 64 

Average number of passengers on cars per single trip 15.2 

Weight of cars 14,500 lbs. 

Est. total weight of cars and persons 15,900 lbs. 

Average weight in motion 45,950 lbs. 

Average electrical horse -power per 1000 lbs. of weight moved. . 0.98 E.H.P. 
Average horse-power developed by engine per 1000 lbs. of weight 

moved 1.52 I.H.P. 

Average watts required per car 11,615 watts 

Average electrical horse-power per car 15.54 E.H.P. 

Average horse-power developed in engine per car 24.25 I.H.P. 

Length of road 10.5 miles. 

Average speed, including all stops, 21 miles in 1 .5 hours = 14 miles per hour. 
Average speed between stops, 21 m. in 1.366 hours = 15.38 miles per hour. 

Proportioning Boiler,Ems;iiie, and Generator for Power- 
stations. — Win. Lee Church (Street Railway Journal, 1892) gives a 
diagram showing the abrupt variations in the current required for an 
electric railway with variable grades. For this case, in which the maximum 
current for a minute or two at a time is 175 amperes, ranging from that to 
zero, and averaging about 50 amperes, he advises that the nominal capacity 
of the generator be 100 amperes. The reason of this is found in the fact 
that an electric generator can stand an overload, or even an excessive over- 
load, provided it does not have to stand it long. The question is simply one 
of heat. The overload here was seen to continue for only about one minute, 
during which time the generator could carry it with ease with no perceptible 
rise of temperature to injure the insulation. Had this load been continuous 
for an hour or so, as would occur in an electric-lighting station, a much 
higher relative generating capacity would be required, approximating the 
maximum load. 

An engine has no such capacity for excessive overload as a generator. In 
other words, the element of time does not enter into the engine problem, 
but it becomes a question of how much the engine can actually lift by main 
strength without taking the governor to an extreme which shall slow down 
the speed. In general terms, the engine should not be called to perform, 
even for a short time, more than 20$, or possibly 25$, above its rating. 

The engine capacity, therefore, would have a nominal rating greater 
than that of the generator, say about 25$ greater. 

The capacity of the engine should be determined without reference to 
condensation. This is for the obvious reason that a condenser may become 
choked, or disabled, or leaky, and the vacuum may be poor, or lost entirely 
under sudden fluctuations. 

The boiler has to deal only with the average of the total load. In this 
particular electric railways exactly resemble rolling-mills, ^aw - mills, 



ELECTRIC LIGHTING. 1051 

and kindred industries, where the load is spasmodic, with variations 
lasting but a few seconds, or at most but a few minutes. The stored heat 
in the water of a boiler is enormous in quantity, and responds instantly to a 
release of pressure. That is to say, the boiler is an immense reservoir of 
power, and provided the drain upon it is not continued too Jong, it will 
stand exactions far beyond its nomiual capacity, and without any effect 
whatever upon the firing. 

The actual size of the boiler will depend upon the type of engine. With 
the compound engine described by Mr. Church, running non-condensing, an 
allowance of 30 pounds of water actually evaporated per I. H. P. per hour will 
give a margin for all contingencies. The engine duty under an average uni- 
form load is a very different tiling from the duty under a variable load rep- 
resented by the average. Under the uniform load, 23 pounds of water 
would be the actual engine performance, and the boiler could be propor- 
tioned with reference to this figure. Under the violent fluctuations of rail- 
way service, the average duty of the engine will rise to about 28 pounds, 
and if the maximum average load is taken, and the boiler proportioned for 
30 pounds, there will be a sufficient margin. Other compound engines not 
possessing the feature which secures uniformity of duty will range up to at 
least 45 pounds under light loads, and often to 60 pounds, and represent an 
average duty not better than 35 to 40 pounds. The same is true of every 
form of non-compounded engine, whether high speed or low speed, both of 
which show a tremendous falling back of fuel duty under variable load. 

ELECTRIC LIGHTING. 

Quantity of Energy required to produce Light.— Accord- 
ing to Mr. Preece, the quantity of energy, measured in watts, required to pro- 
duce light equivalent to one candle-power, measured by the light given out 
by the standard candle, is as follows for different light-giving substances: 

Tallow 124 watts Coal gas 68 watts. 

Wax 94 " Cannel gas 48 " 

Spermaceti 86 " Incandescent lamp.. 15 " 

Mineral oils 80 " Arc lamp 3 " 

Vegetable oils 57 " 

And the relative costs of production are about 1 for the arc lamp; 6 for the 
incandescent lamp ; 5 for the mineral-oil lamp ; 10 for the gas-light; 67 for the 
spermaceti candle. 

Life of Incandescent Lamps. (Eng^g, Sept. 1, 1893, p. 282.)— From 
experiments made by Messrs. Siemens and Halske, Berlin, it appears that 
the average life of incandescent lamps at different expenditure of watts per 
candle-power is as follows: 

Watts per candle-power 1.5 2 2.5 3 3.5 

Life of lamp, hours 45 200 450 1000 1000 

Life and Efficiency Tests of Lamps. (P. G. Gossler, Elec. 
World, Sept. 17, 1892.)— Lamps burning at a voltage above that for which 
they are rated give a much greater illuminating power than 16 candles, but 
at the same time their life is very considerably shortened. It has been ob- 
served that lamps received from the factory do not average the same candle- 
power and efficiency for different invoices; that is, lamps which are received 
in one invoice are usually quite uniform throughout that lot, but they vary 
considerably from lamps made at other times. 

The following figures show the different illuminating-powers of a 16.c.p., 
50-volt, 52-watt lamp, for various voltages from 25 to 80 volts: 
Volts: 
25 34.8 40 48 50 52.5 55.6 59.5 62 68.2 72.5 80 

Amperes * 
.561 .774 '.898 .968 1.055 1.097 1.161 1.226 1.29 1.419 1.484 1.58 
Candles: 
.4 2.47 5.1 12.6 15.8 20.5 28.4 39.3 50.7 74.5 103.2 141 
Watts - 
14.03 26.94 35.92 46.34 52.75 57.57 64.55 72.92 79.98 96.78 107.5 126.4 

Watts per c.p.: 
35.1 10.81 7.04 3.68 3.34 2.81 2.30 1.96 1.58 1.30 1.04 .90 

Street-lighting. (H. Robinson, M I.C.E., Eng'g News, Sept. 12, 1891.) 
—For street-lighting the arc-lamp is the most economical. The smallest 



1052 



ELECTRICAL EHGrlHEERINGr. 



size of arc-lamp at present manufactured requires a current of about 5 
amperes; but for steadiness and efficiency it is desirable to use not less than 
6 amperes. The candle-power of arc-lamps varies considerably, according 
to the angle at which it is measured. The greatest intensity with continuous- 
current lamps is found at an angle of about 40° below the horizontal line. 
The following table gives the approximate candle-power at various angles. 
The height of the lamps should be arranged so as to give an angle of not 
less than 7° to the most distant point it is intended to serve. 



Current 
in Amperes. 

6 
8 
10 



Lighting-power of Arc-lamps. 

Candle-power. ( 

At Angle At Angle At Angle Maximum at 

of 10°. of 20°. Angle of 40°. 

207 322 460 

350 546 780 

495 770 1100 



92 

156 



of 7 
175 
300 
420 



The following data enable the coefficient of minimum lighting-power in 
streets to be determined: 
Let P = candle-power of lamps; 

L = maximum distance from lamp in feet ; 
H = height of lamp in feet ; 
X = a coefficient. 
The light falling on the unit area of pavement varies inversely as the square 
of the distance from the lamp, and is directly proportional to the angle at 
which it falls. This angle is nearly proportional to the height of the lamp 
divided by the distance. Therefore 



A - i» X L 



or X 



PH 

'' LP' 



The usual standard of gas-lighting is represented by the amount of light 
falling on the unit area of pavement 50 feet away from a 12-c.p. gas-lamp 9 
feet high, which gives a coefficient as follows: 



12X9 

50 3 ~ 



0.000864. 



The minimum standard represents the amount of light on a unit area 50 
feet away from a 24-c.p. lamp, 9 ft. high, and gives the coefficient .001728. 

Adopting the first of the above coefficients. Mr. Robinson calculates that 
the before-mentioned sizes of arc-lights will give the same standard of 
light at the heights and distances stated in Table A. Table B gives the 
corresponding distances, assuming the minimum standard to be adopted. 



Table A. 


Table B. 


Hgt. of Lamps. 


30 ft.|s5 ft.|30 ft.|35 ft. 


Height 


20 ft. | 25 ft. 1 30 ft. 1 35 ft. 


Current in 
Ampei-es. 


Max. distances served 
from lamp, in ft. 


Amperes. 


Max. distances served 
from Lamp. 


6 

8 
10 


160 

185 
205 


175 
202 
225 


190 
220 
243 


202 
235 
260 


6 
8 
10 


130 
150 
170 


144 

165 
190 


155 

180 
205 


166 
193 
220 



The distances the lamps are apart would, of course, be double the dis- 
tances mentioned in Tables A and B. One arc-lamp will take the place of 
from 3 to 6 gas-lamps, according to the locality, arrangement, and standard 
of light adopted. A scheme of arc-lighting, based on the substitution of one 
arc-light on the average for 3L£ to 4 gas-lamps, would double the minimum 
standard of light, while the average standard would be increased 10 or 12 
times. 

Candle-power of the Arc-light. (Elihu Thomson, El. World, 
Feb. 28, 1891.)— With the long arc the maximum intensity of the light is from 
40° to 60° downward from the horizontal. The spherical candle-power is 
only a fraction of the rated c.p., which is generally taken at the maximum 
obtainable in the best direction. For this reason the term 2000 c.p. has little 



ELECTRIC WELDING. 1053 

significance as indicating the illuminating-power of an arc. It is now gener- 
ally taken to mean an arc with 10 amperes and not le^s than 45 volts between 
the carbons, or a 450- watt arc. The quality of the carbons will determine 
whether the 450 watts are expended in obtaining the most light or not, or 
whether that light will have a maximum intensity at one angle or another 
within certain limits. The larger the current passing in an arc, the less is 
its resistance. Well-developed arcs with 4 amperes will have about 11 ohms, 
with 10 amperes 4.5 ohms, and with 100 amperes .45 ohm. 

It is not unusual to run from 50 to 60 lights in a series, each demanding 
from 45 to 50 volts, or a total of, say, 3000 volts. In going beyond this the 
difficulties of insulation are greatly increased. 

Reference Books on Electric Lighting.— Noll, How to Wire 
Buildings, $1.00; Hedges, Continental Electric-light Central Stations, $6.00; 
Fleming, Alternating Current Transformers in Theory and Practice, 2 vols., 
$8.00; Atkinson, Elements of Electric Lighting, $1.50; Algave and Boulard, 
Electric Light: its History, Production, and Application, $5.00. 

ELECTRIC WELDING. 

The apparatus most generally used consists of an alternating-current 
dynamo, feeding a comparatively high-potential current to the primary coil 
of an induction-coil or transformer, the secondary of which is made so 
large in section and so short in length as to supply to the work currents 
not exceeding two or three volts, and of very large volume or rate of flow. 
The welding clamps are attached to the secondary terminals. Other forms 
of apparatus, such as dynamos constructed to yield alternating currents 
direct from the armature to the welding-clamps, are used to a limited 
extent. 

The conductivity for heat of the metal to be welded has a decided influ- 
ence on the heating, and in welding iron its comparatively low heat conduc- 
tion assists the work materiallj\ (See papers by Sir F. Bramwell, Proc. 
Inst. C. E., part iv., vol. cii. p. 1; and Elihu Thomson, Trans. A. I. M.E., xix. 
877.) 

Fred. P. Royce, Iron Age, Nov. 28, 1892, gives the following figures show- 
ing the amount of power required to weld axles and tires: 

AXLE-WELDING. 

Seconds. 

1-inch round axle requires 25 H.P. for 45 

1-inch square axle requires 30 H.P. for 48 

1^4-inch round axle requires 35 H.P. for 60 

lJ4-inch square axle requires 40 H.P. for 70 

2-inch round axle requires 75 H.P. for 95 

2-inch square axle requires 90 H.P. for 100 

The slightly increased time and power required for w r elding the square 
axle is not only due to the extra metal in it, but in part to the care which it 
is best to use to secure a perfect alignment. 

TIRE-WELDING. 

Seconds. 

1 X 3/16-inch tire requires 11 H.P. for 15 

1J4 X %-inch tire requires 23 H.P. for ....?». 25 

Wz X %-inch tire requires 20 H.P. for 30 

1^2 X J^-inch tire requires 23 H.P, for 40 

2 X J^-inch tire requires 29 H.P. for 55 

2 X %-inch tire requires 42 H.P. for 62 

The time above given for welding is of course that required for the actual 
application of the current only, and does not include that consumed by 
placing the axles or tires in the machine, the removal of the upset and 
other finishing processes. From the data thus submitted, the cost of welding 
can be readily figured for any locality where the price of fuel and cost of 
labor are known. 

In almost all cases the cost of the fuel used under the boilers for produc- 
ing power for electric welding is practically the same as the cost of fuel 
used in forges for the same amount of work, taking into consideration the 
difference in price of fuel used in either case. 

Prof. A. B. W. Kennedy found that 2^-inch iron tubes J4 incn thick were 
welded in 61 seconds, the net horse-power required at this speed being 23.4 
(say 33 indicated horse-power) per square inch of section. Brass tubing re- 



1054 Electrical ekgineerihg. 

quired 21 . 2 net horse-power. About 60 total indicated horse-power would be 
required for the welding of angle-irons 3x3x^ inch in from two to three 
minutes. Copper requires about 80 horse-power per square inch of section, 
and an inch bar can be welded in 25 seconds. It takes about 90 seconds to 
weld a steel bar 2 inches in" diameter. 

ELECTRIC HEATERS. 

Wherever a comparatively small amount of heat is desired to be auto- 
matically and uniformly maintained, and started or stopped on the instant 
without waste, there is the province of the electric heater. 

The elementary form of heater is some form of resistance, such as coils 
of thin wire introduced into an electric circuit and surrounded with a sub- 
stance, which will permit the conduction and radiation of heat, and at the 
same time serve to electrically insulate the resistance. 

This resistance should be proportional to the electro-motive force of the 
current used and to the equation of Joule's law : 

H = C*Rt X 0.24, 

where Cis the current in amperes; R. the resistance in ohms; t, the time in 
seconds; and h, the heat in gram -centigrade units. 

Since the resistance of metals increases as their temperature increases, a 
thin wire heated by current passing through it will resist more, and grow 
hotter and hotter until its rate of loss of heat by conduction and radiation 
equals the rate at which heat is supplied by the current. In a short wire, 
before heat, enough can be dispelled for commercial purposes, fusion will 
begin; and in electric heaters it is necessary to use either long lengths of 
thin wire, or carbon, which alone of all conductors resists fusion. In the 
majority of heaters, coils of thin wire are used, separately embedded in 
some substance of poor electrical but good thermal conductivity. 

The Consolidated Car-heating Co.'s electric heater consists of a galvanized 
iron wire wound in a spiral groove upon a porcelain insulator. Each heater 
is 30% in. long, 8% in. high, and 6% in. wide. Upon it is wound 625 ft. of 
wire. The weight of the whole is 23^ lbs. 

Each heater is designed to absorb two amperes of a 500-volt current. Six 
heaters are the complement for an ordinary electric car. For ordinary 
weather the heaters may be combined by the switch in different ways, so 
that five different intensities of heating- surface are possible, besides the 
position in which no heat is generated, the current being turned entirely off. 

For heating an ordinary electric car the Consolidated Co. states that 
from 2 to 12 amperes on a 500-volt circuit is sufficient. With the outside 
temperature at 20° to 30°, about 6 amperes will suffice. With zero or lower 
temperature, the full 12 amperes is required to heat a car effectively. 

Compare these figures with the experience in steam-heating of railway- 
cars, as follows- : 

1 B.T.U. = 0.29084 watt-hours. 

6 amperes on a 500-volt circuit = 3000 w r atts. 

A current consumption of 6 amperes will generate 3000 -4- 0.29084 = 10,315 
B.T.U. per hour. 

In steam- car heating, a passenger coach usually requires from 60 lbs. of 
steam in freezing weather to 100 lbs. in zero weather per hour. Supposing 
the steam to enter the pipes at 20 lbs. pressure, and to be discharged at 200° 
F., each pound of steam will give up 983 B.T.U. to the car. Then the 
equivalent of the thermal units delivered by the electrical-heating system in 
pounds of steam, is 10,315 -h 983 = 10V£, nearly. 

Thus the Consolidated Co.'s estimates for electric-heating provide the 
equivalent of lOJ^j lbs. of steam per car per hour in freezing weather and 21 
lbs. in zero weather. 

Suppose that by the use of good coal, careful firing, well designed boilers, 
and triple-expansion engines we are able in daily practice to generate 
1 H.P. delivered at the fly-wheel with an expenditure of 2>£ lbs. of coal per 
"hour. 

We have then to convert this energy into electricity, transmit it by wire 
to the heater, and convert it into heat by passing it through a resistance-coil. 
We may set the combined efficiency of the dynamo and line circuit at 8b%, 
and will suppose that all the electricity is converted into heat in the resist- 
ance-coils of the radiator. Then 1 brake H.P. at the engine = 0.85 electrical 
H.P. at the resistance-coil = 1,683,000 ft. -lbs. energy per hour = 2180 heat- 
units. But since it required 2% lbs. of coal to develop 1 brake H.P., it fol- 



ELECTRICAL ACCUMULATORS OR STORAGE-LATTERIES. 1055 



lows that the heat given out at the radiator per pound of coal burned in the 
boiler furnace will be 2180 ■+- 2^ = 872 H.U. An ordinary steam-heating 
system utilizes 9652 H.U. per lb. of coal for heating; hence the efficiency 
of the electric system is to the efficiency of the steam-heating svstem as 872 
to 9652, or about 1 to 11. (Eug'g News, Aug. 9, '90; Mar. 30, '92;'May 15, '93.) 

ELECTRICAL, ACCUMULATORS OR STORAGE- 
BATTERIES. 

Storage-batteries may be divided into two classes: viz., those in winch the 
active material is formed from the substance of the element itself, either 
by direct chemical or electro-chemical action, and those in which the 
chemical formation is accelerated by the application of some easily reduci- 
ble salt of lead. Elements of the former type are usually called Plante, and 
those of the latter " Faure," or " pasted." 

Faraday when electrolyzing a solution of acetate of lead found that per- 
oxide of lead was produced at the positive and metallic lead at the negative 
pole. The surfaces of the elements in a newly and fully charged Plante cell 
consists of nearly pure peroxide of lead, Pb0 2 , and spongy metallic lead, 
Pb, respectively on the positive and negative plates. 

During the discharge, or if the cell be allowed to remain at rest, the sul- 
phuric acid (H 2 S0 4 ) in the solution enters into combination with the per- 
oxide and spongy lead, and partially converts it into sulphate. The acid 
being continually abstracted from the electrolyte as the discharge proceeds, 
the density of the solution becomes less. In the charging operation this 
action is reversed, as the reducible sulphates of lead which have been 
formed are apparently decomposed, the acid being reinstated in the liquid 
and therefore causing an increase in its density. 

The difference of potential developed by lead and lead peroxide immersed 
in dilute H2SO4 is, as nearly as may be, two volts. 

A lead-peroxide plate gradually loses its electrical energy by local action, 
the rate of such loss varying according to the circumstances of its prepara- 
tion and the condition of the cell. Various forms of both Plante and Faure 
batteries are illustrated in " Practical Electrical Engineering." 

In the Faure or pasted cells lead plates are coated with minium or 
litharge made into a paste with acidulated water. When dry these plates 
are placed in a bath of dilute H 2 S0 4 and subjected to the action of the 
current, by which the oxide on the positive plate is converted into peroxide 
of lead and that on the negative plate reduced to finely divided or porous 
lead. 

Gladstone and Tribe found that the initial electro-motive force of the 
Faure cell averaged 2.25 volts, but after being allowed to rest some little 
time it was reduced to about 2.0 volts. The following tables show the size 
and capacity of two types of Faure cells, known as the E. P. S. cells. (Eng- 
lish.) 

66 E. P. S." Storage-cells, Li Type. 



Description of 
Cell. 


^ ? 

"h "- 

H 


Working Rate. 


£ 

< 


Approximate Exter- 
nal Dimensions. 


= £ 


No. of 
Plates. 


Material of 
Box. 


Charge 


Dis- 
charge. 


a 


S 
? 


60 
'5 

w 




^ D-d 


"1 

23 1 


Wood 

Glass 

Wood 

Glass 

Wood 

Glass 

Wood 

Glass 

Wood 

Glass 


lbs. 
18 
25 
25 
35 
35 
47 
53 
67 
70 
88 


Amper. 
10 to 13 
10 " 13 
16 " 22 
16 " 22 
25 " 30 
25 " 30 
38 " 46 
38 " 46 
50 " 60 
50 " 60 


Amper. 
1 to 13 
1 " 13 
1 " 22 
1 " 22 
1 " 30 
1 " 30 
1 " 46 
1 " 46 
1 " 60 
1 " 60 


130 
130 
220 
220 
330 
330 
500 
500 
660 
660 


in. 

?* 

Wi 

141-4 
19M 

1X1, 


in. 

13' 4 

1314 

1^ 
11 ->4 
I3L, 

n H 


in. 

I8I4 

133, 

I8I4 

1% 
1st. x 

13 : s 
ISI4 
13?4 

isi 4 

loo 4 


in. 

2oy 2 

15% 
20^ 

20^ 

15% 
201., 


lbs. 
74 
68 
107 
101 
143 
12S 
228 
211 
286 
265 



1056 



ELECTRICAL ENGINEERING. 



4 E. P. S.» Cells, T Type. 



Description of Cell. 


. 


Working Rate 


i 


Approx. External 
Dimensions. 


~5 




o £ 




■-* § 
















O fe . 






So 
















«M ffl" 






















°-!r;' G 


No. of 


Material of 




Charge 


Dis- 


- 

£ 


,c5 






+j c3 


**£< 


Plates. 


Box. 


^H 


charge. 




-3 


'5 


bt,<D 


A P. 
.&f g 

5 o 












«i 


s 


5 


w 


M° 


^ o 






lbs. 


Amper. 


Amper. 




in. 


in. 


in. 


in. 


lbs. 


( 


Wood (no lid) ... 


10 


16 to 20 


1 to 20 


66 






11% 


l*\i 


3? 


n 1 


" (with lid).. 


10 


16 " 20 


1 " 20 


66 






13% 


38 


} 


Ebonite (no lid).. 


10 


16 " 20 


1 " 20 


66 


6 




11 


V2% 


30 




Wood (no lid) 


14 


24 " 28 


1 " 30 


95 






- 


i3y 8 


52 


15 1 


" (with lid).. 


14 


24 " 28 


1 " 30 


95 






11% 


1S% 


53 


( 


Ebonite (no lid).. 


14 


24 " 28 


1 " 30 


95 


8 




11 


12% 
13^ 


42 


\ 


Wood (no lid) . . . 


18 


30 " 35 


1 " 40 


120 


11 






65 


19 1 


" (with lid).. 


18 


30 " 35 


1 " 40 


120 


11 






13% 


66 


1 


Ebonite (no lid).. 


18 


30 " 35 


1 " 40 


120 






11 


mi 


54 


( 


Wood (no lid) 


22 


38 " 42 


1 " 50 


145 








isy 8 


79 


23 1 


" (with lid) . 


22 


38 " 52 


1 " 60 


145 




. 




13% 


80 


\ 


Ebonite 


22 


38 " 42 


1 " 50 


145 






11 


im 


66 



For a very full description of various forms of storage-batteries, see 
" Practical Electrical Engineering, 1 ' part xii. For theory of the battery and 
practice with the Julien battery, see paper on Electrical Accumulators by 
P. G. Salom. Trans. A. I. M. E., xviii. 348. 

Use of Storage-batteries in Power and Light Stations. 
(Iron Age, Nov. 2, 1893.)— The storage-batteries in the Edison station, in 
Fifty -third Street, New York, relieve the other stations at the hours of heavy 
load, by delivering into the mains a certain amount of current that would 
otherwise have to come, and at greater loss or " drop, 1 ' from one or another 
of the stations connecting with the network of mains. Hence the load may 
be varied more or less arbitrarily at these stations according to the propor- 
tion of load that the larger stations are desired or able to carry. 

The battery consists of 140 cells each of about 1000 ampere-hour capacity, 
weighing some 750 lbs., and of about 48 inches in length, 21 inches in width, 
and 15 inches in depth. The battery has a normal discharge rate of about 
200 amperes, but can be discharged, if necessary, at 500 amperes. 

A test made when the station was running only 12 hours per day, from 
noon to midnight, showed that the battery furnished about 23.2$ of the total 
energy delivered to the mains. The maximum rate of discharge attained 
by the battery was about 270 amperes. Thus, in this case, we have an ex- 
ample of a battery which is used for the purpose: 1. Of giving a load to 
station machinery that would otherwise be idle. 2. Utilizing the stored 
energy to increase the rate of output of the station at the time of heavy 
load, which would otherwise necessitate greater dynamo capacity. 

The Working Current, or Energy Efficiency, of a storage- 
cell is the ratio between the value of the current or energy expended in the 
charging operation, and that obtained when the cell is discharged at any 
specified rate. 

In a lead storage cell, if the surface and quantity of active material be 
accurately proportioned, and if the discharge be commenced immediately 
after the termination of the charge, then a current efficiency of as much as 
98$ may be obtained, provided the rate of discharge is low and well regu- 
lated. In practice it is found that low rates of discharge are not economical, 
and as the current efficiency always decreases as the discharge rate in 
creases, it is found that the normal current efficiency seldom exceeds 90$, 
and averages about 85$. 

As the normal discharging electro-motive force of a lead secondary cell 
never exceeds 2 volts, and as an electro-motive force of from 2.4 to 2.5 volts 
is required at its poles to overcome both its opposing electro-motive force 
and its internal resistance, there is an initial loss of 20$ between the energy 
required to charge it and that given out during its discharge. 

As the normal discharging potential is continually being reduced as the 
rate of discharge increases, it follows that an energy efficiency of 80$ can 



ELECTROLYSIS. 



1057 



never be realized. As a matter of fact, a maximum of 75% and a mean of 
60# is the usual energy efficiency of lead-sulphuric-acid storage-cells. 

ELECTRO-CHEMICAL EQUIVALENTS. 



Electro-positive. 

Hydrogen 

Potassium... 

Sodium 

Aluminum 

Magnesium 

Gold 

Silver 

Copper (cupric) 

(cuprous) . 

Mercury (mercuric).. . 
" (mercurous). 

Tin (stannic) 

" (stannous) 

Iron (ferric) 

" (ferrous) 

Nickel 

Zinc 

Lead 

Electro -negative. 

Oxygen 

Chlorine 

Iodine 

Bromine 

Nitrogen 



1.00 
39.04 
22.99 
27.3 
23.94 
196.2 
107.66 
63.00 
63.00 



117.8 
117.8 
55.9 
55.9 
58.6 
64.9 



15.96 
35.37 
126.53 
79.75 
14.01 



1.00 
39.04 
22.99 
9.1 
11.97 
65.4 
107.66 
31.5 
63.00 



58.9 
18.64$ 



32.45 
103.2 



126.53 
79.75 
4.67 



3 a £ 



.010384 

.40539 

.23873 

.09449 

.12430 

.67911 

1.11800 

.32709 

.65419 

1.03740 

2.07470 

.30581 

.61162 

.19356 



.36728 
1.31390 



96293.00 
2467.50 
4188.90 
1058.30 

804.03 
1473.50 

894.41 
3058.60 
1525.30 

963.99 

481.99 
3270.00 
1635.00 
5166.4 
3445.50 
3286.80 
2967.10 



|| 



0.03738 
1.45950 
0.85942 
3.40180 
4.47470 
2.44480 
4.02500 
1.17700 
2.35500 
3.73450 
7 J 6900 
1.10090 
2.20180 
0.69681 
1.04480 
1.09530 
1.21330 
3.85780 



* Valency is the atom-fixing or atom -replacing power of an element com- 
pared with hydrogen, whose valency is unity. 

t Atomic weight is the weight of one atom of each element compared with 
hydrogen, whose atomic weight is unity. 

% Becquerel's extension of Faraday's law showed that the electro-chemical 
equivalent of an element is proportional to its chemical equivalent. The 
latter is equal to its combining weight, and not to atomic weight -h valency, 
as defined by Thompson, Hospitalier, and others who have copied their 
tables. For "example, the ferric salt is an exception to Thompson's rule, as 
are sesqui-salts in general. 

ELECTROLYSIS. 

The separation of a chemical compound into its constituents by means of 
an electric current. Faraday gave the nomenclature relating to electroly- 
sis. He called the compound to be decomposed the Electrolyte, and the pro- 
cess Electrolysis. The plates or poles of the battery he called Electrodes. 
The plate where the greatest pressure exists he called the Anode, and the 
other pole the Cathode. The products of decomposition he called Ions. 

Lord Rayleigh found that a current of one ampere will deposit 0.017253 
grain, or 0.001118 gramme, of silver per second on one of the plates of a sil- 
ver voltameter, the liquid employed being a solution of silver nitrate con- 
taining from 15$ to 20% of the salt. 

The weight of hydrogen similarly set free by a current of one ampere is 
,00001038 gramme per second. 



1058 ELECTRICAL ENGINEERING. 

Knowing the amount of hydrogen thus set free, and the chemical equiva- 
lents of the constituents of other substances, we can calculate what weight 
of their elements will be set free or deposited in a given time by a given 
current. 

Thus the current that liberates 1 gramme of hydrogen will liberate 8 
grammes of oxygen, or 107.7 grammes of silver, the numbers 8 and 107.7 
being the chemical equivalents for oxygen and silver respectively. 

To find the weight of metal deposited by a given current in a given time, 
And the weight of hydrogen liberated by the given current in the given 
time, and multiply by the chemical equivalent of the metal. 

Thus: Weight of silver deposited in 10 seconds by a current of 10 amperes 
= weight of hydrogen liberated per second X number seconds X current 
strength X 107.7 = .00001038 X 10 x 10 X 107.7 = .11178 gramme. 

Weight of copper deposited in 1 hour by a current of 10 amperes = 

.00001038 X 3600 X 10 X 31.5 = 11.77 grammes. 

Since 1 ampere per second liberates .00001038 gramme of hydrogen, 
strength of current in amperes 

_ weight in grammes of H. liberated per seco nd 
~ .00001038 

weight of element liberated per second 



.00001038 x chemical equivalent of element" 

The table on page 1057 (from "Practical Electrical Engineering' 1 ) is cal- 
culated upon Lord Rayleigh's determination of the electro-chemical equiva- 
lents and Roscoe's atomic weights. 

EL.ECTRO-M 1GNETS. 
Units of Electro-magnetic Measurements. 

C.G.S. unit of force = 1 dyne = 1.01936 milligrammes in localities in which 
the acceleration due to gravity is 981 centimetres, or 32.185 feet, per second. 

CCS. unit of energy = 1 erg = energy required to overcome the resist- 
ance of 1 dyne at a speed of 1 centimetre per second. 1 watt = 10 7 ergs. 

Unit magnetism = that amount of magnetic matter which, if concentrated 
in a point, will repel an equal amount of magnetic matter concentrated in 
another point one centimetre distant with the force of one dyne. 

Unit strength of field = that flow of magnetic lines which will exert unit 
mechanical force upon unit pole, or a density of 1 line per square centi- 
metre. 

The following definitions of practical units of the magnetic circuit are 
given in Houston and Kennelly's "Electrical Engineering Leaflets." 

Gilbert, the unit of magneto-motive force; such a M.M.F. as would be 

produced by — or 0.7958 ampere-turn. 

If an air-core solenoid or hollow anchor-ring were wound with 100 turns 
of insulated wire carrying a current of 5 amperes, the M.M.F. exerted would 
be 500 ampere-turns = 628.5 gilberts. 

Weber, the unit of magnetic flux; the flux due to unit M.M.F. when the 
reluctance is one oersted. 

Gauss, the unit of magnetic flux-density, or one weber per normal square 
centimetre. 

The flux-density of the earth's magnetic field in the neighborhood of 
New York is about 0.6 gauss, directed downwards at an inclination of about 
72°. 

Oersted, the unit of magnetic reluctance; the reluctance of a cubic centi- 
metre of an air-pump vacuum. 

Reluctance is that quantity in a magnetic circuit which limits the flux 
under a given M.M.F. It corresponds to the resistance in the electric cir- 
cuit. 

The reluctivity of any medium is its specific reluctance, and in the C.G.S. 
system is the reluctance offered by a cubic centimetre of the body between 
opposed parallel faces. The reluctivity of nearly all substances, other than 
the magnetic metals, is sensibly that of vacuum, is equal to unity, and is 
independent of the flux density. 

Permeability is the reciprocal of magnetic reluctivity. 



BLECTRO-MAGKETS. 1059 

The fundamental equation of the magnetic circuit is 

TTT , gilberts 

Webers = -p ; 

oersteds 

, magnetic flux = magneto-motive force -=- magnetic reluctance. 
From this equation we have 

Gilberts = webers X oersteds; oersteds = gilberts -5- webers. 
There are therefore two ways of increasing the magnetic flux: 1. by in- 
casing the M.M.F. ; 2. by decreasing the reluctance. 
Lines and Loops of Force.— In discussing magnetic and electrical 

• enomena it is conventionally assumed that the attractions and repulsions 

I shown by the action of a magnet or of a conductor upon iron filings are 
e to " lines of force " surrounding the magnet or conductor. The '* num- 
r of lines " indicates the magnitude of the forces acting. As the iron 

. ngs arrange themselves in concentric circles, we may assume that the 
•ces may be represented by close curves or " loops of force." The follow- 
* assumptions are made concerning the loops of force in a conductive 

[rcuit: 
I. That the lines or loops of force in the conductor are parallel to the axis 
the conductor. 

I. That the loops of force external to the conductor are proportional in 
unber to the current in the conductor, that is, a definite current generates 
lefinite number of loops of force. These may be stated as the strength of 
Id in proportion to the current. 
3. That the radii of the loops of force are at right angles to the axis of 

Be conductor. 
The magnetic force proceeding from a point is equal at all points on the 
rface of an imaginary sphere described by a given radius about that 
int. A sphere of radius 1 cm. has a surface of 4n square centimetres. If 
= total field strength, expressed as the number of lines of force emanat- 
U from a pole containing M units of magnetic matter, 

F=4nM; M=--F+4n. 

Magnetic moment of a magnet = product of strength of pole M and its 

lgth, or distance between its poles L. Magnetic moment = — — . 

If B = number of lines flowing through each square centimetre of cross- 
ction of a bar-magnet, or the " specific induction," and A = cross-section, 

LAB 

Magnetic moment = — — •. 

[f the bar-magnet be suspended in a magnetic field whose induction is H, 
d so placed that the lines of the field are all horizontal and at right angles 
the axis of the bar, the north pole will be pulled forward, that is, in the 
rection in which the lines flow, and the south pole will be pulled in the 
posite direction, the two forces producing a torsional moment or torque, 

Torque = MLH = LABH -=- 4n, in dyne-centimetres. 

Magnetic attraction or repulsion emanating from a point varies inversely 
the square of the distance from that point. The law of inverse squares, 
iwever, is not true when the magnetism proceeds from a surface of ap- 
eciable extent, and the distances are small, as in dynamo-electric 
achines. (For an analogy see " Radiation of Heat," page 467.) 
Strength of an Electro-magnet.— In an electric magnet made by 
iling a current-carrying conductor around a core of soft iron, the space 
: which the loops of force have influence is called the magnetic field, and 
is convenient to assume that the strength of the field is proportional to 

i,e number of loops of magnetic force surrounding the magnet. Under 

is assumption, if we take a given current passing through a given number 

conductor-turns, the number of magnetic loops will depend upon the 

sistance of the magnetic circuit, just as the current with a given press- 

■ -e in the conductive circuit depends upon the resistance of the circuit. 
The following laws express the most important principles concerning 
ectro-magnets : 

(1) The magnetic intensity (strength) of an electro-magnet is nearly pro- 
>rtional to the strength of the magnetizing current, provided the core is 
>t saturated. 



1060 ELECTRICAL EHGIKEERING. 

(2) The magnetic strength is proportional to the number of turns of wire * 
in the magnetizing coil ; that is, to the number of ampere turns. 

(3) The magnetic strength is independent of the thickness or material of 
the conducting wires. 

These laws may be embraced in the more general statement that the 
strength of an electro-magnet, the size of the magnet being the same, is 
proportional to the number of its ampere turns. 

Force in the Gap between Two Poles of a Magnet.— If 
P = force exerted by one of the poles upon a unit pole in the gap, and m = 
density of lines in the field (that is, that there are m absolute or C.G.S. units 
on each square centimetre of the polar surface of the magnet), the polar 
surface being large relative to the breadth of the gap, P — 2nm. The total 
force exerted upon the unit pole by both nortli and south poles of the 
magnet is 2P — 4nm, in dynes = B, or the induction in lines of force per 
square centimetre. If 8 = number of square centimetres in each polar 
surface, SB = total flow of force, or field strength = F; Sm = total pole 
strength = M, spread over each of the polar surfaces. We then have F = 
4nM, as before; that is, the total field is 4n times the total pole strength. 

Total attractive force between the two opposing poles of a magnet, when 

the distance apart is small, = — - — , in dynes. 

This formula may be used to determine the lifting-power of an electro- 
magnet, thus: 

A bent magnet provided with a keeper is 3 cm. square on each pole, and 
the induction B — 20,000 lines per square centimetre. The attractive force 

9 X 20000 2 
of each limb on the keeper in dynes = , or in kilogrammes for 

9 X 400 X 10 6 
b0th limbS ' 25.12 x 981000 X 2 = 292 kilogrammes. 

The Magnetic Circuit.— In the conductive circuit we have C = — ; 

_ electro-motive force volts 

Current = r— = -= . 

resistance ohms 

In the magnetic circuit we have 
Number of lines, or loops, of force, or magnetism 

Current X conductor turns _ Ampere turns 

"~ Resistance of magnetic circuit _ Resistance of magnetic circuit' 

.l ,. i gilberts 

Or, in the new notation, webers = - — . 

Let N = No. of lines of force, Em - total magnetic resistance, At = 
At 
ampere turns, then N = — — . 

4 
The magnetic pressure due to the ampere turns = — irTO = 1.2577c, 

„ Air TO 1.257 TO 
where T= turns and C = amperes, whence N= Rm - - Rm — . 

If Rm = total magnetic resistance, and Ra, Ra. Rf the magnetic resist- 
ances of the air-spaces, the armature, and the field-magnets, respectively, 

Rrn = Ra + R A + R F ; and N = ^ ^ ° + R ^ 

Determining the Polarity of Electro-magnets.— If a wire 
is wound around a magnet in a right-handed helix, the end at which the 
current flows into the helix is the south pole. If a wire is wound around an 
ordinary wood screw, and the current flows around the helix in the direc- 
tion from the head of the screw to the point, the head of the screw is the 
south pole. If a magnet is held so that the south pole is opposite the eye of 
the observer, the wire being wound as a right-handed helix around it, the 
cuirent flows in a right-handed direction, with the hands of a clock. 



DYNAMO-ELECTRIC MACHINES. 1061 

DYNAMO-ELECTRIC MACHINES. 

There are four classes of dynamo-electric machines, viz.: 

1. The dynamo, in which mechanical energy of rotation is converted into 
the energy of a direct current. 

2. The alternator, in which mechanical energy of rotation is converted into 
the energy of an alternating current. 

3. The motor, in which the energy of a direct current is converted into 
mechanical energy of rotation. 

4. The alternate-current motor, in which the energy of one or more alter- 
nating currents is converted into mechanical energy of rotation. 

For a steady direct current the product of the potential difference and the 
current strength is a true measure of the energy given off. With alternat- 
ing currents the product of voltage into current strength is greater than the 
true energy, since the conductor has the property of reacting upon itself, 
called "self-induction." 

Kinds of Dyiiaino-electric Machines as regards Man- 
ner of Winding. (Houston's Electrical Dictionary.) 

1. Dynamo electric Machine. — A machine for the conversion of mechan- 
ical energy into electrical energy by means of magneto-electric induction. 

2. Compound-wound Dynamo.— The field-magnets are excited by more 
than one circuit of coils or by more than a single electric source. 

3. Closed-coil Dynamo.— The armature-coils are grouped in sections com- 
municating with successive bars of a collector, so as to be connected con- 
tinuously together in a closed circuit. 

4. Open-coil Dynamo. — The armature-coils, though connected to the suc- 
cessive bars of the commutator, are not connected continuously in a closed 
circuit. 

5. Separate-coil Dynamo.— The field-magnets are excited by means of 
coils on the armature separate and distinct from those which furnish cur- 
rent to the external circuit. 

6. Separately-excited Dynamo. — The field-magnet coils have no connec- 
tion with the armature-coils, but receive their current from a separate 
machine or source. 

7. Series-ivound Dynamo.— The field-current and the external circuit are 
connected in series with the armature circuit, so that the entire armature 
current must pass through the field-coils. 

Since in a series-wound dynamo the armature-coils, the field, and the ex- 
ternal-series circuit are in series, any increase in the resistance of the ex- 
ternal circuit will decrease the electro-motive force from the decrease in 
the magnetizing currents. A decrease in the resistance of the external cir- 
cuit will, in a like manner, increase the electro-motive force from the in- 
crease in the magnetizing current. The use of a regulator avoids these 
changes in the- electro-motive force. 

8. Series and Separately-excited Compotind-wound Dynamo. — There are 
two separate circuits in the field-magnet cores, one of which is connected 
in series with the field-magnets and the external circuit, and the other with 
some source by which it is separately excited. 

9. Shunt-wound Dynamo.— The field-magnet coils are placed in a shunt 
to the armature circuit, so that only a portion of the circuit generated 
passes through the field magnet coils, but all the difference of potential of 
the armature acts at the terminals of the field-circuit. 

In a shunt-dynamo machine an increase in the resistance of the external 
circuit increases the electro-motive force, and a decrease in the resistance 
of the external circuit decreases the electro-motive force. This is just the 
reverse of the series-wound dynamo. 

In a shunt-wound dynamo a continuous balancing of the current occurs. 
The current dividing at the brushes between the field and the external cir- 
cuit in the inverse proportion to the resistance of these circuits, if the resist- 
ance of the external circuit becomes greater, a proportionately greater 
current passes through the field-magnets, and so causes the electro-motive 
force to become greater. If, on the contrary, the resistance of the external 
circuit decreases, less current passes through the field, and the electro- 
motive force is proportionately decreased. 

10. Series- and Shunt-wound Compound-wound Dynamo.— The field-mag- 
nets are wound with two separate coils, one of which is in series with the 
armature and the external circuit, and the other in shunt with the arma- 
ture. This is usually called a compound-wound machine. 

11. Sliunt and Separately-excited Compound-wound Dynamo.— The field 



1062 ELECTKICAL ENGINEERING. 

is excited both by means of a, shunt to the armature circuit and by a cur- 
rent produced by a separate source. 
Current Generated toy a Dynamo-electric Machine.— Unit 

current in the C.G.S system is that current which, flowing in a thin wire 
forming a circle of one centimetre radius, acts upon a unit pole placed in 
the centre with a force of 2n dynes. One tenth of this unit is the unit of 
current used in practice, called the ampere. 

A. wire through which a current passes has, when placed in a magnetic 
field, a tendency to move perpendicular to itself and at right angles to the 
lines of the field. The force producing this tendency is P = IcB dynes, in 
which I = length of the wire, c = the current in C.G.S. units, and B the in- 
duction in the field in lines per square centimetre. 

If the current C is taken in amperes, P = WB10 • 

If P k is taken in kilogrammes, 

P k = ~^ = 10.1937ZC£10-8 kilogrammes. 

Example. — The mean strength of field, B, of a dynamo is 5000 C.G.S. lines; 
a current of 100 amperes flows through a wire; the force acts upon 10 centi- 
metres of the wire = 10.1937 X 10 X 100 X 5000 X 10" 8 = .5097 kilogrammes. 

In the "English" or Kapp's system of measurement a total flow of 6000 
C.G.S. lines is taken to equal one English line. Calling Be the induction in 
English, or Kapp's, lines per square inch, and B the induction in C.G.S. lines 
per square centimetre, Be = B -4- 930.04; and taking I" in inches and Pp in 

pounds, P p = 531 Gl"B E 10" 6 pounds. 

Torque of an Armature.— Pp in the last formula, = the force tending 
to move one wire of length I", which carries a current of C amperes through 
the field whose induction is Be English lines per square inch. The current 
through a drum-armature splits at the commutator into two branches, 
each half going through half of the wires or bars. The force exerted 
upon one of the wires under the influence of a pole-piece — y^Pp. If t = the 
number of wires under the pole-pieces, then the total force = \^Ppt. If r = 
radius of the armature to the centre of the conductors, expressed in feet, 
then the torque = l^Pptr, = % X 531 X Cl"B E X 10~ 6 X tr foot-pounds of 
moment, or pounds acting at a radius of 1 foot. 

Example.— Let the length I of an armature = 20 in., the radius = 6 in. or 
.5 ft., number of conductors = 120, of which t = 80 are under the influence 
of the two pole- pieces at one time, the average induction or magnetic flux 
through the armature-field Be = 5 English lines per square inch, and the 
current passing through the armature = 400 amperes; then 

Torque = Y 2 X 531 X 400 X 20 X 5 X 80 X .5 X 10~ 6 = 424.8. 

The work done in one revolution = torque X circumference of a circle of 
1 foot radius = 424.8 X 6.28 = 2670 foot-pounds. 
Let the revolutions per minute = 500, then the horse-power 



33000 



= 40.5 H.P. 



Electro-motive Force of the Armature Circuit.— From the 

horse-power, calculated as above, together with the amperes, we can obtain 
the E.M.F., for CE = H.P. X 746, whence E.M.F. or E = H.P. X 746 -*- C. 

40 5 V 746 

If H.P, as above, = 40.5, and C = 400, E = ^ = 75.5 volts. 

The E.M.F. may also be calculated more directly by the following formulae 
given by Gisbert Kapp: 

C — Total current through armature; c, current through single armature 

conductor; 
e a = E.M.F. in armature in volts; 

t = Number of active conductors counted all around armature; 

p = Number of pairs of poles (p = 1 in a two-pole machine); 

n — Speed in revolutions per minute; 
F = Total induction in C.G.S. lines; 
Z ~ Total induction in English lines. 



DYNAMO-ELECTiUC MACHINES. 



1063 



Electro- motive ' 
force 



1 



e a - ZrnlO' 



j- for two-pole machines. 



for multipolar machines with 
series-wound armature. 



e a = pZmlO~ 

| Kilogramme-metres = 1.615FTC10" 10 \ for two-pole ma- 
i Foot-pounds =:7.05ZtC10' 6 ' chines. 



Torque -j Foot-pounds 

Kilogramme-metres = 3.2SFrcp 10 



) for multipolar ma- 
L Foot-pounds = U.lOZrcplO' 6 ) chines. 



Example.— t = 120, n — 500, length of armature I : 



in., diameter 



d = 12 in., cross-section = 20 x 12 = 240 sq. in., induction per sq. in. B E = 
sq. in., total induction Z = 240 X 5 = 1200; then 



5 lines per sq. 

E= ZmlO- 



: 1200 X 120 X 500 X 10- « =.- 72 volts. 



A formula for horse-power given by Kapp is 

H.P. = 1/746 ZNtnlO- *Ca 

- 1/746 2abmNtnlO- 6 Ca. 

Ca = current in amperes, n = revs, per min., 2ab = sectional area of arm- 
ature-core, m — average density of lines per sq. in. of armature-core, Nt — 
total number of external wires counted all around the circumference, t = 
number of wires correspondirg to one plate in the commutator, N — num- 
ber of plates, Z = 2abm = total number of English lines of force. 

Kapp says that experience bas shown that the density of lines m in the 
core cannot exceed a certain limit, which is reached when the core is satu- 
rated with magnetism. This value is reached when m = 30. A fair average 
value in modern dynamos and motors is m = 20, and the area ab must be 
taken as that actually filled by iron, and not the gross area of the core. 2u 
English lines per sq. in. = 18,600 C.G.S. lines per square centimetre. Sil- 
vanus P. Thompson says it is not advisable in continuous-current machines 
to push the magnetization further than B = 17,000 C.G.S. lines per square 
centimetre. 

Thompson gives as a rough average for the magnetic field in the gap-space 
of a dynamo or motor 6300 lines per sq. cm., or 40,000 lines per sq. in., and 
the drag per inch of conductor .00351 lb. for each ampere of current carried. 
a * H - p - X 33,000 . 

Pounds average drag per conductor = ^ . \ in which Cis the 

it, per min. x o 
number of conductors around the armature. 

Strength, of tne Magnetic Field.— Kapp gives for the total num- 
ber of lines of force (Kapp's lines = C.G.S. lines -4- 6000) in the magnetic cir- 
cuit. Z — — — t—= — , in which Z= number of magnetic lines, X = the 

Ha -r BA -\- RF 
exciting pressure due to the ampere turns = AnTC, Ra, Ra, and Rf. = re- 
spectively the resistances of the air-spaces, the armature, and the field-mag- 
nets. . 

Kapp gives the following empirical values of Ra, Ra, and Rf, for dynamos 
and motors made of well-annealed wrought iron, with a permeability of /* = 
940: 

J^ 
" 06' 



Rf 



-A» 



in which 5 = distance across the span between armature -core and polar 
surface, b = breadth of armature measured parallel to axis, A = length of 
arc embraced by polar surface, so that \b = the polar area out of which 
magnetic lines issue, a = radial depth of armature-core, so that ab = sec- 
tion of armature-core (space actually occupied by iron only being reckoned, 
AB = area of field-magnet core, I = length of magnetic circuit within ar- 
mature, L = length of magnetic circuit in field magnet; all dimensions in 
inches or square inches. 



1064 ELECTRICAL ENGINEERING. 



For cast-iron magnets, Z — ^ — — -, — Qr • 

For double horse-shoe magnets of wrought iron, 

Kb ab AB 

. ' . . Z 0.8X 
and of cast iron, - = — ^ 

These formulae apply only to cases in which the intensity of magnetization 
is not too great— say up to 10 Kapp's lines per square inch. 

Silvanus P. Thompson gives the following method of calculating the 
strength of the field, or the magnetic flux, Mf, or the whole number of 
magnetic lines flowing in the circuit in C.G.S. lines: 

The magnetic resistance of any magnetic conductor is proportional direct- 
ly to its length and inversely to its cross-section and its permeability. 

Magnetic resistance = — , in which L = length of the magnetic circuit 

bp 

passing through any piece of iron, S = section of the magnetic circuit 
passing through any piece of iron. p = permeability of that piece of iron. 

In a dynamo-machine in which the resistances are three, viz.: 1. The field - 
magnet cores; 2. The armature-core; 3. The gaps or air-spaces between 
them,— 

let Lm, Sm, pm refer to the field -magnet part of the circuit; 
Las, Sas, pas refer to the air-space part of the circuit; 
La, Sa, pa refer to the armature part of the circuit; 

the lengths across each of the air-spaces being Las, and the exposed area of 
polar surface at either pole being Sas. 

Total magnetic resistance = ^ m • -f- - Q \- . a . 

Smpm Saspas Safia 

Magnetic flux, or total number of magnetic lines, = 

1. 257 TwO 



Mf = 



Lm Las , La, '■ 



Smpm Saspas Sapa 

Tw — turns of wires, or number of turns in the spiral; 
C = current in amperes passing through spiral. 

Application to Designing of Dynamos. (S. P. Thompson.)— 
Suppose in designing a dynamo it lias been decided what will be a conven- 
ient speed, how many conductors shall be wound upon the armature, and 
tvhat quantity of magnetic lines there must be in the field, it then becomes 
necessary to calculate the sizes of the iron parts and the quantity of excita- 
tion to be provided for by the field-magnet coils. It being known what Mf 
is to be, the problem is to design the machine so as to get the required 
value. Experience shows that in every type of dynamo there is magnetic 
leakage; also, that it is not wise to push the saturation of the armature-core 
to more than 16,000 lines to the square centimetre at the most highly satu- 
rated part, and that the induction in the field-magnet ought to be not 
greater than this, even allowing for leakage. Leakage may amount to 34 
of the whole: hence, if the magnet-cores are made of same quality of iron 
as the armature-cores, their cross-section ought to be at least 5/4 as great 
as that of the armature-core at its narrowest point. If the field-magnets 
are of cast iron, the section ought to be at least twice as great. 

Now, Ba (the induction in the armature-core) = Ma -s- Sa (or magnetic flux 
through armature h- cross-sectional area of the armature ; hence, if this 
is fixed at 16,000 lines per centimetre of cross-section, we at once get Sa — 
Ma -5- Ba. This fixes the cross-section of the armature-core. (Example: If 
Ma — 4,000,000 of lines, then there must be a cross-section equal to 250 

+ . . „ 4,000,000 OKA . 
square centimetres for =250.) 



DYNAMO-ELECTRIC MACHINES. 1065 

Magnetic Length of Armature Circuit.— The size of wires oh the arma- 
ture is fixed by the number of amperes which it must carry without risk. 
Remembering that only half the current (in ring or drum armatures) passes 
through any one coil, and as the number is supposed to have been fixed be- 
forehand, this practically settles the quantity of copper that must be put on 
the armature, and experience dictates that the core should be made so large 
that the thickness of the external winding does not exceed 1/6 of the radial 
depth of the iron core. This settles the size of the armature -core, from 
which an estimate of La, the average length of path of the magnetic lines 
in the core, can be made. 

Length and Section or Surface Area of Air -space.— Experience further 
dictates the requisite clearance, and the advantage of making the pole- 
pieces subtend an arc (in two-pole machines) of at least 135° each, so as to 
gain a large polar area. This settles Las and Sas. 

Length of Field -magnet Iron Cores, etc.— As shown above, the minimum 
value of Sm is settled bj T leakage and materials; Lm therefore remains to 
be decided. It is clear that the magnet-cores must be long enough to allow 
of the requisite magnetizing coils, but should not be longer. As a rule, 
they are made so stout, especially in the yoke part, that they do not add 
much to the magnetic resistance of the circuit, then a little extra length as 
sumed in the calculation does not matter much. It now only remains to 
calculate the number of ampere-turns of excitation for which it will be 
needful to provide. 

It will now be more convenient to rewrite the formula of the magnetic 
circuit as follows: 

i x -kra _j_ o -Las . La ) 

.s. m ,, { Smum ' Sas.pas Sa.n-a \ 
A X Tmw = Ma- -— ; ; 

where A — amperes of current passing through the field-magnet coils; 
Tmw - total turns of the magnet wire; 
\ = leakage coefficient (say 5/4). 
Or, 

4 X Tmtv = Ma ._, ■. 

Or, as before, 

,, i n »„ Ax Tmw 
Ma=1 - 2D7 KR m + R ai -rRa > 

where Rm, Ras, Ra stand for the magnetic resistance of magnets, air- 
space, and armature, respectively. 

But we cannot use this formula yet, because the values of ju. in it depend on 
the degree of saturation of the iron in the various parts. These have to be 
found from the Hopkinson tables, given below; and, indeed, it is preferable 
first to rearrange the formula once more, by dividing it into its separate 
members, ascertaining separately the ampere-turns requisite to force the 
required number of magnetic lines through the separate parts, and then 
add. them together. 

1. Ampere- turns required for magnet-cores = A ~ x ^^-*- 1.257. 

Sm n-m 

2. Ampere-turns required for air-spaces = — -^ x 2— s -=- 1.257. 

Sas pas 

3. Ampere-turns required for armature-core = ^ x — -*- 1.257. 

Sa na 

Now A.--— is the value of B in the magnet-cores, and reference to the table 
of permeability will show what the corresdonding value of /u-m must be. 
Similarly,— a will afford a clue to /u.a. When the total number of ampere- 
turns to be allowed for is thus ascertained, the size and length of wire will 
be determined by the permissible rise of temperature, and the mode of 
exciting the field -magnets, whether in series, or as a shunt machine, or 
with a compound-winding. 



1066 



ELECTRICAL EJSmiNEERLNGL 



Permeability.— Materials differ in regard to the resistance they otter 
to the passage of lines of force; thus iron is more permeable than air. The 
permeability of a substance is expressed by a coefficient fx,, which denotes 
its relation to the permeability of air, which is taken as 1. If H ~ number 
of magnetic lines per square centimetre which will pass through an air- 
space between the poles of a magnet, and B the number of lines which will 
pass through a certain piece of iron in that space, then ju. = B -*- H. The 
permeability varies with the quality of the iron, and the degree of satura- 
tion, reaching a practical limit for soft wrought iron when B =s about 18,000 
and for cast iron when B = about 10,000 C.G.S. lines per square centimetre. 

The following values are given by Thompson as calculated from Hopkin- 
son's experiments: 



Annealed Wrought Iron. 


Gray Cast Iron. 


B 


H 


M 


B 


H 


^ 


5,000 


2 


2,500 


4,000 


5 


800 


9,000 


4 


2,250 


5,000 


10 


500 


10,000 


5 


2,000 


6,000 


21.5 


279 


11,000 


6.5 


1,692 


7,000 


42 


133 


12,000 


8.5 


1,412 


8,000 


80 


100 


13,000 


12 


1,083 


9,000 


127 


71 


14,000 


17 


823 


10,000 


188 


53 


15,000 


28.5 


526 


11,000 


292 


37 


16,000 


52 


308 








17,000 


105 


161 








18,000 


200 


90 








19,000 


350 


54 









Permissible Amperage and Permissible Depth of "Wind- 
ing for Magnets with Cotton-covered Wire. (Walter S. L)ix, 
El. Engineer, Dec. 21, 1892.)— The tables on pp. 1068, 1069, abridged from 
those of Mr. Dix, are calculated from the formula 



"\J\ 



12 XW 



<«W XTXL 
M 
where C = current; 

W = emissivity in watts per square inch; 
co m f = ohms per mil-foot ; 
M = circular mils ; 
T = turns per linear inch ; 
L = number of layers in depth. 
The emissivity is taken at .4 watt per sq. in. for stationary magnets for a 
rise of temperature of 35° C. (63° F.)\ For armatures, according to Esson's 
experiments, it is approximately correct to say that .9 watt per sq. in. will 
be dissipated for a rise of 35° C. 

The insulation allowed is .007 inch on No. to No. 11 B. & S.; .005 inch 
on No. 12 to No. 24 ; and .0045 inch on No. 25 to No. 31 single ; twice these 
values for insulation of double-covered wires. Fifteen per cent is allowed 
for imbedding of the wires. 

The standard of resistance employed is 9.612 ohms per mil-foot at 0°. The 
running temperature of tables is taken at 25° -f 35° = 60° C. The column 
giving the depth for one layer is the diameter over insulation. 

Formulae of Efficiency of Dynamos. 

(S. P. Thompson in " Munro and Jamieson's Pocket-Book.") 
Total Electrical Energy (per second) of any dynamo (expressed in watts^ 
is the product of the whole E.M.F. generated by armature-coils into the 
whole current which passes through the armature. 

Useful Electrical Energy (per second), or useful output of the machine, is 
the product of the useful part of the E.M.F. (i.e., that part which is avail- 
able at the terminals of the machine) into the useful part of the current 
(i.e., that part of the current which flows from the terminals into the exter- 
nal circuit). 



DYNAMO-ELECTRIC MACHINES. 



1067 



Economic Coefficient or "electrical efficiency" of a dynamo is the ratio 
of the useful energy to the total energy. 

Commercial Efficiency of a dynamo is the ratio of the useful energy or 
output to the power actually absorbed by the machine in being driven. 

Let Ea = total E.M.F. generated in armature; 
Ee — useful E.M.F, available at terminals; 
Ca = total current generated in armature; 
C s = current sent round shuut-coils; 
Ce = useful current supplied to external circuit; 
Ra — resistance of armature-coils; 

Rm = resistance of magnet-coils in main circuit (series); 
Rs — resistance of magnet-coils in shunt; 
' R e = resistance of external circuit (lamps, mains, etc.); 
Wa = Watts lost in armature; 
Wm— Watts lost in magnet-coils; 
Vl = lost volts; 

Te = total electrical energy (per second); 
Ue = useful electrical output; 

c — economic coefficient; 

p — commercial efficiency (percentage). 

When only one circuit (series machine) Ce = Ca. 

In shunt machines Cs should not be more than h% of Ce. Also, 

Ca = Ce + Cs. 
In all dynamos, Ra ought to be less than 1/40 as great as the working 

value of Re- 
in series (and compound) machines, Rm should be not greater than R a , 

and preferably only % as great. 
In shunt (and compound) machines, Rs should be not less than 300 times 

as great as Ra and preferably 1000 to 1200 times as great. 





Series Machine. 


Shunt Machine. 


Compound Machine 
(Short Shunt). 


w a 


C\R a 


Cf t R a 


C'a R a 


w m 


ClR m 


C%R s =El+R s 


ClR m +ClR 8 


Vl 


C a R a 


Ca R a 


C a R a -\-C e R m 


T e 


E a C a = 
Cl{R a + R m + R e ) 


E aC a = 

1 R,R P \ 
V R s +R e / 


1 RJR m +R e ) \ 
E a C a = CI R a + S m e 

\ R s +R m +Rj 


v e 


E e C a =ClR e 


E e C e =ClR e 


E e C e =C%R e 




E R e 


C%R e 


C\R e 




E a R a +R m +R e 


C\R e +ClR a +C\R 8 


C 2 e R e +ClR a +ClR s +ClR m 


p 


100xE e C e -+- 

(H.P.X746) 


mxE e c e -+ 

(H.P.X746) 


100x# e <VHH.P.x746) 




N.B. Horse-power 
is converted into 
watts (so as to com- 
pare with electric 
output of the ma- 
chine) by multiply- 
ing by 746. 


* This will be a 
maximum when Re 
is a mean propor- 
tional between Rs 
and R a . 


In well-constructed com- 
pound machines the differ- 
ence between " short shunt" 
and "long shunt" is very 
slight, as Rm is so small. 



1068 ELECTEICAL ENGINEERING. 

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13094 
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DYNAMO-ELECTRIC MACHINES. 



1069 



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1070 ELECTRICAL ENGINEERING. 

Alternating Currents, Multiphase Currents, Trans- 
formers, etc.— The proper discussion of these subjects would take more 
space than can be afforded in this work. Consult S. P. Thompson's " Dy- 
namo-Electric Machinery, 1 ' Bedell and Crehore on " Alternating Currents, 1 ' 
Fleming on " Alternating Currents," and Kapp on "Dynamos, Alternators 
and Transformers." 

The "Electric Motor. — The electric motor is the same machine as 
the dynamo, but with the nature of its operation reversed. In the dynamo 
mechanical energy, such as from a belt, is converted into electric current; 
in the motor the current entering the machine is converted into mechanical 
energy, which may be taken off by a belt. The difference in the action of 
the machine as a dynamo and as a motor is thus explained by Prof. F. B. 
Crocker, (Cassier's Mag., March. 1895): 

In the case of the dynamo there exists only one E.M.F., whereas in the 
motor there must always be two. 

One kilowatt dynamo, C — E -h R; 10 amperes = 100 volts -4- 10 ohms. 

_ . .. .. , „ E-e .. 100 volts - 90 volts. 

One kilowatt motor, C = — = — ; 10 amperes = — = * 

i?i 1 ohm 

C is the current; E, , the direct E.M.F.; e, the counter E.M.F.; R. the total 
resistance of the circuit; R x , the resistance of the armature. The current 
and direct E.M.F. are tiie same in the two cases, but the resistance is only 
one tenth as much in the case of the motor, the difference being replaced 
by the counter E.M.F. , which acts like resistance to reduce the current. In 
the case of the motor the counter E.M.F. represents the amount of the 
electrical energy converted into mechanical energy. The so-called electri- 
cal efficiency or conversion factor = counter E.M.F. -r- direct E.M.F. The 
actual or commercial efficiency is somewhat less than this, owing to fric- 
tion, Foucault currents, and hysteresis. 

For full discussions of the theory and practice of electric motors see S. 
P. Thompson's '"Dynamo-Electric Machinery," Kapp's "Electric Trans- 
mission of Energy," Martin and Wetzler's li The Electric Motor and its 
Applications," Cox's " Continuous Current Dynamos and Motors," and 
Crocker and Wheeler's "Practical Management of Dynamos and Motors." 



LIST OF AUTHORITIES QUOTED IN THIS BOOK. 



When a name is quoted but once or a few times only, the page or pages 
are given. The names of leading writers of text-books, who are quoted fre- 
quently, have the word "various' 1 '' affixed in place of the page-number,. 
The list is somewhat incomplete both as to names and page numbers. 



Abel, F. A., 642 

Abendroth & Root Mfg. Co., 197, 198 

American Screw Co., 209 

Achard, Arthur, 886, 919 

Addy, George, 957 

Addyston Pipe and Steel Co., 187, 188 

Alden, G. I., 979 

Alexander, J. S., 629 

Allen, Kenneth, 295 

Allen, Leicester, 582 

Andrews, Thomas, 384 

Ansonia Brass and Copper Co., 327 

Arnold, Horace L., 959 

Ashcroft Mfg. Co., 752, 7?5 

Atkinson, J. J., 532 

Ayrton and Perry, 1040 . 

Babcock, G. H., 524, 933 

Babcock & Wilcox Co., 538, 636 

Baermann, P. H., 188 

Bagshaw, Walter, 952 

Bailey, W. H., 943 

Baker, Sir Benjamin, 239, 247, 402 

Balch, S. W., 898 

Baldwin, Wm. J., 541 

Ball, Frank H., 751 

Barlow, W. H., 384 

Barlow, Prof., 288 

Barnaby. S. W., 1013 

Barnes, D. L., 631, 861, 863 

Barrus, Geo. H., 636 

Bauer, Chas. A., 207 

Bauschinger, Prof., 239 

Bazin, M., 563, 587 

Beardslee, L. A.. 238, 377 

Beaumont, W. W., 979 

Becuel, L. A., 644 

Begtrup, J., 348 

Bennett, P. D., 354 

Bernard, M. & E., 330 

Birkinbine, John, 605 

Bjorling, P., 676 

Blaine, R. G., 616, 1039 

Blauvelt, W. H., 639, 649 

Blechyuden, A., 1015 

Bodmer, G. R., 753 

Bolland, Simpson, 946 

Booth, Wm. EL, 926 

Box, Thomas, 475 

Brings, Robert, 194, 478, 539, 672 

British Board of Trade, 264, 266, 700 

Brown, A. G., 723, 724 

Brown, E. H., 388 

Brown & Sharpe Mfg. Co., 219, 890 

Browne, Ross E., 597 

Brush, Chas. B., 566 

Buckle, W., 511 



Buel, Richard H., 606, 834 
Buffalo Forge Co., 519, 529 
Builders' Iron Foundry, 374 
Burr. Wm. A., 565 
Burr, Wm. H., 247, 259, 290, 381 

Calvert, F. Crace, 386 

Calvert & Johnson, 469 

Campbell, H. H., 398. 459, 650 

Campredon, Louis, 403 

Carnegie Steel Co., 177, 272, 277, 391 

Carpenter, R. C, 454, 615, 718, etc. 

Chad wick Lead Works, 201, 615 

Chamberlain, P. M., 474 

Chance, H. M., 631 

Chandler, Chas. F., 532 

Chapman Valve Mfg. Co., 193 

Chauvenet, S. H., 370 

Chase, Chas. P., 312 

Chevandier, Eugene, 640 

Christie, James, 394 

Church, Irving P., 415 

Church, Wm. Lee. 784. 1050 

Clapp, Geo. H., 397, 403, 551 

Clark, Daniel Kinnear, various 

Clarke, Edwin, 740 

Claudel, 455 

Clay, F. W.. 291 

Clerk, Dugald, 847 

Cloud, John W., 351 

Codman, J. E., 193 

Coffey, B. H., 810 

Coffin, Freeman C, 292 

Coggswell, W. B., 554 

Cole, Romaine C„ 329 

Coleman, J. J., 470 

Cooper, John H., 876, 900 

Cooper, Theodore, 262, 263, 359 

Cotterill and Slade, 432, 974 

Cowles, Eugene H., 329, 331 

Cox, A. J., 290 

Cox, E. T., 629 

Cox, William, 575 

Coxe, Eckley B., 632 

Craddock, Thomas, 473 

Cramp, E. S.,405 

Crimp, Santo, 564 

Crocker, F. B., 1070 

Cummins, Wm. Russell, 772 

Daelen, R. M., 617 
Dagger, John H. J., 329 
Daniel, Wm., 492 
D'Arcy, 563 
Davenport, R. W., 620 
Day, R. E., 1030 
Dean, F. W., 605, 689 

1071 



1072 



LIST OF AUTHORITIES. 



Decoeur, P., 600 
DeMeritens, A., 386 
Denton, James E , 730, 761, 781, 
Dinsmore, R. E., 963 
Dix, Walter S., 208, 1066 
Dodge Manufacturing Co., 344 
Donald, J. T., 235 
Donkin, B., Jr., 491, 783 
Dudley, Chas. B., 327, 333 
Dudley, P. H., 401, 622 
Dudley, W. D., 167 
Dulong, M.,458, 476 
Dunbar, J. H., 796 
Durand, Prof., 56 
Dwelshauvers-Dery, 662 

Egleston, Thomas, 235, 641 
Emery, Chas. E., 603, 613, 820 
Engelhardt, F. E., 463 
Ellis and Howland, 577 
English, Thos.,753 
Ericsson, John, 286 
Eytelwein, 564 

Fairbairn, Sir Wm, 240, 264, 301 
Fairley, W., 531, 533 
Falkenau, A., 509 
Fanning, J. T., 564, 579 
Favre and Silbermann, 621 
Felton, C. E., 646 
Fernow, B. E., 640 
Field, C. J., 30, 937 
Fitts, James H., 844 
Flather, J. J., 961, 964 
Flynn, P. J., 463, 559 
Foley, Nelson, 700 
Forbes, Prof., 1033 
Forney, M. N., 855 
Forsyth, Wm., 630 
Foster, R. J., 651 
Francis, J. B., 586, 739, 867 
Frazer, Persifor, 624 
Freeman, J. R., 581, 584 
Frith, A. J., 874 
Fulton, John, 637 

Ganguillet & Kutter, 565 

Gantt, H. L., 406 

Garrison, F. L., 326, 331, 409 

Garvin Machine Co., 955 

Gause, F. T., 501 

Gav, Paulin, 966 

Gill, J. P., 657 

Gilmore, E. P., 241 

Glaisher, 483 

Glasgow, A. G., 654 

Goodman, John, 934 

Gordon, F. W., 689, 740 

Gordon, 247 

Goss, W. F. M., 863 

Gossler, P. G., 1051 

Graff, Frederick, 385 

Graham, W., 950 

Grant, George B., 898 

Grant, J. J., 960 

Grashof, Dr., 284 

Gray, J. McFarlane, 661 

Gray, J. M., 958 

Greene, D. M., 567 



Greig and Eyth, 363 
Grosseteste, W., 715 
Gruner, L., 623 

Hadfield, R. A., 391, 409 

Halpin, Druitt, 789, 854 

Halsey, Fred'k A., 490, 817 

Harkness, Wm., 900 

Harrison, W. H., 939 

Hart, F. R., 1047 

Hartig, J., 961 

Hartman, John M., 364 

Hartnell, Wilson, 348, 818, 838 

Hasson, W. F. C, 1047 

Hawksley, T., 485, 513, 564 

Hazen, H. Allen, 494 

Henderson, G. R, 347, 851 

Henthorn, J. T., 965 

Hering, Carl, 1045 

Herschel, Clemens, 583 

Hewitt, G. C, 630 

Hewitt, Wm., 917 

Hildenbrand, Wm., 913 

Hill, John W., 17 

Hiscox, G. D., 968 

Hoadley, John C, 451, 688 

Hobart', J. J., 962 

Hodgkinson, 246 

Holley, Alexander L., 377 

Honey, F. R., 47, 52 

Hoopes & Townsend, 210 

Houston, Edwin J., 1061 

Houston & Kennelly, 1058 

Howard, James E., 242, 382, 385 

Howden, James, 714 

Howe, Henry M., 402, 407, 451, 516 

Howe, Malverd A., 170, 312 

Howland, A. H., 292 

Hudson, John G., 465 

Hughes, D. E., 396 

Hughes, H. W., 909 

Hughes, Thos. E., 917 

Humphreys, Alex. C, 652 

Hunsicker, Millard, 397 

Hunt, Alfred E., 235, 317, 392, 553 

Hunt, Chas. W., 340, 922 

Huston, Charles, 383 

Hutton, Dr., 64 

Huyghens, 58 

Ingersoll-Sergeant Drill Co., 503 
Isherwood, Benj. F., 472 

Jacobus, D. S., 511, 689, 726, 780 

Johnson, J. B., 309, 314 

Johnson, W. B.,475 

Johnsou, W. R., 290 

Jones,* Hoi'ace K., 387 

Jones & Lamson Machine Co., 954 

Jones & Laughlins, 867, 885 

Kapp, Gisbert, 1033 
Keep, W. J., 365, 951 
Kennedy, A. B. W., 355, 525, 764 
KernOt, Prof. 494 
Kerr, Walter C, 781 
Kiersted, W.. 292 
Kimball, J. P., 499, 635, 637 
Kinealy, J. H., 537 



LIST OF AUTHORITIES. 



1073 



Kirk. A. C, 705 
Kirk, Dr., 1004 
Kirkaldy, David, 296 
Kopp, H. G. C, 472 
Kuichling, E , 578 
Kutter, 559 

Landretb, O. H., 712 

Langley, J. W., 409, 410, 412 

Lanza, Gaetano, 310, 369, 864, 977 

La Rue, Benj. F., 248 

Leavitt, E. D., 788 

LeChatelier, M., 452 

Le Conte, J., 565 

Ledoux, M., 981 

Leggett, T. H., 1049 

Leonard, H. Ward, 1026 

Leonard, S. H., 686 

Lewis, Fred. H., 186, 189, 397 

Lewis, I. N.,498 

Lewis, Wilfred, 352, 362, 378, 899 

Linde, G., 989 

Lindenthal, Gustav, 385 

Lloyd's Register, 264, 266, 700 

Loss, H. V., 306 

Love, E. G., 656 

Lovett, T. D., 256 

Lyne, Lewis F., 718 

McBride, James, 974 

MacCord, C. W., 898 

Macdonald, W. R., 956 

Macgovern, E. E., 545 

Mackay, W. M., 542, 544 

Mahler, M., 633 

Main, Chas. T., 590, 780, 790 

Mannesmann, L., 332 

Manning, Chas. H., 675, 823 

Marks, Win. D., 793, 811 

Master Car Builders' Assoc, 376 

Mattes, W. F., 399 

Matthiessen, 1029 

Mayer, Alfred M., 468 

Mehrtens, G. G., 395, 405 

Meier, E. D., 688 

Meissner, C. A., 370 

Melville, Geo. W., 674 

Mendenhall, T. C.,23 

Merriman, Mansfield, 241, 260, 282 

Metcalf, William, 240, 412 

Meyer, J. G. A., 795,856 

Meystre, F. J.. 472 

Miller, Metcalf & Parkin, 412 

Miller, T. Spencer, 344, 927 

Mitchell, A. E., 855, 856 

Molesworth, Sir G. L., 562, 658 

Molyneux and Wood, 736 

Moore, Gideon E., 653 

Morin, 435, 930, 933 

Morison, Geo. S., 381, 393 

Morrell, T. T., 407 

Morris, Tasker & Co., 195, 196 

Mumford, E. R., 1006 

Murgue, Daniel, 521 

Nagle, A. F., 292, 606, 878 
Napier, 471, 669 
Nason Mfg. Co., 4 8, 542 
National Pipe Bending Co., 198 



Nan, J. B., 367, 409 

Newberry, J. S., 624 

Newcomb, Simon, 432 

New Jersey Steel & Iron Co., 253, 310 

Newton, Sir Isaac, 475 

Nichol, BC, 473 

Nichols. 285 

Nonis, R. Van A , 521 

Norwalk Iron Works Co., 488, 504 

Nystrom, John W., 265 

Ordway, Prof., 469 

Paret, T. Dunkin, 967 

Parker, W., 354 

Parsons, H. de B., 361 

Passburg, Emil, 466 

Pattinson, John, 629 

Peclet, M.,471, 478, 731 

Peltou Water Wheel Co., 191, 574, 585 

Pence, W. D., 294 

Pencoyd Iron Works. 179, 232, 868 

Penned, Arthur, 555 

Pennsylvania R. R. Co., 307, 375, 399 

Philadelph'iaEngiueering Works, 526 

Philbrick, P. H., 446 

Phillips, W. B., 629 

Phoenix Bridge Co., 262 

Phoenix Iron Co., 181, 257 

Pierce, C. S., 424 

Pierce, H. M.,641 

Pittsburg Testing Laboratory, 243 

Piatt, John, 617 

Pocock, F. A.. 505 

Porter, Chas. T., 662, 787, 820 

Potter, E. C, 646 

Potts ville Iron & Steel Co., 250 

Pouillet, 455 

Pourcel, Alexandre, 404 

Poupardin, M., 687 

Powell, A. M , 975 

Pratt & Whitney Co., 892, 972 

Price, C. S., 638 

Prony, 564 

Pryibil, P., 977 

Quereau, C. H„ 858, 862 

Ramsey, Erskine, 638 

Rand Drill Co , 490, 505 

Randolph & Clowes, 198 

Rankine, W. J. M., various 

Ransome, Ernest L., 241 

Raymond, R. W., 631, 650 

Reese, Jacob, 966 

Regnault, M., various 

Reichhelm, E. P., 651 

Rennie, John, 928 

Reuleaux, various 

Richards, Frank, 488, 491, 500 

Richards, John, 965, 976 

Richards, Windsor, 404 

Riedler, Prof., 507 

Rites, F. M., 783, 818 

Roberts-Austen, Prof., 451 

Robinson, H., 1051 

Robinson, S. W., 583 

Rockwood, G. J., 781 

John A. Roebling's Sons' Co., 214, 921 



1074 



List OF AUTHORITIES. 



Roelker, C. R., 26.5 
Roney, W. R„ 711 
Roots, P. H. & F. M.,526 
Rose, Joshua, 414, 869* 970 
Rothwell, R. P., 637 
Rowland, Prof.; 456 
Royce, Fred. P , 1053 
Rudiger, E, A., 671 
Ruggles, W. B., Jr., 361 
Russell, S. Bent, 567 
Rust and Coolidge, 290 

Sadler, S. P., 639 

Saint Venant, 282 

Salom, P. G.,406, 1056 

Sandberg, C. P., 384 

Saunders, J. L., 544 

Saunders, W. L., 505 

Scheffler, F. A., 681 

Schroter, Prof., 788 

Schutte, L., &Co., 527 

Seaton, various 

Sellers, Coleman, 890, 953, 975 

Sellers, Wm, 204 

Sharpless, S. P., 311, 639 

Shelton, F. H., 653 

Shock, W. H., 307 

Simpson, 56 

Sinclair, Angus, 863 

Sloane, T. O'Connor, 1027 

Smeaton, Wm., 493 

Smith, Chas. A., 537, 874 

Smith, C. Shaler, 256, 865 

Smith, Hamilton, Jr., 556 

Smith, Jesse M., 1050 

Smith, J. Bucknall, 225, 303 

Smith, Oberlin, 865, 973 

Smith, R. H., 962 

Smith, Scott A., 874 

Snell, Henry I., 514 

Stahl, Albert W., 599 

Stanwood, J. B„ 802, 809, 813, 818 

Stead, J. E., 409 

Stearns, Albert, 465 

Stein and Schwarz, 410 

Stephens, B. F., 292 

Stillman, Thos. B., 944 

Stockalper, E., 493 

Stromeyer, C. E., 396 

Struthers, Joseph, 451 

Sturtevant, B. F., Co., 487, 578 

Stut, J. C. H., 844 

Styffe, Knut, 383 

Suplee, H. H., 769, 772 

Suter, Geo. A., 524 

Sweet, John E„ 826 

Tabor, Harris, 751 
Tatham & Bros., 201 
Taylor, Fred. W., 880 
Taylor, W. J., 646 
Theiss, Emil, 818 
Thomas, J. W., 369 
Thompson, Silvanus P., 1064, 1066 
Thomson, Elihu, 1052 
Thomson, Sir Wm., 461, 1039 
Thurston, R. H, various 
Tilghman, B. F., 966 
Tompkins, C. R , 336 



Torrance, H. C, 401 
Torrey, Joseph, 582, 820 
Tower, Beauchamp, 931, 934 
Towne, Henry R., 876, 907, 911 
Townsend, David, 973 
Trautwine, J. O, 59, 118, 311, 482 
Trautwine, J. C, Jr., 255 
Trenton Iron Co., 216, 223, 230, 915 
Tribe, James, 765 
Trotz, E , 453 
Trowbridge, John, 467 
Trowbridge, W. P., 478, 513, 733 
Tuit, J. E.,616 
Tweddell, R. H., 619 
Tyler, A. H., 940 

Uchatius, Gen'l, 321 

Unwin, W. Cawthorne, various 

Urquhart, Thos., 645 

U. S. Testing Board, 308 

Vacuum Oil Co., 943 
Vair, G. O., 950 
Violette, M , 640, 642 
Vladomiroff, L., 316 

Wade, Major, 321, 374 

Wailes, J. W., 404 

Walker Mfg. Co., 905 

Wallis, Philip, 858 

Wan-en Foundry & Mach. Co., 189 

Weaver, W. D.. 1043 

Webber, Samuel, 591, 963 

Webber, W. O., 608 

Webster, W. R, 389 

Weidemann & Franz, 469 

Weightman, W. H., 762 

Weisbach, Dr. Julius, various 

Wellington, A. M., 290, 928, 935 

West, Chas. D., 916 

West, Thomas D., 328 

Westinghouse & Galton, 928 

Westinghouse El. & Mfg. Co., 1043 

Weston, Edward, 1029 

Whitham, Jay M., 472, 769, 792, 840 

Whitney, A. J., 389 

Willett, J. R., 538, 540 

Williamson, Prof., 58 

Wilson, Robert, 284 

Wheeler, H. A., 908 

White, Chas. F., 714 

White, Maunsel, 408 

Wohler, 238, 240 

Wolcott, F. P., 949 

Wolff, Alfred R , 494, 517, 528, 538 

Wood, De Volson, various. 

Wood, H. A., 9 

Wood, M. P., 386. 389 

Woodbury, C. J. H., 537, 931 

Wootten, J. E., 855 

Wright, C. R. Alder, 331 

Wright, A. W.,289 

Yarrow, A. F., 710 
Yarrow & Co., 307 
Yates, J. A., 287 

Zahner, Robert, 499 
Zeuner, 827 



INDEX. 



Abbreviations, 1 

Abrasive processes, 965 

Abscissas, 69 

Absolute zero, 461 

Absorption refrigerating machines, 

984 
Accelerated motion, 427 
Acceleration, 423 

work of, 430 
Accumulators, electric, 1055 
Air, 481-527 

and vapor, weights of, 484 

compressed, 499 

density and pressure, 481 

-pumps, 839 

-thermometer, 454 
Algebra, 33 
Algebraical signs, 1 
Alligation, 10 
Alloys, 319-338 

aluminum, 328 

aluminum-silicon-iron, 330 

antimony, 336 

bismuth, 332 

caution as to strength, 329 

copper-nickel, 326 

copper-tin, 319 

copper-tin-zinc, 322 

copper-ziuc, 321, 325 

copper-zinc-iron, 326 

for bearings, 333 

fusible, 333 

manganese-copper, 331 

steels, 407 
Alternating currents, 1070 
Altitude by barometer, 483 
Aluminum, 166 

alloys of, 319-338 

brass, 329 

bronze, 328 

bronze wire, 225 

hardened, 330 

properties and uses, 317 

steel, 409 

wire, 225 
Ammonia ice-machines, 983 

vapor, properties of, 993 
Amperage permissible in magnets, 



Analyses of alloys (see Alloys) 

of asbestos, 233 

of coals (see Coal) 

of fire-clay, 234 

of magnesite, 233 

of steel (see Steel) 

of water, 553 
Analytical geometry, 69 
Anemometer, 491 
Angle-bars, sizes and weights, 179 

weight and strength, 279 
Angles, plotting without protract- 
or, 52 

problems in, 37-38 
Angular velocity, 425 
Animal power, 433 
Annealing, effect on conductivity, 
1029 

non-oxidizing, process of, 387 

of steel, 394 

tool-steel, 413 
Annuities, 15-17 
Annuiar gearing, 898 
Anti-friction metals, 932 
Anthracite, analyses of, 624 

gas, 647 

space occupied by, 625 

value of sizes of, 632 
Antimony, 166 

alloys, 336 
Apothecaries' measure and weight, 

18, 19 
Arc lamps, lighting power of, 1052 
Arches, tie-rods for, 281 
Area of circles, 103, 108 
Arithmetic, 2 

Arithmetical progression, 11 
Armature circuit, E. M. F. of, 1062 
Asbestos, 235 

Asymptotes of hyperbola, 71 
Atmosphere, moisture in, 4S3 
Avoirdupois weight, 19 
Axles, steel, specifications for, 401 

strength of, 299 

Babbitt metals, 336 

Bagasse as fuel, 643 

Balance, to weigh on an incorrect, 19 

Ball bearings, 940 

1075 



1076 



Bands and belts, theory of, 876 
Bands for carrying grain, 911 
Barometric readings, 482 
Barrels (see Casks), 64 

No. of in tanks, 126 
Bazin's experiments on weirs, 587 

Formula, flow of water, 563 
Beams and channels, Trenton, 278 
Beams, flexure of, 267 

of uniform strength, 271 

safe load on pine, 1023 

safe loads, 269 

strength of, 268 
Bearing metal alloys, 333 
Bearing-metals, anti-friction, 932 
Bearings, ball, 940 

for high-speeds, 941 

pivot, 939 
Bed-plates of engines, 817 
Belt cement, 887 

conveyors, 911 

dressings, 887 
Belting, 876-887 

strength of, 302 
Belts, open and crossed, 874, 884 
Bends and curves, effect of on flow 

of water, 578 
Bends, valves, etc., resistance of 

672 
Bessemerized cast iron, 375 
Bessemer steel. 391 
Bevel wheels, 898 
Binomials, Theorem, 33, 35 
Birmingham Gauge, 28 
Bismuth, 1(56 

alloys, 332 
Blast-furnace boilers, 689 
Blocks or pulleys, 438 

strength of, 906 
Blowers and fans, 511-526 

experiments with, 514 

for cupolas, 950 

positive rotary, 526 

steam -jet, 526 
Blowing engines, 526 
Blue heat, effect on steel, 395 
Board measure, 20 
Boiling-point of water, 550 
Boiling points, 455 
Bolts and nuts, 209, 211 
Bolts, holding power of, 290 

strength of, 292 
Boiler compounds, 717 

explosions, 720 
Boilers, for steam-heating, 538 
Boiler furnaces, height of, 711 

heads 706 

headsi strength of, 284-286 
Boiler scale, 552 

ship and tank plates, 399 

the steam, 677-741 

tubes, 196 

tubes, holding power of, 307 
Boilers, locomotive, 855 
Brass alloys, 325 

composition of rolled, 203 

sheet and bar, 203 

tubing, 198 
Brass wire and plates, 202 



Brick, fire, sizes of, 233-235 
Brick, strength of, 302, 312 
Bricks, absorption of water by, 312 
Bricks, magnesia, 235 
Brickwork, weight of, 169 
Bridge members, working strain, 
262 

proportioning materials in, 381 

trusses, 443 
Brine, specific gravity, etc., 464, 994 
Bronze (see Alloys), 319 
Bronzes, ancient, 323 
Building construction, 1019 

materials, sizes and weights, 170- 
184 
Buoyancy, 550 
Burr truss, 443 

Cables, electric, insulated, 1033 

wire, 222, 223 
Cable-ways, suspension, 915 
Cadmium, 167 
Calculus, Differential, 72 
Caloric engines, 851 
Calorimeters, steam, 728 
Calorimetric tests of coal, 636 
Cam, the, 438 

Canals, speed of vessels on, 1008 
Canvas, strength of, 302 
Carbon, burned out of steel, 402 

effect of on strength of steel, 389 
Car-heating by steam, 538 
Casks, 64 
Castings, steel, 405 

weight of, from pattern, 952 

iron, analyses of, 373 
Cast iron, 365-375 

and steel mixtures, 375 

bad, 375 

malleable, 375 

specifications, 374 

specific gravitv, 37'4 

strength of, 369, 374 
Catenary, construction of, 51 

the wire rope, 919 
Cement, weight of, 170 

for belts 887 

mortar, strength of, 313 
Centigrade and Fahrenheit table, 449 
Centre of gravity, 418 

of gyration, 420 

of oscillation, 421 

of percussion, 421 
Centrifugal fans, 511 

force, 423 

force in fly-wheels, 820 

tension of belts, 876 
Cera-perduta process, alloys for, 326 
Chain-blocks, 907 

cables, 308, 340 
Chains, crane, 232 

weight and strength, 307, 339 
Channel beams, sizes and weight, 

178, 180 
Channels, steel, strength of, 275 
Charcoal, 640 

making results, 642 

pig iron, 365, 374 

weight of, 170 



1077 



Chemical elements, 163 
Chimneys, 731-741 

brick, 737 

for ventilation, 533 

stability of, 738 

size of, 734 

steel, 740 

slieet iron, 741 

table of sizes of, 735 
Chords of circles, 57 
Chrome steel, 409 
Circle, equation of, 70 

measures of, 57-58 
Circles,- problems, 39-40 

tables of, 103, 108 
Circular arc, length of, 58 

arcs, tables of, 114, 115 

functions in calculus, 78 

measure, 20 

ring, 59 
Circulating pump, 839 
Circumference of circles, 103, 108, 

113 
Cisterns, cylindrical, 121, 126 
Clearance in steam-engines, 752, 792 
Coals, analyses of, 624-631 

calorimetric tests, 636 

evaporative power of, 636 

heating value of, 634 

relative value of, 633 
Coal gas, illuminating, 651 

hoisting, 343 

products of distillation of, 639 

washing, 638 

weathering of, 637 
Coefficient of elasticity, 237 
Coefficients of friction, 928-932 
Coiled pipes, 198 
Coils, heating of, 1036 
Coke, 637 

Coking, experiments in, 637 
Coke manufacture, by products, 639 
Cold drawing steel, 305 
Cold, effect of on iron and steel, 383 
Cold rolling, effect of, 393 
Columns, built, 256 

cast iron, weight of 185 

iron tests of, 305 

strength of, 246-250 
Combined stresses, 282 
Combination. 10 
Combustion, heat of, 456, 621 

gases of, 622 

theory of, 620 
Composition of forces, 415 
Compressed air, 499 

cranes, 912 

motors, 507 

transmission, 488 
Compressed steel, 410 
Compression iw steam-engines, 751 
Compression unit strains, 380 
Compressive strength, 244 
Compressive strength of iron bars, 

304 
Compressors, air, 503 
Compound engines, 701-768 

diameter of cylinder, 768 

economy of, 780 



Compound engines, work of steam 

in, 767 
Compound interest, 14 
Compound numbers, 5 
Compound units of weight and 

measure, 27 
Condenser, evaporative surface, 844 
Condensers, 839-846 
Condensing water, continuous use 

of, 844 
Conduction of heat, 468 
Conductivity, electrical, 1028 
Conductors, electrical, 1029 
Cone, measures of, 61 

pulleys, 874 
Conic sections, 71 
Conoid, parabolic, 63 
Connecting rods, 799 

tapered, 801 
Conservation of energy, 432 
Construction of buildings, 1019 
Convection of heat, 468 
Conveyors, belt, 911 
Co-ordinate axes, 69 
Copper, 167 

at high temperatures, strength of, 
309 

balls, hollow, 289 

round bolt, 203 

strength of, 300 

tubing, 200 

wire and plates, 202 

wire tables of, 218-220 
Copper wire, resistance of hot and 
cold, 1034, 1035 

cost of for long - distance trans- 
mission, 1045 
Cordage, 341,344, 906 
Cork, properties of, 316 
Corrosion of iron, 385 
Corrosion of steam-boilers, 716, 719 
Corrosive agents in atmosphere, 386 
Corrugated iron, 181 
Corrugated furnaces, 266, 709 
Cosecant of an angle, 65 
Cosine of an angle, 65 
Cosines, tables of, 159 
Cost of coal for steam-power, 789 

of steam-power, 790 
Cotangent of an angle. 65 
Counterbalancing engines, 788 

locomotives, 864 

of wiuding engines, 909, 
Couples, 418 

Coverings for steam-pipes, 469 
Cox's formula for loss of head, 575 
Cranes, classification of, 911 

compressed air, 912 

stress in, 440 
Crank angles, 830 

arms, 805, 806 

pins, 801-804 
Crucible steel, 410 

Crushing strength of masonry ma- 
terials, 312 
Cubature of volumes of revolution, 

75 
Cube root, 8 
Cubes and cube roots, table of, 86 



107* 



Cubic measure, 18 
Cupolas, blast-pipes for, 519 

blowers for, 519 

practice, 946 
Current motors, 589 
Currents, electric, 1030 
Cutting stone with wire, 966 
Cycloidal teeth of gears, 892 
Cycloid, construction of, 50 

differential equation of, 79 
Cylinders and pipes, contents of, 120, 
121 

condensation, 752, 753 

engine, dimensions of, 792 

hollow, resistance of, 264 

hollow, strength of, 287-289 

measures of, 61 
Cylindrical ring, 62 

Dangerous steam-boilers, 720 

Dam, stability of a, 417 

D'Arcy's formula, flow of w r ater, 563 

Decimals, 3 

Decimal equivalents of fractions, 

3,4 
Decimals, squares and cubes of, 101 
Deck-beams, sizes and weights, 177 
Delta metal, 326 
Delta-metal wire, 225 
Denominate numbers. 5 
Deoxidized bronze, 327 
Derricks, stresses in, 441 
Diametral pitch, 888 
Differential calculus, 72 

gearing, 898 

pulley, 439 

screw, 439 

screw, efficiency of, 974 

windlass, 439 
Discount and interest, 13 
Draught of chimneys, 731 
Drawing-presses, blanks for, 973 
Drilling: holes, speed of, 956 

machines, electric, 956 
Drop-press, pressure of, 973 
Drums for hoisting-ropes, 917 
Drying and evaporation, 462 
Drying in vacuum, 466 
Dry measure, 18 
Ductility of metals, 169 
Dust explosions, 642 
Dust-fuel, 642 
Durability of iron, 385 
Durand's rule, 56 

Duty trials of pumping-engines, 609 
Dynamo and engine, efficiency of, 
1048 

electric machines, 1061 
Dynamos, designing of, 1064 

efficiency of, 1066 
Dynamometers, 978-980 

Earths, weight of, 170 
Earth-filling, weight of. 170 
Eccentric loading of columns, 255 
Eccentrics, steam engine, 816 
Economizers, fuel, 715 
Edison or circular mil wire gauge, 
30-31 



Effort, definition of, 429 
Elastic limit, 236 

elevation of, 238 
Elasticity, modulus of, 237, 314 
Elastic resilience, 270 
Electric accumulators, 1055 

generator, efficiency of, 1048 

heating, 1054 

lighting, 1051 

motor, 1070 

pumping plant, 1049 

railways, 1050 

transmission, 1038 

transmission, economy of, *1039 

welding, 1053 
Electrical engineering, 1024 

horse -powers, table of, 1042 

resistance, 1028 

standards of measurement, 1024 

units, 1024 
Electric conductivity of steel, 403 
Electricity, analogy with flow of 
water, 1026 

heating by, 546 
Electro-chemical equivalents, 1057 

magnetic measurements, 1058 

magnets, 1058 

magnets, polarity of, 1060 
Electrolysis, 1057 
Elements, chemical, 163 
Elements of machines, 435 
Ellipse, construction of, 45 

equation of, 70 

measures of, 59 
Ellipsoid, 63 

Elongation, measure of, 243 
Emery, grades of, 968 

wheels, 967-969 
Endless screw, 440 
Energy, conservation of, 432 

of recoil of guns, 431 

or stored work, 429 

sources of, 432 
Engine-frames or bed-plates, 817 
Engines, blowing, 526 

gas, 847 

gasoline, 850 

hoisting, 908 

hot-air, 851 

marine, sizes of steam -pipes, 674 

naptha, 851 

petroleum, 850 

steam, 742-847 
Epicycloid, 50 
Equalization of pipes, 492 
Equation of payments, 14 
Equations, algebraic, 34 
Equilibrium of forces, 418 
Equivalent orifice, 533 
Erosion by flow of water, 565 
Evaporating by exhaust-steam, 465 
Evaporation and drying, 462 

by the multiple system, 465 

from reservoirs. 463 

latent heat of, 461 

tabl3 of factors of, 695 
Evaporators, fresh -water, 1018 
Evolution, 7 
Eye-bars, tests of, 304, 393 



1079 



Exhaust steam for heating, 780 
Exhausters, steam-jet. 526 
Expansion by heat, 459 

of iron and steel, 385 

of steam, 742 

of steam, real ratios of, 750 

of wood, 311 
Explosive energy of steam-boilers, 

720 
Exponents, theory of, 36 
Exponential functions, in calculus, 

78 
Factor of safety, 314 

in steam boilers, 700 
Factors of evaporation, tables, 695 
Fahrenheit and centigrade table, 450 
Failures of steel. 403 
Falling bodies. 423-426 
Fans and blowers, 511-526 
Feed-pumps, 843 

water, cold, strains caused by, 727 

water, heater, Weir's, 1016 

water, heaters, 727 

water, purifying, 554 
Fibre-graphite, 945 
Fifth roots and fifth powers, 102 
Fink roof-truss, 446 
Fire brick, sizes of, 233-235 

clay, analysis of 234 

engines, capacities of, 580 
Fireless locomotive, 866 
Fireproof buildings, 1020 
Fire-streams, 580 
Fire temperature of, 622 
Flagging, transverse strength'of, 313 
Flanges, pipe, 192, 193, 676 
Flat plates in steam-boilers, 701, 713 

plar.es, strength of, 283 
Flexure of beams, 267 
Flooring material, weight of, 281 
Floors, strength of, 1019, 1021 
Flowing water, horse-power of, 589 
Flow of air in pipes, 485 

of air through orifices, 484 

of compressed air, 489, 493 

of gas in pipes, 657 

of metals, 973 

of steam in pipes, 669 

of water from orifices, 555, 584 

of water in house service-pipes. 
578 

of water over weirs, 586 
Flues, collapse of, 265 

corrugated. U. S. rules, 709 

(see also Tubes and Boilers.) 
Fly-wheels, 817-824 

arms of, 820 

wire-wound, 824 

wooden, 823 
Flyun's formula, flow of water, 562 
Foot, decimals of, in fractions of 
inch, 112 

pound, unit of work, 428 
Forced draught in marine practice, 

1015 
Force of a blow, 430 

of acceleration, 427 

unit of, 415 
Forces, composition of, 415 



Forces, equilibrium of, 418 

parallel, 417 

parallelogram of, 116 

parallelopipedon of, 416 

polygon of, 416 

resolution of, 415 
Forcing and shrinking fits, 973 
Forging, hydraulic, 618, 620 

tool steel, 413 
Foundry, the, 946*956 
Fractions, 2 

Francis's formula for weirs, 586 
Freezing of water, 550 
French measures and weights, 21-26 
Friction and lubrication, 92S-945 

brakes, 980 

of air in passages, 531 

of steam-engines, 941 

rollers, 940 

work of, 938 
Frictional heads, flow of water, 577 
Fuel, 622-651 

economizers, 715 

gas, 646 

pressed, 633 

theory of combustion of, 620 

weight of, 170 
Fuels, classification of, 623, 624 
Furnace, downward draught, 635, 
712 

kinds of, for different coals, 635 

formulae, 702 
Furnaces, corrugated, 266 

for boilers, 711 

gas-fuel, 651 

use of steam in, 650 
Fusible alloys, 333 

plugs for boilers, 710 
Fusibility of metals, 169 
Fusing disk, Reese's, 966 
Fusion of wires by electric currents, 
1037 

temperatures, 455 

g, value of, 424 

Gallons and cubic feet, table, 122 . 
Galvanized wire rope, 228 
Gas-engines, 847 

fired steam-boilers, 714 

flow of in pipes, 657 

fuel, 646 

illuminating, 651 

illuminating, fuel value of, 656 
Gases, properties of, 479 

waste, use under boilers, 689, 690 
Gasoline-engines, 850 
Gas-pipe sizes and weights, 188 
Gas producers 649, 650 
Gauges, wire and sheet metal, 26-31 
Gearing, efficiency of, 899 

frictional, 905 

of lathes, 955 

speed of, 905 

toothed-wheel, 439, 887-906 
Gear-teeth, strength of, 900-905 
Geometrical problems, 37 

progression, 11 

propositions, 53 
German silver, 326, 332 



1080 



1KDEX. 



German silver, strength of, 300 
Girders for boilers, 703 
Glass, skylight, 184 

strength of, 308 
Gold, 167 

Gordon's formula, 247 
Governors, 836 
Grain, weight of, 170 
Granite, strength of, 312 
Grate and heating surface of a boil- 
er, 678 
Graphite as a lubricant, 945 

paint, 389 
Gravity acceleration due to, 424 

centre of, 418 

specific, 163 
Greatest common measure, 2 
Greenhouse heating by hot water, 
541 

heating by steam, 542 
Green's fuel economizers, 715 
Grindstones, 968, 970 
Gyration, centre and radius of, 420, 
421 

radius of, 247 

Hawley down-draught furnace, 712 
Heads of boilers, 706 
Heat, 448-478 

boiling-points, 455 

conduction of, 468 

convection of, 469 

expansion by, 459 

generated by electric currents, 
1032 

latent, 461 

latent, of evaporation, 462 

latent, of fusion, 461 

melting-points, 455 

of combustion, 456 

radiation of, 467 

specific, 457 

storing of, 789 

unit, 455, 660 
Heaters, feed-water, 727 
Heating a building to 70° F., 545 

and ventilation, 528-546 

blower system of, 545 

by electricity, 546 

by exhaust-steam, 780 

by hot water, 542 

of large buildings, 534 

surface of boilers, 678 
Helix, 60 

Hodgkinson's formula, 246 
Hoisting, 906-916 

coal, 343 

engines, power of, 908 

pneumatic, 909 

rope, 340 
Hooks, hoisting, 907 
Horse-gin, 434 

work of a, 434 
Horse-power, 429 

constants, 757, 758 

of flowing water, 589 

of steam-boilers, 677, 679 

of steam-engines, 755 

power-hours, 429 



Hose, friction losses in, 580 
Hot-air engines, 851 

water heating, 542 
Howe truss, 445 
Humidity in atmosphere, 483 
Hydraulic apparatus, 616 

engine, 619 
Hydraulics, flow of water. 555-588 

forging, 618 

grade-line, 578 

power, 617 

pressure transmission, 616 

ram, 614 
Hydrometer, 165 
Hygrometer, dry and wet bulb, 

483 
Hyperbola, equation of, 71 

construction of, 49 
Hyperbolic logarithms, 156 

curve in indicator diagrams, 759 
Hypocycloid, 50 

I-beams, sizes and weights, 177 

properties of, 274 

spacing of, 273, 280 
Ice and snow, 550 
Ice-making machines, 981-1001 

manufacture, 999 
Illuminating gas, 651 
Impact of bodies, 431 
Incandescent lamps, 1051 
Inches and fractions, decimals of a 

foot, 112 
Inclined plane, 437 

planes, hauling on, 913, 915 

planes, motion on, 428 
Incrustation and scale, 551, 716 
India rubber, tests of, 316 
Indicated horse-power, 755 
Indicator diagrams, 754, 759 

rigs, 759 
Indirect heating surface, 537 
Inertia, 415 

moment of, 247, 419 

of railroad trains, 853 
Injectors, 725 

Inspection of steam boilers, 720 
Insulators, electrical, 1029 
Integral calculus, 79 
Integrals, integration, 73-74 
Intensifier, the Aiken, 619 
Interest and discount, 13 
Involute gear- teeth, 892 

construction of, 52 
Involution, 6 
Iridium, 167 
Iron, 167 

bars, sizes of, 170 

bars, weight of, 171 

corrosion of, 385 

durability of, 385 

and steel, classification of, 362 

and steel, cold rolling of, 393 

and steel, expansion of, 385 

and steel, strength of, 296-300 

and steel, strength at high tem- 
perature, 382 

and steel, strength at low tem- 
perature, 383 



INDEX. 



1081 



Irregular figure, area of, 55 
Isothermal expansion, 742 

Japanese alloys, 326 
Jet propulsion of vessels, 1014 
Jets, vertical water, 579 
Joints, riveted, 354-363 
Joule's equivalent, 456 
Journal bearings, 810-815 

friction, 931 
Journals, engine, 810-815 

Kerosene for scale in boilers, 718 
Keys and set-screws. 977 

for mill-gearing, 975 

holding power of, 977 
Kinetic energy, 429 
King-post truss, 442 
Kirkaldy's tests of materials, 296 
Knot, or nautical mile, 17 
Knots in ropes, 344 
Kutter's formula, 559 

Lacing of belts, 883 

Ladles, foundry, sizes of, 953 

Latent heat, 461 

heat of evaporation, 461 

heat of fusion, 459-461 
Lathe-gearing, 955 

tools, speed of, 953 
Lap and lead of a valve, 829-833 
Lead, 167 

pipe, 200 
Leakage of steam in engines, 761 
Least common multiple, 2 
Leather, strength of, 302 
Leveling by barometer, 482 
Levers, 435 
Lime, weight of, 170 
Limestone, strength of, 313 
Limit gauges for screw-threads, 205 
Lines of force, 1059 
Links, engine, 816 
Link-motion, 834 
Liquation of alloys, 323 
Liquid measure, 18 
Liquids, weight and sp. gr., 164 
Locomotives, 851-866 

dimensions of, 860 

tests of, 863 
Logarithmic curve, 71 

sines, etc., 162 
Logarithms, hyperbolic, 156 

differential of, 77 

of numbers, 127-155 
Logs, lumber, etc., weight, 232 
Loop, the steam, 676 
Loss of head in pipes, 573 
Lubricants, 942 
Lubrication, 942 
Lumber, weight of, 232 

Machines, elements of, 435 
Machine-shop, the, 953-978 
: shop practice, 953 

shops, power used in, 965 

screws, 208, 20'J 

tools, power required for, 960-965 

tools, proportioning sizes of, 975 



Maclaurin's theorem, 76 
Magnesia bricks, 235 
Magnesium, 168 
Magnetic balance, 396 

circuit, 1060 

circuit, units of, 1058 

field, strength of, 1063 
Magnets, winding of, 1068 
Malleability of metals, 169 
Malleable castings, rules for, 376 

cast iron, 375 
Mandrels, sizes of, 972 
Manganese, 168 

bronze, 331 

influence of on cast iron, 368 

influence of on steel, 389 

plating of iron, 387 

steel, 407 
Mannesmann tubes, 296 
Manometer, air, 481 
Man-power, 433 
Manure as fuel, 643 
Man-wheel, 434 
Marble, strength of, 302 
Marine engineering, 1001-1018 

engine, horse-power of, 766 

engine practice, 1015 

engine, ratio of cylinders, 766, 773 
Masonry materials, strength of, 312 

materials, weight and sp. gr., 166 
Materials, 163-235 

strength of, 236-412 
Maxima and minima, 76 
Measures and weights, 17 
Mechanical equivalent of heat, 456 

powers (see Elements of Ma- 
chines), 435 

stokers, 711 
Mechanics, 415-447 
Mekarski compressed air tramway, 

509 
Melting points of substances, 455 
Memphis bridge, strains on steel, 

381 
Mensuration, 54 
Mercury. 168 

bath pivot, 940 
Merrimau's formula for columns, 

260 
Metaline, 945 
Metals, properties of the, 166 

weight and sp. gr., 164 
Metric conversion, tables, 22-26 

measures and weights, 21-22 
Meters, water, 579 
Mil, circular, 18, 29 
Milling cutters, 957 

cutters for gears, 892 

machines, results with, 959 
Mill power, 589 
Miner's inch, 18 

inch measurement, 585 
Mine-ventilating fans, 521 

ventilation, 531 
Modulus of elasticity, 237, 314 
Moisture in steam, 728 
Molesworth's fo.mula, flow of water, 

562 
Moment of force, 416 



1082 



Moment of inertia, 247, 419 

statical, 417 
Momentum, 428 
Morin's laws of friction, 933 
Mortar, strength of, 313 
Motor, electric, 1070 
Motors, compressed air, 507 
Moulding sand, 952 
Moving strut, 436 
Mules, power of, 435 
Multiphase currents, 1070 
Mushet steel, 409 

Nails, 213, 215 

screws, etc., holding power, 289-291 
Naphtha-engines, 851 
Napier's rule for flow of steam, 669 
Natural gas, 649 
Newton's laws of motion, 415 
Nickel steel, 406 
Nozzles, measurement of water by. 

584 
Nuts and bolts, 209, 211 

Ohm's law, 1030 

Oil needed for engines, 943 

Oils and coal as fuel, 646 

lubricating, 944 
Open-hearth steel, 391 
Ores, weight of, 170 
Oscillation, centre of, 421 
Oxen, power of, 435 
Ordinates, 69 

it, value of, 57 

Packing-rings, engines, 796 

Paddle-wheels, 1013 

Painting wood and iron structures, 

388 
Paint, qualities of, 389 
Parabola, construction of, 48 

equation of, 71 
Parabolic conoid, 63 
Parallel forces, 417 
Parallelogram, 54 

of forces, 416 
Parentheses, 33 
Partial payments, 15 
Peat or turf. 643 
Pelton-wheel table, 598 

water-wheel, 597 
Pendulum, 422 

conical, 423 
Percussion, centre of, 421 
Permutation, 10 
Perpetual motion, 432 
Petroleum, 645 

as fuel, 645, 646 

burning locomotive, 865 

distillates of, 645 

engines, 850 

products, specifications, 944 
Phosphor-bronze, 327 

bronze wire, 225 
Phosphorus, influence of on cast 
iron, 367 

influence of on steel, 389 
Piezometer, the, 582 
Pig iron, analysis of, 371 



Pig iron, chemistry of, 370 

grading of, 365 

influence of silicon, etc., 365. 

tests of, 369 
Pillars, strength of. 246 
Pitot tube gauge, 583 
Pipe, lead, 200, 201 

riveted, 197 

sheet-iron hydraulic, 191 

spiral riveted, 197 
Pipes, air-bound, 579 

and cylinders, contents of, 120, 121 

cast-iron, thickness of, 188, 190 

cast-iron, weight of, 185, 186 

coiled, 198 

effect of bends in, 488 

flow of air in, 485 

flow of gas in, 657 

flow of steam in, 669 

flow of water in, 557 

loss of head in, 573 

steam, for steam-heating, 540 

steam, sizes for engines, 673 

water, riveted, 295 

water, wrought-iron and steel, 295 

wrought iron, 194 
Piston-rods, 796-798 

valves, 834 
Pistons, steam-engine, 795 
Pitch, diametral, 888 

of gears, 887 

of rivets, 357-359 

of screw propellers, 1012 
Pivot bearings, 939 
Plane surfaces, mensuration, 54 
Plane, inclined (see Inclined Plane) 
Plate iron, weight of, 175 

steel, classification, 399 
Plates, brass and copper, 202 

square feet in, 123 

strength of for boilers, 705 
Platinum, 168 

wire, 225 
Pneumatic hoisting, 909 

postal transmission, 509 
Polyedrons, 62 
Polygon of forces, 416 
Polygons, construction of 42 

table of, 55 

tables of angles of, 44 
Population of the United States, 12 
Potential energy, 429 
Powell's screw-thread, 975 
Power, animal, 433 

of a fall of water, 588 

rate of work, 429 

of ocean waves. 599 
Power stations, electric, 1050 
Powers of numbers, 7, 33 
Pratt truss, 443 
Pressed fuel, 632 
Presses, punches, etc., 972 
Prism, measures of, 60 
Prismoid, 61 

rectangular, 61 
Prismoidal formula, 62 
Problems, geometrical, 37-52 

in circles, 39, 40 

in lines and angles, 37, 38 



INDEX. 



1083 



Problems in polygons, 42 

in triangles, 41 
Producer, gas, 649 

Progression, arithmetical, and geo- 
metrical, 11 
Prony brake, 979 
Proportion, 5 
Pulley, differential, 439 
Pulleys, 873-875 

arms of, 820 

or blocks, 438 
Pulsometer, 612 
Pumps, 601-614 

air, 841 

air-lift, 614 

boiler-feed, 605, 726 

capacity of, 601 

centrifugal, 606, 609 

circulating, 842 

efficiency of, 603, 608 

horse-power of, 601 

piston speed of, 605 

sizes of, 603 

speed of water through, 602 

steam cylinders of, 602 

suction of, 602 

vacuum, 612 

valves of, 605, 606 
Pumpiug-engine, tests of, 783 
Pumping-engines, duty trials, 609 

leakage test, 611 
Punched plates, strength of, 354 
Punches and dies, 972 
Punching and drilling steel, 395 

steel, effect of, 394 
Purifying feed-water, 554 
Pyramid, 60 
Pyrometers, 451-453 
Pyrometry, 448-454 

Quadratic equations, 35 

Quadrature of surfaces of revolu- 
tion, 75 

Quadruple-expansion engines, 772 

Quantitative measurement of heat, 
455 

Quarter-twist belts, 883 

Queen-post truss, 442 

Radiating surface, rules for. 536 
Radiation of heat, 467 
Radius of gyration, 247, 420, 421 
Railroad trains, resistance of, 851 
Rail-steel, specifications, 401 
Rails, maximum safe load on, 865 

steel, strength of, 298 
Railways, narrow-gauge, 865 
Railway trains, speed of, 859 
Ram, hydraulic, 614 
Ratio and proportion, 5 
Reamers, taper, 972 
Recalescence of steel, 402 
Receiver-space in engines, 766 
Reciprocals of numbers, 80 

use of, 85 
Red- lead as a preservative, 3S9 
Refrigerating machines, 981-1001 
Registers and air-ducts, 539 



Regnault's experiments on steam, 

661 
Resilience, 238 

elastic, 270 
Resistance, electrical, 1028 

of copper wire, 1030 

of ships, 1002 

of trains, 851 

to repeated stresses, 238 
Resolution of forces, 415 
Rhombus and Rhomboid, 54 
Riveted joints, 299, 303, 354, 362 
Riveting-machines, hydraulic, 618 

of steam-boilers. 700 

of steel plates, 394 

pressures, 362 
Rivets, diameter of, 360 

sizes, etc., 211 
Rivet -steel, 401 
Roads, resistance of carriages on, 

435 
Rock-drills, air required by, 506 
Roof-coverings, weight of, 184 

trusses, 446 
Roofing materials, 181, 184 
Rope-driving, 922-927 

wire-, 226-231 
Ropes, 301, 338, 906 

splicing 341, 345 
Rotary blowers, 526 

steam-engines, 791 
Rotation, accelerated, 430 
Rubber-belting, 887 
Rule of three, 6 
Rustless coatings for iron, 386 

Safety, factors of, 721 
Salinometer, strength of brines, 464 
Salt, solubility of, 464 

weight of, 170 
Sand-blast, 966 

moulding, 952 
Sawdust as fuel, 643 
Sawing metal, 966 
Scale and incrustation, 551 

in steam-boilers, 716 
Schiele's anti-friction curve, 50 
Schiele pivot-bearing, 939 
Screw-bolts, efficiency of, 974 

differential, 439 

endless, 440 

the, 437 

thread, Powell's, 975 

threads, 204, 208 

thread, metric, 956 

propeller, 1010 
Screws and screw-threads, 974 

holding power of, 290 

machine, 208, 209 
Secant of an angle, 65 
Sectors and segments, 59 
Sediment in steam-boilers, 717 
Segments of a circle, table, 116 
Segregation in steel ingots, 404 
Separators, steam, 728 
Set-screws, holding-power of, 977 
Sewers, grade of, 566 
Shaft-beariugs, 810 

governor, 838 



1084 



IKDEX. 



Shafting, 867-872 
Shafts, engine, 806-809 

fly-wheel, 809 

propeller, strength of, 815 
Shapes of test-specimens. 243 
Shearing, effect of, on steel, 394 

strength of iron, 306 

strength of woods, 312 

resistance of rivets, 363 

unit strains, 380 
Shear poles, stresses in, 442 
Sheet-iron and steel, weight of, 174 
Shells, spherical, strength of, 286 
Shingles, sizes and weight, 183 
Shippiog measure, 19 
Ships, resistance of, 1002 
Shocks, resistance to, 240, 241 
Shot, lead, 204 
Shrinkage of castings, 951 
Shrinking-fits, 973 

Signs of trigonometrical functions, 
66 

arithmetical, 1 
Silicon-bronze wires, 225, 328 
Silicon, influence of on cast-iron, 3G5 

influence of on steel, 389 
Silver, 168 
Simpson's rule, 56 
Sine of an angle, 65 
Sines, cosines, etc., table of, 159 

etc., logarithmic, 162 
Sinking-funds, 17 
Siphon, the, 581 
Slate, sizes and weights, 183 
Slide-valve, 824-835 
Smoke-prevention, 712 
Snow and ice, 550 
Soapstone as a lubricant, 945 
Softeners, use of in foundry, 950 
Softening hard water, 555 
Solders, 338 
Solid bodies, mensuration of, 60 

of revolution, 62 

measure, 18 
Specifications for axles, steel, 400 

for car-axles, 401 

for cast iron, 374 

for crank -pin steel, 400 

for oils, 944 

for plate steel, 399, 400 

for rail steel, 401 

for rivets, 401 

for spring steel, 400 

for steel, 397 

for steel castings, 406 

for steel rods, 400 
Specific gravity, 163 

gravity of alloys, 320, 323 

gravity of cast iron, 374 

gravity of steel, 403, 411 

heat, 457 

heat of air, 484 
Speed of cutting tools, 953, 954 

of vessels, 1006 
Sphere, measures of, 61 
Spherical polygon, volume of, 61 

shells, strength of, 286 

steam-engine, 792 

triangle, area of, 61 



Spherical zone, 62 

Spheroid, 63 

Spheres, table of, 118 
weights of, 169 

Spikes, sizes and weights, 212, 213 

Spindle, surface and volume, 63 

Spiral, construction of, 50 
gears, 897 
measures of, 60 

Splicing of ropes, 341 
wire ropes, 345 

Spring steel, specifications, 400 
strength of, 299 

Springs for governors, 838 
formula for, 347, 353 
tables of, 349, 353 

Square measure, 18 
root, 8 

Squares and cubes of decimals, 101 
and square roots, table of, 86 

Sui'face condenser, 840, 844 

Sugar manufacture, 643 
solutions, concentration of, 465 

Sulphate of lime, solubility, 464 

Sulphur dioxide refrigerating ma- 
chines, 985 

influence of on cast iron, 367 
influence of on steel, 389 

Suspension cable-ways, 915 

Stability, 417 

Stand-pipes, design of, 292-294 

Statical moment, 417 

Stay-bolt iron, 379 

Stay-bolts for boilers, 710 

Stayed surfaces, strength of, 286 

Stays for boilers, 703 

Steam, 659-676 
boiler, water tube, 688, 689 
boilers, 677-740 

boilers, air space in grate, 681 
boilers, allowable pressures, 706 
boilers, economy, of, 682 
boilers, efficiency of, 683 
boilers, explosive energy of, 720 
boilers, factor of safety, 700 
boilers, forced draught, 714 
boilers, flues and passages, 680 
boilers, gas-fired, 714 
boilers, grate surface of, 678, 680 
boilers, heating air supply to, 687 
boilers, healing surface of, 678 
boilers, horse-power of, 677 
boilers, hydraulic test of, 700 
boilers, incrustation and scale, 716 
boilers, materials for, 700 
boilers, measure of duty of, 678 
boilers, performance of, 681 
boilers, Philadelphia inspection 

rule, 708 
boilers, proportions of, 678 
boilers, rules for construction of, 

700 
boilers, safe working pressure, 707 
boilers, strength of, 700 
boilers, tests of, 685 
boilers, tests, rules for, 690-695 
boilers, tests with different coals, 

688 
boilers, tubulous, 686 



ItfDEX. 



1085 



Steam boilers, using waste gases, 
689 

boilers, use of zinc in, 720 
domes on boilers, 711 
dry, identification of, 730 
engine constants, 757 
engine cylinders, 792-795 
engine, spherical, 792 
engines, 742-847 
engines at Columbian exhibition, 

774 
engines, calculation of mean ef- 
fective pressure, 744 
engines, counterbalancing, 788 
engines, dimensions of parts of, 

792-817 
engines, efficiency of, 749 
engines, feed-water consumption 

of, 753, 760, 775 
engines, friction of, 941 
engines, marine, 1015 
engines, measures of duty, 748 
engines, most economical point of 

cut-off, 777 
engines, performance of, 775-789 
engines, putting on center, 834 
engines, relative economy of, 780 
engines, rotary, 791 
engines, triple expansion, 769 
expansive working of, 747 
flow of, 668 
flow of in pipes, 669 
heating, 536-540 
jet- blowers, 527 
loop, 676 

loss of pressure in pipes, 671 
mean pressures, 743 
moisture in, 728 
pipe coverings, 469 
pipes, copper, 674 
pipes, loss from uncovered, 676 
pipes, marine, 1016 
pipes, overhead, 537 
pipes, size of for engines, 673 
pipes, valves in, 675 
pipes, wire-wound, 675 
superheated, 661 
supply mains, 539 
table of properties of, 659 
temperature, pressure, etc., 659- 

662 
turbines, 791 
vessels, dimensions, horse-power, 

etc., 1009 
vessels, trials of, 1007 
water in, effect of on economy of 

engines, 781 
work of iu a single cylinder, 746, 

749 
Steel, 389-414 
aluminum, 409 

analyses and properties of, 389 
annealing, 413 
Bessemer, 390-392 
castings, 405 
chrome, 409 
compressed. 410 
crucible. 410 
effect of hammering, 412 



Steel, effect of heat on, 412 

effect of nicking, 402 

electric conductivity of, 403 

failures of, 403 

hardening of soft, 393 

manganese, 407 

mushet. 409 

open-hearth, 391, 392 

segregation in, 404 

specific gravity of, 403, 411 

specifications for, 397 

strength of, 389 

tempering 412, 414 

treatment of, 394 

tungsten, 409 

use in structures, 405 

variation in strength of, 398 

working stresses for, 264 
Stone, strength of, 302, 312 
Stoker, under-feed, 712 
Stokers, mechanical, 711 
Storage batteries, 1055 
Storing steam-heat, 789 
Strains in structural iron, 379 
Straw as fuel, 643 
Stream, horse-power of a, 589 
Streams, measurement of, 584 
Strength of boiler-heads, 285 

of bolts, 292 

of columns, 246, 250-261 

compressive, 244 

of flat plates, 283 

of glass, 308 

of materials, 236 

of materials, Kirkaldy's tests, 296 

of stayed surfaces, 286 

of structural shapes, 272 

of timber, 310 

of unstayed surfaces, 284 

torsional, 281 

transverse, 266 

tensile, 242 
Stress and strain, 236 

due to temperature, 283 
Stresses, combined, 282 

effect of. 236 

in framed structures, 440 
Structural iron, strains in, 379 

shapes, elements of, 249 

shapes, properties of, 272 

shapes, sizes and weights, 177 

steel, treatment of, 394 
Structures, framed, 440 
Struts, strength of, 246 
Surface condensers, 840 
Sugar concentration, 465 

manufacture, 643 

Tail-rope system of haulage, 913 
Tan-bark as fuel, 643 
Tangent of an angle, 65 
Tangents, sines, etc., table of, 159 
Tanks, cylindrical, 121, 126 

rectangular, gallons in, 125 
Tannate of soda, for boiler-scale, 718 
Tap drills, 970, 971 
Taper-bolts, pins, reamers, 972 

in lathes, 956 
Tapered wire ropes. 916 



1086 



IXDM. 



Taylor's rules for belting, 880 

Taylor's theorem, 76 

Tee-bars, 280 

Tees, Pencoyd, sizes and weights, 

180 
Teeth of gears, forms of, 892 
Telegraph-wire, 217 
Temperature, effect on tenacity, 
382 

stresses in iron, etc., 283 
Temperatures in furnace's, 451 

judged by color, 454 
Tempering steel, 414 
Tenacity of metals, 169 

of metals at different tempera- 
tures, 382 
Tensile strength. 242 

strength, increase by twisting, 241 
Terra-cotta, 181 
Test-pieces, comparison of small 

and large, 393 
Testing materials, precautions in, 

243 
Thermal unit, British, 455 
Thermodynamics, 478 
Thermometers, 448 
Three-wire currents, 1039 
Tie-rods for brick arches, 281 
Tiles, sizes and weights, 181 
Timber measure, 20, 21 

strength of, 310 
Time, measure of, 20 
Tin. 168 

roofing, 181, 182 
Tires, steel, strength of, 298 
Tobin bronze, 327 
Toggle-joint, 436 
Tool-steel, heating, 412 
Tonnage of vessels, 19, 1001 
Torque of an armature, 1062 
Torsional strength, 281 
Tower spherical engine, 792 
Tractive power of locomotives, 857 
Tractrix, or Schiele's curve, 50 
Trains, resistance of, 851, 853 
Tramways, wire-rope, 914 
Transformers, 1070 
Transmission by hydraulic press- 
ure, 617-620 

by wire rope, 917-922 

eiectric, 1038 

electric, efficiency of, 1047 

of heat, 471-478 

of power by ropes, 922-927 
Transporting power of water, 565 
Transverse strength, 266 
Trapezium, 54 
Trapezoid, 54 
Trapezoidal rule, 55 
Treatment of steel, 394 
Triangle, mensuration of, 54 
Triangles, problems in, 41 

solution of, 68 
Trigonometrical functions, 65-67 

functions, table, 159 
Trigonometry, plane, 65 
Triple effect, multiple system, 463 

expansion engine, 769 
Troy weight, 19 



I Truss, Howe and Warren, 445 

king-post, 442 

queen post, 442 

Pratt or Whipple, 443 
Trusses, roof, 446 
Tungsten-aluminum alloys, 332 

steel. 409 
Tubes for steam-boilers, 704, 709 

Mannesman n, 296 

or flues, collapse of, 265 

weights of, 169 

wiought-iron, 196 
Tubing, brass, 198 
Turbines, steam, 791 
Turbine-wheels, 591 

wheels, tests of, 596 
Turf or peat, 643 
Turnbuckles, sizes, 211 
Twin screw vessels, 1017 
Twist-drills, sizes and speeds, 957 
Type metal, 336 

Unit of heat, 455 

Unstayed surfaces, strength of, 284 

Upsetting of steel, 394 

Vacuum pumps, 612 
Valve-diagrams, 825 

motions, 825 

rods, 815 
Valves, engine-setting, 834 

in st earn -pipes, 675 

of pumps, 605, 606 
Vapors, properties of, 480 

used in refrigerating-machines, 
982 
Velocities, parallelogram of, 426 
Velocity, angular, 425 
Ventila'ting-ducts, discharge of, 530 

fans, 517-525 
Ventilation and heating, 528-546 

blower system of, 546 

by a steam-jet, 526 

of large buildings, 534 
Ventilators for mines, 521 
Venturi meter, 583 
Versed sine of an arc, 66 
Vibrations of engines, preventing, 

789 
Vis-viva, 428 

Warehouse floors, 1019 
Warren girder, 445 
Washers, sizes of, 212 
Water, 547-554 

analyses of, 553 

compressibility of, 551 

erosion by flowing, 565 

expansion of, 547 

flow in channels, 564 

flow of, 555-588 

flow of, experiments, 566 

flow of, in pipes, tables, 558, 559, 
567-570 

gas, 648, 652 

impurities of, 551 

power, 588 

power, value of, 590 

pressure engine, 619 ' 

pressure of, 549 



1087 



Water, softening of hard, 555 

specific heat of, 550 

transporting power of, 565 

weight of, 547 

wheel, the Pelton, 597 
Waves, power of ocean, 599 
Weathering of coal, 637 
Wedge, the, 437 

volume of a, 61 
Weights and measures, 17 

of air and vapor, 484 
Weight of bars, rods, plates, etc., 169 

of brick-work, 169 

of brass and copper, 197-203 

of bolls and nuts, 209-211 

of cast-iron pipes and columns, 
185-193 

of cement, 170 

of flat rolled iron, 172 

of fuel, 170 

of iron bars, 171 

of iron and steel sheets, 174 

of ores, earths, etc., 170 

of plate iron, 175 

of roofing materials, 181-184 

of steel blooms, 176 

of structural shapes, 177-180 

of tin plates, 182 

of wrought-iron pipe, 194-197 

of various materials, 169 
Weir table, 587 
Weirs, flow of water over, 586 
Welding, electric, 1053 

of steel, 394, 396 
Welds, strength of, 300 
Wheel and axle, 439 
Whipple truss, 443 
White-metal alloys, 336 
Whitworth compressed steel, 410 
Wiborgli air-pyrometer 453 
Wind, 493 
Winding engines, 909 

of magnets, 1068 
Windlass, 439 

differential, 439 
Windmills, 495 
Wind pressures, 493 
Wire cables, 222 

copper and brass, 202 

copper tables of, 218-220 

gauges, 28-31 

insulated, 221 



Wire, iron and steel, 217 

iron, size, strength, etc., 216 

rope, 226, 231 

rope haulage, 912 

ropes, durability of, 919 

ropes, splicing, 345 

ropes, strength of, 301 

ropes, tapered, 916 

rope transmission, 917-922 

rope tramways, 914 

strength of, 301, 303 

telegraph, 217 

wound fly-wheels, 824 
Wiring formula for incandescent 

lighting, 1043 
Wires, current required to fuse, 1037 
Wire -table, for 100 and 500 volts, 
1044 

table, hot and cold wires, 1034,1035 
Wohler's experiments, 238 
Wood as fuel, 639 

composition of 640 

expansion of, 311 

heating value of, 639 

strength of, 302, 306, 310, 312 

weight of, 164, 232 
Wooden fly-wheels, 823 
Woodstone or xylolith,316 
Woolf type of compound engine, 

762 
Woot ten's locomotive, 855 
Work, energy, power, 428 

of acceleration, 430 

of men aud animals, 433 

unit of, 428 
Worm-gear, 440 

gearing, 897 
Wrought iron, 377-379 

iron, chemistry of, 377 

iron, specifications, 378 

Xylolith or woodstone, 316 
Yield point, 237 

Z bars, properties of, 276 

sizes and weights, 178 
Zinc, 168 

tubing, 200 

use of, in steam-boilers, 720 
Zeuner valve-diagram, 827 
Zero absolute, 461 



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